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Robust Aspects of Hedging and Valuation in Incomplete Markets and related Backward SDE Theory
DISSERTATION zur Erlangung des akademischen Grades Doctor rerum naturalium (Dr. rer. nat.) im Fach: Mathematik eingereicht an der MathematischNaturwissenschaftlichen Fakult¨at der HumboldtUniversit¨at zu Berlin von M.Sc. Kl´ ebert, Kentia Tonleu Pr¨asident der HumboldtUniversit¨at zu Berlin Prof. Dr. JanHendrik Olbertz Dekan der MathematischNaturwissenschaftlichen Fakult¨at Prof. Dr. Elmar Kulke
Gutachter:
1. Prof. Dr. Dirk Becherer 2. Prof. Dr. Romuald Elie 3. Prof. Dr. Peter Imkeller
Tag der Einreichung: Tag der m¨undlichen Pr¨ufung:
31.07.2015 04.12.2015
Abstract This thesis studies backward stochastic differential equations (BSDEs), and robust notions of dynamic gooddeal valuation and hedging in incomplete financial markets. We start by a mathematical theory, concerning the analysis of BSDEs with jumps driven by random measures that can be of infinite activity with timeinhomogeneous compensators. Under concrete conditions that are easy to verify in practical applications, we provide existence, uniqueness and comparison results for bounded solutions for a class of generator functions that are not required to be globally Lipschitz in the jump integrand. To illustrate the ease of applicability of our results, we solve the exponential and power utility maximization problems with additive and multiplicative liability respectively. The rest of the thesis deals with the more applicationoriented problem of robust valuation and hedging in incomplete markets. We are concerned with the nogooddeal approach, which computes gooddeal valuation bounds by using as pricing measures only a subset of the riskneutral measures satisfying a constraint on the Girsanov kernels described by correspondences with economic meaning. Examples of such constraints are given by bounds on instantaneous Sharpe ratios, optimal growth rates, or expected utilities. Throughout we study a notion of gooddeal hedging that corresponds to gooddeal valuation, and for which hedging strategies arise as minimizers of some dynamic coherent risk measures allowing for optimal risk sharing with the market. Hedging is shown to be at least meanselffinancing in the sense that tracking (hedging) errors satisfy a supermartingale property under suitable apriori valuation measures. The latter is then interpreted as robustness of gooddeal hedging, with respect to the family of valuation measures as generalized scenarios. We derive constructive results on gooddeal valuation and hedging using BSDEs. The results are obtained in a jump framework with unpredictable eventrisk, as well as in a Brownian setting with model uncertainty. In the jump framework we use the theory on BSDEs with jumps, and provide examples in (semi)Markovian models, which are particularly relevant for actuarial applications. In the Brownian setting, we provide new examples for concrete nogooddeal constraints, with closedform expressions for valuations and hedges given via classical option pricing formulas (BlackScholes, Margrabe or Heston). Moreover, under Knightian uncertainty (ambiguity) about the objective realworld probability measure which is not taken to be precisely known, we study robustness of hedging taking into account the investor’s aversion towards ambiguity. Assuming multiple reference priors as candidates for the (uncertain) realworld measure, a worstcase approach leads to gooddeal hedging that is robust with respect to uncertainty in the sense that it is at least meanselffinancing uniformly over all priors. Results are presented for drift uncertainty and volatility uncertainty separately, using classical BSDEs for the former and secondorder BSDEs for the latter. Under drift uncertainty, we also show existence of a worstcase prior with respect to which dynamic valuations and hedges can be computed like in the absence of uncertainty. Here the robust approach yields that gooddeal hedging is equivalent to riskminimization with respect to a suitable measure if drift uncertainty is sufficiently large. In the case of volatility uncertainty, we provide an example for put options in an uncertain volatility model of BlackScholes’ type, where explicit solutions for (robust) gooddeal valuations and hedges are computable under a worstcase prior.
Zusammenfassung Diese Arbeit untersucht stochastische R¨ uckw¨artsdifferentialgleichungen (BSDEs) und robuste Konzepte von dynamischer GoodDealBewertung und Hedging in unvollst¨andigen Finanzm¨arkten. Wir beginnen mit einer mathematischen Theorie zur Analyse von BSDEs mit Spr¨ ungen, getragen von zuf¨alligen Maßen, die von unendlicher Aktivit¨at mit zeitlich inhomogenem Kompensator sein k¨onnen. Unter konkreten Bedingungen, die in praktischen Anwendungen leicht zu verifizieren sind, liefern wir Existenz, Eindeutigkeits und Vergleichsergebnisse beschr¨ankter L¨osungen f¨ ur eine Klasse von Generatorfunktionen, welche nichtnotwendigerweise global Lipschitzstetig im Sprungintegranden sein m¨ ussen. Wir l¨osen das Maximierungsproblem f¨ ur exponentiellen Nutzen bei additiver Verbindlichkeit und f¨ ur PowerNutzen bei multiplikativer Verbindlichkeit, um die Anwendbarkeit unserer Resultate zu veranschaulichen. Der u ¨brige Teil der Arbeit besch¨aftigt sich mit dem eher anwendungsorientierten Problem der robusten Bewertung und des Hedgings in unvollst¨andigen M¨arkten. Wir befassen uns mit dem NoGoodDealAnsatz, welcher GoodDealGrenzen liefert, indem als Bewertungsmaße lediglich eine Teilmenge der risikoneutralen Maße betrachtet werden, die eine Bedingung an den GirsanovKern – beschrieben durch Korrespondenzen mit ¨okonomischer Bedeutung – erf¨ ullen. Beispiele solcher Bedingungen sind Grenzen f¨ ur instantanen SharpeRatio, optimale Wachstumsrate oder erwarteten Nutzen. Durchweg untersuchen wir ein Konzept des GoodDealHedgings, das GoodDealBewertung entspricht und f¨ ur welches Hedgingstrategien als Minimierer geeigneter dynamischer koh¨arenter Risikomaße auftreten, was optimale Risikoteilung mit der Markt erlaubt. Wir zeigen, dass Hedging mindestens imMittelselbstfinanzierend ist. Das heißt, dass Hedgefehler unter geeigneten AprioriBewertungsmaßen eine Supermartingaleigenschaft haben. Dies wird als Robustheit des GoodDealHedgings bez¨ uglich der Familie von Bewertungsmaßen, gesehen als verallgemeinerte Szenarien, interpretiert. Wir leiten konstruktive Ergebnisse zu GoodDealBewertung und Hedging mittels BSDEs her. Die Ergebnisse werden sowohl im Rahmen von Prozessen mit Spr¨ ungen mit unvorhersehbarem Ereignisriungen nutzen siko, als auch im Brown’schen Rahmen mit Modellunsicherheit erzielt. Im Falle von Spr¨ wir die Theorie zu BSDEs mit Spr¨ ungen und liefern Beispiele in (Semi)MarkovModellen, die insbesondere f¨ ur versicherungsmathematische Anwendungen von Bedeutung sind. Im Brown’schen Fall liefern wir neue Beispiele f¨ ur konkrete NoGoodDealBedingungen mit expliziten Formeln f¨ ur Bewertung und Hedging, aufbauend auf klassischen Optionsbewertungsformeln (BlackScholes, Margrabe oder Heston). Unter Knight’scher Unsicherheit bez¨ uglich des nicht genau bekannten objektiven realen Maßes untersuchen wir hier Robustheit des Hedgings unter Ber¨ ucksichtigung der Abneigung des Investors gegen Ungewissheiten. Bei Annahme mehrerer Referenzmaße als Kandidaten f¨ ur das (unsichere) reale Maß f¨ uhrt ein WorstCaseAnsatz zu GoodDealHedging, welches robust bez¨ uglich Unsicherheit, im Sinne von gleichm¨aßig u¨ber alle Referenzmaße mindestens imMittelselbstfinanzierend, ist. Die Ergebnisse zu Drift und Volatilit¨atsunsicherheiten werden separat pr¨asentiert, wobei f¨ ur erstere klassische BSDEs und f¨ ur letztere BSDEs zweiter Ordnung zur Anwendung kommen. Bei Driftunsicherheit zeigen wir außerdem Existenz eines WorstCaseMaßes unter dem sich Bewertungen und Hedging wie bei Abwesenheit der Unsicherheit berechnen lassen. Hier liefert der Robustheitsansatz, dass bei hinreichend großer Driftunsicherheit GoodDealHedging ur ¨aquivalent ist zur Risikominimierung. Im Falle von Volatilit¨atsunsicherheit legen wir ein Beispiel f¨ PutOptionen in einem BlackScholesartigen Modell mit unsicherer Volatilit¨at vor, in dem explizite L¨osungen zur (robusten) GoodDealBewertung und Hedging unter einem WorstCaseAprioriMaß berechnet werden k¨ onnen.
Acknowledgements First, I am extremely grateful to my advisor Dirk Becherer for introducing me to the fascinating areas of stochastic analysis and mathematical finance, and for patiently guiding me throughout the completion of this work. His support, ideas, discerning comments and good sense of mathematical writing were indispensable to the development and writing of this work, and greatly contributed to my scientific training as a whole. In particular I thank him for being such a kind person, for always being willing to share his knowledge with me and for providing advices for my diverse initiatives as well academically as on a personal level. I would like to thank Romuald Elie and Peter Imkeller for immediately agreeing to be coexaminers of this thesis. I also thank my colleagues and friends in Berlin, for the fruitful discussions and numerous extracurricular activities that made the process of completing this thesis quite an agreeable experience. In particular I think about Achref Bachouch, Julio Backhoff, Todor Bilarev, Martin B¨ uttner, Peter Frentrup, Guanxing Fu, Paulwin Graewe, Elena Ivanova, Martin Karliczek, Victor Nzengang, Ludovic Tangpi, and many more that I have forgotten to mention and to whom I am greatly grateful. Special thanks are addressed to Axel Mosch and Guillaume Richard for help with numerical simulations in the examples in Chapter 3. Support from the HumboldtUniversit¨at zu Berlin, and from the German Science Foundation DFG via the Berlin Mathematical School and the Research Training Group 1845 StoA is gratefully acknowledged. They generously provided financial support and a propitious research environment inherent to the completion of this work. Most special thanks go to my parents and siblings for their longterm regular support and for bearing with me leaving home to continue my studies abroad. Finally, I dedicate this thesis to Darl`ene and Keynane for their love and moral assistance. They have been more than a source of motivation and perseverance, especially during the last months preceding the submission of the thesis.
iv
` mes deux amours, Darl`ene et Keynane. A
Contents Introduction
1
1 Concrete criteria for wellposedness and comparison of BSDEs with jumps of infinite activity 19 1.1
Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 19
1.2
Comparison theorems and aprioriestimates . . . . . . . . . . . . . . . . . . . 25
1.3
Existence and Uniqueness of bounded solutions . . . . . . . . . . . . . . . . . 32
1.4
1.3.1
The case of finite activity . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.2
The case of infinite activity . . . . . . . . . . . . . . . . . . . . . . . . 34
Applications to optimal control problems in finance . . . . . . . . . . . . . . . 43 1.4.1
Exponential utility maximization . . . . . . . . . . . . . . . . . . . . . 45
1.4.2
Power utility maximization . . . . . . . . . . . . . . . . . . . . . . . . 52
2 Hedging under generalized gooddeal bounds in jump models with random measures 55 2.1
Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 55
2.2
Gooddeal valuation and hedging . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.3
Case of uniformly bounded correspondences . . . . . . . . . . . . . . . . . . . 65
2.4
2.5
2.3.1
Results for constraint on instantaneous Sharpe ratios (bounded case) . 69
2.3.2
Results for ellipsoidal constraint and uncertainty about jump intensities
Case of nonuniformly bounded correspondences . . . . . . . . . . . . . . . . . 78 2.4.1
Results for constraint on instantaneous Sharpe ratios (unbounded case)
2.4.2
Results for constraint on optimal expected growth rates . . . . . . . . . 81
81
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3 Hedging under generalized gooddeal bounds and drift uncertainty 3.1
76
99
Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 99 viii
3.2
3.3
3.4
3.1.1
Parametrizations in an Itˆo process model . . . . . . . . . . . . . . . . . 102
3.1.2
Gooddeal valuation with uniformly bounded correspondences . . . . . 104
3.1.3
Gooddeal valuation with nonuniformly bounded correspondences . . . 105
Dynamic gooddeal hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.1
Results for ellipsoidal nogooddeal constraints . . . . . . . . . . . . . . 110
3.2.2
Examples for gooddeal valuation and hedging with closedform solutions113
Gooddeal valuation and hedging under model uncertainty . . . . . . . . . . . 120 3.3.1
Model uncertainty framework . . . . . . . . . . . . . . . . . . . . . . . 121
3.3.2
Nogooddeal constraint and gooddeal bounds under uncertainty . . . 122
3.3.3
Robust approach to gooddeal hedging under model uncertainty . . . . 125
3.3.4
Hedging under model uncertainty for ellipsoidal gooddeal constraints . 125
3.3.5
The impact of model uncertainty on robust gooddeal hedging . . . . . 131
3.3.6
Example with closedform solutions under model uncertainty . . . . . . 134
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4 Hedging under gooddeal bounds and volatility uncertainty: a 2BSDE approach144 4.1
4.2
4.3
Mathematical framework and preliminaries . . . . . . . . . . . . . . . . . . . . 144 4.1.1
The local martingale measures . . . . . . . . . . . . . . . . . . . . . . 144
4.1.2
Spaces and norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1.3
Second order backward stochastic differential equations . . . . . . . . . 147
Market model and gooddeal constraint under volatility uncertainty . . . . . . . 155 4.2.1
Financial market with volatility uncertainty . . . . . . . . . . . . . . . 156
4.2.2
Nogooddeal constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Gooddeal bounds and hedging under volatility uncertainty . . . . . . . . . . . 159 4.3.1
Gooddeal bounds under volatility uncertainty . . . . . . . . . . . . . . 160
4.3.2
Robust gooddeal hedging under volatility uncertainty . . . . . . . . . . 163
4.3.3
Example for options on nontraded assets . . . . . . . . . . . . . . . . 169
ix
References
176
List of Figures
190
List of Tables
191
x
Introduction This thesis is concerned with backward stochastic differential equations (BSDEs) and with hedging and valuation of contingent claims in incomplete financial markets. BSDEs have by now found numerous applications in mathematical finance, where they have proved to be suitable tools in describing solutions to many stochastic optimization problems of practical relevance. BSDEs form a common theme for all chapters of the thesis, and will be used throughout in different forms depending on the problem at hand. In particular, Chapter 1 of the thesis is concerned with theoretical foundations of BSDEs with jumps (in short JBSDEs), which are BSDEs driven jointly by a Brownian motion and a random measure. We study wellposedness (existence and uniqueness) and comparison for bounded solutions to this class of BSDEs, for jumps that may have infinite activity with compensators being possibly timeinhomogeneous. Moreover in this chapter, applications of the JBSDE theory will be presented dealing with the utility maximization problem in finance. The remaining chapters of the thesis (Chapters 2 to 4) deal overall with the problem of valuation and hedging of contingent claims in incomplete financial markets. Valuation and hedging are classical topics in mathematical finance for which many approaches have been studied in the literature, especially in the context of incomplete markets where some risks may not be perfectly hedgeable, and valuation and (partial) hedging may involve solving stochastic optimization problems. As far as this thesis is concerned, we will focus on the nogooddeal approach, which does not only prevent arbitrage opportunities from the market, but also excludes an economically meaningful notion of deals that are “too good”. This leads to socalled gooddeal valuation bounds, to which a corresponding concept of hedging will be associated. In a general framework with nogooddeal constraints defined in terms of abstract correspondences (setvalued mappings) for the pricing measures, we will obtain results on gooddeal hedging and valuation in terms of solutions to BSDEs. We will provide examples with explicit formulas that facilitate computations, for specific correspondences associated to more concrete nogooddeal constraints. In particular in Chapter 2 we will apply our theoretical results on JBSDEs from Chapter 1 to gooddeal valuation and hedging in a setup allowing for unpredictable eventrisk, which is modelled by a discontinuous filtration supporting a random measure and a Brownian motion simultaneously. Another topic of central interest in this thesis is robustness. In general, a robust concept will refer to one which remains effective under different admissible market scenarios/variables. In the presence of model uncertainty (ambiguity), the scenarios will correspond to the uncertain priors (models) and we will analyze in Chapters 3 and 4 robust concepts of gooddeal valuation and hedging with respect to model uncertainty. We will focus on continuous filtrations, which will allow us to use the classical theory of BSDEs driven solely by a Brownian motion for the case of uncertainty about the excess return of traded assets (cf. Chapter 3), and the theory of secondorder BSDEs (shortly 2BSDEs) for the case of uncertainty about the volatility (cf. Chapter 4). Before giving a more
1
Page 2 detailed account of the contributions of the thesis, we next explain the necessary background on BSDEs, valuation and hedging in incomplete markets, and model uncertainty. These three general themes are central to the thesis and their connections with different chapters will also be made more precise.
An overview of the theory of backward SDEs BSDEs are studied and used intensively in this thesis. To relate the contributions of the thesis to the historical developments of BSDEs, we present a short overview of advances in the theory that culminated in a wide range of applications to optimal control problems in mathematical finance. Classical BSDEs form a class of stochastic differential equations (SDEs) of the type Z
Yt = ξ + t
T
fs (Ys , Zs ) ds −
Z
T
Zs dWs ,
t ∈ [0, T ],
t
They are described by a semimartingale dynamics for which a terminal condition ξ is given (instead of an initial one as for forward SDEs), and the generator function f (drift of the dynamics) varies with the value process Y of the equation and the control process Z integrated by the driving Brownian motion W under its natural filtration (Ft )t≤T . The solution of a BSDE consists of the couple (Y, Z). Originally BSDEs appeared in [Bis73] with linear generators functions. [PP90] were the first to study existence and uniqueness of square integrable solutions in the classical setting for BSDEs under global Lipschitz assumptions on the generator. Such BSDEs will appear in Chapter 3, under a uniformly boundedness assumptions on the nogooddeal constraint correspondence for the Girsanov kernels of pricing measures. For a detailed exposition on applications of classical BSDEs in mathematical finance and additional results including a comparison principle, we refer to [EPQ97]. Beyond the Lipschitz setting, notable extensions include the case of generators with quadratic growth in the Brownian integrand Z for which [Kob00] has studied bounded solutions (see also [Tev08, BE13]). This has found crucial applications in utility maximization in incomplete markets initiated by [RE00] and [HIM05]. It has been shown in [DHB11] that BSDEs with generators that are of superquadratic growth are typically illposed. Beyond quadratic growth and with generators that are only convex, [DHK13, DHK15] proved existence and uniqueness of minimal supersolutions relying on compactness rather than fixedpoint arguments. This solution concept will be used in Chapter 3, where we will consider nogooddeal constraint correspondences that are not necessarily uniform bounded. Let us mention that by now there exists a plurality of numerical methods for simulation of BSDEs, including MonteCarlo methods which are particularly relevant for higher dimensional problems. For advances in this direction, we refer to [BT04, GLW05, BD07, GT15, BT14]. BSDEs that are driven not only by a Brownian motion W but additionally by a random measure
Page 3 are shortly referred to as JBSDEs (i.e. BSDEs with jumps) and involve a second stochastic integral with respect to the compensated random measure. Their dynamics are of the form Z
Yt = ξ + t
T
fs (Ys− , Zs , Us ) ds −
Z t
T
Zs dWs −
Z TZ
Us (e) µ ˜(ds, de), t
t ∈ [0, T ],
E
with µ ˜ = µ − ν P denoting the compensated random measure of some integervalued random measure µ on a space E for a stochastic basis (Ω, FT , (Ft )t≤T , P ). The solution of a JBSDE is now a triple (Y, Z, U ), where the jump integrand U lives in a possibly infinitedimensional function space and also appears in the generator of the BSDE. For such JBSDEs, [TL94, BBP97] studied square integrable solutions under global Lipschitz conditions in a timehomogeneous setting for Poisson random measures. Bounded solutions to JBSDEs have been studied in [Bec06] for a random measure that is possibly inhomogeneous in time but of finite jump activity, covering a family of generators that satisfy a certain monotonicity property but need not be (globally) Lipschitz in the jump integrand, see also [Par97, Roy06]. A similar study will be considered in Chapter 1 but for possibly infinite activity of jumps, and increased degree of complexity of the generators also allowing for a comparison principle for such JBSDEs. Indeed it appears here (see also [BBP97, Roy06, CE10]) that comparison principles for JBSDEs require more delicate technical conditions than in the Brownian case. These comparison principles will be applied in Chapter 2 to derive JBSDEs for gooddeal valuation bounds and associated hedging strategies. We will consider in Chapter 1 applications of our JBSDE theory to the utility maximization problem in finance, for jumps of infinite activity. Note that JBSDEs with generator of quadratic growth in the Brownian integrand have been studied for a particular generator and infinite activity of jumps in [Mor09, Mor10], in [KTPZ15a] also under timeinhomogeneity, and in [EMN14] in general under finite activity assumptions. For numerical analysis of JBSDEs, see e.g. [BE08]. There is a strong connection between BSDEs and the theory of partial differential equations (PDEs). In fact (firstorder) Markovian BSDEs for which the generator additionally depends on the solution of a forward SDE, hence referred to as forwardbackward SDEs (alternatively FBSDEs), are probabilistic representations `a la FeymannKac for secondorder quasilinear PDEs (i.e. PDEs involving only a linear dependency in the Hessian of the solution). Indeed, the PDE terms depending on the secondorder derivative of the solution can only arise from the quadratic variation of the forward process via Itˆo’s formula. Probabilistic representations of PDEs pave the way to numerical MonteCarlo schemes for simulation of their solutions, which again are more relevant for PDEs with high dimensional statespace. Note that in the case of Markovian JBSDEs, an additional integral appears in the formulation of the PDE, hence yielding a partialintegro differential equations (PIDEs). Due to their importance in practice, one would also like as for quasilinear PDEs to have a probabilistic representation for fully nonlinear PDEs (i.e. PDEs involving a nonlinear dependency in the Hessian of the solution), which are an important class containing e.g. HamiltonJacobiBellman (HJB) equations. It is
Page 4 exactly this fact that motivated [CSTV07] to formally introduce the notion of 2BSDE originally in connexion to the solution to the second order stochastic target problem first introduced by [ST09]. To ease exposition, a 2BSDE in its simplified form is an equation of the type Z
Yt = ξ− t
T
1 bs −Hs (Ys , Zs , Γs ) ds− Γs a 2
Z t
T
Zs dBs +KT −Kt ,
P a.s., t ∈ [0, T ], ∀ P ∈ PH
b is the (ωwise) density of the quadratic variation of the coordinate process B on the where a canonical Wiener space of continuous paths, PH is a subset of (typically mutually singular) local martingale measures for B, and K is a nondecreasing process with K0 = 0. Note that contrary to classical BSDEs, the dynamics of 2BSDEs is required to hold almostsurely under P , for all P in a family PH of reference probability measure, that is to say quasisurely with respect to PH . In this form the solution to the 2BSDE is the triple (Y, Z, Γ), and the generator Fb is the convex conjugate of a nonlinear function H in its third argument Γ, bs − Hs (Ys , Zs , Γs ). For classical BSDEs, the solution components satisfying Fbt (Yt , Zt ) := 12 Γs a Y and Z correspond to the PDE solution and its gradient (firstorder derivative) respectively. For 2BSDEs in a Markovian setting with a canonical forward process, one has an additional unknown variable Γ in the dynamics of the BSDE which essentially corresponds to the Hessian (secondorder derivative) of the solution to a fully nonlinear PDE (justifying the appellation “secondorder” BSDE). For first applications of 2BSDEs in mathematical finance, let us mention among many others [C ¸ ST07, ST09] for the superreplication problem under Gamma constraint and [MPZ15] for the robust utility maximization under volatility uncertainty [ALP95, Lyo95]. In Chapter 4, we extend the list of applications in finance by using 2BSDEs to describe the solutions to gooddeal valuation and hedging problem that are robust (in some sense to be made precise later) with respect to volatility uncertainty. The original formulation of 2BSDEs in [CSTV07] was in a Markovian setting and is somewhat different to the one presented above. The above is a particular case of [STZ12] who used the quasisure analysis of [DM06] to obtain a general formulation for possibly nonMarkovian 2BSDEs and obtained a wellposedness theory for 2BSDEs with Lipschitz generators. Note that in the language of Gstochastic calculus of [Pen10], wellposedness of 2BSDEs with zero generators can be viewed as a martingale representation theorem for Gmartingales (and Gexpectations in particular). The wellposedness theory was later extended by [PZ13] to 2BSDEs with quadratic generators. Subsequently, [MPZ13] and [KTPZ15c, KTPZ15b] studied 2BSDEs reflected on an obstacle and 2BSDEs with jumps respectively. Some numerical schemes for 2BSDEs based on MonteCarlo or/and finite difference methods have been suggested in the literature, e.g. in [CSTV07, FTW08, GZZ15, PT14]; see also [BET09] for a survey on the probabilistic numerical methods for nonlinear PDEs in general.
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Hedging and pricing approaches in mathematical finance Investing in financial markets involves facing some risk that can be synthesized either perfectly (one speaks of replication) or only partially by dynamic trading in liquid assets. The seller of a financial contract (contingent claim) is usually confronted with the following problem: what valuation would she like to sell the claim to the buyer for, to enable a certain form of hedging against the risk of loss at delivery of the claim? A financial market where all contingent claims can be replicated is referred to as complete. The significance of such markets lies in that they allow for pricing by replication so that under the viability of the market the price of a contingent claim is the cost of the replicating portfolio. This was the insight behind [BS73, Mer73] where assuming that asset prices follow a geometric Brownian motion, the authors obtained the price of vanilla call/put options by replicating with the deltahedging strategy and deriving the celebrated BlackScholes formula for option pricing. The BlackScholes formula has been extended for pricing other types of options; for instance, the Margrabe formula [Mar78] is used to price European exchange options, i.e. options to exchange one risky asset for another at a prespecified maturity time. The BlackScholes and Margrabe formulas will play a role in the examples of Chapter 3 (see also example at the end of Chapter 4), where we will derive gooddeal valuations and hedges via these formulas for incomplete models with traded and nontraded assets. The prices resulting from replication are preferencefree and can be computed by taking the expectation of the option’s discounted payoff under an equivalent martingale measure (also called riskneutral measure). The latter is a probability measure equivalent to (i.e. with exactly the same nullsets as) the realworld probability measure and under which asset prices and associated wealth processes discounted at the bank account’s interest rate are martingales, i.e. at each time, the present value is the best prediction for future values given past information. In other words under an equivalent martingale measure, risky assets have zero excess returns, i.e. same meanreturn as with the riskless asset, e.g. as in the BlackScholes model. The connection between riskneutral pricing and martingale theory was first put into rigorous mathematical perspective by [HK79, HP81]. They characterized absence of arbitrage (also called free lunch) in a market with discretetime trading by the existence of an equivalent martingale measure, their result is now known as the fundamental theorem of asset pricing. A consequence of this theorem and the classical predictable representation property of martingales is that completeness of the financial market is equivalent to uniqueness of the equivalent martingale measure. These results were later generalized to continuoustime trading by [DS94, DS98], in the context of asset prices being semimartingales. The latter publications introduced a reasonably general notion of market viability, namely the nofree lunch with vanishing risk (abbreviated NFLVR) condition, and linked it rather to a martingale (resp. local martingale, sigmamartingale) property of bounded (resp. locally bounded, unbounded) asset prices.
Page 6 In this thesis, we concentrate on pricing and valuation in the context of incomplete market where indeed many financial claims carry some inevitable risk. This may include, for instance, claims that are contingent on some nontradeable underlying assets, e.g. weather derivatives or some volatility derivatives. Transaction costs, jumps in the underlying asset price, or unpredictable events in the information are possible reasons for incompleteness of a financial market. Convexity of the set of equivalent martingale measures implies that incomplete arbitragefree markets admit infinitely many pricing measures; yielding notably an interval of riskneutral prices for nonreplicable claims. The issuer of a financial contract striving for robustness with respect to price misspecification and preferencefreeness may therefore wish to sell at the upper bound over all possible noarbitrage prices, socalled upper noarbitrage bound. The buyer’s ideal valuation can be interpreted analogously as the corresponding lower bound. This valuation approach in incomplete markets was rigorously introduced by [EQ95], who showed that the upper noarbitrage bound is exactly the minimal capital that allows the seller to superreplicate the claim in almost every state of the world by dynamically trading in the risky assets. The cost of superreplication being given by the supremum over all the riskneutral prices, the corresponding wealth process was shown to be a supermartingale under any equivalent martingale measures and the associated superreplicating strategy can be derived via the optional decomposition theorem (see [Kra96, FK97, FK98]). Superreplication indeed is an extremely safe concept of hedging, since it excludes the possibility of losses and, while eventually allowing even for intermediate consumption, still ensures the terminal wealth to dominate the liability of the investor at delivery. Unsurprisingly, superreplication is often too costly and may not be appropriate for some practical applications; the minimal capital it requires for instance may be too high to find a buyer. For a long time, research in mathematical finance has been investigating about alternative approaches to (partial) hedging that require lower capital than superreplication would, hence making it more likely to find a buyer. Apart from the nogooddeal approach whose background we describe in the next paragraph and which lies at the heart of Chapter 2, 3 and 4, various solutions have been suggested in this regard: The quantile hedging approach of [FL99] allows the seller of a claim to charge a smaller amount to the buyer but still be able to dominate his liability with some target confidence level. This is more a hedging approach than a pricing one, since its main objective is to limit the risk of loss for the seller to a maximal prespecified level by requiring a minimal capital that will make her position acceptable in this sense. In the same direction, one can consider risk minimization and meanvariance hedging whose objective are the minimization of a quadratic functional of the tracking (hedging) error of trading strategies. In order to achieve this for riskminimization, one relaxes the selffinancing requirement of the replicating strategy (corresponding to vanishing tracking error) and instead requires a notion of meanselffinancing strategy that corresponds to a martingale property of the tracking error. Risk minimization was first introduced by [FS86] in the situation where asset prices are modelled directly as martingales, and later extended in [FS90] to the general semimartingale
Page 7 case where it could only be defined in a local sense (local riskminimization). The local riskminimizing strategy was ultimately derived via the socalled F¨ollmerSchweizer decomposition of the wealth process, which can be viewed as a generalization to the semimartingale case of the wellknown GaltchoukKunitaWatanabe decomposition from martingale theory. Naturally, a pricing concept is attached to riskminimization via the socalled minimal martingale measure. However hedging according to some quadratic criterion has been criticized, mainly because it penalizes gains and losses in the same way. We will provide (cf. Chapter 3) a potential argument against this criticism by showing that if drift uncertainty is sufficiently large in the market, then riskminimization coincides with robust gooddeal hedging (in a suitable sense); the latter is using a nonquadratic hedging criterion by minimizing a dynamic coherent risk measure. If one rather insists on the selffinancing property of the hedging strategy, then a quadratic hedging criterion leads to meanvariance hedging as studied in [BL89, DR91]. Also here one obtains a valuation that is consistent with the hedging criterion and can be computed under the varianceoptimal martingale measure. A comprehensive survey of both approaches can be found in [Sch01]. However it turns out that the minimal and varianceoptimal martingale measures may be only signed measures in general, and hence may lead to negative prices for some positive claims. This is clearly an undesirable feature of these two valuation approaches. However, for a more specific Markovian (incomplete) model of the stock price, for instance the Heston stochastic volatility model [Hes93], the minimal martingale measure can be written explicitly in terms of the market price of risk. In the Heston model the squared volatility process is a CoxIngersollRoss (CIR) process and the price of European call/put options under the minimal martingale measure is given by the Heston formula which is explicit up to the computation of a one dimensional improper integral. Using a single riskneutral measure for valuation is clearly not conservative, as it introduces marktomarket risk that can accumulate due to the necessary regular calibration. In Chapter 3, we suggest a more conservative approach in an example that shows how a robust valuation (and hedging) of volatility risk over a family of riskneutral measures in the Heston model can be obtained, by restricting the meanreversion level of the variance process to be within some confidence interval. Let us mention here also that a further alternative solution to the limitations of the abovementioned approaches is to take into account some utilityrelated preference of the investor or her aversion towards risk; this leads to rational pricing and hedging concepts that are consistent with the maximal expected utility of the investor. The literature in this direction is quite developed, and generally distinguishes between two approaches: the utilityindifference approach (cf. [HH09] for an overview and further references), and the utilitybased approach (cf. [Dav97, HK04]). We do not say more about these approaches. Instead, we will present in Chapter 1 an example, for illustration of our JBDSE theory therein, that will deal with the solution to the classical expected utility maximization problem in incomplete market
Page 8 with additional liability. Note that this approach, somewhat problematically, assumes precise knowledge of the objective realworld probabilities. This is restrictive since model uncertainty, in particular about (highly uncertain) drift and volatility parameters under the realworld measure, is a problem in itself for practical applications. Gooddeal valuation and hedging in the presence of model uncertainty will be studied in Chapters 3 and 4. In this thesis, we are mainly interested with the socalled nogooddeal approach to valuation of contingent claims in incomplete markets; cf. Chapter 2 to 4. As mentioned before, recall that noarbitrage bounds are typically too large for most practical applications involving nonreplicable claims. The nogooddeal approach is a fairly conservative one that lies between using a single measure for pricing and using all equivalent martingale measures. Indeed the main idea is to obtain tighter valuation bounds, called gooddeal bounds, by using as pricing measures only a subset of the equivalent martingale measures preventing some economically meaningful notion of good deals. The latter could be interpreted as trading opportunities that are too favorable and therefore should also be excluded from the market. Inherent to the concept of gooddeal valuation is therefore already a certain notion of robustness (namely with respect to the smaller set of pricing measures as generalized scenarios). Gooddeal bounds were introduced by [CR00] mostly in discrete time, interpreting gooddeals as trading opportunities that admit an instantaneous mean excess return per unit volatility risk (called instantaneous Sharpe ratios) above a certain threshold. Their nogooddeal constraint therefore was imposed as a bound on the instantaneous Sharpe ratios in a financial market that is extended by additional price processes for derivatives. Their results were rigorously extended to continuous time by [BS06] in a Markovian model of asset prices and additional factor processes possibly exhibiting jumps. Using the socalled HansenJagannathan (HJ) bounds (see [HJ91]), both papers showed that the constraint on the instantaneous Sharpe ratios can be obtained by pricing only under equivalent martingale measures satisfying a bound on the norm of their Girsanov kernels. The HJ bounds basically show that the maximal Sharpe ratio over all portfolio strategies cannot exceed the ratio of the standard deviation of a stochastic discount factor (i.e. the RadonNikodym derivative of the pricing measures) to its mean. In continuous (Brownian) filtrations, imposing a bound on instantaneous Sharpe ratios is basically equivalent to imposing a bound on the optimal expected growth rates [Bec09]. Such local nogooddeal constraints for pricing measures are favorable for good timeconsistency properties of the resulting gooddeal bounds; cf. [KS07b]. Following this remark, we will first consider in Chapters 2 and 3 a general theory of gooddeal valuation and hedging for local constraints on Girsanov kernels given in terms of abstract correspondences. This will provide some flexibility as far as the choice of the nogooddeal constraint is concerned (e.g. in the jump setting of Chapter 2 where the Sharpe ratio constraint is no longer equivalent to the optimal growth rate one), but also will prove necessary in the presence of uncertainty, cf. Chapter 3, where the aggregate nogooddeal constraint under uncertainty may be different from any classical
Page 9 one. Note that gooddeal bounds have also been defined by some notion of expected utilities [CH02, Cer03, KS07b]. Gooddeal theory has been developed for a long time as a pure valuation approach (see [BY08, BL09, MMM13, Mur13] in a Brownian setting and [BS06, KS07b, Don11] in a setting with jumps). Contributions about hedging only appeared recently, mostly in the setting of a Brownian filtration. These started from [Bec09] who uses classical BSDEs to derive hedging strategies as minimizers of dynamic coherent risk measures [ADE+ 07] of nogooddeal type yielding the gooddeal bound as the minimal capital for acceptability, i.e. the market consistent risk measure in the spirit of [BE09]. [CT14] studied meanvariance hedging in the context of gooddeal valuations and concluded that both hedging approaches perform reasonably well. Throughout this thesis, we follow the gooddeal hedging approach of [Bec09], for which valuations and hedges will be described by different classes of BSDEs (classical BSDEs, JBSDEs, 2BSDEs), depending on the framework in use. We note that hedging by minimizing a certain risk measure that allows for market consistent valuation is by now standard in the literature [CGM01, BE05, KS07a, BE09]. In addition, dynamic risk measures in general are wellconnected to BSDEs; cf. [Ros06, PR15].
Robustness and model uncertainty in finance Robustness and model uncertainty are important topics in finance and decision theory; cf. [Con06, HS01]. Since definitions of the nogooddeal constraints involve the objective realworld probability measure, model uncertainty is also relevant to gooddeal theory. Chapters 3 and 4 are concerned with robust approaches to uncertainty, in the Knightian sense (cf. [Kni21]), about the objective probability measure with respect to which good deals are defined, and gooddeal bounds and hedging strategies are computed. In economic theory, it has been argued that incorporating uncertainty aversion provides a theoretical ground for explaining some behavioral observations such as the famous Ellsberg paradox [Ell61] or the equity premium puzzle [MP85]. Uncertainty in financial markets is a serious concern for (typically ambiguityaverse) investors who permanently strive for robustness in the valuation and hedging of their financial risks. Diverse mechanisms have been elaborated in the mathematical finance literature to take into account aversion towards uncertainty in financial modeling. In this thesis, we use a multiple prior approach to robustness under uncertainty proposed by [GS89, CE02], where an uncertaintyaverse investor or decisionmaker seeks to protect herself against an eventual misspecification of probabilities by considering the most conservative (worstcase) line of action with respect to some confidence region of subjective probability measures called priors. The mathematical finance literature in this direction is wide and essentially distinguishes between drift uncertainty and volatility uncertainty. Following the same distinction, we will first consider drift uncertainty in Chapter 3 and then volatility uncertainty in Chapter 4, both in the context of gooddeal
Page 10 theory. Drift uncertainty englobes in particular uncertainty about the market price of risk of traded assets, which naturally embeds into a setup where priors are equivalent to each other, i.e. they share the same nullsets and therefore agree about the impossible events in the market. This framework has been considered for instance in [DW92, Que04, GUW07, Sch08] for solving the maxmin expected utility maximization problem. In Chapter 3 we study robustness of gooddeal hedging strategies for a worstcase approach to gooddeal valuation, which yields larger gooddeal bounds under uncertainty. We show the existence of a worstcase prior under which dynamic valuations and hedges can be computed as in the absence of uncertainty. For our results under drift uncertainty, we rely on classical BSDE methods under a fixed reference prior to which all others are equivalent. In the case of volatility uncertainty in Chapter 4, priors may no longer be equivalent to each other and we use 2BSDEs instead, for deriving valuation and hedging results. Historically, [ALP95, Lyo95] introduced the uncertain volatility model as a model of stock prices in the presence of volatility uncertainty, in which pricing and hedging of contingent claims in incomplete markets can be done in an analog way as in the (complete) BlackScholes model. Typically, priors in the uncertain volatility model are mutually singular, since they may have disjoint supports; see e.g. [DM06, EJ13, EJ14]. This model is by now standard in the literature, and consists in modeling asset price dynamics under riskneutral measures on the pathspace that, being viewed as subjective priors, are parametrized by different volatility processes taking values in a prespecified confidence interval of volatility values. [ALP95, Lyo95] (see also [Vor14] for a model of stock prices as geometric GBrownian motions) derived noarbitrage valuation bounds for financial derivatives in terms of the solution to a fully nonlinear PDE called the BlackScholesBarenblatt equation, which is a nonlinear analog of the BlackScholes PDE in the presence of volatility uncertainty. In particular for convex payoff functions they showed that the worstcase model for the upper valuation bound corresponds to the highest volatility under which the two pricing PDEs coincide. In general when priors are nondominated (i.e. they may disagree about the impossible market scenarios), one has to resort to different techniques for dynamic formulations and solutions of robust stochastic optimization problems; see e.g. [EJ14]. A typical difficulty in this case appears when defining the essential supremum of a family of random variables, which in the dominated case is welldefined up to a null set for a dominating prior. However if the priors are mutually singular, the definition of essential supremums necessitates some aggregation procedures for the nullsets of priors, which can then be disjoint. The quasisure analysis of [DM06]) provides a suitable framework for dealing with these technical issues, and is used for example in [DK13a, MPZ15] for maxmin expected utility maximization under volatility uncertainty. For our 2BSDE approach in Chapter 4, the quasisure analysis will be used naturally following the wellposedness theory of Lipschitz 2BSDEs in [STZ12]. In this chapter, robustness of gooddeal hedging strategies for worstcase gooddeal
Page 11 valuation under volatility uncertainty will be shown. Due to the technical issues mentioned above, we are not able to show existence of a worstcase prior for dynamic gooddeal bounds in the general theory. However, in an example for European put options in a (two dimensional) BlackScholes model for a traded and a nontraded asset, we will constructively identify a worstcase prior for dynamic valuations which mimics the relation to convexity of the payoff function as in [ALP95, Lyo95, Vor14]. A closedform expression of the robust hedging strategy will be subsequently given, after explicitly identifying the solution to the 2BSDE in the example. Let us mention that recent developments in robust finance include the driftandvolatility uncertainty framework of [EJ13, EJ14] for formulation of the pricing, hedging and maxmin expected utility maximization problems in a continuous dynamic setting, taking into account the investor’s uncertainty about both volatility and drift. Solutions to the robust utility maximization problem under driftandvolatility uncertainty in continuous time have been investigated recently by [BP15] using PDE methods, focusing on ellipsoidal driftuncertainty for each fixed volatility scenario. Although dealing only with drift uncertainty, Chapter 3 will consider some cases where the confidence set of drift uncertainty is also described by an ellipsoid, which seems natural for drift uncertainty modeling; cf. also [GUW07]. Note that [Nut14] also recently showed existence of an optimal trading strategy for the maxmin utilitymaximization problem, for arbitrary sets of priors and bounded utility functions, but restricting his analysis to discrete time. Driftandvolatility uncertainty is however not the route followed in this thesis, as both types of uncertainty will be considered separately. This is partly motivated by the fact that in our dynamic setting standard conditions for wellposeness of 2BSDEs (in particular regularity and convexity of the generator) as in [STZ12] may not hold for the dynamic gooddeal valuation and hedging problem in Chapter 4 if one considers drift uncertainty in addition.
Contribution of the thesis This thesis is organized in four chapters which are mostly selfcontained and can be read almost independently. The connections between the chapters’ results can be specified as follows: Chapter 1 deals with a theoretical study of wellposedness and comparison for solutions to BSDEs with jumps of infinite activity and timeinhomogeneous compensators; Chapters 2 to 4 are concerned with gooddeal valuation and hedging. More precisely, Chapter 2 applies some results of Chapter 1 to market models that allow for jumps described by abstract random measures; Chapters 3 and 4 finally study robustness (of valuation and hedges) with respect to uncertainty about the drift of traded assets for the former and about their volatility for the latter. A more detailed chapterwise description of the contribution of the thesis will now be given below.
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Chapter 1: Concrete criteria for wellposedness and comparison of BSDEs with jumps of infinite activity This chapter is based on [BBK15] and studies bounded solutions (Y, Z, U ) to JBSDEs. Recall that in comparison to classical BSDEs which are driven only by a Brownian motion W , such JBSDEs are additionally driven by a random measure µ ˜ and involve a second stochastic integral with respect to the compensated random measure µ ˜ = µ − ν P with integrand U on which the generator f (Y− , Z, U ) may also depend. We extend the analysis of JBSDEs beyond classical Lipschitz assumptions (cf. [TL94, BBP97]) on the generator f by concentrating on a family of generators that satisfy a certain monotonicity property, but do not need to be globally Lipschitz in the U component; see also [Bec06, Roy06]. We do not require the compensator ν(dt, de) of µ(dt, de) to be a product measure like λ(de)⊗dt, as it would be natural for random measures of jumps of L´evy type for instance. Instead, ν is only assumed to be absolutely continuous to some reference product measure λ ⊗ dt with a bounded RadonNikodym derivative ζ (implying timeinhomogeneity) where λ is a σfinite measure, hence allowing for infinite activity of the driving jumps. This provides wide scope for stochastic dependencies between W and µ ˜, which can be relevant in applications; cf. examples in [BS05]. Furthermore, it embeds a range of interesting driving processes for BSDEs in addition to Brownian motion, including L´evy processes, Poisson random measures, marked point processes, Markov chains or much more general step processes (as in [HWY92], Chapter 11, including e.g. semiMarkov processes), connecting our analysis to research from [NS01, CE10]. As usual in the BSDE literature, we require a key property on the filtration, namely that µ ˜ and W together have the weak predictable representation property for martingales. In order to deal with the timeinhomogeneous setting, we slightly extend a general but technical comparison theorem on JBSDE from [Roy06] by using a more general (Aγ )condition, in order to derive sufficient conditions for comparison which are easier to verify, since they are formulated in terms of concrete properties of generator functions from our family of interest. This gives rise to apriori estimates of the L∞ norm of the Y component of the JBSDE solution. Additionally we obtain existence and uniqueness results for bounded JBSDE solutions (i.e. wellposedness of such JBSDEs) in the case of jumps with finite activity, as in [Bec06]. These steps enable us to advance to the case of infinite activity. To this end, we first approximate the generator by truncating the activity of the jumps using the σfiniteness of λ. This leads to a monotone sequence of generators for which solutions do uniquely exist. Then using monotone stability arguments like in [Kob00] enables us to obtain wellposedness for the initial JBSDE generator. However, it turns out that such arguments only work at first for terminal conditions ξ which are small in L∞ norm. By pasting solutions for sufficiently small terminal conditions one can show convergence to the bounded solution of the JBSDE for the original data (ξ, f ). For this purpose, we follow the iterative idea of [Mor10] who focused on a particular generator,
Page 13 and elaborate the proof slightly differently and more compactly for our general setting. For results in related but different directions, we mention [CM08] for comparison of JBSDEs with (doubly) reflection for Lipschitz generators, [KMPZ10] for (minimal) solutions with constraints on jumps for Poisson random measures of finite activity, [DI10] for time delayed generators and [KTPZ15a] for a pasting argument for quadratic JBSDEs following the fixedpoint approach of [Tev08]. We note that our results are mainly stated for generators that are Lipschitz continuous in the Zcomponent. For results on generators of quadratic growth in Z we refer e.g. to [Mor10, EMN14, KTPZ15a]. [EMN14] work in finite activity and [Mor10] considers a particular generator function arising from a specific utility maximization problem. As in our setup [KTPZ15a] also works in infinite activity and considers timeinhomogeneous compensators. However, the applicability of their results may be less straightforward than ours due the abstract nature of their assumptions on the generator which are stated in terms of existence of some abstract processes satisfying strong integrability requirements such that certain nonlinear estimates hold without exception of a null set. Another contribution of this chapter is therefore that we provide concrete conditions on generators that are easier to verify in applications. For illustration purposes, we apply our results to the utility maximization problem in finance, for power and exponential utility functions complementing results e.g. from [HIM05, Sek06, Bec06, Mor10, Nut12a]. In Chapter 2, a nonlinear example on gooddeal valuation and hedging in incomplete markets with jumps will be also covered.
Chapter 2: Hedging under generalized gooddeal bounds in jump models with random measures In this chapter, which is based on [BK15c], we study gooddeal hedging and valuation in general jump models driven by random measures. More precisely, we suppose the presence of unpredictable eventrisk in the market in the sense that the information flow (filtration) may be discontinuous, allowing for nontrivial purelydiscontinuous price processes with totally inaccessible jump times. [BS06] first considered gooddeal valuation in a Markovian setting with jumps, where gooddeal bounds were defined by a constraint on the instantaneous Sharpe ratios and derived subsequently as solution to HJB equations. Although [BS06] focused only on valuation, they raised the crucial need for a hedging theory that corresponds to gooddeal valuation. A first attempt to gooddeal hedging was by [Bec09] in a Brownian setting, who derived hedging strategies from minimizing suitable dynamic coherent risk measures that allow for optimal risk sharing with the market through gooddeal valuation. Another attempt, still in a Brownian setting, is based on a quadratic hedging criterion like meanvariance hedging and was developed by [CT14]. In the jump setting, [Del12] studied the approaches of [Bec09, CT14] for pointprocesses with stateindependent jump intensities, restricting to a market with two
Page 14 risky assets (a traded and a nontraded one) and considering solely Sharpe ratio nogooddeal constraints. For a related problem, we study in this chapter a generalization to multiple nonnecessarily Markovian risky assets, more general jump processes driven by abstract integervalued random measures and generalized nogooddeal constraints on Girsanov kernels of pricing measures parametrized by correspondences (i.e. setvalued mappings as studied in [AF90]). Using correspondences provides an abstract framework for incorporating nogooddeal restrictions of different natures (e.g. instantaneous Sharpe ratios, optimal growth rates, instantaneous Sharpe ratios under uncertainty about jump intensities, etc.). We derive generalized gooddeal valuation bounds for possibly pathdependent contingent claims using classical JBSDEs instead of HJB equations (under Markovian assumptions) as in [BS06, Don11, Mur13]. We first consider the case of uniformly bounded correspondences, for which the generators of the resulting JBSDEs are Lipschitz continuous. The resulting valuation JBSDE is derived from the comparison principle of Chapter 1, and its generator is the maximum of a family of linear JBSDE generators parametrized by the Girsanov kernels of nogooddeal pricing measures. This generator in general does not have an explicit form, even for the classical radial (Sharpe ratio) nogooddeal constraint. To obtain more constructive (or even explicit) expressions of the generators, we assume more structure on the contingent claim, the nature of the nogooddeal constraint or the random measure of the jumps. Examples are presented with closedform expressions for the associated gooddeal hedging strategy. The case beyond uniform boundedness of the correspondence is also considered, where the Lipschitz property of the generators is no longer ensured and approximation arguments are used. Using a notion of gooddeal hedging introduced by [Bec09], we contribute results on the existence of hedging strategies for arbitrary bounded correspondences. Moreover we obtain a characterization of the hedging strategies in terms of the solution to the JBSDE describing the gooddeal valuation bound. We show that the tracking errors of hedging strategies, i.e. the dynamic difference between the gooddeal bound and the profit/loss from trading, satisfy a supermartingale property under some apriori valuation measures including the nogooddeal measures. As the martingale property of the tracking errors corresponds to hedging strategies being meanselffinancing in the terminology of [Sch01], this means that the gooddeal hedging strategies can be viewed as being at least meanselffinancing under every apriori valuation measure. The latter can be interpreted as a robustness property of the gooddeal hedging strategy with respect to the set of apriori valuation measures as generalized scenarios in the sense of [ADE+ 07]. For concrete nogooddeal constraints we provide some examples where a gooddeal hedging strategy can be obtained explicitly in terms of solutions to JBSDEs. Hedging is investigated only for bounded correspondences, and in the case beyond uniform boundedness we only present results about gooddeal valuations. In a discontinuous filtration, imposing a bound on the instantaneous Sharpe ratios via a bound on the norms of the Girsanov kernels of pricing measures (as in [BS06]) is not equivalent to
Page 15 imposing a bound on the optimal conditional expected growth rates (as in [Bec09]). We note that the growth rate constraint is mathematically less tractable than the Sharpe ratio one, at least in terms of using Lipschitz BSDEs. Indeed, it turns out that Sharpe ratio constraints fit well with the theory of Lipschitz JBSDEs for arbitrary random measures. For such constraints and for some concrete random measures of jumps, we can even obtain more simplified JBSDE descriptions of gooddeal bounds and hedging strategies. In particular for jumps of a continuoustime Markov chain and without a Gaussian component, we infer from [CS12] that the JBSDE for gooddeal bounds defined from Sharpe ratio constraints, for Markovian European contingent claims depending only on the terminal value of the chain, reduces to a fullycoupled system of ordinary differential equations (ODEs). The latter can be transformed (by reversing time ) into an initial value problem, which can then be solved using any standard numerical ODE solver. For Sharpe ratio constraints, we also present an example for robust hedging under Knightian uncertainty about the intensity of the underlying jump process, linking the result here with those of Chapters 3 and 4. Here robustness of a hedging strategy with respect to uncertainty refers to a property of being at least meanselffinancing under every apriori valuation measure, uniformly with respect to a family of subjective probability measures as candidates for the realworld measure and capturing the uncertainty. On the other hand, optimal growth rate constraints do not fit well with Lipschitz BSDEs since the resulting correspondence may not be uniformly bounded. For such constraints it turns out that we can still rely on the theory of Lipschitz JBSDEs when random measures with finite support of the compensator. Results are then obtained for finitestate semiMarkov processes, which are a flexible class with many practical applications, see e.g. for actuarial applications [BMS14] and references therein.
Chapter 3: Hedging under generalized gooddeal bounds and drift uncertainty This chapter is based on [BK15b] and is concerned with approaches to gooddeal hedging (as in [Bec09]) under ambiguity about the objective probability with respect to which good deals are defined and gooddeal bounds computed. Gooddeal valuations fit into the theory of dynamic monetary convex risk measures (or monetary convex utility functionals) for which results, in particular about dual representations and time consistency, exist in high generality; see e.g. [KS07a, BNN13, DK13b]. We contribute constructive and qualitative results on the (robust, gooddeal) hedging strategies, that facilitate interpretation and are accessible to computation. We pose the gooddeal valuation and hedging problem in a framework with multiple priors, and follow a robust worstcase approach as in [GS89, CE02]. We note that results on gooddeal valuations and hedges under uncertainty in a recent work by [BCCH14] are very different to ours. Indeed, they work mostly in discrete time and study numerical results for a different uncertaintypenalized preference functional, whereas we use dynamic coherent risk measures in continuous time and focus on rather analytical results.
Page 16 After formulating a framework with predictable correspondences as in Chapter 2 but in the Wiener setting, we also describe gooddeal hedging strategies and valuation bounds in terms of solutions to classical BSDEs. In the absence of uncertainty, we obtain results first for uniformly bounded correspondences and then for the case beyond uniform boundedness by approximation. Additionally we characterize the gooddeal valuation bounds in the possibly unbounded case in terms of minimal supersolutions to convex BSDEs (as in [DHK13]). Notably, the abstract generalized constraints are needed to cover relevant examples of uncertainty about the market prices of risk of the assets that are available in the (incomplete) market for dynamic partial hedging. For illustration purposes, we will consider e.g. ellipsoidal correspondences which permit explicit analytic generators in the BSDEs of interest, being efficient for MonteCarlo approximation. In general gooddeal hedging strategies can comprise a speculative bet in the direction of the market price of risk to compensate for unhedgeable risks. In this chapter we also provide new examples on gooddeal valuation and hedging, with closedform formulas for gooddeal bounds and hedging strategies: For an exchange option between tradeable and nontradeable assets, we give a Margrabetype formula [Mar78] for the gooddeal bound, with adjusted input parameters. For the stochastic volatility model by Heston [Hes93], we obtain semiexplicit formulas under gooddeal constraints for pricing measures, which restrict the meanreversion level of the stochastic variance process to be within some confidence interval. A graphical analysis of the dependency on model parameters is also done. An interesting aspect of the latter example shows, how a robust valuation of volatility risk (over a family of nogooddeal pricing measures) can be obtained for an absolutely continuous family of measures. To illustrate to which extend our BSDE solutions could be computed by efficient but generic MonteCarlo methods, complementing numerical approaches to hedging from [CT14], we investigate the errors between the MonteCarlo approximations and our analytic formula in a four dimensional example for an exchange option. In the presence of uncertainty, we derive general results for gooddeal bounds and hedging strategies that are robust with respect to uncertainty, described also by correspondences. Building on a suitable definition of gooddeal bounds in the presence of uncertainty, we note that the problem with multiple priors can be related to a respective problem without uncertainty but with an enlarged gooddeal constraint correspondence, which even in the most natural cases of nogooddeal restriction and uncertainty may easily not have a radial shape; hence the need of a general theory for abstract correspondences in the first place. The worstcase approach naturally leads to a robust notion of valuation by the widest gooddeal bounds that are obtained over all probabilistic models under consideration. We show that there is also a notion for robust hedging, which corresponds to the aforementioned robust gooddeal valuation. Indeed, there exists a unique strategy that is robustly at least meanselffinancing, in the sense that it is at least meanselffinancing (see Chapter 2) uniformly with respect to all priors. By saddle point arguments we derive a minmax identity, that shows how the robust gooddeal hedging strategy
Page 17 is given by the (ordinary) gooddeal hedging strategy with respect to a worstcase measure. Since we rely on BSDEs both are actually identified in a constructive manner. As intuition suggests, a robust approach to uncertainty reduces the speculative component of the gooddeal hedging strategy. As a further contribution, we prove that if the uncertainty is large enough in relation to the nogooddeal constraints, then the robust gooddeal hedging strategy does no longer include any speculative component, but coincides with the (globally) riskminimizing strategy of [FS86]. This offers theoretical support to the commonly held perception that hedging should abstain from speculative objectives (see e.g. [LP00]), and moreover a new justification for riskminimization. Finally, an example with closedform solutions for robust gooddeal bounds and hedging strategies for an option on a nontraded asset illustrates results and graphically analyzes dependencies on parameters. This chapter has built over the Masters’ thesis [Ken11] for preliminary results including those about generalized gooddeal bounds in the absence of uncertainty for uniformly bounded and ellipsoidal correspondences, and part of worstcase valuation in the presence of uncertainty. All remaining results are new; among others, the examples with closedform expressions for the gooddeal bound and hedging strategy, the saddlepoint results on worstcase valuation and hedging in the presence of uncertainty, and the link to riskminimization obtained in the last section.
Chapter 4: Hedging under gooddeal bounds and volatility uncertainty: a 2BSDE approach Chapter 4 is based on [BK15a] and deals with robust gooddeal hedging and valuation with respect to volatility uncertainty. We consider also here an approach under which gooddeal bounds are computed as worstcase valuations over a calibrated class of priors. Contrary to drift uncertainty for which BSDE descriptions can be given under a single dominating prior (see Chapter 3), volatility uncertainty corresponds to priors that are mutually singular, and therefore necessitates a different mathematical framework for valuation and hedging results. In particular a rigorous definition of the dynamic gooddeal bound as essential supremum/infimum of random variables involves additional technical care since it may not be possible to aggregate the nullsets of the different priors. First we present some purely theoretical results on the comparison principle for solutions to 2BSDEs with different generators and terminal conditions, thus extending a result in [STZ12] for which the generators are identical. We use the socalled strong formulation of volatility uncertainty, which considers the uncertain priors as local martingale laws of stock price processes defined on the canonical Wiener space. Our definitions of worstcase gooddeal bounds and hedging strategies are adapted to this framework and we follow a setup by [STZ13], which starting from the canonical space and working with regular conditional probability distributions
Page 18 (in short r.c.p.d.) ensures a timeconsistency property (in the spirit of [NS12]) of the gooddeal bounds as dynamic risk measures under volatility uncertainty. This paves the way for defining gooddeal hedging again as minimization of residual risk from dynamic trading. In this chapter, we concentrate on a nogooddeal restriction imposed as a bound on the instantaneous Sharpe ratios under each reference prior separately. This provides a family of gooddeal bound processes parametrized by priors, and the worstcase upper gooddeal bound arises as the largest among them, i.e. their essential supremum. Building on the intuition from Chapter 3, we derive robust gooddeal bounds and hedging strategies under volatility uncertainty in terms of solutions to Lipschitz 2BSDEs relying on the theory in [STZ12]. Again as in Chapter 3 robustness of the gooddeal hedging strategy with respect to uncertainty is related to the property of being at least meanselffinancing uniformly over all priors. Finally, we contribute an example for put options on nontraded assets under volatility uncertainty, in which a worstcase model can be explicitly computed for the dynamic gooddeal bound and closedform formulas for robust valuations and hedges are derived, like in [ALP95, Lyo95, Vor14]. The latter works focus on robust superhedging in the presence of volatility uncertainty, whereas we focus on gooddeal hedging. As an example demonstrates, our robust gooddeal hedging strategy (and respective valuation bounds) can in general be very different from the superreplicating one.
1. Concrete criteria for wellposedness and comparison of BSDEs with jumps of infinite activity In this chapter, we study JBSDEs for a specific class of generator functions that do not necessarily satisfy global Lipschitz conditions in the jump integrand. The JBSDEs in consideration are driven, additionally to a Brownian motion, by general random measures with compensators that can be inhomogeneous in time and may allow for infinite activity of the jumps of the value process. In this context, we provide in Sections 1.2 and 1.3 concrete conditions that are directly verifiable for existence, uniqueness and comparison of bounded solutions to such JBSDEs, first in the case of finite activity of the jumps and then to the infinite activity case by suitable approximations. Section 1.4 illustrates the range of applicability of our results by solving the utility maximization problem in finance for exponential and power utility functions with additive and multiplicative liability respectively. To make this chapter as selfcontained as possible, we first introduce some useful notations and the mathematical preliminaries.
1.1
Mathematical framework and preliminaries
This section presents the technical framework and sets the notations. We will also summarize the key assumptions on the BSDE generator (1.7) which will play, in varying combinations, a role in our later results. First we recall essential facts on stochastic integration with respect to random measures and on bounded solutions for Backward SDEs which are driven jointly by Brownian motions and a compensated random measure. For notions from stochastic analysis not explained here we refer to [JS03] and [HWY92]. Inequalities between measurable functions are understood almost everywhere with respect to an appropriate reference measure, typically P or P ⊗ dt. Let T < ∞ be a finite time horizon and (Ω, F, (Ft )0≤t≤T , P ) a filtered probability space with a filtration satisfying the usual conditions of right continuity and completeness, assuming FT = F and F0 being trivial (under P ). Due to the usual conditions we can and do take all semimartingales to have right continuous paths with left limits, socalled c`adl`ag paths. Expectations under a probability Q and Conditional expectations given Ft are denoted by E Q [·] and EtQ [·] respectively, or simply E[·] and Et [·] when Q = P . Reference to the probability is omitted if clear from context. Let H be a separable Hilbert space and we denote by B(E) the Borel σfield of E := H\{0}, e.g. H = Rl , l ∈ N or H = `2 ⊂ RN . Then (E, B(E)) is a standard Borel space. In addition, let W be a ddimensional Brownian motion. Stochastic integrals of a vector valued predictable process Z 19
Section 1.1. Mathematical framework and preliminaries
Page 20
with respect to a semimartingale X, e.g. X = W , of the same dimensionality are scalar valued R R semimartingales starting at zero and denoted by (0,t] ZdX = 0t ZdX = Z · Xt for t ∈ [0, T ]. The predictable σfield on Ω × [0, T ] generated by all left continuous adapted processes is e := Ω × [0, T ] × E. denoted by P and Pe := P ⊗ B(E) is the respective σfield on Ω Let µ be an integervalued random measure with compensator ν = ν P (under P ) which is assumed to be absolutely continuous to λ ⊗ dt for a σfinite measure λ on (E, B(E)) satisfying R 2 e E 1 ∧ e λ(de) < ∞ with some Pmeasurable, bounded and nonnegative density ζ, such that ν(dt, de) = ζt (e) λ(de) dt = ζt dλ dt, (1.1) with 0 ≤ ζt (e) ≤ cν P ⊗ λ ⊗ dta.e. for some constant cν > 0. L2 (λ) (resp. L2 (ζt dλ)) R defines the space that of Emeasurable functions γ : E → R with E γ(e)2 λ(de) < ∞ (resp. R 2 2 2 E γ(e) ζt (e)λ(de) < ∞). Note that the Hilbert spaces L (λ) and L (ζt dλ), are separable since the underlying measures are σfinite and the σalgebra E is countably generated (see [Coh13, Proposition 3.4.5]). Hence they admit countable orthonormal bases and are in particular Polish spaces. Since the density ζ can depend on t and ω, the compensating measure ν may be timeinhomogeneous and stochastic. This permits for a richer dependence structure between W and µ; for instance the density ζ and thereby the intensity of the jump measure might fluctuate in dependence of some diffusion process driven by W . Let Q be a probability measures. We denote by Lp (Q), 1 ≤ p < ∞, the space of FT measurable random variable X with kXkpLp (Q) := E Q [Xp ] < ∞, and L∞ (Q) the space of FT measurable random variable kXkL∞ := kXk∞ = ess supQ X < ∞. For a function U : [0, T ] × Ω × E → R we define U ∞ := ess sup(t,e) Ut (e). For stochastic integration with e and W we define sets of Rvalued processes respect to µ ( p
S (Q) :=
!
Y c`adl`ag : Y p := E (
∞
S (Q) :=
Y c`adl`ag : Y ∞
Q
p
sup Yt 
:=
e U Pmeasurable :
for p ∈ [1, ∞) ,
−1 and E(exp(hγ ∗ µ eiT )) = E exp 0T E γ s (e)2 ν(ds, de) 1. ∆(γ ∗ µ < ∞ (see R 2 Theorem 9, [PS08]). In particular, this holds if E γ s (e) ζs (e) λ(de) < const. < ∞ P ⊗ dsa.e. and γ > −1. R
e) ≥ −1 + δ for some δ > 0 and γ ∗ µ e is a BM O(P )martingale. This is due to 2. ∆(γ ∗ µ Kazamaki [Kaz79]. e) ≥ −1 and γ ∗ µ e is a uniformly integrable martingale and E(exp(hγ ∗ µ eiT )) < ∞ 3. ∆(γ ∗ µ (see Theorem I.8, [LM78]). Such a condition is satisfied when γ is bounded and γ ≤ ψ, P ⊗ dt ⊗ dλa.s. for a function ψ ∈ L2 (λ) and ζ ≡ 1. The latter is what is required for instance in the comparison Theorem 4.2 of [QS13]. R
e) for β Note that under above conditions, also the stochastic exponential E( βdW + γ ∗ µ bounded and predictable is a martingale, as it is easily seen by Novikov’s criterion.
In the statement of Proposition 1.4, the dependence of the process γ on the BSDE solutions is not needed for the proof as the same result holds if γ is just a predictable process such that the estimate on the generator f2 and the martingale property (1.9) hold. The further functional dependence is needed for the sequel, as required in the following Definition 1.7. An Rvalued generator function f is said to satisfy condition (Aγ ) if there is a P ⊗ B(Rd+3 ) ⊗ B(E)measurable function γ : Ω × [0, T ] × Rd+3 × E → (−1, ∞) given 0 by (ω, t, y, z, u, u0 , e) 7→ γty,z,u,u (e) such that for all (Y, Z, U, U 0 ) ∈ S ∞ × H2 × (Hν2 )2 with 0 U ∞ < ∞, U 0 ∞ < ∞ it holds for γ := γ Y− ,Z,U,U ft (Yt− , Zt , Ut )−ft (Yt− , Zt , Ut0 ) ≤
Z E
γ t (e)(Ut (e) − Ut0 (e))ζt (e)λ(de), P ⊗ dta.s.
(1.12)
R
e) is a martingale for every bounded and predictable β. and E( βdW + γ ∗ µ
A function f satisfies condition (A0γ ) if the above holds for all bounded U and U 0 with additional e ∈ BM O(P ) and U 0 ∗ µ e ∈ BM O(P ). property that U ∗ µ Clearly, existence and applicability of a suitable comparison result of solutions to JBSDEs implies their uniqueness. In other words assuming there exists a bounded solution for a Lipschitz driver with respect to y and z which satisfies (Aγ ) or (A0γ ), we obtain that such a solution is unique. 0
Example 1.8. A natural candidate γ for drivers f of the form (1.7) is given by γsy,z,u,u (e) := R1 ∂ 0 0 ∂u gs (y, z, tu + (1 − t)u , e) dt 1A (e), assuming differentiability of g. Indeed, we have 0
1
∂ (gs (y, z, tu + (1 − t)u0 , e)) dt 1A (e) 0 ∂t = (gs (y, z, u, e) − gs (y, z, u0 , e))1A (e),
γsy,z,u,u (e)(u − u0 ) =
Z
Section 1.2. Comparison theorems and aprioriestimates
Page 28
and hence the function γ is P ⊗ B(Rd+3 ) ⊗ B(E)measurable since y,z,u,u0
γs
gs (y,z,u,e)−gs (y,z,u0 ,e) 1 (e), A u−u0 (e) = 0,
u 6= u0 u = u0 ,
∂g are P ⊗ B(Rd+3 ) ⊗ B(E)measurable. By the mean value theorem γ has the form and g, ∂u 0
γsy,z,u,u (e) =
∂ g(s, y, z, v, e)1A (e), ∂u
(1.13)
for some v between u and u0 . For generators of the form (1.8) γ simplifies to 0 γsy,z,u,u (e)
Z
= 0
1
∂ gs (tu + (1 − t)u0 , e) dt 1A (e) . ∂u
Definition 1.9. A generator f satisfies condition (Af in ) (on D or for elements in D) or (Ainf i ) if 1. (Af in ): f is of the form (1.7) with λ(A) < ∞, is Lipschitz continuous with respect to y and z, and the map u 7→ g(t, y, z, u, e) is continuously differentiable for all (ω, t, y, z, e) (in D) such that the derivative is strictly greater than −1 (on D ⊆ Ω×[0, T ]×R×Rd ×E) and locally bounded (in u) from above, uniformly in (ω, t, y, z, e). 2. (Ainf i ): f is of the form (1.8), is Lipschitz continuous with respect to y and z, and the map u 7→ gt (u, e) is twice continuously differentiable for all (ω, t, e) with the derivatives being locally (in u) bounded uniformly in (ω, t, e), the first derivative bounded away from ∂g −1 with a lower bound −1 + δ for some δ > 0, and ∂u (t, 0, e) ≡ 0. Example 1.10. Sufficient conditions for (Aγ ) and (A0γ ) are 1. γ is a P ⊗ B(Rd+3 ) ⊗ B(E)measurable satisfying the inequality in (1.12) and 0
C1 (1 ∧ e) ≤ γty,z,u,u (e) ≤ C2 (1 ∧ e), for some C1 ∈ (−1, 0] and C2 > 0 on E = Rd \{0}. In this case exp h βdW + γ ∗ R eiT is clearly bounded and the jumps of βdW + γ ∗ µ e are bigger than −1. Hence µ R e E ( βdW + γ ∗ µ) is a positive martingale ([PS08], Theorem 9). Thus Definition 1.7 generalizes the original (Aγ )condition introduced in [Roy06] for Poisson random measures. R
2. (Af in ) is sufficient for (Aγ ). This follows from Example 1.6.1, (1.13) and λ(A) < ∞. 3. (Ainf i ) is sufficient for (A0γ ). To see this, let u, u0 be bounded by c and γ be the natural candidate in Example 1.8. By the mean value theorem there exist v(e) between u and u0
Section 1.2. Comparison theorems and aprioriestimates
Page 29
and ve(e) between 0 and v(e) such that ∂ g(s, v(e), e)1A (e) ∂u ∂ ∂ ∂2 = g(s, v(e), e) − g(s, 0, e) 1A (e) = v(e) 2 g(s, ve(e), e)1A (e). ∂u ∂u ∂u
0
γsy,z,u,u (e) =
0
So γsy,z,u,u (e) is bounded uniformly in (ω, s, e) by 2 y,z,u,u0 ≤ sup ∂ g(s, y, z, u, e) u + u0  , γ (e) s 2 u≤c
∂u
e is a BMOmartingale by the BMOproperty of U ∗ µ e and U 0 ∗ µ e hence βdW + γ ∗ µ R e) is a martingale by with some lower bound −1 + δ for its jumps. And E( βdW + γ ∗ µ Kazamaki’s criterion of Example 1.6. R
As an application of the above, we can now provide simple conditions for comparison in terms of concrete properties of the generator function, which are much easier to verify than the more general but abstract conditions on the existence of a suitable function γ as in Proposition 1.4 or the general conditions by [CE10]. Note that no convexity is required in the z or u argument of the generator. The result will be applied later to prove existence and uniqueness of JBSDE solutions. Theorem 1.11 (Comparison Theorem). A comparison result between bounded BSDE solutions in the sense of Proposition 1.4 holds true in each of the following cases: 1. (finite activity) f2 satisfies (Af in ). e and U 2 ∗ µ e are BMO(P )martingales 2. (infinite activity) f2 satisfies (Ainf i ) and U 1 ∗ µ 1 1 for the corresponding JBSDE solutions (Y , Z , U 1 ) and (Y 2 , Z 2 , U 2 ).
Proof. This follows directly from Proposition 1.4 and Example 1.10, noting that representation (1.13) in connection with condition (Af in ) resp. (Ainf i ) meets the sufficient conditions in Example 1.6. Unlike classical apriori estimates that offer some L2 norm estimates for the BSDE solution in terms of the data, the next result gives a simple L∞ estimate for the Y component of the solution. Such will be useful for the derivation of BSDE solution bounds and for truncation arguments. Proposition 1.12. Let (Y, Z, U ) ∈ S ∞ × H2 × Hν2 be a solution to the BSDE (ξ, f ) with ξ ∈ L∞ (FT ). Let f be Lipschitz continuous with respect to (y, z) with Lipschitz constant Kfy,z and satisfying (Aγ ) with f. (0, 0, 0) bounded. Then Yt  ≤ exp Kfy,z (T − t) ξ∞ + (T − t)f. (0, 0, 0)∞
Section 1.2. Comparison theorems and aprioriestimates
Page 30
holds for all t ≤ T . Proof. Set (Y 1 , Z 1 , U 1 ) = (Y, Z, U ), (ξ 1 , f 1 ) = (ξ, f ), (Y 2 , Z 2 , U 2 ) = (0, 0, 0) and (ξ 2 , f 2 ) = (0, f ), Then following the proof of Proposition 1.4, equation (1.11) becomes (RY )τ ∧t ≤ (RY )τ ∧T +
Z
τ ∧T
τ ∧t
Rs fs (0, 0, 0) ds − (LτT − Lτt ),
t ∈ [0, T ],
e for all stopping times τ where L := M − hM, N i is in Mloc (Q), M := RZ dW + (RU ) ∗ µ R 2 0,0,U,0 e with γ := γ is in M , N := β dW + γ ∗ µ and the probability measure Q ∼ P is given by dQ := E(N )T dP . Localizing (Lt )0≤t≤T along some sequence (τ n )n∈N ↑ ∞ yields R ∧T EtQ [(RY )τ n ∧t ] ≤ EtQ [(RY )τ n ∧T + ττ∧t Rs fs (0, 0, 0) ds]. By dominated convergence, we conclude that P a.e R
"
Yt ≤
EtQ
RT ξ+ Rt
Z t
T
#
y,z Rs fs (0, 0, 0) ds ≤ eKf (T −t) (ξ∞ + (T − t)f· (0, 0, 0)∞ ). Rt
e with γ e := γ 0,0,0,U , and Q equivalent to P via Analogously, if we define N := β dW + γe ∗ µ dQ := E(N )T dP , we deduce that L := M − hM, N i ∈ Mloc (Q) and R
(RY )τ ∧t ≥ (RY )τ ∧T +
Z
τ ∧T
τ ∧t
τ
τ
Rs fs (0, 0, 0) ds − (LT − Lt ),
t ∈ [0, T ],
for all stopping times τ . This yields the lower bound. Again, we can specify explicit conditions on the generator function that are sufficient to ensure the more abstract assumptions of the previous result. Theorem 1.13. Let (Y, Z, U ) ∈ S ∞ × H2 × Hν2 be a solution to the BSDE (ξ, f ) with ξ in L∞ (FT ), f being Lipschitz continuous with respect to y and z with Lipschitz constant Kfy,z such that f. (0, 0, 0) is bounded. Assume that one of the following conditions holds: 1. (finite activity) f has property (Af in ). e is a BMO(P )martingale. 2. (infinite activity) f has property (Ainf i ) and U ∗ µ
Then Yt  ≤ exp(Kfy,z (T − t))(ξ∞ + (T − t)f· (0, 0, 0)∞ ) holds for all t ≤ T , and in particular Y ∞ ≤ exp(Kfy,z T )(ξ∞ + T f· (0, 0, 0)∞ ). Proof. This follows directly from Proposition 1.12 and Example 1.10, since f satisfies condition (Aγ ) (resp. (A0γ )) using equation (1.13).
Section 1.2. Comparison theorems and aprioriestimates
Page 31
In the last part of this section we apply our comparison theorem for more concrete generators. To this end, we consider a truncation fe of a generator f at truncation bounds a < b (depending on time only), given by fet (y, z, u) := ft (κ(t, y), z, κ(t, y + u) − κ(t, y)),
(1.14)
with κ(t, y) := (a(t) ∨ y) ∧ b(t). Next, we show that if a generator satisfies (Aγ ) within the truncation bounds, then the truncated generator satisfies (Aγ ) everywhere. Lemma 1.14. Let f satisfy (1.12) for Y, U such that a(t) ≤ Yt− , Yt− + Ut (e), Yt− + Ut0 (e) ≤ b(t),
t ∈ [0, T ],
e). and let γ satisfy one of the conditions of Example 1.6 for the martingale property of E(γ ∗ µ e Then f satisfies (1.12). Especially, if f satisfies (Af in ) on the set where a(t) ≤ y, y + u ≤ b(t) then fe is Lipschitz in y and z, locally Lipschitz in u and satisfies (Aγ ).
Proof. Indeed, we have fet (Yt− , Zt , Ut ) − fet (Yt− , Zt , Ut0 ) = ft (κ(t, Yt− ), Zt , κ(t, Yt− + Ut ) − κ(t, Yt− )) − ft (κ(t, Yt− ), Zt , κ(t, Yt− + Ut0 ) − κ(t, Yt− )) ≤ ≤
Z ZE E
γ t (e)(κ(t, Yt− + Ut (e)) − κ(t, Yt− + Ut0 (e))) ζt (e) λ(de) γ t (e)(1{γ≥0,U ≥U 0 } + 1{γ −1 , (t, ω) ∈ [0, T ] × Ω.
(2.10)
Section 2.2. Gooddeal valuation and hedging
Page 62
For a correspondence C satisfying (2.10) we will associate a closedconvexvalued correspondence Ce with values n
Cet (ω) := (γ, β) ∈ L2 (λ) × Rn :
o γ 1{ζt (ω)>0} , β ∈ C¯t (ω) , ζt (ω)
p
(2.11)
in the Hilbert space L2 (λ) × Rn , where C¯t (ω) is the closure of Ct (ω) in L2 (λt (ω)) × Rn , (t, ω) ∈ [0, T ] × Ω. Let P denote the completion of the predictable σfield P under the measure P ⊗ dt. For the application of standard measurable selection arguments in our general setting, we will need the following Assumption 2.9. The correspondence Ce associated to C by (2.11) is Pmeasurable. Let us note at first that Assumption 2.9 will be satisfied (e.g. in Lemma 2.22 and Lemma 2.30) for some concrete examples of nogooddeal constraint correspondences and general measures λ. Also note that assuming Pmeasurability is weaker than assuming predictability. The definition of measurability of a correspondence is in the sense of [AF90]: ofor each closed n 2 n −1 e set F ⊂ L (λ) × R , the set C (F ) := (t, ω) ∈ [0, T ] × Ω : Cet (ω) ∩ F 6= ∅ is measurable. Note that completeness of the underlying σfield is usually required in the theory of measurable correspondences with values in an infinitedimensional spaces. Since our correspondence Ce by definition assumes values in L2 (λ) × R for a possibly infinitelysupported measure λ on R, it appears natural to require Assumption 2.9 for P instead of P. For correspondences taking values in a finite dimensional space, completeness of the underlying σfield is not necessary. For the theory of measurable correspondences and existence of measurable selection, we refer to [AF90, Chapter 8] in infinite dimensional spaces, and to [Roc76] in finite dimensional space. For measures λ that are finitelysupported (e.g. for finite state semiMarkov processes) we will therefore rely on results of [Roc76]. The particularity of Ce in comparison to C is that the range of Ce does not depend on (t, ω), and it will be useful for applying measurable selection arguments (e.g. in the proof of Lemma 2.14). e e) is a positive For predictable β and Pmeasurable γ such that Γγ,β := E (β · W + γ ∗ µ γ,β the probability measure equivalent to uniformly integrable martingale, we denote by Q P with Girsanov kernels (γ, β), i.e. with density process Γγ,β . For riskneutral measures Qγ,β ∈ Me , the martingale condition of Lemma 2.8 additionally requires that β = −ξ + η with ηt (ω) ∈ Ker σt (ω), (t, ω) ∈ [0, T ] × Ω. Hence we define the set Qngd := Qngd (S) of nogooddeal pricing measures as n
o
Qngd := Qγ,β ∼ P : (γ, β) ∈ C, β = −ξ + η, η ∈ Ker σ ⊆ Me ,
(2.12)
where we do (implicitly) require that Γγ,β is a uniformly integrable martingale to define probability measures Qγ,β . Assume (0, −ξ) ∈ C,
(2.13)
Section 2.2. Gooddeal valuation and hedging
Page 63
b γ ,β b = (0, −ξ). In (2.12) we implicitly b = Qb which implies in particular Q ∈ Qngd 6= ∅ since (γb , β) required the γcomponents of Girsanov kernels to satisfy γt ∈ L2 (λt ) for any t ∈ [0, T ]; see (2.10). Note that such a restriction is also made indirectly by [BS06], where the constraint on the size of the instantaneous Sharpe ratios via the HansenJagannathan bounds implicitly requires the L2 (λt )norm of γt to be finite, for any t ∈ [0, T ]. For instance the latter holds e is a locally square integrable local martingale such that γ2 ∗ ν is locally integrable if γ ∗ µ (cf. [JS03, Chapter II, Section 1]). Under uniform boundedness of the correspondence C (see e will be automatically satisfied. Section 2.3), the local square integrability of γ ∗ µ
For sufficiently integrable contingent claims X, the upper and lower gooddeal valuation bounds are defined by πtu (X) := ess sup EtQ [X] Q∈Qngd
and
πtl (X) := ess inf EtQ [X], Q∈Qngd
t ∈ [0, T ].
(2.14)
Because π l (X) = −π u (−X), we focus on studying the upper gooddeal bound. Recall the property of multiplicatively stable (shortly mstable) sets Q of probability measures Q ∼ P : for all Γ1 , Γ2 ∈ Q and τ ≤ T stopping time, the process Γ with Γt := 1{t≤τ } Γ1t + 1{τ ≤t} Γ1τ Γ2t /Γ2τ belongs to Q, or equivalently the random variable ΓT := Γ1τ Γ2T /Γ2τ defines an element of Q. The following result of [Del06] (stated here as in [KS07b, Theorem 2.7] or [Bec09, Proposition 2.6]) provides good dynamic properties for suprema of conditional expectations over an mstable set of equivalent measures. Lemma 2.10. Let Q be a convex and mstable set of measures Q ∼ P and πtu,Q (X) := ess supQ∈Q EtQ [X], t ∈ [0, T ], X ∈ L∞ . Then for all X ∈ L∞ there exists a c`ad`ag version Y of π·u,Q (X) such that Yτ = ess supQ∈Q EτQ [X] =: πτu,Q (X) for any stopping time τ ≤ T . Moreover π·u,Q (·) satisfies the properties of a dynamic coherent risk measure. It is recursive and time consistent: For all σ ≤ τ ≤ T stopping times and for all X1 , X2 ∈ L∞ , πσu,Q (πτu,Q (X 1 )) = πσu,Q (X 1 ) and πτu,Q (X 1 ) ≥ πτu,Q (X 2 ) implies πσu,Q (X 1 ) ≥ πσu,Q (X 2 ). Finally, a supermartingale property holds: for all Q ∈ Q and for all stopping times σ ≤ τ ≤ T , πσu,Q (X) ≥ EσQ πτu,Q (X) . In particular, π·u,Q (X) is a supermartingale under any Q ∈ Q. One can show that the sets Qngd , Me are convex and mstable, enabling an application of ngd Lemma 2.10 to the gooddeal bound π·u (X) = π·u,Q (X). That Me := Me (S) is mstable and convex is a consequence of [Del06, Proposition 5]. The next result shows that Qngd as defined in (2.12) is also convex and mstable; the proof is included in the appendix. Lemma 2.11. The set Qngd is convex and mstable. We use a notion of gooddeal hedging similar to the one in [Bec09] (see also Chapter 3), where a hedging strategy arises as the minimizer of a suitable apriori dynamic risk measure of nogooddeal type, for which the minimal capital to make the position acceptable coincides with
Section 2.2. Gooddeal valuation and hedging
Page 64
the gooddeal valuation bound. The dynamic risk measure to be minimized over all permitted trading strategies is defined for sufficiently integrable contingent claims X, as ρt (X) = ess sup EtQ [X],
t ∈ [0, T ],
Q∈P ngd
where P ngd is a set of apriori valuation measures to be chosen so that, in the spirit of [BE09], the gooddeal bound π·u (X) becomes the market consistent risk measure (valuation) for X that arises from nogooddeal hedging with respect to ρ. An investor holding a liability X and trading parallely in the market according to a permitted strategy φ ∈ Φ would assign to R c her position at every time t ∈ [0, T ] the risk ρt (X − tT φtr s dWs ). For optimal trading, she would like to use a strategy φ¯ ∈ Φ that minimizes her risk at any time t ∈ [0, T ], in such a way that the minimal capital requirement to make her position ρacceptable is πtu (X). This minimum yields the marketbased risk measure (after optimal risksharing with the financial market) in the spirit of [BE09]. In other words, gooddeal valuation should arise from gooddeal hedging by minimizing the dynamic coherent risk measure ρ with respect to the family P ngd of apriori measure as generalized scenarios (see [ADE+ 07]). This yields a hedging notion that corresponds to gooddeal valuation, in that the market based risk measure turns out to be the gooddeal valuation bound π·u (X). The investor’s hedging problem therefore is to find φ¯ ∈ Φ such that
πtu (X) = ρt X −
Z t
T
c φ¯tr s dWs = ess inf ρt X − φ∈Φ
Z t
T
c φtr s dWs , t ∈ [0, T ].
(2.15)
Since 0 ∈ Φ, then (2.15) necessarily requires πtu (X) ≤ ρt (X), t ∈ [0, T ], which in turns hints that P ngd should contain the smaller set Qngd . As in [Bec09] we choose P ngd as the set of probability measures equivalent to P , that are not necessarily martingale measures and satisfy the nogooddeal constraint with C, i.e. such that Qngd = P ngd ∩ Me . More precisely we define n o P ngd := Qγ,β ∼ P : (γ, β) ∈ C . (2.16) In a financial market with no risky asset, i.e. S ≡ 1, any probability measure is a martingale measure and consequently P ngd defined in (2.16) coincides with Qngd (1). Hence P ngd inherits the mstability and convexity of Qngd (1), and thus ρ· (·) is satisfies the properties in Lemma 2.10: ρ is a dynamic coherent timeconsistent risk measure. For a contingent claim X, the tracking error Rtφ (X) from hedging according to a strategy φ ∈ Φ, at time t ∈ [0, T ], is defined by the difference between the capital variations of the claim and the profit/loss from dynamic trading up to time t according to φ, i.e. ct . Rtφ (X) := πtu (X) − π0u (X) − φ · W c , with x0 ∈ R, Remark 2.12. For a selffinancing strategy φ ∈ Φ replicating X = x0 + 0T φtr dW φ the tracking error vanishes, i.e. R (X) = 0. One says that a strategy is a meanselffinancing R
Section 2.3. Case of uniformly bounded correspondences
Page 65
(like riskminimizing strategies studied in [Sch01, Section 2], with EtQ [X] taking the role of πtu (X)) if its tracking error it is a martingale (under P ). We will show (see Theorem 2.19) that the tracking error of a gooddeal hedging strategy φ¯ satisfying (2.15) is a supermartingale under all apriori measures in P ngd . The result will therefore enable us to view φ¯ as being “at least meanselffinancing” under any Q ∈ P ngd . This will be seen as a robustness property of φ¯ with respect to the set of measures P ngd interpreted as generalized scenarios (in the sense of [ADE+ 07]). b
For results on gooddeal valuation and hedging, we shall distinguish two cases, namely the case where the constraint correspondence is uniformly bounded, and the case beyond uniform boundedness. For the first case we will obtain descriptions of gooddeal bounds and hedging strategies in terms of solutions to Lipschitz JBSDEs. In the second case, Lipschitz JBSDE tools are not directly applicable, and we will resort to approximation arguments focusing only on valuation.
2.3
Case of uniformly bounded correspondences
We characterize π u (X) as solution to a JBSDE under the assumption that C is uniformly bounded, which ensures that the resulting JBSDE has a Lipschitz generator function. Indeed, the connection between dynamic coherent risk measures and BSDEs is quite known; cf. e.g. [Ros06, PR15]. We say that a correspondence C satisfying (2.10) is uniformly bounded if Assumption 2.13. sup(t,ω) sup(γ,β)∈Ct (ω) (kγk2L2 (λt (ω)) + β2 ) < +∞. Under Assumption 2.13 one can show as in [QS13, Proposition 3.2] that Γ ∈ S 2 for any Γ density processes of a measure in P ngd . Hence for contingent claims X ∈ L2 ⊃ L∞ that may be pathdependent, πtu (X) and ρt (X) are welldefined as essential suprema of almost surely finitevalued random variables, and one can check (also for X ∈ L2 ) that an analogue of Lemma 2.10 still holds. For each nogooddeal measure Q ∈ Qngd , Lemma 2.4 describes E·Q [X] as the value process of a JBSDE with linear generator. Then using the comparison principle Proposition 2.6, one can describe π·u (X) (and likewise for ρ· (X)) as the value process of a JBSDE whose generator is the supremum of the linear ones. The following Lemma 2.14 (see Appendix 2.5 for its proof using Assumption 2.13) that the maximum is indeed attained. This yields (cf. Theorem 2.16) a worstcase measure under which the gooddeal bound is attained. Obviously such a worstcase measure will usually lie in the L1 closure of the set Qngd of nogooddeal measures. Lemma 2.14. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). Let C¯t (ω) denote the closure of Ct (ω) in L2 (λt (ω)) × Rn , (t, ω) ∈ [0, T ] × Ω and let Z ∈ H2 and U ∈ Hν2 . Then
Section 2.3. Case of uniformly bounded correspondences
Page 66
e a) there exist η¯ = η¯(Z, U ) predictable and γ¯ = γ¯ (Z, U ) Pmeasurable such that for P ⊗ dtalmost all (ω, t) ∈ Ω × [0, T ] holds
(¯ γt (ω), η¯t (ω)) ∈ argmax ηttr (ω)Π⊥ (t,ω) (Zt (ω)) + (γt (ω),ηt (ω))
Z E
Ut (ω, e)γt (ω, e)λt (ω, de), (2.17)
with the supremum taken over all (γt (ω), ηt (ω)) with (γt (ω), −ξt (ω) + ηt (ω)) ∈ C¯t (ω) and ηt (ω) ∈ Ker σt (ω). e ˜ b) there exist β˜ = β(Z, U ) predictable and γ˜ = γ˜ (Z, U ) Pmeasurable such that for P ⊗ dtalmost all (ω, t) ∈ Ω × [0, T ] holds
(˜ γt (ω), β˜t (ω)) ∈
argmax ¯t (ω) (γt (ω),βt (ω))∈C
βttr (ω)Zt (ω)
Z
Ut (ω, e)γt (ω, e)λt (ω, de).
+ E
To (¯ γ , β¯ := −ξ + η¯) ∈ C¯ of Part a) of Lemma 2.14, we associate the probability measure ¯ := Qγ¯,β¯ P , which might not be equivalent to P since γ¯ may take the value −1 on a Q ¯ is possibly not an equivalent local martingale measure, but we now nonnegligible set. So Q show that it belongs to the L1 (P )closure of Qngd . Lemma 2.15. For Z ∈ H2 and U ∈ Hν2 , let (¯ γ , η¯) be as in Part a) of Lemma 2.14. Define ¯ γ ¯ , β b + (1 − 1 )Q, ¯ ¯ ¯ for all n ∈ N. the measures Q = Q P for β := −ξ + η¯ and Qn := n1 Q n n ngd n 1 ¯ in L (P ) as n → ∞. Consequently, it holds Then (Q )n∈N ⊂ Q and Q converges to Q ¯ Q u ¯ πt (X) ≥ E [X], Qa.s., t ∈ [0, T ]. t
Proof. Let n ∈ N. Clearly Qn ∼ P . Moreover dQn /dP = Z n := n1 Zb + (1 − n1 )Z¯ with Zb := b ¯ dQ/dP = E(−ξ · W ) and Z¯ := dQ/dP . Itˆo formula then yields Z n = E((−ξ + η n ) · W + γ n ∗ µ ˜) n n e ¯ n ∈ [0, 1) for η = α¯ η being predictable and γ = α¯ γ is Pmeasurable with α = (1 − n1 )Z/Z thanks to Zb > 0. Therefore η n ∈ Ker σ and γ n > −1 due to γ¯ ≥ −1. Hence (γ n , η n ) ∈ C and n n ¯ in L1 (P ) as n → ∞ is straightforward so Qn = Qγ ,η is in Qngd . Convergence of Qn to Q ¯ by definition of Qn and this implies πtu (X) ≥ EtQ [X] for all t ≤ T . For X ∈ L2 , consider the two JBSDEs: Z
T
Yt = X +
(−ξs + η¯s )tr Zs +
t
−
Z t
Z
Us (e)¯ γs (e)λs (de) ds (2.18)
E T
Zstr dWs
−
Z
T
Z
t
E
e(ds, de), Us (e)µ
and Z
T
Yt = X + t
−
Z t
T
Zstr β˜s +
Zstr dWs
Z
−
Us (e)˜ γs (e)λs (de) ds (2.19)
E
Z t
T
Z E
e(ds, de), Us (e)µ
Section 2.3. Case of uniformly bounded correspondences
Page 67
˜ = (˜ ˜ with (¯ γ , η¯) = (¯ γ (Z, U ), η¯(Z, U )) and (˜ γ , β) γ (Z, U ), β(Z, U )) being by Lemma 2.14. Then we have the following Theorem 2.16. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). Let X ∈ L2 , and ˜ be as in Lemma 2.14. Then (¯ γ , η¯) and (˜ γ , β) ¯P a) the JBSDE (2.18) has a unique solution (Y, Z, U ) in S 2 × H2 × Hν2 and there exists Q ¯ with density dQ/dP = E (−ξ + η¯) · W + γ¯ ∗ µ ˜ such that π·u (X) satisfies ¯
πtu (X) = ess sup EtQ [X] = Yt = EtQ [X],
¯ Qa.s., t ∈ [0, T ].
Q∈Qngd
˜ U ˜ ) in S 2 × H2 × H2 and there exists b) the JBSDE (2.19) has a unique solution (Y˜ , Z, ν e P with dQ/dP e Q = E β˜ · W + γ˜ ∗ µ ˜ such that e ρt (X) = ess sup EtQ [X] = EtQ [X] = Yt , Qa.s., t ∈ [0, T ]. e
Q∈P ngd
For X ∈ L2 and permitted trading strategies φ ∈ Φ, consider the JBSDE T
Z
Yt = X + t
−
T
Z t
− ξstr φs + (Zs − φs )tr β˜s (Z − φ, U ) +
Zstr dWs −
Z t
T
Z
Us (e)˜ γs (Z − φ, U )(e)λt (de) ds
E
Z E
(2.20)
e(ds, de), Us (e)µ
where for φ ∈ Φ the processes γ˜· (Z − φ, U ) and β˜· (Z − φ, U ) are as in Part b) of Lemma 2.14. Then we have the following lemma, whose proof is deferred to Appendix 2.5. Lemma 2.17. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). For X ∈ L2 and φ ∈ Φ, the (2.20) admits a unique solution (Y φ , Z φ , U φ ) ∈ S 2 × H2 × Hν2 that satisfies JBSDE R c Ytφ = ρt X − tT φtr s dWs , t ∈ [0, T ]. Let f and f φ ( for φ ∈ Φ) denote respectively the generators of the JBSDEs (2.18) and (2.20), given for z ∈ Rn , u ∈ L2 (λt ), t ∈ [0, T ] as f (t, z, u) =
ess sup
tr
Z
β z+
u(e)γ(e)λt (de)
¯t (γ,β)∈C β∈−ξt +Ker σt φ
f (t, z, u) =
−ξttr φt
E
+ ess sup β (z − φt ) + tr
¯t (γ,β)∈C
Z
and
u(e)γ(e)λt (de) . E
The following lemma will be used in combination with the comparison theorem for JBSDEs to show existence of a gooddeal hedging strategy φ¯ solution to the hedging problem (2.15). The proof is also deferred to the Appendix 2.5.
Section 2.3. Case of uniformly bounded correspondences
Page 68
Lemma 2.18. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10). Then for all z ∈ Rn , u ∈ L2 (λt ), t ∈ [0, T ] holds (2.21)
f (t, z, u) = ess inf f φ (t, z, u). φ∈Φ
To prove a general existence result for φ¯ solution to the hedging problem (2.15), we will require the additional condition on the abstract correspondence C that there exists ∈ (0, 1) such that {0} × B (−ξt (ω)) ⊆ Ct (ω), for all (t, ω),
(2.22)
where B (−ξ) denotes the (closed or open) ball in Rn centered at −ξ with radius . Condition (2.22) implies in particular (2.13), and will be automatically satisfied for concrete nogooddeal constraint correspondences as in the frameworks of Section 2.3.1 and Section 2.4.2. In addition, (2.22) ensures coercivity of the generators of the JBSDEs (2.20) as functions of φ ∈ Φ, i.e. f φ (t, z, u) → +∞ as φ → ∞ for fixed (t, z, u). Following common arguments in variational analysis, (2.22) will be used in the proof of Theorem 2.19 below to deduce existence of a minimizing strategy φ¯ in (2.15). The precise result is the following, and shows moreover that hedging strategies φ¯ are at least meanselffinancing in the sense that their tracking errors satisfy a supermartingale property with respect to measures in P ngd . The proof is postponed to Appendix 2.5. Theorem 2.19. Let Assumptions 2.9 and 2.13 hold for C satisfying (2.10) and (2.22). For X ∈ L2 , let (Y, Z, U ), (Y φ , Z φ , U φ ) in S 2 × H2 × Hν2 (for φ ∈ Φ) be solutions to the γ , η¯), (˜ γ , η˜) as in Lemma 2.14. Then there exists JBSDEs (2.18), (2.20) respectively, with (¯ ¯ ¯ φ := φ(X) ∈ Φ satisfying ¯
f φ (t, Zt , Ut ) = ess inf f φ (t, Zt , Ut ), t ∈ [0, T ], φ∈Φ
¯ and for such φ¯ hold Yt = ess inf Ytφ = Ytφ , t ∈ [0, T ], and φ∈Φ
πtu (X)
= ess inf ρt X − φ∈Φ
Z t
T
c φtr s dWs
= ρt X −
Z t
T
c φ¯tr s dWs = Yt ,
t ∈ [0, T ].
(2.23)
¯ Moreover the tracking error Rφ (X) of φ¯ is a Qsupermartingale for any Q ∈ P ngd and a ∗ ∗ ¯ U ), β(Z ˜ − φ, ¯ U )). Q∗ martingale, for Q∗ = Qγ ,β ∈ P ngd with (γ ∗ , β ∗ ) := (˜ γ (Z − φ,
Remark 2.20. 1. In accordance with Remark 2.12, Theorem 2.19 shows that gooddeal hedging strategies are robust in the sense that they are at least meanselffinancing with respect to the set P ngd as generalized scenarios. 2. Note that Theorem 2.19 shows only existence of φ¯ and does not claim its uniqueness. The latter may depend on the contingent claim X into consideration. Independently
Section 2.3. Case of uniformly bounded correspondences
Page 69
of the contingent claim, uniqueness may be obtained for particular structures of the correspondence C. We will provide examples (see Section 2.3.2 and last example of Section 2.3.1), where uniqueness is ensured for any claim, and explicit expressions of φ¯ in terms of JBSDE solutions can be obtained. If the values of the correspondence C can be decomposed as Ct = Ctγ × Ctβ for Ctγ ⊂ L2 (λt ) and Ctβ ⊂ Rn , t ∈ [0, T ], i.e. the nogooddeal constraint decouples as a constraint on the unpredictable eventrisk separated from a constraint on the market price of stock risk, then the hedging strategy φ¯ does not depend on the U component of the solution to the JBSDE (2.18). The precise statement is summarized in the following corollary of Theorem 2.19. Corollary 2.21. Let the conditions of Theorem 2.19 hold and assume in addition that Ct = Ctγ × Ctβ for Ctγ ⊂ L2 (λt ) and Ctβ ⊂ Rn , t ∈ [0, T ]. For X ∈ L2 , let (Y, Z, U ) be the solution to the JBSDE (2.18). Then a gooddeal hedging strategy is given by φ¯ ∈ Φ satisfying φ¯t ∈ argmin φ∈Φ
2.3.1
− ξttr φt + ess sup βttr (Zt − φt ) , t ∈ [0, T ]. ¯β βt ∈ C t
Results for constraint on instantaneous Sharpe ratios (bounded case)
We consider a nogooddeal constraint emanating from a bound on the instantaneous Sharpe ratios of investments in the financial market extended by additional derivative price processes. Recall that this case was studied in [BS06] for a Markovian model of asset prices and additional factor processes exhibiting jumps. [BS06, Theorem 2.3] showed an extended form of the Hansen–Jagannathan (HJ) inequality in the sense that SRt  ≤ k(γt , βt )kL2 (λt )×Rn , t ∈ [0, T ], for all (γ, β) Girsanov kernels of measures in Me , where SRt denotes the instantaneous Sharpe ratio at time t. This meant that a bound on the instantaneous Sharpe ratios could be achieved through a bound on the norm of the Girsanov kernels of pricing measures and an application of the HJ inequality. Their nogooddeal constraint was then set as k(γt , βt )k2L2 (λt )×Rn := kγt k2L2 (λt ) + βt 2 ≤ K 2 ,
(2.24)
t ∈ [0, T ],
for some given K ∈ (0, ∞), and they derived the gooddeal bounds in terms of solutions to HJB PIDEs using dynamic programming techniques. Here we rather use JBSDEs to derive gooddeal bounds in a nonnecessarily Markovian model under a nogooddeal constraint of the type (2.24), but for more general K being a positive bounded predictable process. The associated constraint correspondence C is then n
o
Ct = (γ, β) ∈ L2 (λt ) × Rn : γ > −1, kγk2L2 (λt ) + β2 ≤ Kt2 .
(2.25)
Beyond the boundedness of ξ, we assume that for some ε > 0 holds Kt > ξt  + ε,
t ∈ [0, T ],
(2.26)
Section 2.3. Case of uniformly bounded correspondences
Page 70
b ∈ Qngd 6= ∅. Since K is bounded and predictable, the so that (0, −ξ) ∈ C and hence Q nogooddeal restriction in this example fits well into the framework of the current Section 2.3 since C given by (2.25) is then convexvalued, and satisfies (2.10) and Assumption 2.13. Were K not uniformly bounded, then C might fail to satisfy Assumption 2.13, and the framework of the upcoming Section 2.4 would then prevail. Here the closedvalued correspondence C¯ is given by n o C¯t = (γ, β) ∈ L2 (λt ) × Rn : γ ≥ −1, kγk2L2 (λt ) + β2 ≤ Kt2 . (2.27)
Indeed for arbitrary (γ, β) ∈ C¯t (ω), one chooses the sequence (γ k , β k )k∈N ⊂ Ct (ω) with γ k = γ ∨ (−1 + k1 ) and β k = β, so that γ k ≤ γ holds (since γ ≥ −1). By dominated convergence, it then follows that (γ k , β k ) converges to (γ, β) in L2 (λt (ω)) × Rn . The correspondence Ce defined as in (2.11) is then given by n
1/2
Cet = (γ, β) ∈ L2 (λ) × Rn : γ ≥ −ζt
o
, kγk2L2 (λ) + β2 ≤ Kt2 .
(2.28)
Applying the theory of measurable correspondences in [AF90], the following lemma shows that e The proof is relegated to Appendix 2.5. Assumption 2.9 holds for C. Lemma 2.22. The closedconvexvalued correspondence Ce given by (2.28) is Pmeasurable. Now one can apply part a) of Theorem 2.16 to obtain a description of π·u (X) and a worstcase nogooddeal measure for X ∈ L2 in terms of the solution to the JBSDE (2.18). The precise result is the following Theorem 2.23. For X ∈ L2 , the JBSDE (2.18) with (¯ γ , η¯) from (2.17) has a unique solution 2 2 2 ¯ (Y, Z, U ) in S × H × Hν . Moreover there exists Q P in the L1 closure of Qngd (in ¯ the sense of Lemma 2.15), with density dQ/dP = E ((−ξ + η¯) · W + γ¯ ∗ µ ˜) such that the u gooddeal bound π· (X) satisfies ¯
πtu (X) = ess sup EtQ [X] = Yt = EtQ [X]
for all t ≤ T.
Q∈Qngd
(2.29)
Sufficient conditions for explicit formulas for valuation and hedging Let us still consider the nogooddeal constraint on the Sharpe ratio described in terms of the correspondence C in (2.25). We investigate conditions ensuring an explicit form of the maximizer (¯ γ (Z, U ), η¯(Z, U )) in the generator of the JBSDE (2.18), which in turn may also lead to an explicit formula for the gooddeal hedging strategy. Note that the classical KuhnTucker routine may not apply for the maximization problem in (2.17) for C in (2.25), due to the additional constraint {γ ≥ −1}. If one considers the gooddeal valuation problem without this constraint for JBSDEs, then can obtain using KuhnTucker arguments an explicit maximizer
Section 2.3. Case of uniformly bounded correspondences
Page 71
for all t ∈ [0, T ] as 2 1/2 Kt2 − ξt  ⊥ and η¯t = 2 1/2 Πt (Zt ). ⊥ 2 kUt kL2 (λt ) + Πt (Zt )
2 1/2 Kt2 − ξt  γ¯t = 2 1/2 Ut kUt k2L2 (λt ) + Π⊥ t (Zt )
(2.30)
In general the relaxed Girsanov kernels in (2.30) do not induce a measure Qγ¯,−ξ+¯η that is absolutely continuous with respect to P . In addition, (¯ γ , η¯) from (2.30) only give rise to a u,r u relaxed bound π (X) which is clearly larger than π (X), i.e. π u,r (X) ≥ π u (X), for any financial risk X since it is obtained by maximizing EtQ [X] over a set of measure Qr ⊇ Qngd containing eventually signed measures. These facts were already analyzed in [BS06, Section 3.5 and 4.4], where similar relaxed gooddeal bounds was studied using HamiltonJacobiBellman techniques. In terms of JBSDEs, we obtain here the following relaxed version of the JBSDE (2.18), with an explicit generator and value process π·u,r (X) (instead of π·u (X)) by replacing (¯ γ , η¯) in (2.18) by the expressions in (2.30)): Z
T
Yt = X + t
−
Z t
T
− ξstr Πs (Zs ) + Ks2 − ξs 2
Zstr dWs −
T
Z t
1/2
2 kUs k2L2 (λs ) + Π⊥ s (Zs )
Z E
1/2
ds (2.31)
e(ds, de). Us (e)µ
The JBSDE (2.31) has a Lipschitz generator, and hence (by e.g. [Bec06, Proposition 3.2]) admits a unique solution (Y, Z, U ) ∈ S 2 × H2 × Hν2 , for X ∈ L2 , with π·u,r (X) := Y . Note that for a Sharpe ratio bound K conveniently chosen (for example small enough), the relaxed gooddeal bound π·u,r (X) could still be lower than the upper noarbitrage bound. However in general π·u,r (X) may not be a noarbitrage price since (¯ γ , −ξ + η¯) in (2.30) could define a signed measure because γ¯ ≥ −1 may be violated. If for a contingent claim X ∈ L2 one can show that U ≥ 0 for (Y, Z, U ) solving the JBSDE (2.31), then γ¯ in (2.30) satisfies γ¯ ≥ 0 > −1 and the generators of the JBSDEs (2.18) and (2.31) coincide. This would clearly imply that π·u (X) = π·u,r (X), and both are described by the solution of the JBSDE (2.31). In this case, it would even possible to obtain a closedform ¯ Precisely, we have the following expression of gooddeal hedging strategies φ. Proposition 2.24. Assume the Sharpe ratio constraint described by the correspondence C in (2.25). For X ∈ L2 , let (Y, Z, U ) be the unique solution to the JBSDE (2.31). Then 1. If the gooddeal bound and its relaxed version coincide, i.e. π·u (X) = π·u,r (X), then a gooddeal hedging strategy φ¯ ∈ Φ is given by
φ¯t =
2 1/2
kUt k2L2 (λt ) + Π⊥ t (Zt ) Kt2 − ξt 2
1/2
ξt + Πt (Zt ), t ∈ [0, T ].
(2.32)
Section 2.3. Case of uniformly bounded correspondences
Page 72
¯ ¯ = Qγ¯,−ξ+¯η 2. If U ≥ 0, then πtu (X) = πtu,r (X) = EtQ [X] = Yt , t ∈ [0, T ], where Q
is in Qngd with (¯ γt , η¯t ) = Kt2 − ξt 2 satisfying γ¯ ≥ 0 > −1.
1/2
2 −1/2
kUt k2L2 (λt ) + Π⊥ t (Zt )
U t , Π⊥ t (Zt ) ,
Proof. The second claim follows from the preceding discussion. As for the first claim, if π·u (X) = π·u,r (X) then we have for any t ∈ [0, T ] ¯ f φ (t, Zt , Ut ) = −ξttr φ¯t + ess sup β tr (Zt − φ¯t ) + ¯t (γ,β)∈C
Z
γ(e)Ut (e)λt (de) E
2 1/2
≤ −ξttr φ¯t + Kt kUt k2L2 (λt ) + Zt − φ¯t = −ξttr Πt (Zt ) + Kt2 − ξt 2
1/2
2 kUt k2L2 (λt ) + Π⊥ t (Zt )
1/2
= f (t, Zt , Ut ). ¯
By Lemma 2.18 it follows that f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ), t ∈ [0, T ], and therefore applying Theorem 2.19 proves the required claim. A condition similar to U ≥ 0 for obtaining an explicit generator of the JBSDE describing the gooddeal bound was provided in [Del12]. However [Del12] focused on a financial/insurance market with a single traded risky asset modelled by a two dimensional Brownian motion (i.e. d = 1, n = 2), and with the random measure µ associated to jumps of a pointprocess with stateindependent compensator of the form ν(dt) = ζt dt for a predictable process ζ ≥ 0. Our result deals with fairly general jump processes and financial markets (with d ≤ n). A drawback of the condition U ≥ 0 in part 2. of Proposition 2.24 which ensures equality between π·u (X) and π·u,r (X) is that it might not be straightforward to check in general, and also depends on the contingent claim X into consideration. Examples of such claims includes derivatives X that solely on diffusive risk and for which one would expect that U = 0. Also, if µ is the random measure of the jumps of a simple Poisson process and X a claim that pays nothing if the Poisson process does not jump before maturity and a unit if it does, then one U would be nonnegative. Recall that for π·u (X) and π·u,r (X) to coincide, it is sufficient to provide conditions that guarantee that the process γ¯ defined in (2.30) satisfies γ¯ ≥ −1. If the support of the measure λ is finite, one could write a condition on K and λ that does not depend on the claim and, without further hypotheses on U , ensures that γ¯ ≥ −1 holds. Note that in this case, the controls in the optimization problem in Lemma 2.14 would live in a finite dimensional space, simplifying considerably the problem. To explain what happens in this case, we provide an example in a semiMarkov jump setup that includes continuous time Markov chains.
Section 2.3. Case of uniformly bounded correspondences
Page 73
Example for semiMarkov jumpdynamics: We consider a Markov renewal process as (Jn , Tn )n∈N0 , with random variables (Jn )n taking values in the finite state space E = {e1 , . . . , em } ⊂ Rm (w.l.o.g. to fit into our setup from Section 2.1), for ei for i ∈ I := {1, . . . , m} denoting the canonical unit vectors, with m ∈ N, and a nondecreasing sequence (Tn )n , T0 = 0, of Fmeasurable [0, T ]valued random times modeling its renewal times. We assume that the process (Jn , Tn )n starts almostsurely at a prespecified state and has a stochastic semiMarkov kernel Q = (Qij )i,j∈I exogenously given such that Qij t = P [Jn+1 = ej , Tn+1 − Tn ≤ t  Jn = ei ], t ∈ [0, T ], for all n ∈ N. The process J deP fined by Jt := JNt , where Nt := ∞ n=1 1{Tn ≤t} is the counting process associated to the jumps of (Jn , Tn )n , is called the semiMarkov process associated to the Markovrenewal process. Note that (Jn )n is a Markov chain with transition Kernel P = (pij )i,j∈I satisfying ij ij ij pij = Qij T . It is wellknown that the conditional independence relation Qt = p Gt holds, where Gij t = P [Tn+1 − Tn ≤ t  Jn = ei , Jn+1 = ej ] , for all n ∈ N, is the conditional distribution function of the sojourn time Tn+1 − Tn from state ei to state ej . For unexplained notions in the theory of Markov renewal processes, semiMarkov processes and point processes in general, we refer to [C ¸ 75, Bre81]. Let µ be the random measure of the jumps of the semiMarkov process J, identified with P the family of counting processes (N ij )i,j∈I, j6=i through µ([0, t) × D) = i,j∈I, ej −ei ∈D Nti,j , where Ntij =
P∞
n=1 1 Tn ≤t, Jn =ei , Jn+1 =ej
is the number of jumps up to time t from state
ei to state ej of J. Note that Nt = i,j∈I, j6=i Ntij . Let us denote by τt := sup{s ∈ [0, t] : Jt−u = Jt , u ∈ [t − s, t]} the time the process J has spent in its current state Jt . Then the compensator νij of N ij is given by νij (dt) = 1{Jt− =ei } αtij (τt− )dt, with jump intensities P
αtij (u) = lim
h&0
P [Jt+h = ej  Jt = ei , τt = u] , h
u, t ∈ [0, T ], i, j ∈ I, i 6= j,
(2.33)
e ij of N ij is where we set αii = − j6=i αij , i ∈ I. The compensated jump process N R t ij ij ij ij ij e = N − λ ds, where λ := 1 e with the family N {Jt− =ei } αt (τt− ), t ∈ [0, T ]. We identify µ t t t 0 s ij e )i,j∈I, j6=i and assume that (W, µ e) have the weak predictable representation property with (N respect to (F, P ) in the sense that for every local (F, P )martingale M , there exists predictable R R P ij 2 processes Z, U = (U ij )i,j∈I, j6=i with 0T Zs 2 ds < ∞ and i,j∈I, j6=i 0T λij s (Us ) ds < ∞ Rt R P t ij ˜ ij such that Mt = 0 Zs dWs + i,j∈I, j6=i 0 Us dNs , t ∈ [0, T ]. P
For (Z, U ) ∈ H2 × Hν2 , the expressions of γ¯ and η¯ from (2.30) for γ¯ = (¯ γ ij )i∈I,j6=i and ˜ ij become U = (U ij )i∈I,j6=i suitably reparametrized in terms of integrands with respect to N γ¯tij = Kt2 − ξt 2
1/2
2 Π⊥ t (Zt ) +
X
kl 2 λkl t (Ut )
−1/2
Utij ,
−1/2
Π⊥ t (Zt ).
and
k,l∈I, l6=k
η¯t = Kt2 − ξt 2
1/2
2 Π⊥ t (Zt ) +
X k,l∈I, l6=k
kl 2 λkl t (Ut )
Section 2.3. Case of uniformly bounded correspondences
Page 74
Assume that the process K is small enough such that the inequality K 2 − ξ2 1{λij 6=0} ≤ λij , P ⊗ dta.s., for all i, j ∈ I, i 6= j,
(2.34)
holds. Since U ∈ Hν2 , we can assume without loss of generality that U ij = 0 on {λij = 0}. With this, (2.34) implies that for any i, j ∈ I, i 6= j, t ∈ [0, T ] hold Kt2 − ξt 2
1/2
Utij ≥ − Kt2 − ξt 2
1/2
Utij  ≥ − λij t
≥−
1/2
Utij  ij 2 λkl t Ut 
X
1/2
k,l∈I, l6=k
2 ≥ − Π⊥ t (Zt ) +
X
kl 2 λkl t (Ut )
1/2
.
k,l∈I, l6=k
Hence γ¯ ij ≥ −1 for any i, j ∈ I, j 6= i, which in turn ensures π·u (X) = π·u,r (X) = Y for X ∈ L2 , with (Y, Z, U ) solution to the JBSDE (2.31) which now reads Z
T
−
Yt = X + t
−
Z
ξstr Πs (Zs )
+
(Ks2
2 1/2
− ξs  )
2 Π⊥ s (Zs )
+
X
ij 2 λij t (Ut )
1 2
ds
i,j∈I, j6=i T
t
Zstr dWs −
X
Z
i,j∈I, j6=i t
T
˜ ij . Utij dN t
(2.35)
Note that (2.34) implies γ¯ ≥ −1 and not γ¯ > −1 in general. While the latter holds when U ≥ 0, already the former is sufficient to ensure that π·u (X) = π·u,r (X). For gooddeal hedging, applying part 1. of Proposition 2.24 therefore implies that a gooddeal hedging strategy exists for any contingent claim X ∈ L2 and is expressed as
P
φ¯t (X) =
2 1/2
ij ij 2 ⊥ i,j∈I, j6=i λs (Us ) + Πt (Zt )
Kt2 − ξt 2
1/2
ξt + Πt (Zt ), t ∈ [0, T ].
We now provide conditions under which the BSDE (2.35) can be reduced to a system of ODEs, which may then be solved numerically forward in time. To this end, suppose that the semiMarkov process is a continuous time Markov chain. For A = (αij )i,j∈I denoting the deterministic but timedependent rate matrix of the chain J with entries P [Jt+h = ej  Jt = ei ] ≥ 0, t ∈ [0, T ], h&0 h
αij (t) = lim
ij for i 6= j, and j∈I αij = 0 for i ∈ I, it follows that λij t := 1{Jt− =ei } α (t), t ∈ [0, T ]. As in [CS12] and analogously to the boundedness assumption on density ζ in the general setup, we assume that the components of the rate matrix process A are uniformly bounded so that with positive probability the Markov chain does not change state on fixed compact time intervals. For this part, we assume that W and J are independent, and F is the augmentation of FW ∨ FJ .
P
Section 2.3. Case of uniformly bounded correspondences
Page 75
Using J as a factor process, assume that the market price of risk ξ and Sharpe ratio bound K are deterministic functions of the chain, i.e. ξt := ξ(t, Jt− ) and Kt := K(t, Jt− ), t ∈ [0, T ], for K : [0, T ] × Rm → (0, ∞) and ξ : [0, T ] × Rm → Rn measurable. For a contingent claim X = G(JT ), depending solely on the final state of the Markov chain, with G : Rm → R measurable, the Zcomponent of the JBSDE (2.35) vanishes by independence of W and J. Therefore π u (X) = Y holds, for (Y, U ) solution to the JBSDE T
Z
(Ks2
Yt = G(JT ) +
2 1/2
− ξs  )
t
X
ij 2 λij s (Us )
i,j∈I j6=i
12
Z
T
ds − t
X
esij . Usij dN
(2.36)
i,j∈I j6=i
Now the results of [CS12] imply that the solution (Y, U ) to (2.36) is Markovian, i.e. Yt = u(t, Jt ) and Utij = u(t, ej ) − u(t, ei ),
i, j ∈ I,
t ∈ [0, T ],
(2.37)
for a deterministic function u : [0, T ] × Rm → R. Furthermore, u is such that the associated column vector function of time u(t) := (ui (t))i∈I ∈ Rm , with ui (t) := u(t, ei ), i ∈ I, solves the coupled system of ODEs 2 2 1/2 X ij 2 12 X ij dui = − K i (t) − ξ i (t) − α (t) uj (t) − ui (t) , (2.38) α (t) uj (t) − ui (t) dt j∈I
j∈I
i ∈ I, with terminal condition u(T ) = (G(ei ))i∈I , where we use the notation K i (t) := K(t, ei ), ξ i (t) := ξ(t, ei ), i ∈ I, t ∈ [0, T ]. Hence the gooddeal bound is the solution of a system of ODEs, which by reversing the time can be transformed into an initial value problem easily solved by numerical ODE solver. Example with explicit formulas for stronger Sharpe ratio constraints: Instead of considering specific conditions on contingent claims, or on the underlying jump process, let us focus on the nogooddeal constraint itself by considering a stronger Sharpe ratio constraint. Indeed the gooddeal hedging strategy can be obtained explicitly using Corollary 2.21 if the Sharpe ratio constraint in the correspondence defined in (2.25) is reinforced by requiring rather √ max{kγkL2 (λt ) , β} ≤ Kt / 2, t ∈ [0, T ]. (2.39) Recall that the Euclidian norm  · 2 and maximum norm  · ∞ are equivalent in R2 with √  · ∞ ≤  · 2 ≤ 2 · ∞ . Noting this, the upper (resp. lower) gooddeal bounds obtained from the constraint correspondence (2.25) can be estimated from below (resp. above) by those obtained from the stronger Sharpe ratio constraint (2.39). We generalize (2.39) by decoupling the nogooddeal constraint into kγkL2 (λt ) ≤ Ktγ
and β tr At β ≤ (Ktβ )2 , t ∈ [0, T ],
(2.40)
where A is a predictable Rn×n matrixvalued process with symmetric values which are elliptic uniformly in (t, ω) ∈ [0, T ] × Ω, and K γ , K β are positive bounded predictable processes
Section 2.3. Case of uniformly bounded correspondences
Page 76
√ satisfying K β > ξ tr Aξ + ε for some ε > 0. Under this generalized version we obtain below a ¯ The correspondence associated to closedform expression for a gooddeal hedging strategy φ. γ β (2.40) is given for t ∈ [0, T ] by Ct = Ct × Ct with Ctγ = u ∈ L2 (λt ) : u > −1 and kukL2 (λt ) ≤ Ktγ
and Ctβ = x ∈ Rn : xtr At x ≤ (Ktβ )2 .
β Assuming that A−1 αt0 , where αt0 is the ellipticity t (Ker σt ) = Ker σt and that ξt  < Kt constant of A−1 t , t ∈ [0, T ]. It follows from Corollary 2.21 and the upcoming Theorem 3.17 in Chapter 3 that the unique gooddeal hedging strategy φ¯ is given by
p
φ¯t =
tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )
(Ktβ )2 − ξttr At ξt
1/2
At ξt + Πt (Zt ),
1/2
t ∈ [0, T ],
for (Y, Z, U ) solving the JBSDE (2.18).
2.3.2
Results for ellipsoidal constraint and uncertainty about jump intensities
We are also concerned with gooddeal valuation and robust hedging with respect to uncertainty about intensities of jumps in the market. Here the investor faces uncertainty about the market price of jump risk which translates into Knightian uncertainty (ambiguity) about the real world measure. We assume that from empirical/historical data, the investor has isolated a confidence region R of candidate reference measures (subjective priors) centered around a probability measure P and described by n
e)dP R := P ψ ∼ P : dP ψ = E(ψ ∗ µ
o
(2.41)
t ∈ [0, T ].
(2.42)
with ψ ≤ ψ ≤ ψ¯ ,
e functions satisfying where −1 < ψ ≤ 0 ≤ ψ¯ are Pmeasurable
∃K ∈ (0, ∞) s.t.
Z
ψ t (e)2 + ψ¯t (e)2 λt (de) ≤ K,
E
Under each reference measure P ψ ∈ R, we impose an ellipsoidal nogooddeal constraint on the market price of diffusion risk and zero nogooddeal constraint on the market price of jumprisk. In other words, the nogooddeal restriction is only imposed on the βcomponent of the Girsanov kernels (γ, β) of pricing measures in terms of an ellipsoidal correspondence and the γcomponent is set to zero. The resulting set Qngd (P ψ ) ⊆ Me (S, P ψ ) = Me (S, P ) =: Me of nogooddeal measures under P ψ is n
o
Qngd (P ψ ) := Qβ ∼ P ψ : dQβ = E(β · W )dP ψ , β ∈ C β , β ∈ −ξ + Ker σ , n
o
(2.43)
where Ctβ (ω) = x ∈ Rn : xtr At (ω)x ≤ (Ktβ )2 , (t, ω) ∈ [0, T ]×Ω, with A being a predictable Rn×n matrixvalued process with symmetric values that are elliptic and bounded (in operator
Section 2.3. Case of uniformly bounded correspondences
Page 77
norm) uniformly in (t, ω), and K β is a positive bounded predictable process satisfying K β > √ tr ξ Aξ + ε for some ε > 0. The radial case corresponds to A ≡ IdRn . As in Section 3.2.1 of βp 0 Chapter 3, assume the separability condition A−1 αt , t (Ker σt ) = Ker σt and that ξt  < Kt where αt0 is the ellipticity constant of A−1 , t ∈ [0, T ]. In Chapter 3 we will deal with robustness t with respect to uncertainty about the drift of traded assets in a Brownian setting, following a worstcase multiprior approach to ambiguity as in [GS89, CE02]. Here we consider a similar approach for uncertainty about the intensity of the underlying jumps described by the priors P ψ ∈ R. A seller who seeks for robustness can charge the largest valuation bound over all priors, in order to compensate for the eventual misspecification of intensities of the jumps. In this respect, for contingent claims X ∈ L2 , the worstcase approach under uncertainty yields the gooddeal bounds πtu (X) = ess sup EtQ [X], t ∈ [0, T ]. (2.44) Q∈Qngd
where Qngd :=
ngd (P ψ ). ψ≤ψ≤ψ¯ Q
S
Clearly, one can rewrite
πtu (X) = ess sup ess sup EtQ [X]. ψ≤ψ≤ψ¯ Q∈Qngd (P ψ )
By Yor’s formula, it is seen that n
o
(2.45)
Qngd = Qψ,β ∼ P : (ψ, β) ∈ C, β ∈ −ξ + Ker σ , where C = C γ × C β with n
o
Ctγ (ω) := ψ ∈ L2 (λt (ω)) : ψ t (ω) ≤ ψ ≤ ψ¯t (ω) ⊆ L2 (λt (ω)), (t, ω) ∈ [0, T ] × Ω. Hence Qngd is mstable and convex (cf. Lemma 2.11). By (2.42), uniform ellipticity of A and boundedness of K β , the correspondence C satisfies Assumption 2.13. Moreover by standard measurable selection arguments, the associated closedconvexvalued correspondence Ce defined in (2.11) clearly satisfies Assumption 2.9. Hence the set Qngd falls in the general framework of Section 2.2 for a set of nogooddeal measure defined as in (2.12) with the associated correspondence C = C γ × C β satisfying the uniform boundedness and measurability hypotheses of Theorem 2.16. Note that the main difference between the two constraints is that Ctγ (ω) from (2.40) is given in terms of a L2 bound on the integrands γ of the Girsanov kernels, whereas the current one is given in terms of pointwise bounds on the integrands γ. For Z ∈ H2 and U ∈ Hν2 , the optimal Girsanov kernels (¯ γ , η¯) of part a) of Lemma 2.14, can be explicitly derived from the corresponding maximization problem (2.17) and for t ∈ [0, T ] as
γ¯t = ψ t 1{Ut 0} + 01{Ut =0}
and η¯t =
Ktβ
2
− ξttr At ξt
1/2
tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )
−1 ⊥ 1/2 At Πt (Zt ),
Section 2.4. Case of nonuniformly bounded correspondences
Page 78
with γ¯ clearly satisfying γ¯ > −1. Part a) of Theorem 2.16 now applies and the gooddeal ¯ bound π·u (X), for X ∈ L2 , is described by πtu (X) = Yt = EtQ [X], t ∈ [0, T ], for the worstcase ¯ = Qγ¯,−ξ+¯η in Qngd and (Y, Z, U ) ∈ S 2 × H2 × Hν2 solution to the nogooddeal measure Q Lipschitz JBSDE (2.18) which in the present setup rewrites explicitly as T
Z
Yt = X + t
− ξstr Πs (Zs ) + +
T
Z t
Zstr dWs −
Ksβ
2
− ξstr As ξs
1/2
tr −1 ⊥ Π⊥ s (Zs ) As Πs (Zs )
Z Z t
1/2
ψ s (e)1{Us (e)0} Us (e)λs (de) ds
E
−
T
(2.46)
Z E
e(ds, de). Us (e)µ
To show that the correspondence C = C γ × C β satisfies (2.22), note at first that since A is uniformly bounded in the operator norm, there exists a constant a ∈ (0, ∞) such that √ kAt (ω)k ≤ a for a.a. (t, ω). Now if x + ξt  < holds, then by the inequality K β > ξ tr Aξ + ε one has 1/2
xtr At x
≤ (x+ξt )tr At (x+ξt )
1/2
+ ξttr At ξt
1/2
< kAt k1/2 x+ξt −ε+Ktβ < a1/2 −ε+Ktβ .
Now choosing ∈ (0, 1) such that ≤ εa−1/2 implies that (2.22) holds. Hence the correspondence C satisfies the conditions of Corollary 2.21, which together with the results of Section ¯ 3.2.1 in Chapter 3 (cf. Theorem 3.17 therein) implies that the gooddeal hedging strategy φ(X) is uniquely given by
φ¯t (X) =
1/2
tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )
(Ktβ )2 − ξttr At ξt
1/2
At ξt + Πt (Zt ),
t ∈ [0, T ],
for (Y, Z, U ) being solution to the JBSDE (2.46). Note that since P ngd = ψ≤ψ≤ψ¯ P ngd (P ψ ) then, as expected, the gooddeal hedging strategy φ¯ is also robust with respect to uncertainty in S
¯
the sense that its tracking error Rφ (X) satisfies a supermartingale property under all measures in P ngd (P ψ ) uniformly for all reference priors P ψ ∈ R. For similar results in the Brownian setting, we refer to Chapter 3 with uncertainty about market price of (diffusion) risk and to Chapter 4 with uncertainty about the volatility of tradeable assets.
2.4
Case of nonuniformly bounded correspondences
Beyond Assumption 2.13, let us now consider a convexvalued correspondence C that still satisfies (2.10) but may fail to be uniformly bounded. In this case the generator of the JBSDE (2.18) may not be Lipschitz continuous, and results on Lipschitz JBSDEs may not apply as in the case of uniformly bounded correspondences. However for a nonuniformly
Section 2.4. Case of nonuniformly bounded correspondences
Page 79
bounded correspondence C, one can still derive approximations of the gooddeal bound π·u (X) by solutions to Lipschitz JBSDEs arising from truncations of the correspondence C which satisfy Assumption 2.13. Here the density processes Γ of nogooddeal measures may not be in S 2 and X ∈ L2 may no longer imply X ∈ L1 (Q) for all Q ∈ Qngd . For this reason, we shall restrict the study here to financial claims X ∈ L∞ . We consider a sequence Ctk := (γ, β) ∈ Ct : kγk2L2 (λt ) + β2 ≤ k 2 , k ∈ N, of correspondences satisfying Assumption 2.13. Since C is convexvalued and satisfies (2.10), then each C k is also convexvalued and satisfies (2.10). Moreover, since the correspondence Ce given as in (2.11) satisfies Assumption 2.9, one can show using arguments similar to those in the proof of Lemma 2.22 n o −1/2 k k 2 n e ¯ that the correspondences Ct = (γ, β) ∈ L (λ) × R : ζt 1{ζt >0} γ, β ∈ Ct are also k k Pmeasurable, where C¯t (ω) denotes the closure of Ct (ω) in L2 (λt (ω)) × Rn . Since each C k , k ∈ N, satisfies Assumption 2.13, then results of Section 2.3 are applicable if one replaces C by any of the C k , k ∈ N. In addition Ctk (ω) % Ct (ω), as k % ∞. For k ∈ N, denote Qngd the set defined in (2.12) with C k instead of C and consider the associated process k πtu,k (X) = ess sup EtQ [X], t ∈ [0, T ]. Q∈Qngd k
The correspondences C k can be interpreted as describing a nogooddeal constraint consisting of the initial constraint in C in addition to a constraint on the instantaneous Sharpe ratios given by the constant bound K = k ∈ N. Note that the sets Qngd k , k ∈ N, also are convex and mstable (by Lemma 2.11). First we have the following Lemma 2.25. Let X ∈ L∞ . Then the following dynamic principles hold: 1. π u (X) is the smallest adapted c`adl`ag process such that it is a supermartingale under every Q ∈ Qngd with terminal value X. 2. For all k ∈ N, π u,k (X) is the smallest adapted c`adl`ag process such that it is a supermartingale under every Q ∈ Qngd with terminal condition X. k Proof. The supermartingale properties of π u (X) and π u,k (X) respectively under Q ∈ Qngd and Q ∈ Qngd are consequences of mstability and convexity of Qngd and Qngd (see Lemma k k 2.10). That they are respectively the smallest ones follows from definitions as essential suprema of closed martingales of the type E·Q [X]. For nonuniformly bounded correspondences, the following Theorem 2.26 describes in detail the approximation of gooddeal valuation bounds for unbounded correspondences C by solutions to JBSDEs obtained from truncations C k of C, which are uniformly bounded and fit to the setting of Section 2.3. This is an analogue of Theorem 3.7 for a possibly discontinuous filtration. Note however the presence of an additional part (part 5.) here. We mention that both theorems
Section 2.4. Case of nonuniformly bounded correspondences
Page 80
are only concerned with approximations for gooddeal valuation bounds, and not with hedging strategies. It is an interesting question if the hedging strategies associated to the approximating bounds converge in some sense to a process that can somehow be interpreted as hedging strategy. We do not investigate further on this issue. The proof of Theorem 2.26 is postponed to Appendix 2.5. Theorem 2.26. Let C be a correspondence satisfying (2.10) and such that Assumption 2.9 holds. For a contingent claim X ∈ L∞ , hold: 1. πtu,k (X) % πtu (X) a.s. as k → ∞, for all t ∈ [0, T ] 2. For any k ∈ N, π u,k (X) = Y k for (Y k , Z k , U k ) ∈ S ∞ × H2 × Hν2 unique solution to the Lipschitz JBSDE Z
T
Yt = X +
−
t
−
Z
T
t
ξstr Πs (Zs )
Zstr dWs −
Z
T
t
ess sup
+
¯ k +(0,ξs ) (γs ,ηs )∈C s ηs ∈Ker σs
(ηstr Π⊥ s (Zs )
Z
Us (e)γs (e)λs (de)) ds
+ E
Z E
(2.47)
e(ds, de), Us (e)µ
b the DoobMeyer decompositions 3. π u (X) and π u,k (X) for k ≥ kξk∞ admit under Q c +U ∗µ e − A, and π u (X) = π0u (X) + Z · W
(2.48)
c + Uk ∗ µ e − Ak , π u,k (X) = π0u,k (X) + Z k · W
(2.49)
b × H2 (Q) b and A, Ak are nondecreasing predictable processes where (Z, U ) ∈ H2 (Q) ν b Ak = A0 = 0, and with AT , AkT ∈ L2 (Q), 0 k
A =
Z
·
ess sup ¯ k +(0,ξt ) 0 (γt ,ηt )∈C t ηt ∈Ker σt
k ηttr Π⊥ t (Zt )
Z
+ E
Utk (e)γt (e)λt (de) dt.
(2.50)
b Fu ), Z k converges to Z weakly 4. For all u ∈ [0, T ], Aku converges to Au weakly in L2 (Ω, Q, 2 k b ⊗ dt), and U converges to U weakly in L2 (Ω × E × [0, u], P ⊗ ν) in L (Ω × [0, u], Q as k → ∞. ¯
¯ in the L1 closure of Qngd , such that π u (X) = E Q [X], then π·u (X) is a 5. If there exists Q 0 b for any ¯ quasileftcontinuous Qmartingale, and π·u,k (X) converges to π·u (X) in S p (Q) k 1/2 c + (U k − U ) ∗ µ b and e T converges to 0 in L1 (Q), p ∈ [1, ∞). Moreover (Z − Z) · W b b with E Q Ak converges to A in S 1 (Q) [AT ] ≤ 2kXk∞ , with Z k , Z, U k , U from 3..
¯ for π u (X) in In Theorem 2.26, part 5., the hypothesis on existence of a worstcase measure Q · the L1 closure of Qngd is ensured for any contingent claim X ∈ L∞ if the set of densities ZTQ
Section 2.4. Case of nonuniformly bounded correspondences
Page 81
(with respect to P ) of measures Q in Qngd is weakly relatively compact in L1 (i.e. uniformly integrable, by DunfordPettis compactness theorem [DM78, Chapter II, Theorem 25]). This is a consequence of James’ theorem (cf. [AB06, Theorem 6.36]). In case the correspondence C is ¯ is proved in part a) of Theorem 2.16, and the approximations uniformly bounded, existence of Q in Theorem 2.26 are not necessary in the first place. An example of a correspondence that may not satisfy Assumption 2.13 and for which the set of densities of measures in Qngd is uniformly integrable is given by n
Ct = (γ, β) ∈ L2 (λt ) × Rn : γ > −1
and
1 2 β + 2
Z E
o
g˜ 1 + γ(e) λt (de) ≤ K , (2.51)
with K ∈ (0, ∞) and the nonnegative function g˜ defined by g˜(y) = y log y − y + 1, y > 0. Such a correspondence results from a nogooddeal constraint imposed as a dynamic bound K 2 (σ − τ ) on the conditional relative entropy Eτ ΓΓστ log ΓΓστ for stopping times τ ≤ σ ≤ T and density processes Γ from Qngd (see e.g. also [Kl¨o06, Chapter 3]). For the correspondence C, uniform integrability of Qngd is ensured by applying the de la Vall´ee Poussin’s theorem [DM78, ¯ to Chapter II, Theorem 22]. In lack of more general assumptions for the worstcase measure Q exists, its existence could be checked in some specific situations, using the specific structure of the claim X in the model at hand. An example for this will be given in Section 3.2.2 of Chapter 3, where X is a put option in the Heston model, F is the augmented Brownian filtration and C is a radial correspondence modeling an unbounded constraint on the instantaneous Sharpe ratios (as e.g. in Section 2.4.1 below).
2.4.1
Results for constraint on instantaneous Sharpe ratios (unbounded case)
Consider gooddeal bounds from a constraint on a Sharpe ratio described by the radial correspondence n
o
Ct = (γ, β) ∈ L2 (λt ) × Rn : γ > −1, kγk2L2 (λt ) + β2 ≤ Kt2 ,
(2.52)
for a positive predictable process K that could unbounded. Similarly to Section 2.3.1, C is convexvalued, satisfies (2.10), and is such that Assumption 2.9 holds for the associated correspondence Ce defined as in (2.11). However C does not satisfy Assumption 2.13 if K is unbounded, since for βt := Kβ 0 −1 β 0 , with β 0 ∈ Rn \ {0}, one has (0, β) ∈ C but sup(t,ω) βt (ω) ≥ K∞ = ∞. Hence the correspondence C in (2.52) fits the setup of Section 2.4 and therefore the associated gooddeal bounds can be described by Theorem 2.26.
2.4.2
Results for constraint on optimal expected growth rates
For another application, we consider gooddeal bounds emanating from a constraint on the optimal expected growth rates of logreturns. The set Qngd for such a constraint can be
Section 2.4. Case of nonuniformly bounded correspondences
Page 82
formulated in terms of a bound on the conditional reverse relative entropy of nogooddeal measures with respect to the reference measure P . Recall that for stopping times τ ≤ σ and a measure Q equivalent to P with density process Γ, the Fτ conditional reverse relative entropy Hτσ (P  Q) of Q with respect to P is the conditional f divergence (for f = − log) defined as Eτ − log ΓΓστ =: Hτσ (P  Q) ≥ 0, with nonnegativeness following from the P submartingale property of − log Γ (cf. Proposition 2.27). [Kl¨o06, KS07b] studied dual representations of (static) gooddeal bounds and their dynamic properties in a L´evy framework with constraints on the f divergence for diverse choices of the function f corresponding to logarithmic (f (z) = − log(z)), exponential (f (z) = z log z corresponding to constraint (2.51) on the conditional relative entropy) and power (f (z) = z p , p ≥ 1) utility functions. Related pricing (and hedging) approaches from a constraint on generalized relative entropy are considered in [Lei08]. In [Bec09] it was shown in a Brownian setting that a dynamic bound on the reverse relative entropy of riskneutral measures (in Me ) corresponds to a bound on the optimal expected growth rates of (log)returns, for any extension of the financial market by additional derivative price processes that are computed as conditional expectations under nogooddeal pricing measures (in Qngd ). This provides a nogooddeal constraint that, in the Brownian setting, is essentially equivalent to the constraint on the instantaneous Sharpe ratios. We first note that for a discontinuous filtration (presence of jumps), the two nogooddeal constraints are no longer equivalent and the constraint on the Sharpe ratios as in Section 2.3.1 appears mathematically more tractable in terms of JBSDEs. In fact, the correspondence resulting from a constraint on optimal growth rates may fail to satisfy Assumption 2.13, even when the growth rates are bounded by a constant. In this section, we derive gooddeal bounds and show existence of hedging strategies for a nogooddeal constraint on optimal growth rates, using Lipschitz JBSDEs in a setup with jumps of finite state semiMarkov processes, i.e. in particular having a finitelysupported jump compensator. The restriction to finite state space is important as this ensures that the resulting JBSDEs have classical Lipschitz continuous generators and existence of gooddeal hedging strategies can be shown as in Theorem 2.19. Beyond finitely supported compensators, the results in this section may not guarantee existence of a gooddeal hedging strategy for such nogooddeal constraints since the associated correspondence C in (2.57) may be unbounded; yet we do still have result on gooddeal valuation bounds. To be more precise in our current setup, let K be a positive bounded predictable process and define Qngd as consisting of measures Q ∈ Me that satisfy Hτσ (P
1  Q) ≤ Eτ 2
Z τ
σ
Ku2 du ,
for all τ ≤ σ ≤ T,
(2.53)
where τ, σ are stopping times. Let us recall [Bec09, Proposition 2.2] providing some useful properties of the process − log Γ, for Q ∼ P with finite reverse entropy. Proposition 2.27. Let Q ∼ P with density process Γ of Q with respect to P such that log ΓT ∈
Section 2.4. Case of nonuniformly bounded correspondences
Page 83
L1 . Then − log(Γ) is a P submartingale of class (D) with a DoobMeyer decomposition − log(Γ) = N + A, where N is a uniformly integrable P martingale and A a predictable, nondecreasing and P integrable process, with N0 = A0 = 0. Moreover Eτ − log ΓΓστ = Eτ [Aσ − Aτ ] holds for all stopping times τ ≤ σ ≤ T . Using Proposition 2.27, we can reformulate the definition of Qngd in terms of deterministic Q times. To this end, measures υh and κ = κ i on the predictable σfield P hR define positive i RT T 1 2 by υ(B) := 2 E 0 1B Ku du and κ(B) := E 0 1B dAu , B ∈ P, where A is the nondecreasing process in the DoobMeyer decomposition of − log(Γ), for Γ being the density process of a measure Q ∼ P . We have the following equivalent condition for Q ∼ P to be in Qngd (see [Bec09, Proposition 2.3]). Proposition 2.28. For Q ∼ P with density process Γ satisfying
Es − log
Γt 1 ≤ Es Γs 2
Z s
t
Ku2 du for all deterministic times s ≤ t ≤ T,
(2.54)
holds κ(B) ≤ υ(B) for any B ∈ P. In particular, condition (2.54) is equivalent to its stopping time analogue (2.53). Thus a measure Q ∈ Me is element of Qngd if and only if (2.54) holds. One can interpret, as mentioned above, the constraint (2.53) as a bound on the optimal expected growth rates in the financial market extended by additional derivative price processes (see [Bec09, Theorem 3.1]). Using Lemma 2.8, one can formulate a definition of the set Qngd by a condition on the Girsanov kernels of the associated measures. For Q ∈ Me , the following proposition derives N and A from Proposition 2.27 in our setup in terms of the Girsanov kernels of Q. The proof is deferred to Appendix 2.5. Proposition 2.29. Let Q ∼ P with Girsanov kernels (γ, β) and density process Γ = E(M ) e as in part b) of Lemma 2.1, and let also g be defined by (2.3). Then where M = β · W + γ ∗ µ ⊥ 1. If Qγ,β ∈ Me with E[− log Γγ,β T ] < ∞, then β = −ξ + η, with Πt (βt ) = ηt , t ∈ [0, T ]. γ,β γ,β γ,β Moreover the DoobMeyer decomposition − log Γ = N +A is given by
e and N γ,β = −β · W − log(1 + γ) ∗ µ Z · Z 1 βs 2 + g(1 + γs (e))λs (de) ds. Aγ,β = 2 0 E
(2.55)
2. If Qγ,β ∈ Qngd , then β = −ξ + η and 1 ηt 2 + 2
1 1 g(1 + γt (e))λt (de) ≤ Kt2 − ξt 2 , 2 2 E
Z
t ∈ [0, T ].
(2.56)
Section 2.4. Case of nonuniformly bounded correspondences
Page 84
e eintegrable function γ > −1 and any predictable µ 3. Reciprocally, any Pmeasurable R process β with Πt (βt ) = −ξt , t ∈ [0, T ] satisfying 12 βt 2 + E g(1 + γt (e))λt (de) ≤ 1 2 γ,β ∈ Qngd with Girsanov kernels (γ, −ξ + η), where 2 Kt , t ∈ [0, T ], define a measure Q ⊥ ηt = Πt (βt ), t ∈ [0, T ]. b ∈ Qngd 6= ∅. From Proposition Assume that K > ξ + ε, for some ε ∈ (0, 1), ensuring that Q 2.29, the constraint correspondence satisfying (2.10) for the set of nogooddeal measures defined by (2.12) can be chosen as n
Ct := (γ, β) ∈ L2 (λt ) × Rn : γ > −1,
o
k2g(1 + γ)kL1 (λt ) + β2 ≤ Kt2 .
(2.57)
The correspondence C has nonempty convex values (since (0, 0) ∈ C and g is convex). However, it is easily seen that C given by (2.57) does not satisfy Assumption 2.13 in general. Indeed, assume that µ is the random measure of the big jumps of a Gamma L´evy process with parameters a = b = 1, i.e. with E = R \ {0}, ζ ≡ 1 and λ(dx) = exp(−x)x−1 1{x≥1} dx. R Then for the function γ with γ(x) = exp(x/2)1{x≥1} satisfies E g 1 + γ(x) λ(dx) < ∞ and R 2 n E γ(x) λ(dx) = ∞. Therefore for suitably chosen K ∈ (0, ∞) the sequence ((γ , 0))n∈N of n 2 Girsanov kernels with γ = γ1[1,n] is included in C but is not bounded in L (λ). It is for this reason that we present this case as an application of Section 2.4 which works generally beyond Assumption 2.13. Note that around −1 < γ ≤ 1, one has the following Taylor approximation of g up to the 2 (leading) second order: g(1 + γ) = − log(1 + γ) + γ = γ2 + O(γ 3 ). In this sense, the constraint on Sharpe ratios can be viewed as an approximation of that on optimal growth rates, for pricing measures Qγ,β possessing a low market prices of jumprisk γ. It is therefore not surprising that for continuous filtrations (absence of jumprisk, i.e. trivial µ = ν = 0), formally γ = 0 and the two types of nogooddeal constraints are equivalent; cf. [Bec09]. Clearly, a (bounded) constraint on the Sharpe ratios is mathematically more tractable since it naturally leads to standard Lipschitz JBSDEs for gooddeal valuation and hedging. The correspondence C in (2.57) has been defined so that it satisfies (2.10) but, for our theory, we will show that its associated correspondence Ce satisfies Assumption 2.9. First we describe the closure C¯t of the set Ct in L2 (λt ) × Rn . This needs some preparation because the function g and its derivative g 0 explode in the neighborhood of 0. Consider the pointwise approximation (g l )l∈N of g consisting of nonnegative Lipschitz functions ( l
g (y) :=
g( 1l ), g(y),
if 0 ≤ y ≤ if y ≥ 1l .
1 l
The sequence (g l )l is nondecreasing and converges pointwise to g on (0, ∞) as l tends to infinity. In particular for any l ∈ N the function g l satisfies g l (1 + y) ≤ Const y2 for all y ≥ −1 for some Const > 0. This property will be useful later in the proof of the second claim of
Section 2.4. Case of nonuniformly bounded correspondences
Page 85
Lemma 2.30. Note that the function g(1 + ·) is dominated by Const y2 only locally around the origin. Now define for each l ∈ N the correspondence n
o
C¯tl := (γ, β) ∈ L2 (λt ) × Rn : γ ≥ −1, k2g l (1 + γ)kL1 (λt ) + β2 ≤ Kt2 . Since g l is continuous and nonnegative, then by Fatou’s lemma the sets C¯tl are closed in L2 (λt ) × Rn , for each l ∈ N, t ∈ [0, T ]. The related correspondence Ce l according to (2.11) therefore writes n
1/2
Cetl = (γ, β) ∈ L2 (λ) × Rn : γ ≥ −ζt
1
− , 2g l (1 + ζt 2 γ) L1 (λt ) + β2 ≤ Kt2 .
o
T For any l ∈ N since g l ≤ g, then Ct ⊆ C¯tl , which implies C¯t ⊆ C¯tl . Hence Cet ⊆ l∈N Cetl , t ∈ [0, T ]. In fact equality holds in the latter inclusion as claimed by the following lemma, which also infers that Ce satisfies Assumption 2.9. The proof is provided in Appendix 2.5.
Lemma 2.30. For C defined in (2.57), holds Cet = closedvalued correspondence Ce is Pmeasurable.
el l∈N Ct ,
T
t ∈ [0, T ]. In particular the
Overall, the correspondence C defined in (2.57) and describing a nogooddeal constraint as a bound on the optimal expected growth rates in the financial market satisfies the hypotheses of Theorem 2.26. This yields an approximation of the associated gooddeal bound π·u (X) in terms of solutions to Lipschitz JBSDEs for abstract random measures µ and contingent claims X ∈ L∞ . Although the correspondence C in (2.57) might not satisfy Assumption 2.13 for general random measures µ, this assumption apparently holds when the measures λt (with ν(dt, de) = λt (de)dt respect to which the compensator ν is absolutely continuous) are finitely supported. In this case the results of Section 2.3 (on valuation and hedging) are again applicable, and the gooddeal bounds can be directly described as solutions to JBSDEs. Without loss of generality, we elaborate on this by considering the semiMarkov setup of Section 2.3.1. Example for semiMarkov jumpdynamics: Consider again the framework of Section 2.3.1, with a semiMarkov process J on finite state space E = {e1 , . . . , em } ⊂ Rm , and denote P ij of the jumps, compensator I = {1, . . . , m}, and counting process N = i,j∈I, j6=i N P ij ij ij ν(dt) := i,j∈I, j6=i λij t dt, for jump intensities λt = 1{Jt− =ei } αt (τt− ) with α ≥ 0 defined in (2.33) and the time the process has spent at state Jt being τt := sup{s ∈ [0, t] : Jt−u = Jt , u ∈ [t − s, t]}. The constraint in the optimization problem in Lemma 2.14 for C in (2.57) is finite dimensional and given for t ∈ [0, T ] by the set of (γ, η) ∈ (−1, ∞)m×m−1 × Rn satisfying X 1 2 1 2 2 η + g(1 + γ ij )λij t ≤ (Kt − ξt  ), 2 2 i,j∈I, j6=i
(2.58)
Section 2.4. Case of nonuniformly bounded correspondences
Page 86
for g given by (2.3). We can assume without loss of generality γ ij = 0 on the set {λij = 0}. This together with (2.58) imply that g(1 + γ ij ) ≤
Kt2 − ξt 2 2λij t
for all i, j ∈ I, j 6= i.
1{λij 6=0} t
(2.59)
Since g is continuous and limx&−1 g(1 + x) = limx→∞ g(1 + x) = +∞, then (2.59) yields compactness in (−1, ∞)m×m−1 × Rn of the set of values of ((γ ij )i,j∈I, j6=i , η) satisfying (2.58). Furthermore for Z ∈ H2 and U ∈ Hν2 , the objective function F (t, γ, η) := η tr Π⊥ t (Zt ) + P ij ij ij m×m−1 n × R and predictable in (t, ω) ∈ i,j∈I, j6=i Ut γ λt is continuous in (γ, η) ∈ (−1, ∞) [0, T ] × Ω. Hence by the usual direct method of variational analysis (cf. [ET99]) and standard measurable selection theorems (cf. [Roc76], which do not require completeness of the measure space for correspondences with finite dimensional ranges), there exists a predictable (t, ω)wise maximizer (¯ γ , η¯) := (¯ γ (Z, U ), η¯(Z, U )) ∈ (−1, ∞)m×m−1 × Rn of F over the constraint set described by (2.58). Since by part 3. of Lemma 2.32 the function g satisfies x − 2 ≤ (g(1 + x))2 , for all x > −1, then (2.59) implies that γ ij ≤
(Kt2 −ξt 2 )2 1{λij 6=0} 2 4(λij t t )
+ 2. This in turn yields (after squaring and
summing over all states ei , ej ∈ E, j 6= i) 2 ij ij γ λ t ≤
X i,j∈I, j6=i
≤
(K 2 − ξ 2 )2 t t
X i,j∈I, j6=i K4 ∞
4
2
1{λij 6=0} + 2 λij t
2 4(λij t )
∨2
X
t
i,j∈I, j6=i
−2 (λij t ) 1{λij t 6=0}
+1
2
(2.60) λij t ,
with the convention that 0/0 = 0. Now assume that
∃ c¯λ ≥ cλ > 0 s.t. cλ 1{λij 6=0} ≤ λij ≤ c¯λ
for all i, j ∈ I, j 6= i.
(2.61)
Condition (2.61) ensures by (2.60) that the correspondence C defined in (2.57) satisfies the uniform boundedness Assumption 2.13 in the current semiMarkov jump setup with 2
β +
γ ij 2 λij t i,j∈I, j6=i X
≤
K2∞
K4
+ m(m − 1)
∞
4
1
∨2
c2λ
2
+ 1 c¯λ ,
(2.62)
for all (γ, β) ∈ C. Note that (2.61) does not exclude the fact that the intensities λij can vanish on a nonnegligible set. Indeed we only require on the set where they do not vanish, that they are bounded from below by a positive constant, uniformly over all states ei , ej ∈ E and (t, ω) ∈ [0, T ] × Ω. That the intensities of the jumps are bounded from above appears as a nonrestrictive assumption for practical examples, which could be interpreted as a sufficient condition preventing the rate of statechange of the semiMarkov process from exploding.
Section 2.5. Appendix
Page 87
Former calculations suggest that in the limit as m → ∞, the righthand side of in (2.62) would tend to infinity. In the limiting case, therefore, the correspondence C may no longer be uniformly bounded; this shows the importance of restricting to finitely many states. Part a) of Theorem 2.16 applies and yields that the gooddeal bound π·u (X) for X ∈ L2 is ¯ ¯ = Qγ¯,−ξ+¯η and (Y, Z, U ) ∈ S 2 × H2 × Hν2 solution to given by π·u (X) = E·Q [X] = Y for Q the BSDE (2.18) which in the present setup rewrites T
Z
Yt = X +
t
−
Usij γ¯sij λij s ds
i,j∈I, j6=i T
Z
X
(−ξs + η¯s )tr Zs +
t
Zstr dWs
X
−
Z
i,j∈I, j6=i t
T
(2.63)
e ij . Usij dN s
¯ = Qγ¯,−ξ+¯η is in fact a nogooddeal measure, i.e. In addition, the worstcase measure Q ¯ ∈ Qngd , because the optimal Girsanov kernels (¯ Q γ , −ξ + η¯) ∈ C satisfies γ¯ ij > −1 for all i, j ∈ I, j 6= i. It is also possible to obtain a qualitative result about gooddeal hedging in this setting. Indeed, condition (2.22) is clearly satisfied for = ε ∈ (0, 1) since by assumption K > ξ + ε. Hence applying Theorem 2.19 yields in particular existence of a gooddeal hedging ¯ strategy φ¯ = φ(X) (for X ∈ L2 ) with ¯
f φ (t, Zt , Ut ) = ess inf f φ (t, Zt , Ut ),
(2.64)
φ∈Φ
for (Y, Z, U ) solution to the BSDE (2.63) and
f φ (t, Zt , Ut ) = −ξttr φt + ess sup β tr (Zt − φt ) + (γ,β)
X
Utij γ ij λij t ,
i,j∈I, j6=i
where the supremum is taken over all (γ, β) = ((γ ij )i,j∈I, j6=i , β) ∈ (−1, ∞)m×m−1 × Rn satisfying (2.58). One could not expect to obtain for (2.64), in the generality of the present example, an explicit formula for the gooddeal hedging strategy φ¯ solving the minimization problem in (2.64). Yet, approximations might be computed using numerical algorithms for convex optimization problems (cf. e.g. [BV04]), and of Lipschitz JBSDEs (cf. e.g. [BE08] for related but different types of generators).
2.5
Appendix
This appendix collects some proofs and statements of results that were omitted in the main body of the chapter. The order of appearance here is the same as in the main text. ˜ with P characteristics Proof of Lemma 2.1. For part a), apply [JS03, Theorem III.3.24] to X P ˜ has a canonical (B, c, ν) := (0, I, ν ) with respect to the truncation function h. Note that X
Section 2.5. Appendix
Page 88
˜ = W + (IdE − h) ∗ µ + h ∗ µ e in terms of the truncation function h, where representation X IdE is the identity function on E. Part b) is a consequence of part a), the weak predictable e) with respect to (P, F), and [JS03, Proposition III.5.10, representation property (2.2) of (W, µ Theorem III.5.19 and Corollary III.5.22] which apply with Y := 1 + γ, and a = 0, Yˆ = 0 since ν λ ⊗ dt. Proposition 2.31 ([LM78], Theorem II.5). Let M be a quasileftcontinuous local martingale satisfying ∆M ≥ −1 and define T¯ := inf {t : ∆Mt = −1} ∧ T . If the predictable compensator Λ of the process X
D = hM c i·∧T¯ +
∆Ms2 1
s≤·∧T¯
is bounded, then E martingale.
h
[E(M )]T
1/2 i
∆Ms ≤1
+ ∆Ms 1
∆Ms >1
(2.65)
< ∞. In particular E(M ) is a uniformly integrable
We have the following lemma. Being purely analytical, the proof is omitted. Lemma 2.32. For any y ≥ 0, hold 1. (1 − 2. (1 −
√ 2 y) ≤ (y − 1)2 1{y≤2} + y − 11{y>2} ≤ √
1 √ √ (1 − y)2 , 2 (1 − 2)
y)2 ≤ g(y), for the function g defined in (2.3),
3. y − 1 − 2 ≤ (g(y))2 . Proof of Proposition 2.3. By Lemma 2.32 it follows that 0≤
Z E
1−
q
2
1 + γt (e) λt (de) ≤
Z
g 1 + γt (e) λt (de) ≤ K, t ∈ [0, T ].
E
2 √ Hence the process 1 − 1 + γ ∗ ν is locally P integrable. Then applying [JS03, Theorem eintegrability of γ, with II.1.33, d)] (with a = 0 and γb = 0 since ν λ ⊗ dt holds) yields the µ e being a purely discontinuous local martingale. Moreover by (2.5), β · W is welldefined as a γ ∗µ e. continuous local martingale, so that M is a local martingale with M c = β · W and M d = γ ∗ µ e e)t = γ(t, ∆Xt )1 e By definition, the jumps of M are given by ∆Mt = ∆ (γ ∗ µ , t ∈ [0, T ], ∆X 6=0 t
e is the semimartingale X e = W + (IdE − h) ∗ µ + h ∗ µ e with h(e) := e1{e≤1} . Now where X
since γ > −1 then ∆M > −1, and therefore E(M ) is a positive local martingale. By [JS03, Corollary II.1.19] and ν λ ⊗ dt, M is quasileftcontinuous. Hence the process D in (2.65) R· 2 2 can be written as D = 0 βs  ds + γ 1{γ≤1} + γ1{γ>1} ∗ µ. By [JS03, Proposition II.1.28] the predictable P compensator Λ of D is Λ =
R·
2 2 0 βs  ds + γ 1{γ≤1} + γ1{γ>1} ∗ ν. Now
Section 2.5. Appendix
Page 89
using (2.4) and (2.5), Lemma 2.32 yields boundedness of Λ. By Proposition 2.31 this implies that Γ = E(M ) is a positive uniformly integrable martingale. In particular Γ defines a measure Q ∼ P via dQ = ΓdP . Let (β Q , γ Q ) be the actual Girsanov kernels of Q from part a) of e. Hence Lemma 2.1. By part b) of Lemma 2.1, Γ = E(M Q ) holds with M Q = β Q · W + γ Q ∗ µ Q Q E(M ) = E(M ) and by taking stochastic logarithms one obtains M = M , or equivalently e. The left hand side is a continuous local martingale whereas (β Q − β) · W = (γ Q − γ) ∗ µ the right hand side is a purely discontinuous local martingale. By orthogonality both local martingales are equal to zero. Since from (2.5) both local martingales are square integrable, then β = β Q P ⊗ dta.s. and γ = γ Q P ⊗ λ ⊗ dta.s.. i
i
Proof of Lemma 2.11. For (γ i , β i ) ∈ C and β i = −ξ + η i , η i ∈ Ker σ, i = 1, 2, let Qγ ,β be e . in Qngd with density processes Γi with respect to P given by Γi := E (−ξ + η i ) · W + γ i ∗ µ Convexity: Let α ∈ [0, 1] and Γ = αΓ1 + (1 − α)Γ2 . Since Me is convex, then Γ ∈ Me and corresponds to a measure Qγ,β ∼ P with Girsanov kernels (γ, β = −ξ + η), where η ∈ Ker σ. Using Itˆo’s formula and convexity of the values of C one shows that (γt , βt ) = αΓ1t− 1 1 Γt− (γt , βt )
+
(1−α)Γ2t− (γt2 , βt2 ) Γt−
∈ Ct , t ∈ [0, T ]. Hence Qngd is convex.
Mstability: Let τ ≤ T be a stopping time and Γt := 1{t≤τ } Γ1t + 1{τ ≤t} Γ1τ Γ2t /Γ2τ , t ∈ [0, T ]. Since Me is mstable, then Γ ∈ Me and corresponds to a measure Qγ,β ∼ P with Girsanov kernels (γ, β := −ξ + η), where η ∈ Ker σ. We show that (γ, β) ∈ C. It holds that R R et − 0t 21 βsi 2 + E g(1 + γsi (e))λs (de) ds, for i = 1, 2, log Γit = β i · Wt + (log(1 + γ i )) ∗ µ and g being the function given by (2.3). Hence 2
2
eT − log ΓT = β · WT + (log(1 + γ )) ∗ µ
T
Z
1
βs2 2
Z
+
g(1 + γs2 (e))λs (de) ds
2 E 1 eτ + (β − β ) · Wτ + (log(1 + γ ) − log(1 + γ 2 )) ∗ µ Z τ Z 1 12 − (βs  − βs2 2 ) + (g(1 + γs1 (e)) − g(1 + γs2 (e))λs (de) ds. 2 0 E 1
0
2
Equivalently we have Z
eT − β · WT + (log(1 + γ) ∗ µ 1
T
1
2
0
1
eτ − = β · Wτ + (log(1 + γ )) ∗ µ Z
T
+ τ
−
Z
T
βs2 dWs + 1
τ
1
= 1B β + 1B c β −
Z 0
T
βs2 2 +
2 2
T
Z τ
Z E
Z E
2
βs  +
Z
g(1 + γs (e))λs (de) ds E
τ
1
2
0
βs1 2
Z
+ E
g(1 + γs1 (e))λs (de) ds
e(ds, de) (log(1 + γs2 (e))µ
g(1 + γs2 (e))λs (de) ds
· WT + log 1 + 1B (s)γ 1 + 1B c (s)γ 2
1 1B (s)β 1 + 1B c (s)β 2 2 +
2
Z
s
s
Z E
eT ∗µ
g 1 + 1B (s)γs1 (e) + 1B c (s)γs2 (e) λs (de) ds,
Section 2.5. Appendix
Page 90
where B = [0, τ ] = {(t, ω) : t ≤ τ (ω)} ∈ P. Thus (γ, β) = 1B (γ 1 , β 1 ) + 1B c (γ 2 , β 2 ) ∈ C since C has convex values. Proof of Lemma 2.14. Proofs for part a) and b) are analogous, so we only prove part a). Consider the equivalent (to (2.17)) maximization problem (γt∗ (ω), ηt∗ (ω)) =
argmax ηttr (ω)Π⊥ (t,ω) (Zt (ω)) +
(γt (ω),ηt (ω))
Z
Ut (ω, e)γt (ω, e)(ζt (ω, e))1/2 λ(de),
(2.66)
E
where the maximum is taken over (γt (ω), ηt (ω)) ∈ Cet (ω) + (0, ξt (ω)) and ηt (ω) ∈ Ker σt (ω), with Ce given in (2.11). The maximization problem (2.66) is more convenient for measurable selection arguments since the range L2 (λ) × Rn of the associated correspondence Ce does not depend on t nor ω. The corresponding maximizers of (2.17) and (2.66) are related by γ∗ (¯ γt , η¯t ) = √t 1{ζt >0} , ηt∗ , t ∈ [0, T ]. ζt
(2.67)
For all (t, ω) ∈ [0, T ] × Ω, the sets Cet (ω) are closed and convex in L2 (λ) × Rn since C¯t (ω) are closed and convex in L2 (λt (ω)) × Rn . In addition Cet (ω) are bounded in L2 (λ) × Rn (hence weakly compact) since by Assumption 2.13 the sets C¯t (ω) are bounded in L2 (λt (ω)) × Rn . As a consequence Cet (ω) + (0, ξt (ω)) ∩ L2 (λ) × Ker σt (ω) is also weakly compact in L2 (λ) × Rn . Since Z ∈ H2 and U ∈ Hν2 , the objective function of the maximization problem (2.66) is linear and continuous in (γ, η) ∈ L2 (λ) × Rn , t ∈ [0, T ]. Hence by the direct method in variational analysis (see [ET99]), there exists for all (t, ω) ∈ [0, T ] × Ω a maximizer (γt∗ (ω), ηt∗ (ω)) in Cet (ω) + (0, ξt (ω)) ∩ L2 (λ) × Ker σt (ω) . Now we show that one can choose (γ ∗ , η ∗ ) such that η ∗ is Pmeasurable and γ ∗ (hence γ¯ = γ ∗ ζ −1/2 1{ζ>0} ) is P ⊗ Emeasurable (since ζ clearly is). Note that the Hilbert space L2 (λ) is separable (hence is a Polish space) and also that the correspondence Ce + (0, ξ) ∩ L2 (λ) × Ker σ is Pmeasurable since Ce is Pmeasurable by Assumption 2.9 and ξ, σ are predictable processes. Since Z is predictable and e U is Pmeasurable, the objective function is Pmeasurable in (t, ω) ∈ [0, T ] × Ω and hence a Carath´eodory function defined on [0, T ] × Ω × L2 (λ) × Rn . By standard measurable selection [AF90, Theorems 8.1.3, 8.2.11] one obtains (γ ∗ , η ∗ ) satisfying (2.66) for all (t, ω) ∈ [0, T ] × Ω, with η ∗ Pmeasurable and γ ∗ P − B(L2 (λ))measurable. Let us show that γ ∗ defined by γ ∗ (t, ω, e) := γ ∗ (t, ω)(e) is actually P ⊗ Emeasurable. Denote by (un )n∈N an orthonormal P basis of L2 (λ). Then γt∗ (ω) has the decomposition γt∗ (ω) = n∈N hγt∗ (ω), un iL2 (λ) un for any (t, ω) ∈ [0, T ] × Ω. Now since for each n ∈ N the map L2 (λ) 3 γ 7→ hγ, un iL2 (λ) is continuous, then hγ ∗ , un iL2 (λ) is a Pmeasurable process for all n ∈ N. Thus γ ∗ is P ⊗ Emeasurable as a countable sum of P ⊗ Emeasurable functions. Now by approximation of measurable functions e by simple functions, one can make η ∗ predictable and γ ∗ Pmeasurable through modification on a P ⊗ dtnullset. The corresponding (¯ γ , η¯) given by (2.67) then solves (2.17) for P ⊗ dtalmost all (t, ω) ∈ [0, T ] × Ω.
Section 2.5. Appendix
Page 91
Proof of Theorem 2.16. Part a) and b) are analogous, so we only prove part a). Denote by f the generator of the JBSDE (2.18). For all t ∈ [0, T ], ft is, by part a) of Lemma 2.14, the supremum over (γ, −ξ + η) ∈ C¯ and η ∈ Ker σ of a family of linear generators ftγ,η (t, z, u)
tr
Z
:= (−ξt + ηt ) z +
u(e)γt (e)λt (de), E
where coefficients (γ, (−ξ in Rn × L2 (λt (ω)) uniformly in (t, ω) by Kf := + η)) are bounded sup(t,ω) sup(γ,β)∈C(t,ω) kγkL2 (λt ) + β ∈ (0, ∞). The generator f is then Lipschitz contin¯ n uous in (z, u) ∈ R × L2 (λt (ω)), uniformly in (t, ω) ∈ [0, T ] × Ω with Lipschitz constant Kf and satisfies ft (0, 0) = 0. By [Bec06, Proposition 3.2], the JBSDE (2.18) has a unique ¯ the process solution (Y, Z, U ) ∈ S 2 × H2 × Hν2 . Now recall that for each (γ, β) ∈ C, R 2 β is bounded and E γt (e)λt (de) is uniformly bounded in t ∈ [0, T ]. Hence by Lemma 2.4 and the subsequent remark, the JBSDEs with generators f γ,η have unique solutions γ,β (Y γ,η , Z γ,η , U γ,η ) ∈ S 2 × H2 × Hν2 , which satisfy Ytγ,η = EtQ [X], t ∈ [0, T ], for β = −ξ + η. ¯ ¯ Furthermore since f = f γ¯,¯η holds, then one also has Yt = EtQ [X], Qa.s.. From Lemma 2.15 ¯ Q u ¯ it holds that πt (X) ≥ Et [X], Qa.s., and so to conclude the proof, one has to show that πtu (X) ≤ Yt , P a.s.. For all (γ, β := −ξ + η) ∈ C (defining Qγ,β ∈ Qngd ) holds ft (Zt , Ut ) = ftγ¯,¯η (Zt , Ut ) ≥ ftγ,η (Zt , Ut ),
t ∈ [0, T ],
for (Y, Z, U ) solution to the JBSDE (2.18). Moreover since f γ,η are Lipschitz in (z, u) with uniform Lipschitz constants Kf and ftγ,η (Ztγ,η , Utγ,η )
−
ftγ,η (Ztγ,η , Ut )
Z
= E
γt (e)(Utγ,η (e) − Ut (e))λt (de),
t ∈ [0, T ],
e) being a uniformly integrable martingale by Proposition 2.31, then with E ((−ξ + η) · W + γ ∗ µ Proposition 2.6 implies that Yt ≥ Ytγ,η , P a.s., for all (γ, β = −ξ + η) ∈ C. As a consequence Yt ≥ ess sup(γ,η) Ytγ,η = πtu (X), P a.s., where (γ, −ξ + η) range over all Girsanov kernels of measures Q ∈ Qngd .
Proof of Lemma 2.17. Denote by f φ (for φ ∈ Φ) the generator of the JBSDE (2.20). By part b) of Lemma 2.14, the generator f φ is the supremum of a family of affine JBSDE R (φ,γ,β) generators ft (t, z, u) = −ξttr φt + (Zt − φt )tr βt + E Ut (e)γt (e)λt (de), with coefficients (γ, β) ∈ C¯ bounded in Rn × L2 (λt (ω)) uniformly in (t, ω) ∈ [0, T ] × Ω by the constant Kf := sup(t,ω) sup(γ,β)∈C(t,ω) kγkL2 (λt ) + β ∈ (0, ∞). Hence for all φ ∈ Φ, f φ is Lipschitz ¯ continuous in (z, u) ∈ Rn × L2 (λt (ω)) uniformly in (t, ω) ∈ [0, T ] × Ω with Lipschitz constant Kf , and satisfies ftφ (0, 0) ∈ H2 since ξ is bounded and φ ∈ H2 . By [Bec06, Proposition 3.2], the JBSDE (2.20) has a unique solution (Y φ , Z φ , U φ ) ∈ S 2 × H2 × Hν2 .
Section 2.5. Appendix
Page 92
c and Z e = Z − φ, so that using (2.20) gives Now let Ye := Y φ − φ · W
−dYet = − dYtφ + ξttr φt dt + φtr t dWt
e U) + = Zettr β˜t (Z,
Z E
Z tr e e e(dt, de). Ut (e)˜ γt (Z, U )(e)λt (de) dt − Zt dWt − Ut (e)µ E
e solves the JBSDE (2.19) with terminal value YeT = X − φ · W cT ∈ L2 . This means that (Ye , Z) cT . Finally translation invariance Hence part b) of Theorem 2.16 implies that Yet = ρt X − φ · W
ct = ρt X − of ρ yields Ytφ = Yet + φ · W
RT
c φtr s dWs , t ∈ [0, T ].
t
Proof of Lemma 2.18. Consider deterministic (and timeindependent) parameters z ∈ Rn , u ∈ L2 (ζλ), σ ∈ Rd×n , ξ ∈ Im σ tr , and for a convex closed and bounded set C¯ ⊆ L2 (ζλ) × Rn , consider the function L : Rn × (L2 (ζλ) × Rn ) → R defined by tr
tr
(φ, (γ, β)) 7→ L(φ, (γ, β)) := −ξ φ + β (z − φ) +
Z
u(e)γ(e)ζ(e)λ(de). E
Clearly for any fixed φ ∈ Rn the function (γ, β) 7→ L(φ, (γ, β)) is linear and bounded, and for any fixed (γ, β) the function φ 7→ L(φ, (γ, β)) is linear and continuous. Since the set C¯ is convex closed and bounded, it is weakly compact in L2 (ζλ) × Rn . Now since Im σ tr is convex and closed, then by [ET99, Proposition 2.3, Chapter VI] the minimax identity inf
sup L(φ, (γ, β)) = sup
φ∈Im σ tr (γ,β)∈C ¯
inf
¯ φ∈Im σ (γ,β)∈C
tr
L(φ, (γ, β))
(2.68)
holds. Plus, the right hand side of (2.68) is equal to sup
inf
¯ φ∈Im σ (γ,β)∈C
L(φ, (γ, β)) = sup tr
Z
tr
β z+
u(e)γ(e)ζ(e)λ(de) +
¯ (γ,β)∈C
=
sup ¯ (γ,β)∈C Π(β)=−ξ
E
β tr z +
inf
φ∈Im σ
φtr (ξ + Π(β)) tr
Z
u(e)γ(e)ζ(e)λ(de), E
since inf φ∈Im σtr φtr (ξ + Π(β)) equals 0 if Π(β) = −ξ and −∞ otherwise. Now extending the arguments to random and timedependent parameters clearly gives (2.21). Proof of Theorem 2.19. Consider deterministic (and timeindependent) parameters z ∈ Rn , u ∈ L2 (ζλ), σ ∈ Rd×n , ξ ∈ Im σ tr , and for a convex bounded set C¯ ⊆ L2 (ζλ) × Rn satisfying ¯ consider the following analog of f φ as a function of φ: {0} × B (−ξ) ⊆ C, Rn ⊇ Im σ tr 3 φ 7→ F (φ) := −ξ tr φ + ess sup β tr (z − φ) + ¯ (γ,β)∈C
Z
u(e)γ(e)ζ(e)λ(de) . E
The function F is clearly convex and continuous. Moreover F is coercive on Im σ tr , i.e. ¯ one gets the F (φ) → ∞ as φ → ∞ for φ ∈ Im σ tr . Indeed, using {0} × B (−ξ) ⊆ C,
Section 2.5. Appendix
Page 93
estimate F (φ) ≥ −ξ tr φ + ess supβ∈B (−ξ) β tr (z − φ) = −ξ tr z + z − φ, which clearly implies coercivity of F . Hence by [ET99, Chapter II, Proposition 1.2], the function F admits a minimizer in Im σ tr . By extending the arguments to random and timedependent parameters, ¯ existence of φ¯ ∈ Φ satisfying f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ) follows by standard measurable selection arguments. Recall by Lemma 2.18 that it holds in particular that ¯ f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ) = f (t, Zt , Ut ) for all t ∈ [0, T ]. As a consequence of ¯ uniqueness of the solution of the JBSDE (2.18), we obtain that Y φ = Y . Now by Lemma 2.17 R ¯ c and part a) of Theorem 2.16 follows πtu (X) = Yt = Ytφ = ρt X − tT φ¯tr s dWs , t ∈ [0, T ]. To conclude that (2.23) holds, it remains to show that for any φ ∈ Φ one has Y ≤ Y φ . Let φ be in Φ. Then f φ (Z φ , Ut ) − f φ (Z φ , U φ ) =(Z φ − φt )tr β˜t (Z φ − φ, U ) − β˜t (Z φ − φ, U φ ) t
t
t
t
t
t
Z
Ut (e)˜ γt (Z φ − φ, U )(e)λt (de)
+ ZE
−
E
(2.69)
Utφ (e)˜ γt (Z φ − φ, U φ )(e)λt (de).
By part b) of Lemma 2.14, the couple γ˜t (Z φ − φ, U φ ), β˜t (Z φ − φ, U φ ) is the maximizer of R the expression (Ztφ − φt )tr βt + E Utφ (e)γt (e)λt (de) over all (γt , βt ) ∈ C¯t . Now by the fact that γ˜t (Z φ − φ, U ), β˜t (Z φ − φ, U ) ∈ C¯t holds, it follows
(Ztφ
−
φt )tr β˜t (Z φ
φ
− φ, U ) +
Z E
≥
(Ztφ
−
Utφ (e)˜ γt (Z φ − φ, U φ )(e)λt (de)
φt )tr β˜t (Z φ
− φ, U ) +
Z E
Utφ (e)˜ γt (Z φ − φ, U )(e)λt (de).
Using this inequality in (2.69) implies ftφ (Ztφ , Ut )
−
ftφ (Ztφ , Utφ )
≤
Z E
γ˜t (Z φ − φ, U )(e)(Ut (e) − Utφ (e))λt (de).
(2.70)
By Assumption 2.13, (˜ γt (Z φ − φ, U ), β˜t (Z φ − φ, U )) is bounded in Rn × L2 (λt (ω)) uniformly in (t, ω). With this at hand, one shows using Proposition 2.31 and following the arguments in the ˜ φ − φ, U ) · W + γ˜ (Z φ − φ, U ) ∗ µ e proof of Proposition 2.3 that the stochastic exponential E β(Z is a uniformly integrable martingale. Now applying Proposition 2.6 (plus Remark 2.7) to the JBSDEs with parameters (f1 , X1 ) := (f, X) and (f2 , X2 ) := (f φ , X) yields Yt ≤ Ytφ , t ∈ [0, T ]. To show the second claim of the theorem, let Qγ,β ∈ P ngd . The tracking error is given by ¯ ct . Using part a) of Theorem 2.16 and a change of measure Rφ (X) = πtu (X) − π0u (X) − φ¯ · W to Qγ,β , one obtains for all t ∈ [0, T ] that ¯ −dRtφ (X)
= f (t, Zt , Ut )dt − Zt dWt −
Z
ct ¯tr dW e(dt, de) + φ Ut (e)µ t
E
= f (t, Zt , Ut ) +
ξttr φ¯t
− (Zt − φ¯t )tr βt −
Z
Ut (e)γt (e)λt (de) dt E
Qγ,β
− (Zt − φ¯t )dWt
−
Z E
γ,β
eQ Ut (e)µ
(dt, de).
Section 2.5. Appendix
Page 94 ¯
Since by Lemma 2.18 we have f φ (t, Zt , Ut ) = ess infφ∈Φ f φ (t, Zt , Ut ) = f (t, Zt , Ut ), then ¯ the finite variation part under Qγ,β of Rφ (X) is nondecreasing and vanishes for (γ, β) = (γ ∗ , β ∗ ). By Assumption 2.13 k(γt (ω), βt (ω))kL2 (λt (ω))×Rn is bounded uniformly in (t, ω), ¯ e) ∈ S 2 . Since Rφ (X) ∈ S 2 , then H¨ hence dQγ,β /dP = E(β · W + γ ∗ µ older’s inequality implies ¯ ¯ Rφ (X) ∈ S 1 (Qγ,β ), and therefore Rφ (X) is a Qγ,β supermartingale and a martingale under ∗ ∗ Q∗ = Qγ ,β . Proof of Lemma 2.22. One rewrites Cet (ω) = {(γ, β) ∈ Ht (ω) : G(γ, β) ∈ It (ω)} , where the map G : L2 (λ) × Rn → R is defined by G(γ, β) := kγk2L2 (λ) + β2 , and I and H are closedconvexvalued correspondences with values
q
It (ω) = 0, Kt2 (ω) ⊆ R and Ht (ω) = γ ∈ L2 (λ) : γ ≥ − ζt (ω) × Rn ⊆ L2 (λ) × Rn .
Since K is a predictable process, then I is predictable (and in particular Pmeasurable). Since G(·) is continuous, then applying two times [AF90, Theorem 8.2.9] (noting that L2 (λ) is a separable Hilbert space, hence a Polish space), one obtains first that H is Pmeasurable and then that Ce is Pmeasurable (since I is). Proof of Theorem 2.26. Without loss of generality, we argue for X ≥ 0, since otherwise one can use a translation argument with X + kXk∞ ≥ 0. Part 1: For t ∈ [0, T ], since Ctk (ω) ⊆ Ctk+1 (ω) ⊆ Ct (ω) for all k ∈ N, then πtu,k (X) ≤ πtu,k+1 (X) ≤ πtu (X), for any k ∈ N. Since X is bounded, then the monotone a.s. limit Jt := limk%∞ πtu,k (X) is finite and Jt ≤ πtu (X). It remains to show that πtu (X) ≤ Jt holds. To this end we show that J is a c`adl`ag Qsupermartingale for all Q ∈ Qngd and use Lemma 2.25. First J is a c`adl`ag Qγ,β supermartingale for any Qγ,β ∈ Qngd with Girsanov kernel (γ, β) satisfying kγk2L2 (λt ) + β2 ≤ c, t ∈ [0, T ], for some constant c > 0. Indeed for such a measure Qγ,β , there exists k0 ∈ N such that (γ, β) ∈ C k for all k ≥ k0 . Since Jt = limk%∞,k≥k0 πtu,k (X) and π u,k (X) is a bounded c`adl`ag Qγ,β supermartingale for every k ≥ k0 , then J is a c`adl`ag Qγ,β supermartingale as the increasing limit of c`adl`ag Qγ,β supermartingales of class D (see e.g. [Doo01, Section 2.IV.4]). This is in particular valid when Qγ,β ∈ Qngd for some k ∈ N. k γ,β ngd Now for an arbitrary Q ∈ Q , i.e. generally satisfying (γ, β = −ξ + η) ∈ C, define k k , t ∈ [0, T ]. Then the sequence (γ , η )k∈N with (γtk , ηtk ) := (γt , ηt )1k(γt ,ηt )k 2 n ≤k k k Qγ ,β
Qngd k .
L (λt )×R
= ∈ and hence := ∈ Moreover limk γ k = γ, P ⊗ λ ⊗ dta.e. and limk η k = η, P ⊗ dta.e. By the above argument, since ξ and X are bounded, then J b is a bounded c`adl`ag Qsupermartingale and hence admits a DoobMeyer decomposition which, c, µ b F) (see e.g. [HWY92, e) under (Q, using weak predictable representation property of (W Theorem 3.22] or Part 2. in Example 1.1 of Chapter 1) and the fact that ν Qb = ν, reads c +U ∗µ b × H2 and A a nondecreasing predictable e − A, for (Z, U ) ∈ H2 (Q) J = J0 + Z · W ν
(γ k , β k
−ξ + η k )
Ck
Qk
Section 2.5. Appendix
Page 95
b processes with A0 = 0 and AT being Qintegrable since J is bounded (cf. [DM82, Inequality b to Qk on one hand and to Qγ,β on (15.1), Section VII.15]). By a change of measure from Q the other hand, one rewrites k
k
eQ + J = J0 + Z · W Q + U ∗ µ
Z · 0
Zttr ηtk +
Z
(2.71)
Ut (e)γtk (e)λt (de) dt − A,
E
and γ,β
J = J0 + Z · W Q
eQ +U ∗µ
γ,β
+
Z · 0
Zttr ηt +
Z
(2.72)
Ut (e)γt (e)λt (de) dt − A.
E
Now since (γ k , η k ) ∈ C k then J is a bounded c`adl`ag Qk supermartingale for any k ∈ N. Hence R tr k k from (2.71), it follows that dAt ≥ Zt ηt + E Ut (e)γt (e)λt (de) dt. Taking the limit as k
goes to ∞ and using the dominated convergence theorem (since η k ≤ η and γ k ≤ γ R a.s.) one obtains dAt ≥ Zttr ηt + E Ut (e)γt (e)λt (de) dt, t ∈ [0, T ] and hence the process R· R A − 0 Zttr ηt + E Ut (e)γt (e)λt (de) dt is nondecreasing and in particular nonnegative. Now γ,β γ,β eQ since X is nonnegative, so is J and this implies from (2.72) that J0 + Z · W Q + U ∗ µ is a nonnegative local Qγ,β martingale and is therefore a Qγ,β supermartingale. Finally, Qγ,β R R integrability of 0T Zttr ηt + E Ut (e)γt (e)λt (de) dt − AT follows, by boundedness of J, and thus J is a Qγ,β supermartingale. Part 2: For k ∈ N, the process π u,k can be seen as the gooddeal bound associated to the correspondence C k satisfying the hypotheses of Theorem 2.16, which implies the required result, after using the apriori estimates of [Bec06, Proposition 3.3] to obtain Y ∈ S ∞ . b ∈ Qngd ⊂ Qngd . Hence by Lemma 2.25, π u (X) and Part 3: For any k ≥ kξk∞ holds Q k b π u,k (X) (for k ≥ kξk∞ ) are bounded c`adl`ag Qsupermartingales since X is bounded. Thus they admit DoobMeyer decompositions (2.48) and (2.49) respectively. Now since the triple (π u,k (X), Z k , U k ) solves the JBSDE (2.47), one obtains that Ak satisfies (2.50). Moreover b × H2 (Q) b and AT , Ak ∈ L2 (Q) b follows from arguments in the proof of Part 1. (Z, U ) ∈ H2 (Q) ν T b Fu ) as k → ∞ for Part 4: From Part 3., that Aku converges to Au weakly in L2 (Ω, Q, all u ∈ [0, T ] follows Theorem VII.18 and subsequent remark]. This applies from [DM82, u,k since the sequence π (X) is uniformly bounded by kXk∞ (and hence uniformly k≥kξk∞
of class D), and therefore Part 1. and dominated convergence imply that πuu,k (X) converges b Fu ) as k → ∞ for all u ∈ [0, T ]. Furthermore the convergences of the to πuu (X) in L2 (Ω, Q, R R k cu + u e(ds, de) converges sequences (πuu,k (X))k∈N and (Aku )k∈N imply that Z k · W 0 E Us (e)µ RuR 2 cu + b e to Z · W (e) µ (ds, de) weakly in L (Ω, Q, F ) for all u ∈ [0, T ]. By the weak U u 0 E s c b e) under Q, strong orthogonality and isometry, the predictable representation property of (W , µ required result follows. Part 5: By Lemma 2.10, π u (X) is a Qsupermartingale with terminal value X, for all Q ∈ Qngd . ¯ is in the L1 closure of Qngd , then π u (X) is also a Qsupermartingale ¯ Since by assumption Q
Section 2.5. Appendix
Page 96 ¯
with terminal value X. This together with π0u (X) = E Q [X] implies that π·u (X) has constant ¯ ¯ Qexpectation and is therefore a Qmartingale. Further, quasileftcontinuity of π u (X) is clear ¯ since ν λ⊗dt implies ν Q λ⊗dt by part a) of Lemma 2.1. Since X ∈ L∞ , then to show that b for any p ∈ [1, ∞), it suffices by dominated convergence π u,k (X) converges to π u (X) in S p (Q) u,k u to show that supt∈[0,T ] πt (X)−πt (X) converges to 0 in probability. By part 1. we know that πtu,k (X) % πtu (X) a.s. for all t ∈ [0, T ]. Moreover, denoting by p Y the predictable projection of an integrable process Y relative to the filtration F, it holds that p π u,k (X)t % p π u (X)t for any t ∈ [0, T ]. Recall that for every uniformly integrable martingale M holds p M = M− , and that for a predictable process K one has p K = K. Now because π u,k (X) is a bounded u,k b and Ak is continuous, then holds p π u,k (X)t = πt− (X). Moreover by Qsupermartingale u quasileftcontinuity of π (X), one has that A is a continuous process (by [HWY92, Theorem u (X). As a consequence one has π u,k (X) % π u (X) a.s. for 5.50]) and hence p π u (X)t = πt− t− t− all t ∈ [0, T ]. Now by the extended Dini’s Lemma in [DM82, Page 185] uniform convergence in time follows, i.e. supt∈[0,T ] πtu,k (X) − πtu (X) & 0. To prove the remaining claims, note that for all k ∈ N holds E Qb [AkT ] ≤ E Qb [πTu,k (X) − π0u,k (X)] ≤ 2kXk∞ , which implies that supk∈N E Qb [
RT 0
dAkt ] < ∞. Finally by [BP90, Corollary 2], the required claims follow.
Proof of Proposition 2.29. Part 1: If Qγ,β ∈ Me , then by Lemma 2.8 it follows that β = −ξ+η with the required properties, P and moreover − log Γγ,β = −Mt + 12 hM c it − s≤t (log(1 + ∆Ms ) − ∆Ms ), holds for t ∈ [0, T ]. t P Define the process V by Vt := s≤t (log(1 + ∆Ms ) − ∆Ms ). Since E[− log Γγ,β T ] < ∞ then by Proposition 2.27 the process − log Γγ,β is a P submartingale of class (D) with a DoobMeyer R decomposition − log Γγ,β = N γ,β + Aγ,β . Hence M , hM c i = 0· βs 2 ds and V are locally P integrable with et + g(1 + γ) ∗ νt , −Vt = (− log(1 + γ) + γ) ∗ µt = g(1 + γ) ∗ µt = g(1 + γ) ∗ µ
where the third equality is obtained from [JS03, Proposition II.1.28] and g ≥ 0. Hence one has et + 12 hM c it + g(1 + γ) ∗ νt . Now because M = M c + M d − log Γγ,β = −Mt + g(1 + γ) ∗ µ t e, it follows that with M c = β · W and M d = γ ∗ µ et + − log Γt = −β · Wt − log(1 + γ) ∗ µ 
{z
:=Ntγ,β
}
Z t 1 0
2
2
βs  +
Z
g(1 + γs (e))λs (de) ds, E
{z
:=Aγ,β t
}
e is locally P integrable thanks to the local P integrability of V where the process log(1 + γ) ∗ µ e, and an application of [JS03, Proposition II.1.28]. and γ ∗ µ
Part 2: For Q ∈ nQngd ⊂ Me , Proposition 2.28 implies that o(κQ − υ)(B) ≤ 0, for any B ∈ P. R Choosing B = 12 β2 + E g(1 + γ· (e))ζ· (e)λ(de) > 21 K 2 , the Fubini’s theorem (see e.g. [Coh13, Proposition 5.2.1]) gives B ∈ P and hence (κQ − υ)(B) ≤ 0 holds, which implies by
Section 2.5. Appendix
Page 97
definition of κQ and υ that B is a P ⊗dtnullset. Thus 12 β2 + E g(1+γ· (e))ζ· (e)λ(de) ≤ 12 K 2 holds, which is equivalent to (2.56), since by Part 1. one has Πt (βt ) = ξt and Π⊥ t (βt ) = ηt . R
e eintegrable function, and let β be a predictable Part 3: Let γ > −1 be a Pmeasurable and µ ⊥ process with Πt (βt ) = ξt and Πt (βt ) = ηt , t ∈ [0, T ], such that (γ, β) satisfies the inequality R R 2 1 1 2 1 2 2 β + E g(1 + γ· (e))ζ· (e)λ(de) ≤ 2 K . Then E g(1 + γ· (e))ζ· (e)λ(de) ≤ 2 K and 2 2 β ≤ K . By boundedness of K, the couple (γ, β) defines from Proposition 2.3 the Girsanov kernels of a measure Q ∼ P . That Q ∈ Me follows from Lemma 2.8. For such a measure Q the integrability condition on (γ, β) directly implies from the last claim of Proposition 2.27 that (2.54) is satisfied; hence Q ∈ Qngd .
Proof of Lemma 2.30. From the discussion preceding the statement of the lemma, one already T T −1/2 has Cet ⊆ l∈N Cetl , t ∈ [0, T ]. Now let t ≤ T and (γ, β) ∈ l∈N Cetl , i.e. ζt 1{ζt >0} γ, β ∈ −1/2 1{ζ >0} γ, β ∈ C¯t . Define the sequence C¯tl , for all l ∈ N. It is to be shown that ζ t
−1/2
γ k , β k , k ∈ N such that ζt
t
−1/2
1{ζt >0} γ ∨ (−1 + 1/k) and β k := β, k ∈ −1/2 −1/2 √ N. One has for all k ∈ N that ζt 1{ζt >0} γ k ≤ ζt 1{ζt >0} γ (since γ ≥ − ζt ) and
1{ζt >0} γ k := ζt −1/2
so dominated convergence implies that ζt
−1/2
1{ζt >0} γ k , β k converges to ζt 1{ζt >0} γ, β −1/2 2 n in L (λt ) × R as k → ∞. To conclude that ζt 1{ζt >0} γ, β ∈ C¯t it remains to show −1/2
that ζt
1{ζt >0} γ k , β ∈ Ct holds for any k ∈ N. For this purpose, one has to show
−1/2 −1/2 that 2g(1 + ζt 1{ζt >0} γ k ) L1 (λt ) + β2 ≤ Kt2 holds. Note that g 1 + ζt 1{ζt >0} γ k =
−1/2
g k 1+ζt
when 1 +
1{ζt >0} γ
−1/2
λa.e. since 1+ζt
−1/2 ζt 1{ζt >0} γ
−1/2
1{ζt >0} γ k either takes the value 1+ζt
1{ζt >0} γ −1/2 ≥ 1/k or 1/k when 1 + ζt 1{ζt >0} γ ≤ 1/k. This implies that
k
−1/2 −1/2 k
ζt 1{ζt >0} γ ) L1 (λt ) = 2g (1 + ζt 1{ζt >0} γ) L1 (λt ) . Since
for all k ∈ N, 2g(1 + −1/2 ζt 1{ζt >0} γ, β ∈ C¯tk , for all k ∈ N, this concludes the first claim of the lemma.
To prove the second claim, recall that for any l ∈ N the function g l satisfies g l (1 + y) ≤ Const y2 for all y ≥ −1 for some Const > 0. We first show that the map Gl : [0, T ] ×
−1/2 Ω × L2 (λ) × Rn → R defined by Gl (t, ω, γ, β) := β2 + 2g l (1 + ζt (ω)γ) L1 (λt (ω))
is continuous in (γ, β) ∈ L2 (λ) × Rn . For this purpose, it suffices to show that for a R sequence (γ n )n ⊂ L2 (λ) which converges in L2 (λ) to γ, the sequence of integrals E g l (1 + R −1/2 n −1/2 ζt γ (e))λt (de) converges to E g l (1 + ζt γ(e))λt (de). If the measure λ is finite, this follows straightforwardly from the Lipschitz property of the function g l . For infinite λ, note that there exist by [AB06, Theorem 13.6] a subsequence (γ nk )k of (γ n )n and a function ψ ∈ L2 (λ) such that γ nk  ≤ ψ λa.e. and γ nk converges λa.e. to γ. Renaming the R −1/2 nk γ (e))λt (de) converges to subsequence if necessary we can assume that E g l (1 + ζt R l R −1/2 n lim supn E g (1+ζt γ (e))λt (de) as k → ∞. Dominated convergence implies that E g l (1+ R R −1/2 nk −1/2 ζt γ (e))λt (de) converges to E g l (1 + ζt γ(e))λt (de), and therefore lim supn E g l (1 + R −1/2 n −1/2 ζt γ (e))λt (de) = E g l (1 + ζt γ(e))λt (de). By similar arguments one can also show
Section 2.5. Appendix
Page 98 −1/2
−1/2
that lim inf n E g l (1 + ζt γ n (e))λt (de) = E g l (1 + ζt γ(e))λt (de), and this concludes l the continuity of G . Now by Fubini’s theorem ([Coh13, Proposition 5.2.1]), the map Gl is e in addition Pmeasurable in (t, ω) ∈ [0, T ] × Ω since ζ is Pmeasurable and g l is continuous. Hence Gl is a Carath´eodory map and following the same arguments as those of the proof of Lemma 2.22 one shows that Ce l is Pmeasurable for any l ∈ N. Now Pmeasurability of Ce follows by [AF90, Theorem 8.2.4]. R
R
3. Hedging under generalized gooddeal bounds and drift uncertainty In this chapter, we study gooddeal valuation and hedging as in Chapter 2 but in a Brownian setting (i.e. continuous filtration), focusing on constructive examples with closedform solutions and on robustness with respect to uncertainty (ambiguity) about the market price of risk (excess returns) of hedging assets. We describe robust gooddeal bounds and hedging strategies by solutions to standard BSDEs, and show that robust gooddeal hedging is equivalent to riskminimization (with respect to a specific nogooddeal pricing measure depending on the claim to be hedged) if uncertainty is very large. Section 3.1 formulates a framework for gooddeal constraints which are described by predictable correspondences sufficiently general for all later sections to incorporate the natural radial (Sharpe ratio) constraints that are predominant in the good deal literature, but also extensions to ellipsoidal constraints. Section 3.2 studies hedging strategies and provides new examples with closedform formulas for gooddeal bounds and hedging strategies. In the presence of model uncertainty, good deal bounds and hedging strategies that are robust with respect to uncertainty are derived the Section 3.3, with the link to riskminimization made more precise. In Appendix 3.4 we provide statements of intermediary results and proofs that are omitted in the main body of the chapter.
3.1
Mathematical framework and preliminaries
We work on a filtered probability space (Ω, F, F, P ) with time horizon T < ∞; the filtration F = (Ft )t≤T generated by an ndimensional Brownian motion W , augmented with P nullsets, satisfying the usual conditions. Let F = FT . Inequalities between random variables (processes) are meant to hold almost everywhere with respect to P (resp. P ⊗ dt). For stopping times τ ≤ T , the conditional expectation given Fτ under a probability measure Q is denoted by EτQ · . We write Eτ = EτP if there is no ambiguity about P . Lp (Rm , Q), p ∈ [1, ∞), (or L∞ (Rm , Q)) denotes the space of FT measurable Rm valued random variables X with kXkpLp (Q) = E Q [Xp ] < ∞ (resp. X Qessentially bounded). P denotes the predictable σfield on [0, T ] × Ω. Stochastic integrals of predictable integrands H with respect to semimartingales R p (Rm , Q) denote the space of predictable Rm S are denoted H · S = 0· Httr dSt . Let H h i valued processes Z with kZkpHp (Q) = E Q semimartingales Y with kY kS p (Q)
RT
Zs 2 ds
0
= supt≤T Yt 
p 2
< ∞, and S p (Q) that of c`adl`ag
< ∞ If the dimension is clear, we just
Lp (Q) Hp and
write and and if Q = P just S p , for p ∈ [1, ∞]. The Euclidean norm of a matrix M ∈ Rn×d is M  := (Tr M M tr )1/2 and its usual operator norm is denoted by kM k. Lp (Q)
Hp (Q),
Lp ,
99
Section 3.1. Mathematical framework and preliminaries
Page 100
We will make use of classical theory of BSDEs [PP90, EPQ97]. BSDEs are stochastic differential equation of the type (3.1)
−dYt = f (t, Yt , Zt )dt − Zttr dWt , for t ≤ T, and YT = X,
where the terminal condition X is an FT measurable random variable and the generator function f : Ω×[0, T ]×R1+n → R a P ⊗B(R1+n )B(R)measurable function. They are well established in mathematical economics. A pair (f, X) constitutes standard parameters (also called data) for a BSDE (3.1) if X ∈ L2 , f (·, 0, 0) is in H2 and f is uniformly Lipschitz in y and z, i.e. there exists L < ∞ such that f (ω, t, y, z) − f (ω, t, y 0 , z 0 ) ≤ L(y − y 0  + z − z 0 ) holds for all t, y, z, y 0 , z 0 . A solution of the BSDE (3.1) is a couple (Y, Z) of processes such that Y is R realvalued continuous, adapted, and Z is Rn valued predictable and satisfies 0T Zt 2 dt < ∞. For standard parameters (f, X) there exists a unique solution (Y, Z) ∈ S 2 × H2 to the BSDE (3.1), [EPQ97, Theorem 2.1]. Let us refer to BSDEs with standard parameters as classical and to the solution to such BSDEs as standard. A comparison theorem [EPQ97, Proposition 3.1] is very useful for optimal control problems stated in terms of classical BSDEs: Given standard BSDE solutions (Y, Z), (Y a , Z a )a∈A for a family of standard parameters (f, X), (f a , X a )a∈A , if there exists a ¯ ∈ A such that f (t, Yt , Zt ) = ess inf f a (t, Yt , Zt ) = f a¯ (t, Yt , Zt ), t ≤ T , and a∈A
X = ess inf X a = X a¯ , then Yt = ess inf Yta = Yta¯ holds for all t ≤ T. a∈A
a∈A
Section 3.1.1 will specify a financial market with d risky assets whose discounted price processes S i (i ≤ d) with respect to a fixed num´eraire asset (with unit price S 0 = 1) are nonnegative locally bounded semimartingales. The set of equivalent local martingale measures (risk neutral pricing measures) is denoted by Me := Me (S) and we assume Me 6= ∅, i.e. there is no free lunch with vanishing risk in the sense of [DS94]. The market is incomplete with Me being of infinite cardinality if d < n. We will define generalized gooddeal bounds by using abstract predictable correspondences C defined on [0, T ] × Ω with nonempty compact and convex values Ct (ω) ⊂ Rn , with predictability in the sense of [Roc76], i.e. for each closed set F ⊂ Rn , the set C −1 (F ) := {(t, ω) ∈ [0, T ] × Ω : Ct (ω) ∩ F 6= ∅} is predictable. More specific examples, e.g. for ellipsoidal constraints, will exhibit (semi)explicit solutions for optimizers. We write C : [0, T ] × Ω Rn with “ ” to emphasize that C is a setvalued mapping, and λ ∈ C to mean that the predictable function λ is a selection of C, i.e. λt (ω) ∈ Ct (ω) holds on [0, T ] × Ω. In the sequel a standard correspondence will refer to a predictable one, whose values are nonempty, compact and convex. Let C : [0, T ] × Ω Rn be a fixed standard correspondence with 0 ∈ C. The set Qngd := Qngd (S) of (equivalent) nogooddeal measures is given by n
o
Qngd (S) := Q ∈ Me dQ/dP = E (λ · W ) , λ predictable, bounded, λ ∈ C .
(3.2)
In the definition (3.2) and in subsequent definitions of sets of equivalent measures, we tacitly assume that Girsanov kernels λ are such that the stochastic exponentials E (λ · W ) are uniformly
Section 3.1. Mathematical framework and preliminaries
Page 101
integrable martingales. For gooddeal valuation and hedging results later, concrete assumptions (e.g. Assumption 3.3) ensure that such holds for all selections λ of C. We remark that for good timeconsistency properties, gooddeal constraints should be specified locally in time ([KS07b]). For contingent claims X in L2 , upper and lower gooddeal valuation bounds πtl (X) := ess inf EtQ [X] Q∈Qngd
and
πtu (X) := ess sup EtQ [X],
t ∈ [0, T ].
Q∈Qngd
(3.3)
are defined over a suitable (yet abstract) set of no good deal pricing measures Qngd . Hence πtu (X) (respectively πtl (X)) can be seen as the highest (lowest) valuation that does not permit too good deals to the seller (buyer). Since π·l (X) = −π·u (−X), further analysis can be restricted to π·u (X). As mentioned already in the introduction, the definition (3.3) in itself could already be viewed as a robust representation in a sense (over Q’s). For our purpose here however, the correspondence C and the respective set Qngd of no good deal measures are (at first) given with respect to one objective real world measure P (cf. remarks after (3.4)). To be clear in our use of terminology, we will in the sequel restrict our use of terms model uncertainty, ambiguity or robust hedging/valuation to situations with Knightian model uncertainty about P . Note that the use of terminology in some literature (e.g. [Del12]) is different, where the terms may instead refer to representations like (3.3). Definition (3.2) implies that density processes of measures Q ∈ Qngd are in S p , p ∈ [1, ∞). Hence X ∈ L2 = L2 (P ) ⊂ L1 (Q). In particular for X ∈ L∞ ⊂ L2 , we will show (cf. Theorem 3.7 and Proposition 3.5) that πtu (X) = ess supQ∈Qngd EtQ [X], where n
o
Qngd := Q ∈ Me dQ/dP = E (λ · W ) , λ predictable and λ ∈ C .
(3.4)
is a larger set than Qngd , containing measures with Girsanov kernels that are not necessarily bounded. We recall that for radial constraints C (like in (3.21) with A ≡ IdRn and constant h ∈ (0, ∞)), common in the gooddeal literature, one has a known financial justification. By a direct duality argument, one can see (e.g. [Bec09, Section 3] in a semimartingale framework) that any (arbitragefree) extension S¯ = (S, S 0 ) of the market S by derivative price processes S 0 := EtQ [X] for contingent claims X (with Q ∈ Me , X − ∈ L∞ , X + ∈ L1 (Q)) does permit ¯ whose expected only wealth processes V > 0 from selffinancing trading strategy (in S) growth over h rates i (log utilities) h i any time period 0 ≤ t < τ ≤ T satisfy the (sharp) estimate EtP log VVτt ≤ EtP − log ZZτt , where Z is the density process of Q. For Q ∈ Qngd with radial constraint, this estimate is bounded by h2 (τ − t)/2, ensuring a bound h2 /2 to expected growth rates (good deals) for any market extension (ideas going back at least to [CR00, CH02]). For the gooddeal bounds to have nice dynamic properties, multiplicative stability (mstability) of the set of nogooddeal measures is important. Mstability of dominated families of probability measures in dual representations (like e.g. (3.3)) for dynamic coherent risk measures ( see e.g. [ADE+ 07]) ensures in particular time consistency (recursiveness) and has been studied
Section 3.1. Mathematical framework and preliminaries
Page 102
in a general context by [Del06]. In economics, it is known as rectangularity [CE02]. A set Q of measures Q ∼ P is called mstable if for all Q1 , Q2 ∈ Q with density processes Z 1 , Z 2 and for all stopping times τ ≤ T , the process Z := I[0,τ ] Z·1 + I]τ,T ] Zτ1 Z·2 /Zτ2 is the density process of a measure in Q, where [0, τ ] := {(t, ω) ∈ [0, T ] × Ω  t ≤ τ (ω)} denotes the stochastic interval and IA is the indicator function on a set A. As noted in [Del06, Rem. 6], by closure this definition extends to sets of measures that are absolutely continuous but not necessarily equivalent; such is formally achieved by setting ZT2 /Zτ2 = 1 on {Zτ2 = 0}. The role of mstability shows in results due to [Del06], stated in Lemma 3.1, Part a); for details cf. [KS07b, Theorem 2.7] or [Bec09, Proposition 2.6]. Proof for part b) is provided in the appendix. Lemma 3.1. Let Q be a convex and mstable set of probability measures Q ∼ P and πtu,Q (X) := ess supQ∈Q EtQ [X], for X ∈ L∞ . a) There exists a c`adl`ag version Y of π·u,Q (X) such that for all stopping times τ ≤ T , Yτ = ess supQ∈Q EτQ [X] =: πτu,Q (X). Moreover π·u,Q (·) has the properties of a dynamic coherent risk measure. It is recursive and stopping time consistent: For stopping times u,Q 1 u,Q u,Q 1 u,Q σ ≤ τ ≤ T holds πσ (X ) = πσ πτ (X ) , and πτ (X 1 ) ≥ πτu,Q (X 2 ) for X 1 , X 2 ∈ L∞ implies πσu,Q (X 1 ) ≥ πσu,Q (X 2 ). Finally, a supermartingale property holds: For all stopping times σ ≤ τ ≤ T and Q ∈ Q, πσu,Q (X) ≥ EσQ πτu,Q (X) , and π·u,Q (X) is a supermartingale under any Q ∈ Q. b) The sets Me and Qngd are mstable and convex and hence for Q = Qngd , π·u (X) = π·u,Q (X) satisfies the properties of Part a).
3.1.1
Parametrizations in an Itˆ o process model
This section describes the Itˆo process framework for the financial market, and details the parametrizations for dynamic trading strategies and for the nogooddeal constraints. The latter are specified at this stage by abstract correspondences (3.2) such that respective dynamic nogooddeal valuation bounds for contingent claims can be conveniently described in terms of (super)solutions to BSDEs (Sections 3.1.23.1.3) within a convenient framework sufficiently general for all later Sections 3.23.3. We consider models for financial markets where prices (S i )i=1...d of d risky assets evolve according to a stochastic differential equation (SDE) ct , t ∈ [0, T ], dSt = diag(St )σt (ξt dt + dWt ) =: diag(St )σt dW
S0 ∈ (0, ∞)d ,
for predictable Rd  and Rd×n valued coefficients ξ and σ, with d ≤ n. This includes basically all examples of continuous price and state evolutions in (typically incomplete) markets of the gooddeal literature, and permits also for nonMarkovian evolutions. Risky asset prices S are
Section 3.1. Mathematical framework and preliminaries
Page 103
given in units of some riskless num´eraire asset whose discounted price S 0 ≡ 1 is constant. We assume that σ is of maximal rank d ≤ n (i.e. det(σt σttr 6= 0, that means no locally redundant assets) and that the market price of risk process ξ, satisfying ξt ∈ Im σttr , is bounded. This ensures that market is free of arbitrage but typically incomplete (if d < n) in the sense that b given by dQ b = E (−ξ · W ) dP (see Me 6= ∅, as the minimal local martingale measure Q [Sch01]) is in Me , which however is typically not a singleton. Trading strategies are represented by the amount of wealth ϕ = (ϕit )i invested in the risky assets (S i )i . A selffinancing trading strategy is described by a pair (V0 , ϕ), where V0 is the initial capital while ϕ = (ϕit )i describes the amount of wealth invested in the risky assets (S i )i at any time t. Theh set Φϕ of permitted i R 2 strategies consists of Rd valued predictable processes ϕ satisfying E P 0T ϕtr t σt  dt < ∞. For an permitted strategy ϕ, the associated wealth process V from initial capital V0 has c dynamics dVt = ϕtr t σt dWt . To ease notation, we reparametrize strategies in Φϕ in terms of c . Indeed, equalities φ = σ tr ϕ and ϕ = (σ tr )−1 φ, where integrands φ := σ tr ϕ with respect to W (σ tr )−1 := (σσ tr )−1 σ is the pseudoinverse of σ tr , provide a onetoone relation between ϕ and φ. Define the correspondences Γt (ω) := Im σttr (ω) and Γ⊥ t (ω) := Ker σt (ω),
(t, ω) ∈ [0, T ] × Ω,
(3.5)
where Im σttr and Ker σt denote the range (image) and the kernel of the respective matrices. n Clearly, Rn = Γt ⊕ Γ⊥ t and any z ∈ R decomposes uniquely into its orthogonal projections as ⊥ z = ΠΓt (z) ⊕ ΠΓ⊥ (z) =: Πt (z) ⊕ Πt (z). Let t
(
Φ = Φφ :=
hZ φ φ is predictable, φ ∈ Γ and E
T
i
)
φt 2 dt < ∞
0
denote the (reparametrized) set of permitted trading strategies. Proving the claims of the next proposition is routine, using [Roc76] for the first. Proposition 3.2. dictable.
1. The correspondences Γ and Γ⊥ are closedconvexvalued and pre
2. Q ∈ Me if and only if Q ∼ P with dQ = E(λ · W )dP , where λ is predictable and λ = −ξ + η, with −ξt = Πt (λt ) ∈ Im σttr and ηt = Π⊥ t (λt ) ∈ Ker σt ∀t. By Part 2 of Proposition 3.2, the set Qngd defined in (3.2) can be written as n
o
Qngd = Q ∼ P dQ/dP = E (λ · W ) , λ predictable, bounded and λ ∈ Λ ,
(3.6)
where Λ : [0, T ] × Ω Rn is defined by Λt (ω) := Ct (ω) ∩ (−ξt (ω) + Ker σ). By Part 1 of Proposition 3.2 and [Roc76, Corollary 1.K and Theorem 1.M], Λ is a compactconvexvalued predictable correspondence. Slightly beyond the nofreelunch with vanishing risk condition, b or equivalently −ξ ∈ C. This implies that Λ is we assume that Qngd contains the measure Q, nonempty valued, hence standard.
Section 3.1. Mathematical framework and preliminaries
3.1.2
Page 104
Gooddeal valuation with uniformly bounded correspondences
We here consider the case where the nogooddeal constraint is described by a uniformly bounded correspondence; a more general case is studied afterwards. We say that a correspondence C is uniformly bounded if it satisfies Assumption 3.3. sup(t,ω) supx∈Ct (ω) x < ∞. Let C : [0, T ] × Ω Rn be a standard correspondence satisfying Assumption 3.3 and 0 ∈ C. Under Assumption 3.3, selections of C are uniformly bounded processes. In particular, the Girsanov kernels of nogooddeal measures are uniformly bounded, and hence boundedness in the definition (3.2) (see also (3.6)) of Qngd is not necessary. The gooddeal valuation bound πtu (X) := ess supQ∈Qngd EtQ [X] is welldefined for a contingent claim X ∈ L2 ⊃ L∞ , that may be pathdependent, and one can check that in this case an analog of Part a) of Lemma 3.1 still holds. Though Assumption 3.3 fits well with the classical theory, it would be too restrictive to impose it in general since it may not hold in some interesting practical situations; see for instance the example in Section 3.2.2. Let us recall a fact about linear BSDEs (cf. [EPQ97]) which explains their role for valuation purposes. Lemma 3.4. For Q ∼ P with bounded Girsanov kernel λ, the linear BSDE −dYt = Zttr λt dt − Zttr dWt , t ≤ T,
with YT = X in L2 ,
(3.7)
has a unique standard solution (Y λ , Z λ ) with Ytλ = EtQ [X] = Y0λ + Z · WtQ , and W Q := R W − 0· λt dt. If X ∈ L∞ then Y is bounded. Boundedness of λ in Lemma 3.4 clearly implies that the parameters of the BSDE (3.7) are standard. For unbounded λ, the classical BSDE theory no longer applies and one needs different results to characterize the gooddeal bounds in terms of BSDEs. Under Assumption 3.3, Λ is uniformly bounded and thus Girsanov kernels λQ for all Q ∈ Qngd are bounded by the same constant. One has the following Proposition 3.5. Let Assumption 3.3 hold. ¯ := λ(Z) ¯ 1. For any predictable Rn valued process Z, there exists a predictable process λ = tr tr ¯ ¯ (λt (Zt ))t≤T ∈ Λ such that λt Zt = ess sup λt Zt , t ∈ [0, T ]. λt ∈Λt
2. For X ∈ L2 , let (Y λ , Z λ ) (for λ = λQ ∈ Λ, Q ∈ Qngd ) and (Y, Z) be respectively standard solutions to the classical BSDEs (3.7) and ¯ t (Zt )dt − Z tr dWt , t ≤ T, and −dYt = Zttr λ t
YT = X,
(3.8)
¯ ¯ = λ(Z) ¯ with λ from Part 1. Then πtu (X) = ess supQ∈Qngd EtQ [X] = EtQ [X] = Yt holds ¯ λ λ ¯ · W )dP , Yt = ess sup ¯ ∈ Qngd given by dQ ¯ = E(λ for Q λ∈Λ Yt = Yt .
Section 3.1. Mathematical framework and preliminaries
Page 105
Proof. Part 1 follows by a direct application of the measurable maximum theorem [Roc76, Theorem 2.K] and measurable selection theorem [Roc76, Theorem 1.C]. As for Part 2, by ¯ Assumption 3.3 the parameters of the BSDEs (3.8) and (3.7) are standard. Moreover Q ngd ¯ ∈ Λ. The remaining of Part 2 hence follows from existence and is clearly in Q since λ uniqueness results as well as the comparison theorem for classical BSDEs, cf. [EPQ97, Section 23]
3.1.3
Gooddeal valuation with nonuniformly bounded correspondences
To relax the Assumption 3.3 of uniform boundedness, we now admit for a nonuniformly bounded standard correspondence C, with 0 ∈ C, which satisfies ∃ R predictable with
sup x ≤ Rt (ω) ∀(t, ω) and x∈Ct (ω)
Z
T
Rt 2 dt < ∞.
0
(3.9)
It is relevant to look beyond Assumption 3.3, because examples of practical interest require to do so, see Section 3.2.2 where quasiexplicit formulas of gooddeal bounds are obtained in a stochastic volatility model, with C not being uniformly bounded but satisfying (3.9). Classical BSDE results do not apply as before to characterize gooddeal bounds directly by standard BSDE solutions. Yet, we can still (cf. Theorem 3.7) approximate π·u (X) for X ∈ L∞ by solutions to classical BSDEs for suitable truncations of C, and prove that πtu (X) coincides with the essential supremum over the larger set Qngd ⊆ Me given in (3.4). We show, under condition (3.9), that π·u (X) is the minimal supersolution of the BSDE (3.8). Finally, we ¯ for π u (X) exists. show that π·u (X) is the minimal solution to (3.8) if a worstcase measure Q 0 ¯ may be attained rather in the larger set Qngd . Obviously, a maximizing Q To this end, let Ctk (ω) = {x ∈ Ct (ω) : x ≤ k} for (t, ω) ∈ [0, T ] × Ω with k ∈ N be a sequence of correspondences. Since C is standard with 0 ∈ C, the same holds for each C k . Clearly, any C k satisfies Assumption 3.3 and Ctk (ω) % Ct (ω) as k % ∞. For each k ∈ N, let Qngd := Qngd k k (S) denote the set n
Qngd := Q ∼ P dQ/dP = E (λ · W ) , with λ predictable and λ ∈ Λk k
o
(3.10)
of nogooddeal measures (for S) corresponding to C k with Λk : [0, T ] × Ω Rn given by Λkt (ω) := Ctk (ω) ∩ (−ξt (ω) + Γ⊥ t (ω)) and hence also satisfying Assumption 3.3. For 2 X ∈ L , we define analogously the bounds π·u,k (X) associated to the sets Qngd k , k ∈ N as u,k Q ngd πt (X) := ess supQ∈Qngd Et [X], t ∈ [0, T ]. The sets Qk , k ∈ N are mstable and convex k as well.
Section 3.1. Mathematical framework and preliminaries
Page 106
Lemma 3.6. (Dynamic principle): Let Q be a convex and mstable set of probability measures Q ∼ P and πtu,Q (X) := ess supQ∈Q EtQ [X], for X ∈ L∞ . Then π·u,Q (X) is the smallest adapted c`adl`ag process that is a supermartingale under any Q ∈ Q with terminal value X. Proof. The supermartingale properties of π·u,Q (X) under every Q ∈ Q hold by Part a) of Lemma 3.1. Let Y be another process satisfying the same properties. Then for all Q ∈ Q one has Yt ≥ EtQ [X], t ∈ [0, T ], and taking the essential supremum over Q ∈ Q then yields Yt ≥ πtu,Q (X).
ngd are convex and mstable, Lemma 3.6 holds in particular for Note that since Qngd k , Q u,Qngd
ngd
π·u (X) = π·u,Q (X) and π·u,k (X) = π· k (X), k ∈ N. Theorem 3.7 is analogous to Parts 1.4. of Theorem 2.26 in Chapter 2. We still include its proof in Appendix 3.4, because it seems more instructive in the absence of jumps. Theorem 3.7. For any contingent claim X ∈ L∞ it holds 1. πtu,k (X) % ess sup EtQ [X] = πtu (X) P a.s. as k % ∞, for all t ∈ [0, T ]. 2. For any k
Q∈Qngd ∈ N, π·u,k (X)
= Y k for (Y k , Z k ) being standard solution to the BSDE
tr −dYt = (ess sup λtr t Zt )dt − Zt dWt , t ≤ T, with
YT = X.
λt ∈Λkt
(3.11)
3. π·u (X) and π·u,k (X) for k ≥ kξk∞ admit DoobMeyer decompositions c −A π·u (X) = π0u (X) + Z · W
and
c − Ak , π·u,k (X) = π0u,k (X) + Z k · W
(3.12)
b , where Z, Z k ∈ H2 (Q) b and A, Ak are nondecreasing predictable processes with under Q k 2 k b AT , AT ∈ L (Q), A0 = A0 = 0 and
Ak =
Z · 0
k ξttr Ztk + ess sup λtr t Zt dt. λt ∈Λkt
(3.13)
b Fu ) and Z k → Z weakly in L2 (Ω×[0, u], Q⊗dt). b 4. For all u ≤ T , Aku → Au weakly in L2 (Ω, Q,
Let g be the function defined by gt (z) := ess sup λtr t z, λt ∈Λt
t ∈ [0, T ], z ∈ Rn .
(3.14)
Since g may not be Lipschitz if C does not satisfy Assumption 3.3, then π·u (X) cannot directly be characterized by classical BSDEs. But one can still obtain a characterization by the minimal supersolution to the BSDE with data (g, X).
Section 3.1. Mathematical framework and preliminaries
Page 107
Definition 3.8. (Y, Z, K) is a supersolution of the BSDE with parameters (f, X) if −dYt = f (t, Yt , Zt )dt − Zttr dWt + dKt
for t ≤ T, and YT = X,
with K nondecreasing c`adl`ag adapted, , K0 = 0, and 0T Zt 2 dt < ∞. A supersolution with K ≡ 0 is a BSDE solution. A (super)solution (Y, Z, K) is minimal if Yt ≤ Y¯t , t ∈ [0, T ] holds ¯ K). ¯ for any other (super)solution. (Y¯ , Z, R
Note that a minimal supersolution when it exists is unique, as minimality implies uniqueness of the Y components; since continuous local martingales of finite variation are trivial, identity of the Z and Kcomponents follows. Existence of the minimal supersolution is sometimes investigated under the condition that there exists at least one supersolution to the BSDE (cf. [DHK13]). This condition is satisfied for the BSDE with parameters (g, X), X ∈ L∞ since g(·, 0) = 0 and thus (Y, Z, K) := (X∞ − (X∞ − X)I{T } , 0, (X∞ − X)I{T } ) is a supersolution. Note that g satisfies gt (z) ≥ −ξttr z, t ∈ [0, T ] and moreover (g, X) satisfies the hypotheses of [DHK13, Theorem 4.17] which implies existence of the minimal supersolution to the BSDE with parameter (g, X). We show that π·u (X) can be identified with the Y component R of this minimal supersolution. Condition (3.9) ensures that the process 0· gt (Zt )dt for g in R (3.14) and Z satisfying 0T Zt 2 dt < ∞ is realvalued, since CauchySchwarz inequality would R R 1 1 R imply 0T gt (Zt )dt ≤ ( 0T Zt 2 dt) 2 ( 0T Rt 2 dt) 2 < ∞. b and a nondecreasing Theorem 3.9. Let (3.9) hold and X ∈ L∞ . There exists Z ∈ H2 (Q) u predictable process K with K0 = 0 such that (π· (X), Z, K) is the minimal supersolution to the BSDE for data (g, X) with g from (3.14), and π·u (X) ∈ S ∞ .
The proofs for this theorem and for the next corollary are given in Appendix 3.4. ¯ ∈ Qngd such Corollary 3.10. Let (3.9) hold and X ∈ L∞ . If there exists a measure Q ¯ u Q Q u ¯ that π0 (X) = supQ∈Qngd E [X] = E [X], then π· (X) is a Qmartingale and there exists 2 u b Z ∈ H (Q) such that (π· (X), Z) is the minimal solution to the BSDE with parameters (g, X) ¯ of Q ¯ tr Zt , for all ¯ satisfies ess supλ ∈Λ λtr Zt = λ for g defined in (3.14). The Girsanov kernel λ t t t t t ∈ [0, T ]. ¯ ∈ Qngd as in Corollary 3.10 may be shown by For concrete case studies, existence of Q direct considerations, see Section 3.2.2 for examples. If one could formulate the nogood¯ would exist for deal restriction so that the set Qngd becomes weakly compact in L1 , then Q ∞ any X ∈ L from maximizing a bounded linear objective functional over a weakly compact subset of L1 . Note that Assumption 3.3 only implies (by DunfordPettis compactness theorem [DM78, Chapter II, Theorem 25]) that Qngd is weakly relatively compact in L1 . If Qngd is not weakly relatively compact in L1 , then by James’ theorem (cf. [AB06, Theorem 6.36]) there exists X ∈ L∞ such that the supremum in π0u (X) = supQ∈Qngd E Q [X] is not attained in the
Section 3.2. Dynamic gooddeal hedging
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L1 closure of Qngd (since Qngd is convex) and in particular also not in Qngd . Let us give ¯ does not exist in Qngd for some contingent claim and C does neither an example where Q satisfy Assumption 3.3 nor (3.9). Section 3.2.2 will furthermore give an example in a stochastic ¯ exists and C is not uniformly bounded but satisfies (3.9). volatility model where Q Example 3.11. Let n = 2 with W = (W 1 , W 2 ), d = 1 with dSt = St σ S dWt1 , S0 > 0, R σ S > 0, and ξ = 0. Let h > 0 be a deterministic predictable process with 0T ht dt = ∞ and Ct (ω) := {0} × [−ht , ht ], (t, ω) ∈ [0, T ] × Ω. Now let X := I{W 2 ≥0} ∈ L∞ , then T π0 := supn∈N Qn [{WT2 ≥ 0}] ≤ π0u (X) ≤ 1, where dQn = E(λn · W 2 )dP with λnt = R ht ∧ n, t ∈ [0, T ], n ∈ N. The process W 2,n := W 2 − 0· λnt dt is a Qn Brownian motion. R R Hence WT2,n ∼ N (0, T ) under Qn . We have 0T λnt dt % 0T ht dt = ∞ as n % ∞. Hence R π0 = supn∈N Qn [{WT2,n ≥ − 0T λnt dt}] = 1. Therefore π0u (X) = 1. But there exists no ¯ ∈ Qngd such that π u (X) = E Q¯ [X]. Indeed for such a measure, one would have measure Q 0 0 2 ≥ 0}] = 1 which is not possible since Q ¯ ¯ ∼ P. Q[{W T
3.2
Dynamic gooddeal hedging
Let again C be a standard correspondence satisfying 0 ∈ C, and define the family of apriori valuation measures n
P ngd := Q ∼ P dQ/dP = E (λ · W ) , λ predictable, bounded, λ ∈ C
o
(3.15)
which satisfy the same nogooddeal constraint as those in Qngd , except that the local martingale condition for S is omitted. One could view P ngd as the nogooddeal measures for a market only consisting of the riskless asset S 0 ≡ 1, i.e. P ngd = Qngd (1). It is natural to define (3.15) as apriori valuation measures, as the concept of nogooddeal valuation is to consider those risk neutral valuation measures Q, for which any extension of the financial market by additional derivatives price processes (being Qmartingales) would not give rise to ’good deals’; see e.g. [BS06, KS07a, Bec09] for rigorous detail in continuous time for Sharpe ratios, utilities or growth rates; for concepts cf. [Cer03]. Like Qngd , the set P ngd clearly is again mstable and convex. Just as in (3.3), we define the apriori dynamic coherent risk measure ρt (X) := ess sup EtQ [X], Q∈P ngd
t ∈ [0, T ],
(3.16)
for contingent claims X ∈ L2 . Note that ρt (X) is welldefined as the essential supremum of finitely valued random variables since measures in P ngd have bounded Girsanov kernels and hence density processes in S p (P ) for any p ∈ [1, ∞). Elements Q of P ngd or Qngd can be considered as generalized scenarios (as in [ADE+ 07]). Since P ngd ∩ Me = Qngd clearly holds, then ρt (X) ≥ πtu (X) for all t ≤ T . An investor holding
Section 3.2. Dynamic gooddeal hedging
Page 109
a liability X and trading in the market according to a permitted trading strategy φ, would R c assign at time t a residual risk ρt (X − tT φtr s dWs ) to his position. The investor’s objective is to hedge his position by a trading strategy φ¯ that minimizes his residual risk at any time t ≤ T . To justify a premium π·u (X) for selling X, the minimal capital requirement to make his position ρacceptable should coincide with π·u (X). Thus, his hedging problem is to find a trading strategy φ¯ ∈ Φ such that
πtu (X) = ρt X −
Z t
T
c φ¯tr s dWs = ess inf ρt X − φ∈Φ
Z t
T
c φtr s dWs ,
(3.17)
t ∈ [0, T ].
The gooddeal hedging strategy will be defined as a minimizer φ¯ in (3.17), and the gooddeal valuation π·u (·) becomes the market consistent risk measure corresponding to ρ, in the spirit of [BE09]. For a contingent claim X, the tracking error at time t ∈ [0, T ] ct Rtφ (X) := πtu (X) − π0u (X) − φ · W
(3.18)
of a hedging strategy φ ∈ Φ is defined as the difference between the dynamic variations in the capital requirement and the profit/loss from trading (hedging) according to φ up to time t. Proposition 3.12. For X ∈ L2 , let the strategy φ¯ ∈ Φ solve (3.17). Then the tracking error ¯ Rφ (X) is a Qsupermartingale for all Q ∈ P ngd . ¯
Proof. By the first equality of (3.17) and the definition of the tracking error it holds Rtφ (X) = R c −π0u (X) + ess supQ∈P ngd EtQ [X − 0T φ¯tr s dWs ], t ∈ [0, T ]. The claim then follows from mstability and convexity of P ngd , applying Lemma 3.1, Part a) extended to X ∈ L2 by some BSDEs arguments. From Remark 2.12 in Chapter 2, let us point out that by Proposition 3.12 we can view the gooddeal hedging strategy as being at least meanselffinancing under Q ∈ P ngd . The latter is a property that we again interpret as robustness of φ¯ with respect to the set of measure P ngd as generalized scenario (in the sense of [ADE+ 07]). To describe solutions to the hedging problem (3.17), we will often assume that C has further structure and is uniformly bounded. Section 3.2.2 also contains an example for a correspondence C that is not uniformly bounded but satisfies (3.9) in the Heston model, where the hedging problem can be solved in a semiexplicit manner. For a correspondence C satisfying Assumption 3.3, one can describe ρ· (X) (like π·u (X) in Proposition 3.5, proof being analogous) by solutions to classical BSDEs: e and (Y λ , Z λ ) (for λ ∈ C) Proposition 3.13. Let Assumption 3.3 hold. For X ∈ L2 , let (Ye , Z) be the respective standard solutions to the BSDEs
˜ t dt − Z tr dWt , t ≤ T, with YT = X, and −dYt = Zttr λ t
(3.19)
−dYt = Zttr λt dt − Zttr dWt , t ≤ T, with YT = X,
(3.20)
Section 3.2. Dynamic gooddeal hedging
Page 110
tr ˜ = λ(Z) ˜ ˜ tr Zt = ess sup where λ ∈ C is a predictable process satisfying the equality λ λt ∈Ct λt Zt t e e with Girsanov kernel λQ ˜ is in P ngd , and ρt (X) = for t ∈ [0, T ]. Then the measure Q =λ
e ess sup Ytλ = EtQ [X] = Y˜t , t ∈ [0, T ]. λ∈C
3.2.1
Results for ellipsoidal nogooddeal constraints
This section derives more explicit BSDE results to describe the solution to the valuation and the hedging problem (3.17) for (predictable) ellipsoidal nogooddeal constraints. Such generalization includes the important special case of radial constraints (as e.g. in [Bec09]), which is common to the gooddeal literature and justified by bounds (uniform in (t, ω)) on optimal growth rates or instantaneous Sharpe ratios, while still permitting comparably explicit results. The generalization could be interpreted as imposing different bounds on growth rates (or Sharpe ratios) for the risk factors associated to the principal axes. While such might appear as rather technical at this stage, in the subsequent context of model uncertainty (cf. Remark 3.24 b)) nonradial constraints will appear naturally. To this end, let h be a positive bounded predictable process, and A be a predictable Rn×n matrixvalued process with symmetric values and uniformly elliptic i.e. Atr = A and xtr Ax ≥ c x2 , for all x ∈ Rn and some c ∈ R+ . The common radial case is achieved by choosing A ≡ IdRn . We define the standard (see [Roc76, Corollary 1.Q]) correspondence n
o
Ct (ω) = x ∈ Rn  xtr At (ω)x ≤ h2t (ω) ,
(t, ω) ∈ [0, T ] × Ω,
(3.21)
that satisfies Assumption 3.3 due to ellipticity and boundedness of h. Assume that the kernel of the volatility matrix σ is spanned by eigenvectors of A, i.e. A−1 t (Ker σt ) = Ker σt ,
(3.22)
t ∈ [0, T ].
As the eigenvectors of A are orthogonal and (Ker σ)⊥ = Im σ tr , then (3.22) can be interpreted as separability of Im σ tr and Ker σ in the sense that each of these subspaces has a basis of eigenvectors of A. Given (3.22), the subspaces Im σ tr and Ker σ are orthogonal under the scalar product defined by A, one can rewrite
n
o
Qngd = Q ∼ P dQ/dP = E (λ · W ) , λ predictable, λ = −ξ + η, η ∈ C ξ ∩ Ker σ , with Ctξ (ω) = x ∈ Rn  xtr At (ω)x ≤ h2t (ω) − ξt (ω)tr At (ω)ξt (ω) , also satisfying Assumption 3.3. The correspondence C ξ is standard if
h2 > ξ tr Aξ,
(3.23)
The separability condition (3.22) ensures that −ξ + η ∈ C is equivalent to η ∈ C ξ , for η ∈ Ker σ. This way the ellipsoidal constraint on the Girsanov kernels transfers to one on their
Section 3.2. Dynamic gooddeal hedging
Page 111
ηcomponent, which permits to formulate the nogooddeal constraint only with respect to nontraded risk factors in the market. In this setup, it is straightforward to obtain an expression ¯ from Part 1 of Proposition 3.5 via for λ Lemma 3.14. For z ∈ Rn \ {0}, h > 0 and a symmetric positive definite n × nmatrix A, the unique maximizer of y tr z subject to y tr Ay ≤ h2 is y¯ = h(z tr A−1 z)−1/2 A−1 z. For X ∈ L2 , since C satisfies Assumption 3.3, there exists a unique standard solution (Y, Z) to the BSDE with terminal condition YT = X and
dYt =
ξttr Πt (Zt )
q
−
h2t
ξttr At ξt
−
q
tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )
dt + Zttr dWt .
(3.24)
¯ from Part 1 of We will see that π·u (X) = Y holds, and that the optimal Girsanov kernel λ ¯ Proposition 3.5 takes the form λ = −ξ + η¯ with η¯ ∈ Ker σ given by q
h2t − ξttr At ξt
η¯t = q
tr
Π⊥ t (Zt )
⊥ A−1 t Πt (Zt )
⊥ A−1 t Πt (Zt ),
t ∈ [0, T ].
(3.25)
In particular when h tends to ξ tr Aξ P ⊗ dta.s., then η¯ tends to 0 and the gooddeal bound b is the minimal local martingale measure. By Lemma πtu (X) converges to EtQ [X], for Q tr ⊥ 3.14 and using (3.22), one obtains η¯ttr Π⊥ t (Zt ) = ess supηt ∈C ξ ∩Ker σt ηt Πt (Zt ), and hence b
t
¯ tr Zt = −ξ tr Πt (Zt ) + h2 − ξ tr At ξt 1/2 Π⊥ (Zt )tr A−1 Π⊥ (Zt )1/2 , t ≤ T . Therefore Part 2 λ t t t t t t t of Proposition 3.5 yields Theorem 3.15. Assume (3.22) and (3.23) hold. For X ∈ L2 , let (Y, Z) be the standard ¯ ¯ ∈ Qngd is solution to the BSDE (3.24). Then πtu (X) = Yt = EtQ [X], t ∈ [0, T ], where Q ¯ = E ((−ξ + η¯) · W ) dP with η¯ given explicitly by (3.25). given by dQ The observation of the following lemma is straightforward. Lemma 3.16. The matrices A−1 t (ω), for (t, ω) ∈ [0, T ] × Ω, are positivedefinite and satisfy 0 (ω) x2 for all x, t, where α0 (ω) = ckA (ω)k−2 > 0 for c being the constant xtr A−1 (ω)x ≥ α t t t t of uniform ellipticity of A. Moreover kAk ≥ c holds. tr ˜ = h(Z tr A−1 Z)−1/2 A−1 Z satisfies λ ˜ tr Zt = ess sup tr By Lemma 3.14, λ t λt At λt ≤h2t λt Zt , ˜ tr Zt = ht (Z tr A−1 Zt )1/2 , t ∈ [0, T ]. Hence Proposition 3.13 gives ρt (X) = t ∈ [0, T ], with λ t t t Yt , t ∈ [0, T ], where (Y, Z) uniquely solves the classical BSDE with terminal condition YT = X and 1/2 −dYt = ht (Zttr A−1 dt − Zttr dWt . (3.26) t Zt )
Section 3.2. Dynamic gooddeal hedging
Page 112
Thanks to Lemma 3.16, a sufficient condition to ensure (3.23) is √ ξ < h α0 .
(3.27)
In addition it is used to verify for Lemma 3.35 the KuhnTucker conditions before applying the KuhnTucker theorem (see [Roc70, Section 28]), after which comparison results for BSDE yield the result of Theorem 3.17 below. The proof is omitted as it is analogous to that of [Bec09, Theorem 5.4 and Lemma 6.1], using now Lemma 3.35 instead of Lemma 6.1 there. For φ ∈ Φ, let (Y φ , Z φ ) denote the standard solution to the BSDE with terminal condition YT = X and, for t ≤ T , −dYt =
− ξttr φt + ht (Zt − φt )tr A−1 t (Zt − φt )
1/2
dt − Zttr dWt .
(3.28)
Theorem 3.17. Assume (3.22),(3.27) hold. For X ∈ L2 , let (Y, Z) and (Y φ , Z φ ) (for φ ∈ Φ) R c be standard solutions to the BSDEs (3.24),(3.28). Then Ytφ = ρt (X − tT φtr s dWs ), t ≤ T , and the strategy q tr −1 ⊥ Π⊥ t (Zt ) At Πt (Zt )
φ¯t =
q
h2t − ξttr At ξt
(3.29)
At ξt + Πt (Zt )
¯
is in Φ and satisfies Ytφ = ess inf Ytφ = Yt for any t ∈ [0, T ], that is φ∈Φ
πtu (X) = ess inf ρt X − φ∈Φ
Z t
T
c φtr s dWs = ρt X −
Z t
T
φ¯ c φ¯tr s dWs = Yt .
¯
Moreover, the tracking error Rφ (X) is a supermartingale under all measures Q ∈ P ngd and a martingale under the measure Qλ ∈ P ngd with Girsanov kernel ¯ −1/2 A−1 Zt − φ¯t , λt := ht (Zt − φ¯t )A−1 t t (Zt − φt )
t ∈ [0, T ].
One could interpret the dynamics of the nogooddeal valuation (3.24) as follows. By dYt =: ct + Π⊥ (Zt )dW c (cf. Section 3.1.1) it −at dt + Πt (Zt )ξt dt + Zt dWt = −at dt + Πt (Zt )dW t c , that is dynamically decomposes into a hedgeable part Πt (Zt )(ξt dt + dWt ) = Πt (Zt )dW c spanned by tradeable assets, an orthogonal part Π⊥ t (Zt )dW , being a martingale under P (and b and a remaining part being an absolutely continuous (finite variation) process whose rate Q), at ≥ 0 may be seen as a premium inherent to the upper good deal bound to compensate the seller of the claim for nontradeable risk. Note that a > 0 on {(ω, t) : Π⊥ (Zt ) 6= 0}. The summands in the expression (3.29) for the strategy φ¯ play different roles from the perspective of hedging. The second summand is a nonspeculative component that hedges locally tradeable risk by replication, while the first is a speculative component that compensates (“hedges”) for unspanned nontradeable risk by taking favorable bets on the market price of risk. Clearly, good deal bounds fit into the rich theory of gexpectations and marketconsistent risk measures (cf. [BE09] and more references therein). See [Lei07] for closely related ideas about instantaneous measurement of risk.
Section 3.2. Dynamic gooddeal hedging
3.2.2
Page 113
Examples for gooddeal valuation and hedging with closedform solutions
Explicit formulas, if available, facilitate intuition and enable fast computation of valuations, hedges and comparative statics. To this end, several concrete case studies are provided, starting with European options with monotone payoff profiles (e.g. call options) on nontraded assets in a multidimensional model of BlackScholes type, in which tradeable assets only permit for partial hedging. In parallel to [CT14, Proposition 3, Section 5.3] and [BY08], who employ SDE respectively PDE methods, this demonstrates how previous BSDE analysis can be applied in concrete case studies and we contribute some slight generalizations as well (e.g. higher dimensions, ellipsoidal constraints). As a further example, we contribute new explicit formulas for an option to exchange (geometric averages of) nontraded assets into traded assets. As before, the nogooddeal approach here gives rise to a familiar option pricing formula (by Margrabe) but suitable adjustments of parameter inputs are required, showing the difference to a simple noarbitrage valuation approach that uses only one (given) single risk neutral measure. A further example derives semiexplicit gooddeal solutions for the stochastic volatility model by Heston, for nogooddeal constraints on market prices of (unspanned) stochastic volatility risk which impose an interval range on the mean reversion level of the stochastic variance process under any valuation measure Q ∈ Qngd . Technically, this corresponds to imposing bounds on the instantaneous Sharpe ratio which are inversely proportional to the stochastic volatility. This is different to a related result by [BL09], in that their example imposes no good deal constraints in terms of bounds on simultaneous changes in the level of meanreversion combined with opposite changes in reversion speed. We emphasize that, in addition to valuation formulas, all our examples provide explicit results for gooddeal hedging strategies as well. Detailed derivations of the formulas in Sections 3.2.23.2.2 are given in Appendix 3.4 Closedform formulas for options in a generalized BlackScholes model The market information F = (Ft )t≤T is generated by an ndimensional P Brownian motion W := (W 1 , . . . , W n )tr with W S = (W 1 , . . . , W d )tr , d < n for n, d ∈ N, and is augmented by nullsets. The financial market consists of d ≤ n (incomplete if d < n) stocks with (discounted) prices S = (S k )dk=1 and further n − d nontraded assets with values H = (H l )n−d l=1 . We consider e b a risk neutral model (P = Q ∈ M , ξ = 0) where the processes S and H evolve as dSt = diag(St )σ S dWtS
and dHt = diag(Ht ) γdt + βdWt ,
t ∈ [0, T ],
S) d×d invertible, with S0 ∈ (0, ∞)d , H0 ∈ (0, ∞)n−d , constant coefficients σ S = (σki k,i ∈ R β = (βli )l,i ∈ R(n−d)×n and γ ∈ Rn−d . The volatility matrix of S is σ := (σ S , 0) ∈ Rd×n and is clearly of maximal rank d ≤ n. For z ∈ Rn , we have Π(z) = (z 1 , . . . , z d , 0, . . . , 0)tr ∈ Rn
and Π⊥ (z) =
0, . . . , 0, z d+1 , . . . , z n
tr
∈ Rn . We assume the ellipsoidal framework of
Section 3.2. Dynamic gooddeal hedging
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Section 3.2.1, with h ≡ const > 0 and A ≡ diag(a), with a ∈ (0, ∞)n . Clearly A satisfies the assumption (3.22). By convention,set 0/0 = 0. From Theorem 3.15 we know that ¯ Q u ¯ ¯ πt (X) = Yt = Et [X] with dQ/dP = E λ · W where ¯t = h λ
n X
(Zti )2 /ai
−1/2
(0, . . . , 0, Ztd+1 /ad+1 , . . . , Ztn /an )tr ,
i=d+1
for (Y, Z) solving the classical BSDE n X
−dYt = h
(Zti )2 /ai
1/2
dt − Zttr dWt , t ≤ T,
and YT = X.
(3.30)
i=d+1
By Theorem 3.17 the gooddeal hedging strategy is φ¯t = Π(Zt ), t ≤ T . Define the 1/d Qd ˜ t = Qn−d Htl 1/(n−d) . Then one can rewrite geometric averages S˜t = Stk and H k=1 l=1 ˜ 2 t , where ˜t = H ˜ 0 exp β˜tr Wt + γ˜ − 1 β S˜t = S˜0 exp σ ˜ tr WtS + µ ˜ − 12 ˜ σ 2 t and H 2 ˜ 2 − 1 β2 , σ 2 − 1 σ S 2 , β˜ = 1 β tr 1 and γ˜ := 1 γ tr 1+ 1 β σ ˜ := 1 (σ S )tr 1, µ ˜ := 1 ˜ d
2
2d
n−d
with 1 = (1, . . . , 1)tr . We treat the following two examples.
n−d
2
2(n−d)
˜ T ) in European option on nontraded assets: Consider a European option X = G(H 2 ˜ L on the geometric average H, where x 7→ G(x) is a nondecreasing measurable payoff function of polynomial growth in x±1 , i.e. G(x) ≤ k(1 + xn + x−n ) for all x > 0, ¯ is a constant process given by λ ¯t = for some k > 0 and n ∈ N. We claim that λ −1/2 Pn tr 2 ˜ ˜ ˜ h (0, . . . , 0, βd+1 /ad+1 , . . . , βn /an ) . Let (Y, Z) be the standard solution i=d+1 βi /ai ¯ For λ ¯ constant, the FeynmanKac formula yields to the linear classical BSDE (3.7) with λ = λ. ¯ ˜ T )] = u(t, H ˜ t ∂x u(t, H ˜ t ) and Zt = H ˜ t )β˜ for a function u ∈ C 1,2 [0, T ) × (0, ∞) Yt = EtQ [G(H solution to a BlackScholes type PDE (after coordinate transformations that reduce the PDE into the heat equation using [KS06, Section 4.3]). Since G is nondecreasing, ∂x u ≥ 0 holds. 1/2 i 2 ¯ tr Zt = h Pn ¯ is Because of this one actually has λ , t ∈ [0, T ]. Hence λ t i=d+1 (Zt ) /ai ¯ ˜ T )]. The process H ˜ satisfies indeed the constant process given above, and πtu (X) = EtQ [G(H 1/2 Pn 1 ˜2 α t tr 2 ˜ ˜ + ˜ ˜ ¯ ¯ Ht = H0 e exp β Wt − 2 β t , t ∈ [0, T ], where α± := γ˜ ± h and W i=d+1 βi /ai ¯ is an ndimensional QBrownian motion. Specifically for G(x) := (x − K)+ , X is a call option ˜ with strike K and maturity T . The upper gooddeal bound is given for t ∈ [0, T ] by a on H BlackScholes type formula (with “vol” abbreviating volatility) ˜ t eα+ (T −t) − KN (d− ) πtu (X) = N (d+ )H ˜ , (3.31) ˜ t , strike: Ke−α+ (T −t) , vol: β =eα+ (T −t) ∗ B/Scallprice time: t, spot: H √ ˜ 2 (T − t) β ˜ T − t−1 and N is the cdf of the ˜ t /K + α+ ± 1 β where d± := ln H 2 standard normal law. Analogously, the lower gooddeal bound turns out as ˜ . ˜ t , strike: Ke−α− (T −t) , vol: β πtl (X) = eα− (T −t) ∗ B/Scallprice time: t, spot: H
Section 3.2. Dynamic gooddeal hedging
Page 115
The difference between the gooddeal valuation formulas above and the standard BlackScholes b b for a call option (H ˜ T − K)+ formula for riskneutral valuation EtQ [X] under measure P = Q ˜ t , which reduce to the riskneutral shows in the factors eα± (T −t) multiplying the spot price H P case eγ˜(T −t) if h = 0, i.e. α± = γ˜ . The difference α± − γ˜ = ±h( n β˜2 /ai )1/2 when i=d+1 i at π·u (X)
h > 0 translates into an additional premium an option trader (selling or buying at π·l (X)) would require, if using the nogooddeal approach instead of the arbitragefree valuation b (being an element of Qngd ). The gooddeal hedging under a given risk neutral measure P = Q strategy for the seller of X in terms of parametrizations of Section 3.1.1 is ˜ t β˜1 , . . . , β˜d , 0, . . . , 0 tr , t ∈ [0, T ], φ¯t = eα+ (T −t) N (d+ )H
(3.32)
b for the call option which coincides with the delta hedging strategy (as computed under P = Q) + ˜ T − K) only if α+ is zero and the risky asset H ˜ is tradeable. The hedging strategy of the (H buyer is derived analogously.
Exchange option of traded and nontraded assets: Consider an European option to ˜ at maturity T with payoff X = exchange the traded asset S˜ for the nontraded asset H ¯ ˜ T − S˜T )+ ∈ L2 . The upper bound π u (X) = E Q [X] can be explicitly derived (see Appendix) (H t t using arguments from the previous example in combination with a change of num´eraire. We thereby obtain a Margrabe type formula ˜ t eα+ (T −t) − N (d− )S˜t eµ˜(T −t) πtu (X) = N (d+ )H (3.33) α (T −t) µ ˜ (T −t) ˜ te + = B/Scallprice time: t, spot: H , strike: S˜t e , vol: δ , −1 √ 2 ˜ t /S˜t + (α+ + µ where d± := ln H ˜ ± δ2 )(T − t) δ T − t . Analogously, the corresponding lower gooddeal bound is
˜ t eα− (T −t) , strike: S˜t eµ˜(T −t) , vol: δ . πtl (X) = B/Scallprice time: t, spot: H
The gooddeal hedging strategy φ¯t for the seller of the exchange option equals tr ˜ t eα+ (T −t) β˜1 , . . . , β˜d , 0, . . . , 0 tr − N (d− )S˜t eµ˜(T −t) σ N (d+ )H ˜1 , . . . , σ ˜d , 0, . . . , 0 .
Again, the difference between the gooddeal valuation formula and the classical Margrabe b for formula, as computed by standard noarbitrage valuation under risk neutral measure P = Q, + α (T −t) ˜ T − S˜T ) shows by the presence of the factors e ± the exchange option (H involving the Pn 2 1/2 ˜ term ±h( i=d+1 βi /ai ) , which depends only on the parameters A and h for nogooddeal restrictions.
Section 3.2. Dynamic gooddeal hedging
Page 116
Computational results by Monte Carlo To demonstrate that gooddeal bounds and hedging strategies can be computed numerically in moderately high dimensions by generic simulation methods available for classical BSDE, we apply the (generic) multilevel Monte Carlo algorithm from [BT14] (that builds on [GT15]) to approximate the solution (Y, Z) of the BSDE (3.30) in dimension n = 4, and compare with ˜ T − S˜T )+ . Using parameters the known analytical solution for the exchange option X := (H d = 2, T = 1 and !
H0 = β=
!
1 , 1
0.3 0.5
S0 =
0.4 0.7
0.2 0.3
1 , 1
σS =
0.5 0
!
0.2 , 0.4
γ = (0.1, 0.3)tr ,
!
0.5 , 0.4
h = 0.3,
and A = diag(0.5, 0.65, 0.8, 0.95),
we compare the approximate values at time t = 0 to the known theoretical values obtained from Section 3.2.2. The exact value of the gooddeal bound at time t = 0 according to the formula (3.33) is then π0u (X) = 0.5494, up to four digits, while for the hedging strategy it is φ¯0 = (0.3049, 0.4440, 0, 0), the exact value of Z0 being (0.3049, 0.4440, 0.2792, 0.5025). We use a 4level algorithm on an equidistant time grid with N = 24 steps, a number of sample paths M = 3 × 106 and with K = 504 regression functions, being indicator functions on a hypercube partition of R4 , the state space of the forward process (S, H). Table 3.1 provides the numerical simulation results, summarized by the approximation means for the gooddeal bound and the hedging strategy at time 0, the empirical rootmeansquare errors (RMSE) computed coordinatewise and the corresponding relative values (Rel.RMSE), based on 80 independent simulation runs. Simulation in Matlab for one run took 153sec on a corei7 cpu laptop, showing relative errors (in terms of maximal coordinates in Rel.RMSE) of about 0.07% for valuation and 0.34% for hedging. Y0 approx
Z0 approx
φ¯0 approx
Mean
0.5499
(0.3052, 0.4462, 0.2852, 0.5137)
(0.3052, 0.4462, 0, 0)
RMSE
10−4 × 4
10−4 × (10, 13, 12, 13)
10−4 × (10, 13, 0, 0)
Rel.RMSE
10−4 × 7
10−4 × (34, 29, 41, 27)
10−4 × (34, 29, 0, 0)
Table 3.1: Mean and (relative) rootmeansquare errors of approximations
Section 3.2. Dynamic gooddeal hedging
Page 117
Semiexplicit formulas in the Heston stochastic volatility model The market information is generated by a 2dimensional P Brownian motion W = (W S , W ν ), and is augmented by nullsets. We are going to consider a European put option X = (K − ST )+ on S with strike K in the Heston model q √ √ a dSt = St νt dWtS and dνt = b( − νt )dt + β νt ρdWtS + 1 − ρ2 dWtν , t ≤ T, b b with S0 , ν0 > 0, a, b, β > 0 and that is specified directly under a risk neutral measure P = Q, ρ ∈ (−1, 1). Here the variance process ν is a CIR process with b representing the meanreversion speed, a/b the meanreversion level and β/2 the volatility of the variance. Assume that the condition β 2 ≤ 2a is satisfied, such that by the Feller’s test for explosions (cf. [KS06, Theorem 5.5.29]) applied to the process ln(ν) the variance process ν is strictly positive. In the sequel we refer to this condition (i.e. β 2 ≤ 2a) for a CIR process as the Feller condition. The equivalent local martingale measures Q ∈ Me in this model are specified by Girsanov kernels λ such that dQ/dP = E(λ · W ν ) is a uniformly integrable martingale. Indeed, we parametrize the pricing measures only by the second component of their Girsanov kernels (i.e. with respect to W ν ) since the first component is always zero. We consider the nogooddeal constraint correspondence q
Ct (ω) = x ∈ R2 : x ≤ ε/ νt (ω)
(t, ω) ∈ [0, T ] × Ω,
(3.34)
for a constant ε > 0. One observes that C is standard with 0 ∈ C, nonuniformly bounded √ and satisfies (3.9) for R = ε/ ν (since ν > 0 is continuous). Hence gooddeal valuation results for uniformly bounded correspondences may not apply. Using [CFY05], we can obtain a convenient Hestontype formula (semiexplicit, computation requiring only 1dim. integration) for the gooddeal bound of the put option X = (K − ST )+ , πtu (X)
q
= Hestonputprice(time: t, a ¯ := a + βε 1 − ρ2 , b, β),
(3.35)
just like the ordinary Heston put price, associated to parameters (t, a, b, β), but where the p parameter a has to be adjusted to a ¯ := a + βε 1 − ρ2 . The formula for the lower bound πtl (X) p is similar, but with a ¯ replaced by a := a − βε 1 − ρ2 , for which the Feller condition β 2 ≤ 2a ¯ is still satisfied if ε ≤ 12 β −1 (2a − β 2 )(1 − ρ2 )−1/2 . In particular, πtu (X) = EtQ [X] holds with √ ¯ dQ/dP = E (ε/ ν) · W ν . By Corollary 3.10 this yields Y¯ = π·u (X) for the minimal solution ¯ ∈ S ∞ × H2 of the BSDE (Y¯ , Z) ε −dYt = √ Zt2 − Zttr dWt , t ∈ [0, T ], νt
YT = (K − ST )+ .
(3.36)
The (seller’s) gooddeal hedging strategy φ¯ is given by the semiexplicit formula √ βρ φ¯t = St νt ∆t + Vt , 2
(3.37)
Section 3.2. Dynamic gooddeal hedging
Page 118
where ∆t and Vt denote the delta and the vega of the put option at time t in the Heston model with parameters (¯ a, b, β). Derivations are provided in Appendix 3.4. We note that (3.37) coincides (cf. [PSHE09]) with the riskminimizing strategy (in the sense of [Sch01]) for the put ¯ in a Heston model, not with respect to the probability P but with respect to the measure Q (derived just before) under which also Heston dynamics but with modified parameters prevail. This shows, how the strategy (3.37) differs from the standard risk minimizing strategy under P (as in [PSHE09, HPS01]). Gooddeal valuation bounds for a put option in the Heston model are thus given by a Heston type formula but for a meanreversion level increased by p βε 1 − ρ2 /b > 0. Similar to earlier examples, this difference constitutes an increase in the premium that an issuer selling at π·u (X) would require according to gooddeal valuation, in b comparison to a standard arbitrage free valuation under one given risk neutral measure P = Q, when S is the only risky asset available for hedging and stochastic volatility risk is otherwise taken to be unspanned. Figures 3.1,3.2,3.3 graphically illustrate this, showing the gooddeal valuations π0u (X), π0l (X) (at t = 0) for a longdated put option with maturity T = 10 in relation to the underlying S0 , to the correlation coefficient ρ and to the nogooddeal constraint parameter ε (for bound √ on optimal growth rate h = ε/ ν) respectively. Other global parameters are K = 100, a = 0.12, b = 3, β = 0.3, ν0 = 0.04. Computations of the Heston formula have been done in Matlab following the algorithm of [KJ05]. Figure 3.1 is a plot of π0u (X), π0l (X) as function of initial stock price S0 for values of ε in {0.15, 0.25}. Similarly Figure 3.2 provides a plot illustrating the variation with ρ for ε ∈ {0.1, 0.2}, while Figure 3.3 illustrates the dependence on ε. The largest value 0.35 for ε in Figure 3.3 has been chosen as the maximal one allowing p for the Feller condition β 2 ≤ 2a = a − βε 1 − ρ2 for the lower bound π0l (X) to be satisfied, i.e. ε ≤ 12 β −1 (2a − β 2 )(1 − ρ2 )−1/2 ≈ 0.35 for the chosen parameters. Because the values of ε are close to zero, the lines in Figure 3.3 may look straight at the first impression, but by having a closer look the reader can convince himself that the lines are indeed not straight as expected. We could have plotted the upper bound π0u (X) for larger values of ε, but we simply chose to use on the same plot the same range of ε as that for the lower bound π0l (X). The standard Heston price computed directly under a given risk neutral (minimal martingale) b (i.e. for ε = 0) lies between the upper and lower gooddeal bounds, whose measure P = Q spread increases with ε > 0. The monotonicity in ε is intuitively obvious since as increases, the correspondence C maps to larger sets, yielding weaker nogooddeal constraints which then imply wider gooddeal valuation bounds. That the bounds in Figure 3.2 coincide for perfect correlation ρ ∈ {−1, 1} is also intuitively clear. Indeed since for ρ = 1 volatility risk is entirely spanned by the tradeable asset, then the former can be perfectly hedged such that the Heston model becomes complete and π0u (X) = π0l (X) = E Qb [X] holds for all contingent claims X.
Section 3.2. Dynamic gooddeal hedging
Page 119
100
Heston price under MinMartMeasure: ǫ=0
90
upper & lower bounds for ǫ=0.15 upper & lower bounds for ǫ=0.35
80
70
π u0 , π l0
60
50
40
30
20
10
0 0
50
100
150
S0
Figure 3.1: Dependence of π0u (X), π0l (X) on S0 for ρ = −0.7 and T = 10.
42
40
Heston price under MinMartMeasure: ǫ=0 upper & lower bounds for ǫ=0.1 upper and lower bounds for ǫ=0.2
38
36
π u0 , π l0
34
32
30
28
26
24
22 1
0.8
0.6
0.4
0.2
ρ
0
0.2
0.4
0.6
0.8
1
Figure 3.2: Dependence of π0u (X), π0l (X) on ρ for S0 = 100 and T = 20.
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 120
26
upper bound: π u0(X)
24
lower bound: π l0(X)
π u0 , π l0
22
20
18
16
14
12 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
ǫ
Figure 3.3: Dependence of π0u (X), π0l (X) on ε for S0 = 100, ρ = −0.7 and T = 10.
3.3
Gooddeal valuation and hedging under model uncertainty
In preceding sections, gooddeal bounds and hedging strategies have been described by classical BSDEs under the probability measure P , expressing the objects of interest in terms of the market price of risk ξ with respect to P . In reality, the objective real world probability measure is not precisely known, hence there is ambiguity about the market price of risk. To include model uncertainty (ambiguity) into the analysis, we follow a multiple priors approach in spirit of [GS89, CE02, ES03], by specifying a confidence region of reference probability measures {P θ : θ ∈ Θ} (multiple priors, interpreted as potential real world probabilities of equal right), centered around some measure P0 . In practice, an investor facing model uncertainty may first extract an estimate P0 for the true but uncertain P from data, but then consider a class R of potential reference measures in some confidence region around P0 to acknowledge the statistical uncertainty of estimation. Starting point for gooddeal valuation approach under uncertainty is then to associate to each model P θ its own family of (apriori) nogooddeal measures Qngd (P θ ) (resp. P ngd (P θ )). A robust worstcase approach requires the seller of a derivative to consider ¯ the (worstcase) model P θ that provides the largest upper gooddeal valuation bound, to be conservative against model misspecification (see (3.58)). Such leads to wider gooddeal bounds, corresponding to a larger overall set of nogooddeal measures under uncertainty. Notably, it will simultaneously also give rise to a suitable robust notion of gooddeal hedging, which is uniform with respect to all P θ , by means of a saddle point result that ensures a minmax identity
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 121
(see Theorem 3.30). We associate to each model P θ a correspondence C θ that defines the set of nogooddeal measures in this model. The aggregate set of nogooddeal measures will be e which incorporates also the uncertainty. Technically, described then by single correspondence C, this makes is possible to apply analysis obtained in the framework of previous sections of this chapter, with P0 taking the role of P .
3.3.1
Model uncertainty framework
Let (Ω, F, P0 , F) be a probability space with a usual filtration F = (Ft )t≤T generated by an ndimensional P0 Brownian motion W 0 . We assume that all reference measures P θ are equivalent to P0 with corresponding Girsanov kernels θ evolving in some given confidence region Θ. More precisely, we define
n
o
R := P θ ∼ P0 dP θ /dP0 = E θ · W 0 , with θ predictable and θ ∈ Θ , where Θ : [0, T ] × Ω Rn is a standard correspondence satisfying Assumption 3.3 and 0 ∈ Θ, hence P0 ∈ R 6= ∅. A similar framework has been considered for example in [CE02, Que04] for solving the robust utility maximization problem under Knightian uncertainty about drift coefficients. We do write θ ∈ Θ for θ being a predictable selection of Θ. The financial market consists of d ≤ n tradeable risky assets whose discounted prices (S i )di=1 under P θ (for θ ∈ Θ) evolve as Itˆo processes, solving the SDEs cθ, dSt = diag(St )σt (ξtθ dt + dWtθ ) =: diag(St )σt dW t
(3.38)
t ≤ T,
with S0 ∈ (0, ∞)d , for Rn valued predictable ξ θ and Rd×n valued predictable volatility σ of R full rank, and W θ := W 0 − 0· θs ds a P θ Brownian motion. Noting that market prices of risk, ξtθ and ξt0 , canonically take values in Im σttr , we assume that market prices of risk ξ θ (under P θ for θ ∈ Θ) have the form (3.39)
ξtθ = ξt0 + Πt (θt ) ∈ Im σttr , t ∈ [0, T ],
and that ξ 0 is bounded. By (3.39), the solutions of the SDEs (3.38) coincide P0 a.s. for all θ ∈ Θ. The process ξ θ (for θ ∈ Θ) is the market price of risk in the model P θ and is also bounded (since ξ 0 is bounded and Θ satisfies Assumption 3.3). Hence, the minimal b θ with respect to P θ is dQ b θ = E(−ξ θ · W θ )dP θ . In addition martingale measure [Sch01] Q R b θ = E Π⊥ (θ) · W c 0 dQ b 0 and W cθ = W c 0 − · Π⊥ (θt )dt, for all θ ∈ Θ. We recall from dQ 0 t Section 3.1.1 how dynamic trading strategies are defined and reparametrized in terms of c 0 . The set of permitted trading strategies is integrands (φi )di=1 with respect to W n
Φ := φ φ is predictable, φ ∈ Im σ tr and E P0
hZ 0
T
i
o
φt 2 dt < ∞ .
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 122
Since φtr Π⊥ (θ) = 0 for θ ∈ Θ, the wealth process V φ of strategy φ ∈ Φ with initial capital V0 c θ = V0 + φ · W c 0 , for all θ ∈ Θ. Let Me (P θ ) := Me (S, P θ ) denote the set is V φ = V0 + φ · W of equivalent local martingale measures for S in the model P θ . Noting P θ ∼ P 0 and recalling Proposition 3.2 one easily obtains e θ e e θ Proposition 3.18. M (P ) = M (P0 ) for allθ ∈ Θ. In addition, every Q ∈ M (P ) satisfies dQ = E λθ · W θ dP θ and dQ = E λ0 · W 0 dP0 , with λθ = −ξ θ + η θ and λ0 = −ξ 0 + η 0 ,
where Π⊥ (λθ ) = η θ ,
Π⊥ (λ0 ) = η 0 and η θ = η 0 − Π⊥ (θ).
Thus, we simply write Me = Me (S) for the set of equivalent martingale measures.
3.3.2
Nogooddeal constraint and gooddeal bounds under uncertainty
Let {C θ  θ ∈ Θ} be a family of standard correspondences satisfying −ξ θ ∈ C θ
for all θ ∈ Θ.
(3.40)
In the model P θ , θ ∈ Θ, let the nogooddeal constraint be such that the Girsanov kernels of measures in Me are selections of C θ . The resulting set Qngd (P θ ) of nogooddeal measures is equal to n
o
Q ∼ P θ dQ/dP θ = E λ · W θ , λ predictable, bounded, λ ∈ (−ξ θ + Ker σ) ∩ C θ .
b θ ∈ Qngd (P θ ) 6= ∅ for all θ ∈ Θ. By Proposition 3.18 holds By (3.40), then Q n
o
Qngd (P θ ) = Q ∼ P0 dQ/dP0 = E λ · W 0 , λ ∈ −ξ 0 + (Ce θ ∩ Ker σ)
(3.41)
where λ is predictable and bounded, and for all θ ∈ Θ the correspondences Ce θ are given by Ce θ := C θ + ξ θ + Π⊥ (θ) = C θ + ξ 0 + θ.
(3.42)
Following a worstcase approach, we take the (robust) upper gooddeal valuation π·u (·) under uncertainty as being the largest of all gooddeal bounds π·u,θ (·) over all models P θ , θ ∈ Θ. The respective set Qngd of nogooddeal valuation measures corresponding to π·u (·) can be described in terms of the sets Qngd (P θ ), θ ∈ Θ. At first, one might guess that Qngd should be the union of all Qngd (P θ ). However, to have mstability and convexity of Qngd for good dynamic properties of the resulting gooddeal bounds (as in Lemma 3.1), one has to define Qngd as the smallest mstable and convex set containing all Qngd (P θ ), θ ∈ Θ. Definition 3.19. Qngd is the smallest mstable convex subset of Me containing all Qngd (P θ ), θ ∈ Θ. For sufficiently integrable claims X (e.g. in L∞ ), the worstcase upper gooddeal bound under uncertainty is πtu (X) := ess supQ∈Qngd EtQ [X].
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 123
We characterize the set Qngd from Definition 3.19 using a suitable single correspondence Ce which is derived from all C θ , θ ∈ Θ. To this end, we impose the Assumption 3.20. The correspondence with values compactvalued and predictable.
eθ θ∈Θ Ct (ω),
S
(t, ω) ∈ [0, T ] × Ω, is
The theory of measurable correspondences is welldeveloped for closedvalued correspondences (see [Roc76]). Assumption 3.20 ensures closedvaluedness and predictability of Ce for the proposition below. If all C θ (θ ∈ Θ) are equal to some given C 0 , as in the following example, such an assumption will automatically hold in the setting required for Section 3.3.4, where Ce θ (θ ∈ Θ) are ellipsoidal. Example 3.21. For a standard correspondence C 0 with ξ θ ∈ C 0 , θ ∈ Θ, let C θ := C 0 , θ ∈ Θ. S Then Ce θ = C 0 + ξ 0 + θ and θ∈Θ Ce θ = C 0 + ξ 0 + Θ satisfies Assumption 3.20. Proposition 3.22. Let Assumption 3.20 hold. Then Qngd equals n
o
Q ∼ P0 dQ/dP0 = E λ · W 0 , λ = −ξ 0 + η predictable, bounded, η ∈ Ce ,
(3.43)
S for the standard correspondence Cet (ω) := Ker σt (ω) ∩ Conv θ∈Θ Cetθ (ω) .
Proof. With Assumption 3.20, [Roc76, Theorem 1.M and Proposition 1.H] imply that Ce is standard. Note that Ce is nonemptyvalued since −ξ 0 ∈ C 0 and hence 0 ∈ Cet0 (ω)∩Ker σt (ω) ⊂ Cet (ω). Denote by Q the set in (3.43). By definition Cet (ω) ⊂ Ker σt (ω), implying Q ⊆ Me . We first prove that Qngd ⊆ Q. Applying [Del06, Theorem 1] or following the steps of the proof for Lemma 3.1, Part b), one sees that Q is mstable and convex. By (3.41) and since Cetθ (ω) ∩ Ker σt (ω) ⊆ Cet (ω) for all θ ∈ Θ, then Q contains the union of all Qngd (P θ ), θ ∈ Θ. By definition Qngd is the smallest mstable convex subset of Me with this property, hence Qngd ⊆ Q. Let us show Q ⊆ Qngd . The L1 closure of Qngd is an mstable closed and convex set of measures Q P0 , and Qngd comprises exactly those elements of its closure that are equivalent to P0 . Closeness and convexity of the closure of Qngd are clear. We now show its mstability. To this end, let ZT1 , ZT2 be in the closure of Qngd , τ ≤ T be a stopping time and ZT := Zτ1 ZT2 /Zτ2 I{Zτ2 >0} + Zτ1 I{Zτ2 =0} . There exist ZT1,n n , n ZT2,n n ⊆ Qngd such that ZT1,n → ZT1 and ZT2,n → ZT2 in L1 . By mstability of Qngd holds ZTn := Zτ1,n ZT2,n /Zτ2,n ∈ Qngd for each n ∈ N. Now E[ZTn ] = 1 for all n ∈ N, and ZTn → ZT in probability as n → ∞. In addition, E[ZT ] = E[Zτ1 ZT2 /Zτ2 I{Zτ2 >0} ] + E[Zτ1 I{Zτ2 =0} ] = E Eτ [ZT2 /Zτ2 ] Zτ1 I{Zτ2 >0} + E[Zτ1 I{Zτ2 =0} ] = E[Zτ1 ] = 1.
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 124
By Scheff´e’s lemma one obtains ZTn → ZT in L1 as n → ∞, and mstability of the closure of Qngd follows. As W 0 is a continuous P0 martingale with the predictable representation property, it satisfies the hypotheses of [Del06, Theorem 2], implying by Definition 3.19 the existence of a closedconvexvalued predictable correspondence C 1 such that the nogooddeal measure set Qngd is equal to
n
o
Q ∼ P0 dQ/dP0 = E λ · W 0 , λ = −ξ 0 + η predictable, η ∈ C 1 ∩ Ker σ .
To prove the claim, it suffices to show that all predictable selections of Ce are also predictable selections of C 1 ∩ Ker σ. To this end it suffices to show that for all θ ∈ Θ, any predictable selection of Ce θ ∩ Ker σ is a predictable selection of C 1 ∩ Ker σ. Assume the contrary that there exists θ ∈ Θ and a predictable process η such that η ∈ Ce θ ∩ Ker σ and η is not selection of C 1 ∩ Ker σ. Then E (−ξ 0 + η) · W 0 is in Qngd (P θ ) but not in Qngd , which contradicts Qngd (P θ ) ⊆ Qngd .
Using the characterization of Qngd in Proposition 3.22 we can apply the results of Sections 3.13.2 in order to derive worstcase gooddeal bounds and hedging strategies under uncertainty like in the absence of uncertainty, with the center P0 of the set of reference measures R taking the role of P (in Sections 3.13.2) and the enlarged correspondence Ce taking the role of C there. Example 3.23. For C θ , θ ∈ Θ, as in Example 3.21 holds Ce = (C 0 + ξ 0 + Θ) ∩ Ker σ and n
Qngd = Q ∼ P0 dQ/dP0 = E λ · W 0 , λ ∈ (−ξ 0 + Ker σ) ∩ (C 0 + Θ)
o
(3.44)
with λ denoting bounded predictable selections, by Proposition 3.22. Moreover the union S ngd (P θ ) is convex, mstable (cf. Lemma 3.26) and equals Qngd . θ∈Θ Q Remark 3.24. a) Equation (3.43) shows, how the gooddeal valuation and hedging problem under model uncertainty can technically be embedded into the mathematical framework of Sections 3.13.2 without uncertainty, by considering an enlarged nogooddeal constraint correspondence C as Conv(∪θ∈Θ (C θ +θ)) in (3.6) with P0 taking the role of P . In Example 3.23, (3.44), it simply means to take C as C 0 + Θ. b) Typical examples for gooddeal constraints are radial, i.e. C 0 is a ball. This case is predominant in the literature and justified from a finance point of view by ensuring a constant bound on instantaneous Sharpe ratios (or growth rates). But typical examples for uncertainty (ambiguity) constraints Θ can well be nonradial (see [CE02, ES03]). For instance, Θ may arise from a confidence region for some unknown drift parameters in a multivariate (log)normal model; such would in general be ellipsoidal but not radial, and the sum C 0 + Θ can even be
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 125
nonellipsoidal. To offer a suitable framework for such and other examples, Section 3.1 treats abstract correspondences. A constructive method to solve for such a typical parametrization of C 0 + Θ is described in Remark 3.31.
3.3.3
Robust approach to gooddeal hedging under model uncertainty
ngd θ As in Sectionn3.2 (cf. (3.15) and the definition of Q (P )), we define for θ ∈ o Θ the set θ θ θ θ ngd θ P (P ) := Q ∼ P  dQ/dP = E λ · W , λ ∈ C predictable, bounded in order to
introduce a robust notion of gooddeal hedging. Let P ngd denote the smallest mstable convex set of measures Q ∼ P0 containing all P ngd (P θ ), θ ∈ Θ. Then ρt (X) := ess sup EtQ [X],
t ∈ [0, T ], X ∈ L2 (P0 ),
Q∈P ngd
defines a timeconsistent dynamic coherent risk measure by Lemma 3.1. Like in Section 3.2, the gooddeal hedging problem under uncertainty is posed as a minimization problem (3.45) of apriori risk measures ρ of hedging errors: for a contingent claim X, find a strategy φ∗ ∈ Φ such that for all t ∈ [0, T ] holds
πtu (X) = ρt X −
T
Z t
c 0 = ess inf ρt X − φ∗s tr dW s φ∈Φ
Z t
T
c0 φtr s dWs .
(3.45)
The gooddeal hedging strategy under uncertainty is defined as this minimizer (if it exists) φ∗ ∈ Φ. For X ∈ L2 (P0 ), one can prove (as in Proposition 3.12) that the tracking error ∗ Rφ (X) (defined as in (3.18)) of the strategy φ∗ is a supermartingale under every measure in P ngd : Proposition 3.25. For X ∈ L2 (P0 ), let φ∗ be the strategy solving (3.45). Then the tracking ∗ error Rφ (X) of this strategy is a Qsupermartingale for all Q ∈ P ngd . A strategy solving the gooddeal hedging problem under uncertainty and whose tracking error satisfies the supermartingale property under all measures in P ngd (as in Proposition 3.25) will be qualified as robust with respect to uncertainty. Note that this is a different notion of robustness compared to the one in Remark 2.12, because the supermartingale property has S to hold for measures in P ngd (P θ ) uniformly for all models P θ ∈ R (since θ∈Θ P ngd (P θ ) is a subset of P ngd ). More concrete results under uncertainty will be derived next under additional conditions.
3.3.4
Hedging under model uncertainty for ellipsoidal gooddeal constraints
In this section we consider ellipsoidal gooddeal constraints. To this end, let n
o
Ct0 (ω) = x ∈ Rn xtr At (ω)x ≤ h2t (ω) ,
(t, ω) ∈ [0, T ] × Ω,
(3.46)
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 126
where A is a uniformly elliptic and predictable matrixvalued process, and h some positive bounded and predictable process. We assume that A satisfies the separability condition (3.22) with respect to σ. Let Θ be an arbitrary standard correspondence satisfying the uniform boundedness Assumption 3.3 and 0 ∈ Θ. As in Example 3.21, we let C θ := C 0 , for all θ ∈ Θ, yielding by (3.42) that tr
n
Cetθ ∩ Ker σt = x ∈ Rn xtr At x ≤ h2t − ξtθ At ξtθ
o\
Ker σt + Π⊥ t (θt ).
Clearly, C θ is standard and satisfies Assumption 3.3 for θ ∈ Θ. Similarly to (3.27), to derive explicit BSDE formulations for solving the hedging problem we will assume √ (3.47) ξ θ  < h α0 for all θ ∈ Θ, where the process α0 is the constant of ellipticity of A−1 as in Lemma 3.16. Recall that, thanks to Lemma 3.16, the inequality (3.47) implies in particular that −ξ θ ∈ C 0 , θ ∈ Θ; hence (3.40) holds and the correspondences Ce θ ∩ Ker σ are standard, θ ∈ Θ. Note that condition (3.47) ensures applicability of Lemma 3.35 in our current setup for any model P θ . Since C θ is equal to C 0 and satisfies Assumption 3.3, one has n
o
P ngd (P θ ) = Q ∼ P0 dQ/dP0 = E λ · W 0 , λ predictable, λ ∈ C 0 + θ .
(3.48)
The following lemma is proven in the Appendix. Lemma 3.26. 2. The set
1. The set
S
ngd (P θ ) θ∈Θ Q
S
θ∈Θ P
ngd (P θ )
is mstable, convex and equal to P ngd .
is mstable, convex and equal to Qngd .
Thanks to Lemma 3.26, the dynamic risk measure ρ satisfies for X ∈ L2 (P0 ) ρt (X) := ess sup EtQ [X] = ess sup ρθt (X), Q∈P ngd
t ∈ [0, T ],
θ∈Θ
with ρθt (X) := ess supQ∈P ngd (P θ ) EtQ [X]. The worstcase upper gooddeal bound πtu (X) for X ∈ L2 (P0 ) rewrites from Definition 3.19 as πtu (X) := ess sup ess sup EtQ [X] = ess sup πtu,θ (X), t ∈ [0, T ], θ∈Θ
Q∈Qngd (P θ )
θ∈Θ
(3.49)
where πtu,θ (X) = ess supQ∈Qngd (P θ ) EtQ [X]. The corresponding lower bound π·l (X) is obtained via π·l (X) = −π·u (−X). For a worstcase approach to uncertainty we will investigate valuation of claims according to π·u (·) and hedging with the optimal trading strategy solution to (3.45). We employ results from Section 3.2.1 (under P = P θ ) to characterize π·u,θ (X) as well as the associated hedging strategies φ¯θ in Φ. For θ ∈ Θ and φ ∈ Φ let us consider the classical BSDEs −dYt = f φ,θ (t, Zt )dt − Zttr dWt0 , −dYt = f θ (t, Zt )dt − Zttr dWt0 ,
t ≤ T, t ≤ T,
YT = X YT = X,
and
(3.50) (3.51)
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 127
with generators tr
f φ,θ (t, z) = θttr (z − φt ) − ξt0 φt + ht (z − φt )tr A−1 t (z − φt )
1/2
tr ⊥ 0 tr Π⊥ t (θt ) Πt (z) − ξt Πt (z) 1/2 ⊥ tr −1 ⊥ 1/2 tr h2t − ξtθ At ξtθ Πt (z) At Πt (z) .
θ
f (t, z) = +
(3.52) (3.53)
It is straightforward to derive the BSDE descriptions for π·u,θ (X) and ρθ· (X) stated in the subsequent proposition. The proof is analogous to that for Theorem 3.17, using (3.47) instead of (3.27), replacing P by P θ and changing measure from P θ to P0 . Proposition 3.27. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), θ ∈ Θ and φ ∈ Φ, let (Y φ,θ , Z φ,θ ) and (Y θ , Z θ ) be the standard solutions to the BSDEs (3.50) and (3.51) ¯θ ¯ θ ∈ Qngd (P θ ) given by respectively. Then πtu,θ (X) = Ytθ = EtQ [X], t ∈ [0, T ], holds with Q ¯ θ /dP0 = E (−ξ 0 + η¯θ ) · W 0 for dQ tr
η¯tθ = h2t − ξtθ At ξtθ
Moreover Ytφ,θ = ρθt X −
1/2
tr
−1/2
−1 ⊥ θ θ Π⊥ t (Zt ) At Πt (Zt )
⊥ θ ⊥ A−1 t Πt (Zt ) + Πt (θt ).
RT
c0 ¯θ φtr s dWs holds, and the strategy φ (in Φ)
t
−1/2 tr θ tr −1 ⊥ θ 1/2 2 φ¯θt := Πt (Ztθ ) + Π⊥ ht − ξtθ At ξtθ At ξtθ t (Zt ) At Πt (Zt )
satisfies
πtu,θ (X)
=
ρθt
X−
Z
T
t
c 0 = ess inf ρθ X − (φ¯θs )tr dW s t
Z
φ∈Φ
t
T
c0 φtr s dWs .
By Proposition 3.27, we can write π·u (X) from (3.49) as πtu (X)
= ess sup θ∈Θ
ess inf ρθt φ∈Φ
X−
Z t
T
c0 φtr s dWs ,
t ∈ [0, T ].
(3.54)
This permits to describe π·u (X) and the associated hedging strategy φ¯ in the next theorem by the solution to the classical BSDE −dYt = f (t, Zt )dt − Zttr dWt0 , t ≤ T
and
YT = X,
(3.55)
with generator f (t, Zt ) := ess supθ∈Θ f θ (t, Zt ), for f θ given in (3.53). The theorem moreover ¯ identifies by θ¯ the worstcase model P θ ∈ R. Theorem 3.28. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), let (Y, Z) be the standard ¯ solution to the BSDE (3.55). Then there exists a unique predictable selection θ¯ := θ(X) of Θ θ ¯ satisfying θt = argmaxθ∈Θ f (t, Zt ) such that for all t ∈ [0, T ] ¯
πtu (X) = ρθt X −
Z t
T
u,θ¯ c0 φ¯tr s dWs = πt (X) = Yt
(3.56)
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 128
¯ holds with φ¯ = (φ¯t )t∈[0,T ] := φ¯θ (X) ∈ Φ given by tr
1/2 2 ¯ ¯−1/2 ¯ tr −1 ⊥ φ¯t = Πt (Zt ) + Π⊥ ht − ξtθ At ξtθ At ξtθ ]. t (Zt ) At Πt (Zt )
(3.57)
¯ c 0 of the strategy φ ¯ is a supermartingale The tracking error Rφ (X) := π·u (X) − π0u (X) − φ¯ · W ¯ ¯ ¯ 0 ¯ in P ngd (P θ ) given by dQ/dP ¯ under any Q in P ngd (P θ ), and is a martingale under Q 0 = E(λ·W ) with ¯ t := ht (Zt − φ¯t )A−1 (Zt − φ¯t )−1/2 A−1 Zt − φ¯t + θ¯t , t ∈ [0, T ]. λ t t
Proof. Pointwise existence and uniqueness of θ¯ ∈ Θ follow by the continuity and strict concavity of f θ as a function of θ ∈ Rn , and the uniform boundedness of Θ. Predictability of θ¯ follows by [Roc76]. The claims (3.56) and (3.57) are corollaries of Proposition 3.27. The remaining claims are similar to those of Theorem 3.17, hence their proof goes likewise, making again use of Lemma 3.35 (instead of [Bec09, Lemma 6.1]) and (3.22) and (3.47). ¯ The process φ¯ := φ¯θ in Theorem 3.28 is the gooddeal hedging strategy of X for the worst¯ ¯ case model P θ ∈ R which yields that highest gooddeal valuation with π·u (X) = π·u,θ (X). ¯ The tracking error of φ¯ is therefore a supermartingale under any measure in P ngd (P θ ) (cf. ¯ Proposition 3.12), i.e. φ¯ is “at least meanselffinancing” under any measure in P ngd (P θ ). However, it is not clear at this stage whether the supermartingale property of the tracking error of φ¯ holds simultaneously under all measures in P ngd (P θ ) for all models R = {P θ : θ ∈ Θ}. We will show that this is the case, and that φ¯ and its associated valuation bound π·u (X) are indeed robust with respect to uncertainty. The idea is first to find an alternative bound π·u,∗ (·) and an associated strategy φ∗ that satisfy the supermartingale property of the tracking error S simultaneously under all measures in θ∈Θ P ngd (P θ ) and are therefore robust. After this, we show that π·u,∗ (X) coincides with the worstcase bound π·u (X), and that the same holds for ¯ the hedging strategies φ(X) and φ∗ (X) for any contingent claim X. In general the gooddeal bound π·u (X) is dominated by π·u,∗ (X), but thanks to a saddle point result (Theorem 3.30) one can actually prove that the two bounds are identical. Exchanging the order between ess sup and ess inf in the expression (3.54) for π·u (X), we define for X ∈ L2 (P0 ) and t ≤ T
πtu,∗ (X)
:= ess inf φ∈Φ
ess sup ρθt θ∈Θ
X−
Z t
T
c0 φtr s dWs .
(3.58)
From this it is clear that in general πtu,∗ (X) ≥ πtu (X), for all X ∈ L2 (P0 ). We will show that in fact the minimax identity holds in the sense that the expressions in (3.54) and (3.58) coincide, and that a saddle point exists, giving equality of π·u (X) and π·u,∗ (X). To this end, we describe π·u,∗ (X) and φ∗ in terms of the standard solution (Y, Z) for the BSDE −dYt = f ∗ (t, Zt )dt − Zttr dWt0 ,
t≤T
and
YT = X,
(3.59)
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 129
where f ∗ (t, Zt ) := ess infφ∈Φ f φ (t, Zt ), with f φ (t, Zt ) := ess supθ∈Θ f φ,θ (t, Zt ) for f φ,θ from (3.52). Indeed tr
f φ (t, Zt ) := −ξt0 φt + ess sup θttr (Zt − φt ) + ht (Zt − φt )tr A−1 t (Zt − φt )
1/2
θt ∈Θt
(3.60)
holds and we can identify the robust gooddeal hedging strategy φ∗ by Proposition 3.29. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), let (Y, Z) be the standard solution to the BSDE (3.59). Then there exists a unique φ∗ ∈ Φ satisfying φ∗t = argminφ∈Φ f φ (t, Zt ) for t ∈ [0, T ] such that πtu,∗ (X)
= ess inf ρt X − φ∈Φ
Z t
T
c0 φtr s dWs
= ρt X −
Z t
T
c 0 = Yt , t ∈ [0, T ]. φ∗s tr dW s
(3.61)
∗
c 0 is a Qsupermartingale for all Q ∈ P ngd , Moreover Rφ (X) := π·u,∗ (X) − π0u,∗ (X) − φ∗ · W ∗ ∗ ngd ∗ and a Q martingale for Q ∈ P with dQ /dP0 = E(λ∗ · W 0 ), where −1/2
λ∗ = h (Z − φ∗ )tr A−1 (Z − φ∗ )
A−1 (Z − φ∗ ) + θ∗ ,
with θt∗ := θt∗ (φ∗ ) = argmaxθ∈Θ θttr (Zt − φ∗t ) such that f ∗ (t, Zt ) = f φ
∗ ,θ ∗
(t, Zt ).
Proof. It is clear that for any φ ∈ Φ, there exists θ∗ (φ) ∈ Θ such that θt∗ (φ)tr (Zt − φt ) = ∗ ess supθt ∈Θt θttr (Zt − φt ) and f φ (t, Zt ) = f φ,θ (φ) (t, Zt ). Consider the convex continuous 1/2
tr
function Rn 3 φ 7→ F (φ) := −ξ 0 φ + ess supθ∈Θ θtr (z − φ) + h (z − φ)tr A−1 (z − φ) , for 0 constant h, φ,z, ξ ,σ and A satisfying the notations of Lemma 3.35 and for a compact set Θ ⊂ Rn containing the origin. The function F is also coercive on Im σ tr , i.e. F (φ) → +∞ √ as φ → +∞ for Π⊥ (φ) = 0 because ξ 0 < h α0 and ess supθ∈Θ θtr (z − φ) ≥ 0. Hence existence of φ∗ ∈ Φ follows from [ET99, Chapter 1.2]. Uniqueness of φ∗ follows n II, Proposition o ⊥ from the fact that F is strictly convex over Π (φ) = 0 if Π⊥ (z) 6= 0 and strictly convex at φ = z if Π⊥ (z) = 0 because (3.47) holds. Finally, predictability of φ∗ follows from [Roc76, Theorem 2.K] via Part 1 of Proposition 3.2. c0 From Proposition 3.27, for φ ∈ Φ and θ ∈ Θ, Y φ,θ = ρθ· X − ·T φtr s dWs is the Y component of the solution to the classical BSDE (3.50). As a consequence for every φ ∈ Φ it holds ∗ ess supθ∈Θ f φ,θ (t, Zt ) = f φ,θ (φ) (t, Zt ) = f φ (t, Zt ), t ∈ [0, T ]. The generators f φ are standard, so that by the comparison theorem for classical BSDEs, (Y φ , Z φ ) with Ytφ := ess supθ∈Θ Ytφ,θ is the standard solution to the BSDEs (under P0 ) with parameters (f φ , X), for φ ∈ Φ. The ∗ ∗ ∗ generator f ∗ is also standard because f ∗ (t, Zt ) = f φ ,θ (φ ) (t, Zt ) = ess infφ∈Φ f φ (t, Zt ). Now the comparison theorem yields (3.61) from (3.58). R
∗
The supermartingale property of Rφ (X) can be proved from (3.61) using arguments in the proof of Proposition 3.12. A BSDE proof for the supermartingale property can also be
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 130
given along the same line as the following for the martingale property. By (3.48) it holds ∗ Q∗ ∈ P ngd (P θ ) ⊂ P ngd since λ∗ ∈ C 0 + θ∗ and θ∗ ∈ Θ. Because π u,∗ (X) is the value process of the BSDE (3.59), then after changing measures from P0 to Q∗ one obtains ∗
tr
∗
−dRtφ (X) = f ∗ (t, Zt ) + ξt0 φ∗t − λ∗t tr (Zt − φ∗t ) dt − (Zt − φ∗t )tr dWtQ .
∗
(3.62)
∗
Furthermore the finite variation part of (3.62) vanishes since f ∗ (t, Zt ) = f φ ,θ (t, Zt ). Because ∗ Rφ (X) ∈ S 2 (P0 ) and dQ∗ /dP0 ∈ Lp (P0 ) for all p < ∞ (since λ∗ is bounded), then H¨older’s ∗ ∗ inequality implies that Rφ (X) ∈ S 2− (Q∗ ) for ∈ (0, 1). Thus Rφ (X) is a Q∗ martingale.
Proposition 3.29 shows that the tracking error of the hedging strategy φ∗ with respect to valuation according to π·u,∗ (X) has the supermartingale property simultaneously under all S measures in P ngd = θ∈Θ P ngd (P θ ). The next theorem shows that a minimax identity holds: the supinf representation of π·u (·) in (3.54) is equal to the infsup representation of π·u,∗ (·) in (3.58); see also (3.63). Moreover, the gooddeal hedging strategy φ¯ with respect to the ¯ that gives the highest gooddeal valuation bound π u (·), is worstcase model (given by θ) · identical with the robust gooddeal hedging strategy φ∗ from Proposition 3.29. Theorem 3.30. Assume (3.22) and (3.47) hold. For X ∈ L2 (P0 ), let (Y, Z) be standard solution of the BSDE (3.59). Then fφ
∗ ,θ ∗
¯¯
(t, Zt ) = ess inf ess sup f φ,θ (t, Zt ) = ess sup ess inf f φ,θ (t, Zt ) = f φ,θ (t, Zt ) (3.63) φ∈Φ
θ∈Θ
θ∈Θ
φ∈Φ
¯ θ), ¯ (φ∗ , θ∗ ) from Theorem 3.28 and Proposition 3.29. Moreover (Y, Z) coincides holds with (φ, with the standard solution to the BSDE (3.55) and πtu (X) = πtu,∗ (X) = Yt
and φ∗t (X) = φ¯t (X),
t ∈ [0, T ].
(3.64)
Proof. Let X ∈ L2 (P0 ). By an application of Lemma 3.36, the generator f φ,θ of the BSDE (3.50) for θ ∈ Θ and φ ∈ Φ satisfy the minimax relation (3.63). By Theorem 3.28 and ∗ ∗ ¯¯ Proposition 3.29 it holds f (t, Zt ) = f φ,θ (t, Zt ) and f ∗ (t, Zt ) = f φ ,θ (t, Zt ), t ∈ [0, T ], for f, f ∗ respectively generators of the BSDEs (3.55), (3.59). Also, πtu (X) = πtu,∗ (X) = Yt , t ∈ [0, T ], since by uniqueness of BSDE solutions (Y, Z) also solves the BSDE (3.55). Hence ¯ θ) ¯ and (φ∗ , θ∗ ) are both saddle points of the function (φt , θt ) 7→ f φ,θ (t, Zt ). Now for any (φ, 1/2 tr θ ∈ Θ and z ∈ Rn , the function φ 7→ F (φ, θ) := θtr (z − φ) − ξ 0 φ + h (z − φ)tr A−1 (z − φ) is strictly over {Π⊥ (φ) = 0} if Π⊥ (z) 6= 0, and strictly convex at φ = z if Π⊥ (z) = 0, convex √ θ since ξ < h α0 . [ET99, Chapter VI, Proposition 1.5] implies that the φcomponents of the saddle points are identical, yielding φ¯ = φ∗ .
Section 3.3. Gooddeal valuation and hedging under model uncertainty
3.3.5
Page 131
The impact of model uncertainty on robust gooddeal hedging
In the framework of Section 3.3.4, results have so far been stated for an arbitrary standard correspondence Θ without further structural assumptions, and ellipsoidal correspondences were only assumed for the nogooddeal constraints C θ , θ ∈ Θ. Recall (cf. Theorem 3.17 and subsequent remarks) that in the absence of uncertainty the gooddeal hedging strategy contains a speculative component in the direction of the market price of risk. This already indicates that under uncertainty one should expect to see relevant differences by a robust approach to hedging. To investigate the effect of uncertainty about the market price of risk θ on robust gooddeal hedging, we assume in addition (noting that θ ∈ Im σ tr is natural) that for all (t, ω) ∈ [0, T ] × Ω, the set Θt (ω) is a subset of Im σttr (ω) in the sense that (3.65)
Θt (ω) = Θ0t (ω) ∩ Im σttr (ω)
holds for some standard correspondence Θ0 with 0 ∈ Θ0 satisfying the uniform boundedness Assumption 3.3. With (3.65), one clearly has Π⊥ (θ) = 0 for all θ ∈ Θ. This leads to the following simplified expressions of the BSDE generators f φ,θ , f θ : tr
1/2
, and
1/2
.
f φ,θ (t, z) = θttr (Πt (z) − φt ) − ξt0 φt + ht (z − φt )tr A−1 t (z − φt ) f θ (t, z) =
tr −ξt0 Πt (z)
+ h2t −
1/2 tr ξtθ At ξtθ
tr
−1 ⊥ Π⊥ t (z) At Πt (z)
¯ As a consequence, the process θ¯ = θ(X) does actually not depend on the contingent claim 2 X ∈ L (P0 ) under consideration, and solves the minimization problem ¯ tr
tr
¯
ξtθ At ξtθ = min ξtθ At ξtθ , θt ∈Θt ¯
t ∈ [0, T ].
(3.66)
In addition in this case, one has Qngd (P θ ) = θ∈Θ Qngd (P θ ) = Qngd . To obtain even more explicit results one may assume e.g. ellipsoidal uncertainty n
S
o
for all (t, ω) ∈ [0, T ] × Ω,
Θ0t (ω) := x ∈ Rn  xtr Bt (ω)x ≤ δt2 (ω)
(3.67)
with δ being a positive bounded and predictable process, and B being a uniformly elliptic and predictable matrixvalued process, satisfying the separability condition (3.22) with respect to σ. Clearly f φ (t, Zt ) from (3.60) in this case is equal to tr
1/2
−ξt0 φt + δt (Πt (Zt ) − φt )tr Bt−1 (Πt (Zt ) − φt )
+ ht (Zt − φt )tr A−1 t (Zt − φt )
1/2
.
In terms of φ∗ and the solution (Y, Z) to the BSDE (3.59), the process θ∗ = θ∗ (φ∗ ) of Proposition 3.29 is given by θt∗ (X) = δt (Πt (Zt ) − φ∗t )tr Bt−1 (Πt (Zt ) − φ∗t )
−1/2
Bt−1 (Πt (Zt ) − φ∗t ).
(3.68)
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 132
Remark 3.31. Let us recall Remark 3.24 b). In the present context of Section 3.3.5 with constraints of ellipsoidal type for gooddeals (3.46) and for model uncertainty (3.67), results as explicit as in Section 3.2.1 can be obtained in particular cases, as elaborated subsequently, but not in general. Indeed, using ξ θ = ξ 0 + θ for θ ∈ Θ, to find the minimizer θ¯ (the worstcase) in (3.66) requires to compute the projection of −ξt0 onto the ellipsoid Θt with respect to the norm induced by the matrix At . In the radial case A ≡ IdRn the projection is Euclidian. While there is no closed formula for the projection in general, the solution is described by a parametric formula in terms of a Lagrangian multiplier that solves a 1dimensional equation, and it can be computed by efficient algorithms (see [Kis94]) even if this operation may be required frequently (as in Monte Carlo simulation, cf. Section 3.2.2). It is instructive to look at the special case where in addition the matrices A and B are related through B = A/r for some scalar r > 0; in other words, B basically equals A up to a change √ of δ to rδ. In this case (3.66) is solved by √ rδt 0 ¯ (3.69) θt = −ξt I{ξ0 tr At ξ0 ≤rδ2 } − ξt0 I{ξ0 tr At ξ0 >rδ2 } , t ∈ [0, T ], tr 0 t t t t t t (ξt At ξt0 )1/2 and replacing φ∗ = φ¯ in the formula of θ∗ in (3.68) by its expression from (3.57) in terms of θ¯ ¯ Note that (3.69) implies that θ¯tr Aθ¯ is equal to ξ 0 tr Aξ 0 on {ξ 0 tr Aξ 0 ≤ rδ 2 } one obtains θ∗ = θ. tr and equal to rδ 2 on {ξ 0 Aξ 0 > rδ 2 }. In other words, the worstcase Girsanov kernel −θ¯ is equal to the market price of risk ξ 0 of the center P0 of the confidence set R of reference measures, being truncated such that θ¯tr Aθ¯ = rδ 2 holds for large values of ξ 0 outside of the ellipsoidal set {x ∈ Rn : xtr Ax ≤ rδ 2 }. To obtain an intuition about the impact that model uncertainty may have on robust gooddeal hedging, let us look at the behavior of the worstcase Girsanov kernel θ¯ = θ∗ obtained in (3.69) and the hedging strategy φ¯ = φ∗ in (3.57) for varying scaling constant r: As r becomes large, the worstcase Girsanov kernel −θ¯ becomes close to the market price of risk ξ 0 and φ∗ = φ¯ close to Π(Z). This shows that as uncertainty becomes overwhelming, the robust gooddeal hedging strategy ceases to comprise a speculative component in the direction of the market price of risk. In such a situation one can show that the hedging strategy is the riskminimizing ¯ strategy under the worstcase nogooddeal measure in the worstcase model P θ . More precisely, for an arbitrary shape of the correspondence Θ0 , if uncertainty is big enough for the confidence set R of reference measures to contain some risk neutral pricing measure from Me , then robust gooddeal hedging for any claim X does not comprise a speculative component and the holdings φ∗ of the hedging strategy in risky assets coincide with those of the globally riskminimizing strategy by [FS86] (cf. also [Sch01, Section 2]) under worstcase nogooddeal ¯ ¯ ¯ ¯ ¯ = Q(X, ¯ measure Q P θ ) ∈ Qngd (P θ ), i.e. satisfying πtu (X) = πtu,θ (X) = EtQ [X] for any ¯ t ∈ [0, T ], for the worstcase model P θ . Note that here riskminimization is under a riskneutral ¯ that could also depend on the contingent claim into consideration, and not under measure Q
Section 3.3. Gooddeal valuation and hedging under model uncertainty
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b as in the original works [FS86, Sch01]. The eventually the minimal martingale measure Q nonspeculative nature of the robust gooddeal hedging strategy under (large) uncertainty offers new theoretical support for the quadratic hedging objective of risk minimization, which may be criticized for giving equal weighting for upside and downside risk. More broadly, such gives support to a common perception (see e.g. [LP00]) that speculative objectives should be avoided in hedging, in addition to more practical arguments like simplifications for markingtomarket (uses risk neutral valuation). To make the said statement precise, consider the classical BSDE tr
tr
1/2
−1 ⊥ −dYt = − ξt0 Πt (Zt ) + ht Π⊥ t (Zt ) At Πt (Zt )
dt − Zttr dWt0 ,
(3.70)
for t ∈ [0, T ] with YT = X. First we prove the following Proposition 3.32. Assume (3.22) and (3.47) hold and that Θ satisfies (3.65). For any X ∈ L2 (P0 ), let (Y X , Z X ) denote the standard solution of the BSDE (3.70). Then π·u (X) = Y X and φ∗ (X) = Π(Z X ) for all X ∈ L2 (P0 ) e
R ∩ M (S) 6= ∅.
holds, if and only if
(3.71) (3.72)
Proof. Let X ∈ L2 (P0 ). Recall that for Θ defined in (3.65), θ¯ from Theorem 3.28 does not vary with X and solves the minimization problem (3.66). Now if (3.72) holds, then there exists b θ = P θ , and therefore ξ θ = 0. This implies that θ ∈ Θ such that P θ ∈ R ∩ Me (S) 6= ∅, i.e. Q ¯ 0 0 θ = θ¯ = −ξ and hence ξ ∈ Θ. As a consequence, the generator f = f θ of the BSDE (3.55) coincides with that of the BSDE (3.70). By uniqueness of standard BSDE solutions follows π·u (X) = Y X . Now from Theorem 3.28 and Theorem 3.30 one obtains that φ∗ = φ¯ = Π(Z X ). ¯ Conversely, suppose that (3.71) holds. Then the generator f = f θ for the BSDE (3.55) and the one for (3.70) are equal everywhere by [CHMP02, Theorem 7.1 and Rmk. 4.1]. This implies ¯ b θ¯ = P θ¯, and hence R ∩ Me (S) 6= ∅. (since Π⊥ (θ) = 0 for all θ ∈ Θ) that ξ θ = 0, i.e. Q
Now we can make the previously described relation between global risk minimization and gooddeal hedging under (large) uncertainty precise. Theorem 3.33. Let the assumptions of Proposition 3.32 and (3.72) hold. For X in L2 (P0 ), let (Y, Z) be the standard solution of the BSDE (3.70). Then π·u (X) = Y has the GKW c 0 (and S = diag(S)σ · W c 0 , cf. Section 3.1.1) decomposition with respect to σ · W ∗
c 0 + Rφ , πtu (X) = π0u (X) + φ∗ · W t t
t ∈ [0, T ], ¯
∗
(3.73)
Q ¯ with φ∗ = Π(Z). The tracking error Rφ (X) = Π⊥ · (Z) · W is a Qmartingale orthogonal to 0 c , for Q ¯ ∈ Qngd given by dQ/dP ¯ σ·W ¯) · W 0 with 0 = E (−ξ + η tr
−1 ⊥ η¯t = ht Π⊥ t (Zt ) At Πt (Zt )
−1/2
⊥ A−1 t Πt (Zt ),
t ∈ [0, T ].
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 134
Proof. From Proposition 3.32 we have φ∗ = Π(Z) and π·u (X) = Y . By the definitions of η¯ ¯ Yt = Y0 + Z · W Q¯ holds for all t ∈ [0, T ]. As a consequence, one obtains πtu (X) = and Q, t c 0 +Π⊥ (Z· )·W Q¯ , t ∈ [0, T ]. Thus (3.73) holds with Rφ∗ (X) = Π⊥ (Z)·W Q¯ being π0u (X)+φ∗ · W t · · t R ¯ ¯ a Qmartingale orthogonal to S = S0 + 0· diag(St )σt dWtQ since σ(Π⊥ (Z)) = 0. Furthermore · ¯ ⊥ ∗ ∗ 0 ∗ Q φ∗ c c 0 under since φt ⊥Πt (Zt ), and φ · W = φ · W , then R (X) is also orthogonal to φ∗ · W ¯ Therefore (3.73) is the GKW decomposition of π ¯ Q. ¯·u (X) under Q. c 0 , any GaltchoukKunitaRemark 3.34. Note that using Section 3.1.1 and dS/S = σdW Watanabe (GKW) decomposition (see [Sch01]) of a continuous local Qmartingale M for Q in c 0 gives a GKW decomposition with respect to S and vice versa. In Qngd with respect to σ · W this sense, Theorem 3.33 shows that the robust gooddeal hedging strategy φ∗ for X coincides with the (global) riskminimizing strategy of [FS86] (cf. [Sch01, Section 2]) with respect to a b 0 ) under which π u (X) is equal to E Q¯ [X] ¯ = Q(X) ¯ specific measure Q ∈ Qngd (instead of Q t t b0 Q 0 b ¯ (instead of E [X]), t ∈ [0, T ]. Note that Q(X) is not equal to Q in general unless h = 0, in t
which case equality holds for any contingent claim X ∈ L2 (P0 ), or X is replicable. A seminal notrade result by [DW92] shows that a utility optimizing agent abstains from taking any position in a tradeable risky asset if uncertainty is too large. In comparison, the above theorem shows that a gooddeal hedger keeps dynamically trading according to the risk minimizing component Π(Z) but ceases to comprise any speculative component. [BCCH14] demonstrate by numerical computation in an example, in a setting quite different to ours, that the relative benefit of dynamic hedging compared to static hedging could decrease if uncertainty increases. This is intuitive, since (see e.g. [Con06]) static hedges can be less exposed to model risk. Proposition 3.32 likewise addresses how increasing uncertainty affects dynamic hedging, but is different in that it offers theoretical conditions under which dynamic gooddeal hedging φ∗ ceases to comprise speculative components in order to compensate for exposures to nonspanned risk.
3.3.6
Example with closedform solutions under model uncertainty
The usual filtration is generated by a twodimensional Brownian motion W 0 = (W 0,S , W 0,H )tr under P0 . We consider a single traded risky asset with price S and a nontraded asset with value H modelled under P0 for t ∈ [0, T ] by dSt = St σ
S
ξ
0,S
dt +
dWt0,S ,
dHt = Ht γdt +
β(ρdWt0,S
+
q
1 − ρ2 dWt0,H )
with S0 , H0 > 0, scalars σ S , β > 0, γ, ξ 0,S ∈ R and correlation coefficient ρ ∈ [−1, 1]. We derive robust gooddeal bounds and hedging strategies in closedform, for European call options on the nontraded asset and for nogooddeal constraint and uncertainty modelled (as in
Section 3.3. Gooddeal valuation and hedging under model uncertainty
Page 135
Section 3.3.5) using the radial sets C 0 = {x ∈ R2 : x ≤ h} and Θ0 = {x ∈ R2 : x ≤ δ} for scalars h, δ ≥ 0. Here one has Θ = Θ0 ∩ Im σ = [−δ, δ] × {0}, for σ = (σ S , 0), and hence ξ θ = (ξ θ,S , 0)tr := (ξ 0,S + θS , 0)tr ∈ Im σ, for models P θ with θ = (θS , 0)tr , where θS ∈ [−δ, δ]. ¯ From (3.69), with A = B ≡ IdR2 and r = 1, the worstcase model P θ corresponds to ξ 0,S θ¯S = −ξ 0,S I{ξ0,S ≤δ} − δ 0,S I{ξ0,S >δ} . ξ 
(3.74)
By Theorems 3.28,3.30, the robust gooddeal bound and hedging strategy for a call option tr 2 ¯ θ,S ¯ , X := (HT − K)+ are given by π·u (X) = Y and φ(X) = Z 1 + √ 2Z θ,S ξ 0 , for ¯ 2 h −ξ

¯ (Y, Z := (Z 1 , Z 2 )tr ) solving the BSDE (3.55), equaling the BSDE (3.51) for θ = θ: ¯
¯
¯
−dYt = − ξ θ,S Zt1 + (h2 − ξ θ,S 2 )1/2 Zt2  dt − Zttr dWtθ
and YT = X,
(3.75)
¯ b θ¯ and using (3.74), with Wtθ := (Wt0,S − θ¯S t, Wt0,H )tr , t ∈ [0, T ]. Writing (3.75) under Q arguments analogous to those in the derivation of (3.31) yield
πtu (X) = N (d+ )Ht eα˜ + (T −t) − KN (d− ) =: eα˜ + (T −t) ∗ B/Scallprice time: t, spot: Ht , strike: Ke−α˜ + (T −t) , vol: β ,
πtl (X) = eα˜ − (T −t) ∗ B/Scallprice time: t, spot: Ht , strike: Ke−α˜ − (T −t) , vol: β ,
with
√ 1 d± := ln Ht /K + α ˜ + ± β 2 (T − t) / β T − t , 2 q
˜ 1 − ρ2 α ˜ ± := γ + β − ρξ 0,S ± h
and 1/2 ¯ 2 1/2 ˜ = h(δ) ˜ h := h2 − ξ θ,S 2 = hI{ξ0,S ≤δ} + h2 − ξ 0,S − δ I{ξ0,S >δ}
. Analogously to the derivation of (3.32), note that Z = eα˜ + (T −t) N (d+ )Ht β(ρ, Hence the (seller’s) robust gooddeal hedging strategy is obtained as α ˜ + (T −t)
φ¯t (X) = e
N (d+ )Ht β ρ +
p
1 − ρ2 ξ 0,S e 0,S  hξ
tr
ξ 0,S  − δ 1{ξ0,S >δ} , 0
1 − ρ2 )tr .
p
, t ∈ [0, T ].
¯ For ξ 0,S  > δ, the speculative nature of φ(X) is reflected by the presence of the second ¯ summand in the first component of φ(X) above. For ξ 0,S  ≤ δ, this summand vanishes and the function δ 7→ α ˜ + is constant on δ ∈ [ξ 0,S , ∞]. In this case robust gooddeal hedging is ¯ = Qλ¯ ∈ Qngd (P0 ) with Girsanov then globally riskminimizing with respect to the measure Q ¯ := − ξ 0,S , htr and nonspeculative as proved in Theorem 3.33. Note that for kernel λ b 0 in absence of uncertainty), we recover δ = ξ 0,S = 0 (i.e. riskneutral setting under P0 = Q formulas of Section 3.2.2 for n = 2 and d = 1.
Section 3.4. Appendix
Page 136
Figure 3.4 illustrates the dependence of the bounds π0u (X) and π0l (X) in the presence of uncertainty, on the correlation coefficient ρ, uncertainty size δ and nogooddeal constraint (optimal growth rate bound) h, and for global parameters γ = 0.05, β = 0.5, K = 1, H0 = 1 and T = 1. Figures 3.4a,3.4b are plots of π0u (X) and π0l (X) as functions of ρ for fixed δ = 0 (i.e. absence of uncertainty) and ξ 0,S ∈ {0, 0.2}, showing how the gooddeal bounds vary for different values of h. Figure 3.4d contains a similar plot for fixed h = ξ 0,S = 0.2, showing how the bounds vary with ρ for different values of δ. One can observe that the maximum of π0u (X) and minimum of π0l (X) are attained at ρ = 0 only for ξ 0,S = 0 (cf. Figure 3.4a). b 0 ), then the largest In other words, if the market price of risk ξ 0,S is zero (hence P0 = Q gooddeal bounds are obtained when the traded and nontraded assets are uncorrelated (i.e. ρ = 0). On the other hand if ξ 0,S > 0 (as e.g. in Figures 3.4b,3.4d), the plots are tilted so that the maximum of π0u (X) (resp. minimum of π0l (X)) is reached at ρ < 0 (resp. ρ > 0). For π0u (X), this is explained by the fact that if the market price of risk ξ 0,S is positive, the ¯ = Qλ¯ ∈ Qngd (P0 ) with supremum in (3.3) is maximized by the nogooddeal measure Q ¯ := − ξ 0,S , h ˜ tr under which the upward drift α Girsanov kernel λ ˜ + of the underlying price process H is maximized, clearly at a negative correlation ρ. The explanation for π0u (X) is similar, with α ˜ − being minimal at a positive correlation, for ξ 0,S > 0. For ξ 0,S < 0 the tilt of the plots occurs in the other direction. That the gooddeal bounds in Figures 3.4a,3.4b,3.4d coincide for perfect correlation ρ = ±1 is clear, because in this case derivatives X on H are attainable and admit unique noarbitrage prices, implying π·u (X) = π·l (X). Finally, Figure 3.4c illustrates the evolution with respect to δ of the gooddeal bounds at time t = 0 for ρ = 0.6, ξ 0,S = 0.2 and different values of h, with ξ 0,S  chosen as the smallest value h0 of h. One observes that for each given h, the gooddeal bound curves become flat for δ ≥ ξ 0,S  (as predicted by Proposition 3.32), and match (i.e. π0u (X) = π0l (X)) for δ = ξ 0,S  − h0 = 0 (as might be expected in the absence of uncertainty for a degenerate expected growth rate bound h = ξ 0,S ).
3.4
Appendix
This appendix includes lemmas and proofs omitted from the main body of the chapter. For the convenience of the reader, some derivations are detailed as well. Lemma 3.35. For d < n, let σ ∈ Rd×n be of fullrank, A ∈ Rn×n be symmetric and positive definite, and h > 0, Z ∈ Rn , ξ ∈ Im σ tr . Let α0 > 0 be a constant of ellipticity √ of A−1 and assume that ξ < h α0 and A−1 (Ker σ) = Ker σ. Then the vector φ¯ := 1/2 2 −1/2 Π(Z) + Π⊥ (Z)tr A−1 Π⊥ (Z) h − ξ tr Aξ Aξ is the unique minimizer of the function φ 7→ F (φ) := −ξ ∗ φ + h (Z − φ)tr A−1 (Z − φ)
1/2
on Im σ tr .
Proof. Since A−1 (Ker σ) = Ker σ, then φ¯ ∈ Im σ tr . The KuhnTucker optimality conditions
Section 3.4. Appendix
(a)
h0 = ξ0,S = 0, δ = 0
(c)
h0 = ξ0,S = 0.2, ρ = 0.6
Page 137
(b)
h0 = ξ0,S = 0.2, δ = 0
(d)
h = ξ0,S = 0.2
Figure 3.4: Dependence of π0u (X), π0l (X) on ρ, h and/or δ ¯ The function F is convex and differentiable at every φ 6= Z, where its are satisfied by φ. −1/2 −1 ¯ = gradient is ∂F (φ) = −ξ − h (Z − φ)tr A−1 (Z − φ) A (Z − φ). This yields ∂F (φ) 1/2 −1/2 6 Z, using A−1 (Ker σ) = Ker σ. − h2 − ξ tr Aξ Π⊥ (Z)tr A−1 Π⊥ (Z) A−1 Π⊥ (Z) for φ¯ = √ At φ = Z the subgradient is welldefined and the inclusion E := x ∈ Rn x ≤ h α0 −ξ ⊆ ¯ = − h2 − ξ tr Aξ 1/2 Π⊥ (Z)tr A−1 Π⊥ (Z)−1/2 A−1 Π⊥ (Z) for ∂F (φ) holds. Overall ∂F (φ) ¯ for φ¯ = Z. In any case, φ¯ satisfies the KarushKuhnTucker φ¯ 6= Z and 0 ∈ E ⊆ ∂F (φ) conditions and since F is convex and the minimization constraint φ ∈ Im σ tr is linear, optimality of φ¯ follows from the KuhnTucker theorem (cf. [Roc70, Section 28]). Uniqueness of φ¯ is implied by the fact that F is strictly convex over Im σ tr if Π⊥ (Z) 6= 0 and strictly convex at φ¯ √ if Π⊥ (Z) = 0 since ξ < h α0 .
Lemma 3.36. Let d < n, h > 0 be constant, Z ∈ Rn , A ∈ Rn×n a symmetric positive definite matrix, σ ∈ Rd×n a full (d)rank matrix, and ξ 0 ∈ Φ := Im σ tr . Let Θ ⊂ Rn be a convex1/2 tr compact set, and F : Rn × Rn 3 (φ, θ) 7→ θtr (Z − φ) − ξ 0 φ + h (Z − φ)tr A−1 (Z − φ) . Then the minmax identity inf φ∈Φ supθ∈Θ F (φ, θ) = supθ∈Θ inf φ∈Φ F (φ, θ). holds. Proof. For all φ ∈ Rn , the function θ 7→ F (φ, θ) is concave, continuous. For all θ ∈ Rn the
Section 3.4. Appendix
Page 138
function φ 7→ F (φ, θ) is convex and continuous. As Θ ⊂ Rn is convex and compact, and Φ = Im σ tr is convex and closed, a minimax theorem [ET99, Chapter VI, Proposition 2.3] applies and the minmax identity holds.
Proof of Lemma 3.1. Part a) is classical (see [Del06] and cf. previously given other references). As for Part b), mstability and convexity of Me follow from [Del06, Proposition 5]. Convexity of Qngd follows from that of Me and the values of C. To show mstability of Qngd , let Z i = E(λi · W ) ∈ Qngd , i = 1, 2, τ ≤ T be a stopping time and Z = I[0,τ ] Z·1 + I]τ,T ] Zτ1 Z·2 /Zτ2 . Since Me is mstable, then Z ∈ Me and one has Z = E(λ · W ) for some predictable process λ. It remains to show that λ is bounded and that λ ∈ C. From the expression of Z, writing the densities Z, Z 1 , Z 2 as ordinary exponentials by distinguishing t ≤ τ and t ≥ τ , and taking the 2 2 R logarithm yields (λ − I[0,τ ] λ1 − I]τ,T ] λ2 ) · W = 12 0· λs 2 − I[0,τ ] (s) λ1s − I]τ,T ] (s) λ2s ds. Since F is the augmented Brownian filtration, then [0, τ ] and ]τ, T ] are predictable and so is 1 2 1 2 λ − I[0,τ ] λ − I]τ,T ] λ . Hence λ − I[0,τ ] λ − I]τ,T ] λ · W is a continuous local martingale of finite variation and is thus equal to zero. As a consequence λ = IB λ1 + IB c λ2 is bounded since λ1 , λ2 are, and satisfies λ ∈ C since C is convexvalued.
Proof of Theorem 3.7. Without loss of generality, we argue only for X ≥ 0; otherwise one can use translation invariance with X + kXk∞ ≥ 0. et (X) := ess sup Part 1: Let t ∈ [0, T ] and define π E Q [X]. We have the inclusions Q∈Qngd t
Ctk (ω) ⊆ Ctk+1 (ω) ⊆ Ct (ω) for all (t, ω) and for all k ∈ N, and hence the chain of inet (X) holds for k ∈ N. Since the sequence equalities πtu,k (X) ≤ πtu,k+1 (X) ≤ πtu (X) ≤ π u,k u,k (πt (X))k∈N is nondecreasing and πt (X) ≤ kXk∞ , for all k, the monotone a.s. limit et (X) ≥ Jt . It remains to show the reverse inequality, Jt := limk%∞ πtu,k (X) is finite and π et (X) = πtu (X), using Part 2 of Proposition 3.5 to π·u,k (X) to obtain a sequence which implies π ¯ k ∈ Qngd ⊆ Qngd satisfying π u (X) ≥ E Q¯ k [X] = π u,k (X) % π et (X) as k → ∞. of measures Q t t t k To this end, it suffices to show that J is a c`adl`ag Qsupermartingale for all Q ∈ Qngd and ngd
e· (X) = π·u,Q (X) since Qngd is also convex and mstable (arthen apply Lemma 3.6 to π gument being analogous to that for Qngd in Lemma 3.1) b). First notice that J is a c`adl`ag Qsupermartingale for any Q ∈ Qngd with Girsanov kernel λQ = λ. Indeed for such measures Q, there exists k0 ∈ N such that λ ∈ Λk for all k ≥ k0 . Since Jt = limk πtu,k (X) and π·u,k (X) is a bounded c`adl`ag Qsupermartingale for every k ≥ k0 , then J is a c`adl`ag Qsupermartingale as the increasing limit of c`adl`ag Qsupermartingales of class D (cf. [Doo01, Section 2.IV.4]). Now let Q ∈ Qngd with λQ = λ = −ξ + η ∈ Λ not necessarily bounded. Then λn := −ξ + η n with η n := ηI{η≤n} ∈ Ker σ forms a sequence of bounded Girsanov kernels for measures
Section 3.4. Appendix
Page 139
Qn ∈ Qngd such that limn→∞ λn = λ P ⊗ dt − a.e.. By the above arguments, since ξ and X b are bounded, then J is a bounded c`adl`ag Qsupermartingale (hence of class D) that admits a c with respect DoobMeyer decomposition which, by the predictable representation property of W b F) (cf. [HWY92, Theorem 13.22]), reads J = J0 + Z · W c − A, where Z ∈ H2 (Q) b and A (Q, 2 b is a nondecreasing predictable processes with A0 = 0 and AT ∈ L (Q) because J ∈ S ∞ is bounded (cf. [DM82, Inequality (15.1), Section VII.15, page 202]). One rewrites n
J = J0 + Z · W Q + J = J0 + Z · W Q +
·
Z
Z 0
0 ·
Zttr ηtn dt − A, and
Zttr ηt dt − A.
(3.76) (3.77)
Since the Girsanov kernels λn are bounded, J is a c`adl`ag supermartingale under Qn for all n. Hence from (3.76) one has dAt ≥ Zttr ηtn dt, t ∈ [0, T ], for all n ∈ N. By dominated convergence, taking the limit as n → ∞ implies dAt ≥ Zttr ηt dt, t ∈ [0, T ]. Now since X is nonnegative, R then so is J and positivity of − 0· Zttr ηt dt + A implies Z · W Q ≥ const from (3.77). Being bounded from below, the local Qmartingale Z · W Q is therefore a Qsupermartingale. Finally R because J is bounded, then 0T Zttr ηt dt − AT is Qintegrable and thus J is a Qsupermartingale. Part 2: For k ∈ N, the process π u,k is the gooddeal bound associated to the constraint correspondence C k satisfying Assumption 3.3. Hence applying Part 2 of Proposition 3.5 with C replaced by C k yields the result. b ∈ Qngd ⊂ Qngd . Hence by Lemma 3.6, π u (X) and π·u,k (X) Part 3: For all k ≥ kξk∞ holds Q · k b b as are bounded c`adl`ag Qsupermartingales, admitting DoobMeyer decompositions under Q k in (3.12), and by Part 2 A satisfies (3.13). By arguments similar to those for Part 1 follows b and A, Ak ∈ L2 (Q). b Z, Z k ∈ H2 (Q) b Fu ) for all u ≤ T follows Part 4: From Part 3, that Aku converges to Au weakly in L2 (Ω, Q, from [DM82, Theorem VII.18 and subsequent remarks]. These apply since the sequence (π·u,k (X))k≥kξk∞ is uniformly bounded by kXk∞ , and hence Part 1 and dominated convergence b Fu ), for all u ∈ [0, T ]. Furthermore the imply that πuu,k (X) converges to πuu (X) in L2 (Ω, Q, u,k k cu → Z · W cu weakly in L2 (Ω, Q, b Fu ) convergences of (πu (X))k and (Au )k imply that Z k · W for all u ∈ [0, T ]. By the predictable representation property and Itˆo’s isometry, follows b ⊗ dt) for any u. Z k → Z weakly in L2 (Ω × [0, u], Q
b the DoobMeyer decomposition Proof of Theorem 3.9. By Theorem 3.7, π·u (X) admits under Q R · c − A = π u (X) + Z · W + ξ tr Zt dt − A, where Z ∈ H2 (Q) b π·u (X) = π0u (X) + Z · W 0 0 t and A is a nondecreasing predictable process with A0 = 0. Alternatively one rewrites R R −dπtu (X) = gt (Zt )dt − Zttr dWt + dKt , with K := A − 0· ξttr Zt dt − 0· ess supλt ∈Λt λtr t Zt dt u being finitevalued and predictable. For (π· (X), Z, K) to be a supersolution to the BSDE with
Section 3.4. Appendix
Page 140
parameters (g, X) it suffices to show that K is nondecreasing. For any λ = −ξ + η ∈ Λ, one can construct the sequence of λn = −ξ + η n ∈ Λ Girsanov kernels of measures Qn ∈ Qngd with η n = ηI{η≤n} such that λn → λ P ⊗ dta.s. as n → ∞. For each Qn it holds R n π·u (X) = π0u (X) + Z · W Q + 0· Zt tr ηtn dt − A. Since π·u (X) is a bounded Qn supermartingale, then dAt − ξttr Zt dt ≥ Zttr λnt dt, for all n ∈ N. Taking the limit as n → ∞ and using dominated convergence one obtains dAt − ξttr Zt dt ≥ Zttr λt dt. Now taking the essential supremum over all λ ∈ Λ yields dKt ≥ 0. To show that the supersolution (π·u (X), Z, K) is minimal, it suffices (by Lemma 3.6) to show that the Y component of any other supersolution is a c`adl`ag Qsupermartingale for every ¯ K) ¯ be a supersolution of the BSDE with parameters (g, X), with Q ∈ Qngd . Let (Y¯ , Z, ∞ ¯ Y ∈ S . By change of measure, the dynamics of Y¯ under some measure Q ∈ Qngd with Girsanov kernel λQ ∈ Λ is
Q ¯ ¯ tr Q ¯ tr ¯ −dY¯t = ess sup λtr t Zt − Zt λt dt − Zt dWt + dKt ,
t ∈ [0, T ].
λt ∈Λt
(3.78)
¯ is nondecreasing, it holds that Since K
¯ t + ess sup λtr ¯ ¯ tr Q dK t Zt − Zt λt dt ≥ 0,
t ∈ [0, T ].
λt ∈Λt
(3.79)
From (3.79), (3.78) and boundedness of Y¯ , the local martingale Z¯ · W Q is bounded from below, hence is a supermartingale. Again since Y¯ ∈ S ∞ , then the integral of (3.79) in [0, T ] is Qintegrable and therefore Y¯ is a Qsupermartingale.
Proof of Corollary 3.10. By mstability and convexity of Qngd , Lemma 3.6 and Part 1. of ¯ Theorem 3.7 imply that π·u (X) is a c`adl`ag Qsupermartingale with terminal value X since ngd ¯ ∈ Q . We have Q ¯
¯
¯
¯
¯
E Q [X] = π0u (X) ≥ E Q [πtu (X)] ≥ E Q [EtQ [X]] = E Q [X], t ≤ T. b from Theorem 3.9 with ¯ Hence π·u (X) is a Qmartingale. Let Z ∈ H2 (Q)
K := A −
Z 0
·
ξttr Zt dt −
Z 0
·
ess sup λtr t Zt dt λt ∈Λt
such that (π·u (X), Z, K) is the minimal supersolution to the BSDE with parameters (g, X). R ¯ t + ξt )dt − A. Since ¯ as π·u (X) = π u (X) + Z · W Q¯ + · Zttr (λ One writes π·u (X) under Q 0R 0 · ¯ t + ξt )dt = 0. Therefore since λ ¯ ∈ Λ, one ¯ π·u (X) is a bounded Qmartingale, then A − 0 Zttr (λ R · tr ¯ t + ξt )dt = 0. Thus K = 0 and hence (π u (X), Z) is a BSDE obtains 0 ≤ K ≤ A − 0 Zt (λ · solution. Any solution being a supersolution, minimality follows from Theorem 3.9. Finally
Section 3.4. Appendix
Page 141
¯ tr Zt dt − Z tr dWt , with π u (X) = X, then ¯ since the Qmartingale π·u (X) satisfies −dπtu (X) = λ t t T ¯ tr Zt holds. ess supλt ∈Λt λtr Z = λ t t t
˜ T , S˜T ) ∈ L2 for a payoff function Derivation of (3.33). Consider a European option X = G(H (0, ∞)2 3 (x, y) 7→ G(x, y) ∈ R being measurable, nondecreasing in x and at most of polynomial growth in x±1 , i.e. G(x, y) ≤ k(1 + xn + x−n ) for all (x, y) ∈ (0, ∞)2 , for some k > 0 and n ∈ N. Again following the arguments of the proof in the example of an option on ˜ one can show that H, ¯=h λ
n X
−1/2 β˜i2 /ai (0, . . . , 0, β˜d+1 /ad+1 , . . . , β˜n /an )tr
i=d+1 ¯ ˜ t , S˜t ) for u ∈ C (0, ∞) × (0, ∞)2 with ∂x u ≥ 0. ˜ T , S˜T )] = u(t, H and πtu (X) = EtQ [G(H Moreover one obtains for all t ∈ [0, T ] that
Z i = t Z i = t
˜ t ∂x u(t, H ˜ t , S˜t ) + σ ˜ t , S˜t ), for i ≤ d and β˜i H ˜i S˜t ∂y u(t, H ˜ t ∂x u(t, H ˜ t , S˜t ), β˜i H for i ≥ d + 1
˜ T − S˜T )+ ∈ L2 . Denoting Lt := For the specific case G(x, y) := (x − y)+ , one has X = (H ˜ t /S˜t , t ∈ [0, T ], gives X = S˜T (LT − 1)+ . A change of num´eraire dQ/d ˜ Q ¯ = e−˜µt S˜t /S˜0 H Ft ˜ ˜t −σ ˜ S − 1 δ2t , yields π u (X) = eµ˜(T −t) S˜t E Q (LT − 1)+ . Now Lt = L0 e(α+ −˜µ)t exp β˜tr W ˜ tr W t
t
t
2
1/2 Pn P ˜2 1/2 , δ := β ˜ 2 + ˜ ˜ an nwith α± := γ˜ ± h σ 2 − 2 di=1 σ ˜i β˜i , and W i=d+1 βi /ai ˜ dimensional QBrownian motion. Now the formula (3.33) follows from the classical Margrabe formula for exchange options.
√ Derivation of (3.35),(3.37). The stochastic exponential E (ε/ ν) · W ν is a uniformly in¯ ∈ Qngd ⊇ Qngd (see (3.4) for definition of tegrable martingale which defines a measure Q √ √ e ¯ := ε/ ν, i.e. dQ/dP ¯ Qngd ⊂ M ) with Girsanov kernel λ = E (ε/ ν)·W ν . Indeed, applying √ √ [CFY05, Theorem 2.4 and Section 6] one gets that E (ε/ ν) · W ν and S = S0 E ν · WS ¯ is are uniformly integrable P  respectively Qmartingales. The variance process ν under Q
again a CIR process with parameters (¯ a, b, β, ρ) where a ¯ := a + βε 1 − ρ2 > a and the Feller ¯ condition β 2 ≤ 2¯ a still holds. For a put option X = (K − ST )+ ∈ L∞ , Y¯t := EtQ [X] are ¯ (instead of P ). Since the Heston given by the Heston formula (cf. [Hes93]), applied under Q price is nondecreasing in the mean reversion level of the variance process ([OA11, Proposition ¯ 5.3.1]) one expects that πtu (X) = Y¯t = EtQ [X]. Let us make this precise. For Q ∈ Qngd with √ Girsanov kernel λ satisfying λ ≤ ε/ ν, one has YTQ = Y¯T = X with YtQ = EtQ [X]. Using p
Section 3.4. Appendix
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FeynmanKac, Y¯t = u(t, St , νt ) for a function u ∈ C 1,2,2 ([0, T ] × R+ × R+ ) with ∂u ∂ν ≥ 0 (see [OA11, Theorem 5.3.1, Corollary 5.3.1]). By Itˆo’s formula and change of measure follows q q √ √ ∂u ε ∂u dY¯t =β 1 − ρ2 νt λt − √ (t, St , νt )dt + β 1 − ρ2 νt (t, St , νt )dWtQ,ν νt ∂ν ∂ν √ ∂u √ ∂u (t, St , νt ) + βρ νt (t, St , νt ) dWtS , t ∈ [0, T ]. (3.80) + St νt ∂S ∂ν
Since X is bounded, then Y¯ is in S ∞ (Q) and a Qsupermartingale by (3.80) . Hence YtQ ≤ Y¯t for all Q ∈ Qngd , which by Part 1. of Theorem 3.7 implies the claim and thus we obtain the Heston type formula (3.35). ¯
¯ ∈ Qngd and π u (X) = E Q [X] with X ∈ L∞ , Corollary 3.10 implies that the gooddeal Since Q 0 b of the ¯ ∈ S ∞ × H2 (note P = Q) bound is the Y component of the minimal solution (Y¯ , Z) √ ¯ t z 2 = εz 2 / νt , for z = (z 1 , z 2 ), and terminal condition BSDE (3.36) with generator gt (z) = λ X. Now consider the strategy √ ∂u √ ∂u √ βρ φ¯t = Z¯t1 = St νt (t, St , νt ) + βρ νt (t, St , νt ) = St νt ∆t + Vt . ∂S ∂ν 2 Clearly φ¯ is in the set Φ = H2 (R) of permitted trading strategies since Z¯ ∈ H2 (R2 ). Recall that √ P ngd consists of dQ/dP = E (λS , λν ) · W such that (λS , λν ) ≤ ε/ ν with (λS , λν ) being bounded. For Q ∈ P ngd , any wealth process φ · W S , φ ∈ Φ, is thus in S 1 (Q). As Qngd ⊆ P ngd R holds, clearly πtu (X) ≤ ρt (X − tT φs dWsS ) for any strategy φ ∈ Φ. To prove that φ¯ is a R gooddeal hedging strategy, we show the reverse inequality πtu (X) ≥ EtQ X − tT φ¯s dWsS for all Q ∈ P ngd . Let Q ∈ P ngd with Girsanov kernel (λS , λν ). Like in (3.80), we obtain for any stopping time τ that Y¯τ ∧T −
Z
τ ∧T
τ ∧t
φ¯s dWsS = Y¯τ ∧t +
Z
τ ∧T
τ ∧t
q √ ε ∂u β 1 − ρ2 νs λνs − √ (s, Ss , νs )ds νs ∂ν
(3.81)
+ Lτ ∧T − Lτ ∧t , R p √ Q,ν for the local Qmartingale L := 0· β 1 − ρ2 νs ∂u ∂ν (s, Ss , νs )dWs . By the inequalities R √ τ ∧T ∂u ν S ¯ ¯ ¯ ∂ν ≥ 0 and λ ≤ ε/ ν follows that Yτ ∧T − τ ∧t φs dWs is less than Yτ ∧t + Lτ ∧T − Lτ ∧t . Localizing L along a sequence of stopping times τn ↑ ∞ and taking conditional Qexpectations R ∧T ¯ S ∈ S 1 (Q), the claim then yields EtQ [Y¯τn ∧T − ττnn∧t φ¯s dWsS ] ≤ Y¯τn ∧t . Using X ∈ L∞ and φ·W ∂u follows by dominated convergence. Hence (3.37) holds for Vt := ∂σ (t, St , νt ) = 2σt ∂u ∂ν (t, St , νt ) √ and volatility σt = νt .
Proof of Lemma 3.26. Part 1: We use (3.48) to show that the set θ∈Θ P ngd (P θ ) is mstable and convex. Let κ ∈ [0, 1], τ ≤ T be a stopping time and Z i = E(λi · W 0 ), with λi selection S
Section 3.4. Appendix
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of C 0 + θi , θi ∈ Θ, i = 1, 2. From the proof of the second part of Lemma 3.1 the process Z := I[0,τ ] Z 1 + I]τ,T ] Z 2 Zτ1 /Zτ2 satisfies Z = E(λ · W 0 ) with λ = I[0,τ ] λ1 + I]τ,T ] λ2 . By convexity of the values of C 0 it follows that λ ∈ C 0 + θ for θ := I[0,τ ] θ1 + I]τ,T ] θ2 ∈ Θ S by convexity of the values of Θ. Hence Z is in P ngd (P θ ), and therefore θ∈Θ P ngd (P θ ) is mstable. To show convexity, consider the density process Z˜ = κZ 1 + (1 − κ)Z 2 . Then 2 ˜ · W 0 ) with λ ˜ = 1 κZ 1 2 λ1 + (1−κ)Z Z˜ = E(λ λ2 . Again by convexity of the values κZ +(1−κ)Z κZ 1 +(1−κ)Z 2 2 2 ˜ ∈ C 0 + θ, ˜ for θ˜ := 1 κZ 1 2 θ1 + (1−κ)Z of C 0 , λ 1 2 θ ∈ Θ since Θ is convexvalued. κZ +(1−κ)Z
κZ +(1−κ)Z S Concerning Part 2: Mstability and convexity of θ∈Θ Qngd (P θ ) S S ngd (P θ ) T Me . 1, and θ∈Θ Qngd (P θ ) = θ∈Θ P
follow from that of Me , Part
4. Hedging under gooddeal bounds and volatility uncertainty: a 2BSDE approach In this chapter, we study gooddeal bounds defined from a bound on the instantaneous Sharpe ratios in the economy and a notion of robust hedging (as in Chapter 3) in the presence of volatility uncertainty. We describe worstcase gooddeal bounds and robust hedging strategies in terms of solutions to 2BSDEs. In Section 4.1 we clarify the canonical setup incorporating volatility uncertainty and provide some preliminary results about 2BSDEs. Then in Section 4.2 we describe a model of the financial market under volatility uncertainty, together with a parametrization of the nogooddeal restriction in this model. Section 4.3 is devoted to the main results of the chapter, namely a 2BSDE characterization of gooddeal bounds and associated hedging strategies and the fact that the latter are at least meanselffinancing uniformly over all priors (robustness). It includes in addition an example for European put options on nontradeable assets in a BlackScholes model with uncertain volatility, where worstcase valuations can be computed explicitly from a BlackScholes’ type formula under a worstcase prior. Robust gooddeal hedging strategies are also obtained in closedform in this example, and it is shown that they are in general not superreplicating under volatility uncertainty.
4.1
Mathematical framework and preliminaries
We consider a canonical setting with filtered probability space (Ω, F, P 0 , F). Here Ω := {ω ∈ C([0, T ], Rn ) : ω(0) = 0} denotes the space of continuous paths starting at 0 and equipped with the norm kωk∞ := supt∈[0,T ] ω(t). The canonical process B is defined by Bt (ω) := ω(t), for ω ∈ Ω and its law is P 0 , the Wiener measure. The underlying filtration F = (Ft )t∈[0,T ] is generated by B and F+ = (Ft+ )t∈[0,T ] denotes its rightlimit, with Ft+ = Ft+ . For a probability measure Q, the conditional expectation given Ft will be denoted by EtQ [·].
4.1.1
The local martingale measures
A probability measure P is called a local martingale measure if B is a local martingale with respect to (F, P ). From [Kar95] (see also [F¨ol81]), it follows that there exists a Fprogressively R R measurable process denoted by 0· Bstr dBs which coincides with the P Itˆo integrals (P ) 0· Bs dBstr P a.s. for all local martingale measures P . In particular, this yields pathwise definition of the R b with respect quadratic variation hBi of B as hBi := BB tr − 2 0· Bs dBstr and of its density a
144
Section 4.1. Mathematical framework and preliminaries
Page 145
to the Lebesgue measure dt as bt (ω) := lim sup a &0
hBit (ω) − hBit− (ω) ,
b is welldefined We denote by P W the set of all local martingale measures P for which a >0 n×n and takes values P almost surely in the space Sn ⊂ R of positive definite symmetric n × nmatrices. As mentioned in [STZ11], the measures in P W can be typically mutually singular. In particular, there is no dominating measure in P W and this can be illustrated by the following example of [STZ11]: √ Example 4.1. For n = 1, P = P 0 , P 0 = P 0 ◦ ( 2B)−1 , A = hBit = t, for all t ∈ [0, T ] and A0 = hBit = 2t, for all t ∈ [0, T ] , it holds P, P 0 ∈ P W , P (A) = P 0 (A0 ) = 1 and P (A0 ) = P 0 (A) = 0. Hence P ⊥P 0 . −1
bt 2 dBt , t ∈ [0, T ] is a Note that for any P ∈ P W , the process W P defined by WtP := (P ) 0t a P P Brownian motion under P (by L´evy characterization and since hW it = t, P a.s.). Similarly to [STZ12], we will use the socalled strong formulation of volatility uncertainty according to which we consider only on the local martingale measures induced by the laws of solutions to 1/2 SDEs dXt = at (X)dBt , P a.s.. More precisely, uncertainty will be considered only over the subclass P S ⊂ P W consisting of all probability measures R
α
0
α −1
P := P ◦ (X )
where
,
Xtα
Z
:= 0
t
αs1/2 dBs ,
P 0 a.s. t ∈ [0, T ],
T with S>0 n −valued Fprogressively measurable diffusion coefficient α satisfying 0 αt dt < ∞, P 0 a.s.. The subscript S in P S stands for “strong” as in strong formulation, as opposed to W in P W which stands for “weak”. The consequence of restricting oneself to the subclass P S is the aggregation property it possesses in the sense that the following lemma (see [STZ11, Lemma 8.1, Lemma 8.2]) holds.
R
P
Lemma 4.2. For P ∈ P W , let F and FW P P FW .
P
denote respectively the P augmentations of the P
P
filtrations F and Then P S = P ∈ P W : F = FW P , and B has the martingale representation property simultaneously with respect to all P ∈ P S . In addition, every P ∈ P S satisfies the Blumenthal ZeroOne law.
Remark 4.3. 1. For any P α ∈ P S one has P α ◦ B −1 = P 0 ◦ (X α )−1 , i.e. the distribution of B under P α coincides with the distribution of X α under P 0 . In particular with the filtration characterization of P S in Lemma 4.2, this implies that the density of the b(B) = α ◦ βα (B), P α ⊗ dta.s., for quadratic variation of B under P α is equal to a some Fprogressively measurable map βα : [0, T ] × Ω → Rn (see [STZ13, Lemma 2.2])
Section 4.1. Mathematical framework and preliminaries
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2. Note that for any P ∈ P S , the Blumenthal ZeroOne law in Lemma 4.2 implies EtP [X] = E P [XFt+ ] P a.s. for any X ∈ L1 (P ), t ∈ [0, T ]. In particular, any Ft+ measurable random variable has a Ft measurable P version. We will work with an even more restricted set of local martingale measures by considering for fixed a, a ∈ S>0 n , the subclass PH of P S defined by n
o
(4.1)
b ≤ a, P ⊗ dta.e. . PH = P ∈ P S : a ≤ a
κ (1 < κ ≤ 2) given Remark 4.4. The definition of PH is slightly different from the one for PH in [STZ12, Definition 2.6] as
κ b ≤ aP , P ⊗ dta.e., PH = P ∈ P S : ∃aP , aP ∈ S>0 n s.t. aP ≤ a
E
P
h Z
T
0
κ
2 i
bt ) dt Ft (0, 0, a
κ
(4.2)
0 n ,
(4.3)
n×n is such that and we set Ft (ω, y, z, a) := +∞ for a ∈ Rn×n \ S>0 n , where DH ⊆ R n×n Ht (ω, y, z, γ) = +∞ for γ ∈ R \ DH , and DH is assumed to notdepend on (t, ω, y, z) and b bt ) and Fbt0 := Fbt (0, 0) for P ∈ PH . Let to contain the origin. We denote Ft (y, z) := Ft (y, z, a DFt (y,z) be the domain of F in a for fixed (t, ω, y, z). To obtain existence and uniqueness of solutions to 2BSDEs with generators F we need the following combination of Assumption 2.8 and Assumption 4.1 of [STZ12]:
Assumption 4.6. (i) PH = 6 ∅ and DFt (y,z) = DFt is independent of (ω, y, z), for all t ∈ [0, T ], (ii) F is Fprogressively measurable in (t, ω) for any fixed (y, z, a), (iii) F is uniformly continuous in ω with respect to the supremum norm k · k∞ , (iv) Fb is PH q.s. uniformly Lipschitz in (y, z), in the sense that ∃ C > 0 s.t. PH q.s. for all y, y 0 ∈ R, z, z 0 ∈ Rn , 1/2
bt (z − z 0 ) , t ∈ [0, T ]. Fbt (y, z) − Fbt (y 0 , z 0 ) ≤ C y − y 0  + a
(v) Fb 0 satisfies
R
T b0 2 0 Fs  ds
1/2
∈ L2H , i.e.
sup E P ∈PH
P
h
P
ess sup t∈[0,T ]
EtH,P
Z 0
T
Fbs0 2 ds
i
< ∞.
(4.4)
Remark 4.7. 1. Assumption 4.6, (iii) is less standard, and its importance lies in the proof of existence of solutions to 2BSDEs. In fact it provides the additional regularity of the generator needed to use the regular conditional probability distributions (shortly r.c.p.d.), which exist in the present canonical Wiener setting (see e.g. [SV79]). Using r.c.p.d. ensures a pathwise construction of solutions to 2BSDEs, i.e. without exception of negligible sets, hence avoiding any issue caused by singularity of measures in PH . 2. Assumption 4.6, (v) implies in particular that sup E P P ∈PH
Z 0
T
Fbs0 2 ds < ∞,
(4.5)
2 in [STZ12]. Recall which in turn yields the integrability condition in the definition of PH that we have originally omitted (cf. Remark 4.4) this condition in the definition (4.1) of PH . For H such that Fb 0 is bounded PH quasisurely, (4.4) automatically follows.
Section 4.1. Mathematical framework and preliminaries
Page 149
Existence and uniqueness of solutions to 2BSDEs Following [STZ12], a secondorder BSDE is a stochastic integral equation of the type Yt = X −
Z
T
Z
Fbs (Ys , Zs )ds −
t
t
T
Zstr dBs + KT − Kt ,
t ∈ [0, T ], PH q.s.,
(4.6)
or equivalently −dYt = −Fbt (Yt , Zt )ds − Zttr dBt + dKt ,
t ∈ [0, T ],
YT = X, PH q.s..
The solution to the 2BSDE (4.6) is defined as follows. Definition 4.8. For X ∈ L2H , a couple (Y, Z) ∈ D2H × H2H is called solution to the 2BSDE (4.6) if YT = X PH q.s., the process K P defined for each P ∈ PH by KtP := Y0 − Yt +
t
Z
Z
Fbs (Ys , Zs )ds + 0
0
t
Zstr dBs ,
t ∈ [0, T ], P a.s..
(4.7)
is P a.s. nondecreasing, and the family K P , P ∈ PH satisfies the minimum condition
KtP =
P
ess inf
0
0
P 0 ∈PH (t+ ,P )
EtP [KTP ],
P a.s., for all P ∈ PH , t ∈ [0, T ].
(4.8)
If moreover the family {K P , P ∈ PH } can be aggregated into a universal process K, i.e. K = K P , P a.s. for all P ∈ PH (see [STZ11] for more on aggregation), then one calls (Y, Z, K) solution to the 2BSDE. The pair (F, X) will be called the parameters (generator and terminal condition) of the 2BSDE (4.6). Y will be referred to as value process and Z as the control process. The following proposition is a combination of [STZ12, Theorem 4.3, Theorem 4.6], and provides conditions for existence and uniqueness of solutions to 2BSDEs with Lipschitz generators. In addition it gives a representation of the value process of the 2BSDE in terms of the value processes (under P ∈ PH ) of the associated standard BSDEs. We employ the classical notation for standard BSDEs (as in e.g. [EPQ97]) according to which the generator of the BSDE (4.10) is −Fb (i.e. with a minus sign). The notation for the generator of the associated 2BSDEs however remains unchanged. Proposition 4.9. Let Assumption 4.6 hold. Then 1. Assume that X ∈ L2H and that (Y, Z) ∈ D2H × H2H is a solution to the 2BSDE (4.6). Then for any P ∈ PH , Y has the representation Ys =
P
ess sup
P 0 ∈PH (s+ ,P )
0
YsP (t, Yt ), P a.s., s ≤ t ≤ T,
(4.9)
Section 4.1. Mathematical framework and preliminaries
Page 150
where for each P ∈ PH , the couple (Y P (τ, ξ), Z P (τ, ξ)) is the unique solution to the standard BSDE with parameters (−Fb , ξ): YtP = ξ −
Z t
τ
Fbs (YsP , ZsP )ds −
Z t
τ
(ZsP )tr dBs ,
t ≤ τ, P a.s.,
(4.10)
for a F+ stopping time τ and Fτ+ measurable random variable ξ ∈ L2 (P ). In particular, the 2BSDE (4.6) has at most one solution in D2H × H2H . 2. For X ∈ L2H , the BSDE (4.6) admits a unique solution (Y, Z) ∈ D2H × H2H . Remark 4.10. 1. From the dynamics of the value process Y , the control process Z of the 2BSDE (4.6) is uniquely given by dhY, Bit = Zt dhBit , PH q.s.. As a consequence, one can obtain Z from Y as Zt = lim sup &0
hY, Bit − hY, Bit− , hBit − hBit−
t ∈ [0, T ].
2. Note that the representation (4.9) in Proposition 4.9 naturally follows from the minimum condition (4.8); this is a key step in deriving the probabilistic representation of solutions to fully nonlinear PDEs via 2BSDEs. With this at hand, uniqueness of the solution to the 2BSDE is a direct consequence of uniqueness for standard BSDEs. 3. As can be seen in part 2 of Proposition 4.9, sufficient conditions for existence and uniqueness of solutions to Lipschitz 2BSDEs crucially rely on the terminal X being in L2H . Clearly a trivial example of random variables X that lie in L2H are those in UCb (Ω), i.e. that are uniform continuous and bounded. These include e.g. constants and also random variables that can be written as X := g(Bt1 , . . . , Btk ), with t1 , . . . , tk ∈ [0, T ] and some bounded uniformly continuous function g : Rn×k → R, k ∈ N. This is true because since the function ω → ω(t) is Lipschitz continuous in the norm k·k∞ for any t ∈ [0, T ], then X would be uniformly continuous as it is the composition of uniformly continuous functions. In particular, it is sufficient for this purpose that the function g be Lipschitz and bounded.
Comparison theorems for 2BSDEs As theoretical results for 2BSDEs in this chapter, we now state and prove some comparison theorems for 2BSDEs with different generators. Unlike the wellknown comparison theorem for standard BSDEs, and because of the presence of the nondecreasing processes in the 2BSDEdynamics, comparison of the generators of the 2BSDEs at one of the solutions does not suffice to imply comparison of the value processes. We distinguish two approaches: The first one leads to Proposition 4.12 and assumes that the generators of the two 2BSDEs at the
Section 4.1. Mathematical framework and preliminaries
Page 151
solutions of one of the associated standard BSDEs are quasisurely comparable. The second approach (cf. Theorem 4.13) rather assumes that the generators at the solution of one of the 2BSDEs are comparable and that an additional monotonicity condition on the difference of the nondecreasing processes holds, and obtain partly as a result that this difference also satisfies the minimum condition (4.8). For consistency with the current setup, our results are stated with respect to the family PH of mutually singular measures, but the proofs would be κ of [STZ12] as defined in (4.2). For two analogous for the more general families of measures PH 2BSDEs with the same generator but different terminal conditions, a comparison theorem was stated in [STZ12, Corollary 4.4] as a byproduct of the representation result (4.9). However in applications, one can sometimes be concerned with 2BSDEs with different generators. In [PZ13, Proposition 3.1], a comparison theorem is proved (as generalization of [Tev08, Theorem 2]) assuming that the difference of the nondecreasing components is also nondecreasing. Their focus is on quadratic 2BSDEs, but they also state the result for 2BSDEs with same generators. Here we prove general comparison theorems for 2BSDEs with possibly different generators. Our proofs rely on a classical linearization procedure (also used in [STZ12]) coupled with a change of measure argument. Instead of imposing specific conditions on the generators which imply existence of solutions, we only insist that we have solutions and impose conditions on the generators and other processes of interest quasisurely. To this end, the following intermediate result will be needed. Lemma 4.11. Let X ∈ L2H , λ and η be bounded Fprogressively measurable R and Rn valued processes respectively and ϕ ∈ H2H . Let (Y, Z) ∈ D2H × H2H be a solution to the BSDE Z
Yt = X + t
T
b1/2 ϕs + λs Ys + ηstr a s Zs ds −
Z t
T
Zstr dBs + KTP − KtP , t ∈ [0, T ], P a.s., (4.11)
with K P nondecreasing and K0P = 0, for all P ∈ PH . If X ≥ 0 and ϕt ≥ 0, t ∈ [0, T ], P a.s. for all P ∈ PH and, then Yt ≥ 0, t ∈ [0, T ], P a.s. for all P ∈ PH . If in addition Y0 = 0, then X = 0, ϕt = 0 and Yt = 0, t ∈ [0, T ], P a.s. for all P ∈ PH . R
−1/2
bs Proof. Let P ∈ PH and M be defined by Mt := exp 0t ηstr a dBs + P a.s., t ∈ [0, T ]. Applying Itˆo’s product rule between t and T gives
Mt Y t = MT X −
Z t
T
−1
bs 2 ηs )tr dBs + Ms (Zs + Ys a −1
Z t
T
Ms dKsP +
Z t
T
Rt
1 2 0 (λs + 2 ηs  )ds ,
ϕs Ms ds, P a.s.. (4.12)
bs 2 )tr dBs is a P martingale. Indeed, under P one can The process N := 0· Ms (Zs + Ys ηs a Rt 1/2 bs Zs + Ys ηs )tr dWsP , t ∈ [0, T ], where W P is a P Brownian motion. write Nt = 0 Ms (a Hence (BDG) inequality it suffices to show that the P expectation h Rby BurkholderDavisGundy 1/2 i 1/2 T P 2 2 b E is finite. Because λ and η are bounded, Z ∈ H2H and 0 Ms as Zs + Ys ηs  ds R
Section 4.1. Mathematical framework and preliminaries
Page 152
Y ∈ D2H , it follows by BDG inequality that E
P
h Z
T
0
2 b1/2 Ms2 a s Zs + Ys ηs  ds
h
≤ E P sup Ms
Z
T
2 b1/2 a s Zs + Ys ηs  ds
0
s≤T
≤ E P sup Ms2
1/2
EP
1/2
hZ
T
0
s≤T
≤ E P sup Ms2
1/2 i
EP
2 b1/2 a s Zs + Ys ηs  ds T
Z 0
s≤T
1/2 i
2 b1/2 a s Zs  ds
1/2
i1/2
+ kηk∞ T 1/2 E P sup Ys 2
1/2
< ∞,
s≤T
where the second and third inequalities are obtained using H¨older’s and Minkowski’s inequalities respectively. Therefore N is a true P martingale and taking the conditional expectation in (4.12) yields Yt =
Mt−1 EtP MT X
T
Z
+ t
Ms dKsP
T
Z
ϕs Ms ds ,
+ t
P a.s., for all P ∈ PH .
(4.13)
Now if X ≥ 0 and ϕ ≥ 0, then it follows from (4.13) that Y ≥ 0 (since M > 0 and K P is nondecreasing). If moreover Y0 = 0 then E
P
Z
MT X + 0
T
Ms dKsP
Z
T
ϕs Ms ds = 0,
+ 0
for all P ∈ PH .
(4.14)
Finally since the random variable inside the expectation in (4.14) is nonnegative and M > 0, then X = 0, ϕ = 0 and K P = 0. Therefore Y = 0. Note n that the proofoof Lemma 4.11 does not require the minimum condition (4.8) to be satisfied for K P , P ∈ PH . Lemma 4.11 will be used to prove the second comparison Theorem 4.13, the first being the following Proposition 4.12. Let X i be in L2H and F i be the generator associated by (4.3) to a nonlinear function H i (for i = 1, 2) and satisfying (4.5) and Assumption 4.6, (i),(ii),(iv). Let (Y i , Z i ) ∈ D2H × H2H be a solution to the 2BSDE with parameters (F i , X i ), having the representation (4.9). Suppose X1 ≥ X2
and
Fbt1 Yt2,P , Zt2,P ≤ Fbt2 Yt2,P , Zt2,P , P a.s., for all t ∈ [0, T ], P ∈ PH ,
where (Y i,P , Z i,P ) denotes the solution of the standard BSDE with parameters (−Fb i , X i ) under P , for P ∈ PH (for i = 1, 2). Then Yt1 ≥ Yt2 , t ∈ [0, T ], P a.s. for all P ∈ PH . Proof. Applying the comparison principle [EPQ97, Theorem 2.2] for standard BSDEs, one obtains Yt1,P ≥ Yt2,P , P a.s., for all t ∈ [0, T ], for all P ∈ PH . Now for any fixed P ∈ PH ,
Section 4.1. Mathematical framework and preliminaries
Page 153
taking the essential supremum over all P 0 ∈ PH (t+ , P ) yields Yt1 =
P
ess sup
P 0 ∈P
H
(t+ ,P )
Yt1,P
0 ,X 1
≥
P
ess sup
P 0 ∈P
H
(t+ ,P )
Yt2,P
0 ,X 2
= Yt2 ,
P a.s., t ∈ [0, T ],
which proves the required result by using (4.9). If an hypothesis on Fb 1 and Fb 2 as in Proposition 4.12 is satisfied PH quasisurely at (Y 2 , Z 2 ), instead for all Y 2,P , Z 2,P , P ∈ PH , then by imposing an additional monotonicity condition on the differences of the nondecreasing components of the associated 2BSDEs, we obtain the following similar result. Theorem 4.13. Let X i be in L2H and F i be the generator associated to a nonlinear function H i (i = 1, 2) and satisfying (4.5) and Assumption 4.6, (i),(ii),(iv). Let (Y i , Z i ) ∈ D2H × H2H be a solution to the 2BSDE with parameters (F i , X i ). Suppose X 1 ≥ X 2,
Fbt1 (Yt2 , Zt2 ) ≤ Fbt2 (Yt2 , Zt2 ), for all t ∈ [0, T ], P a.s. for all P ∈ PH ,
and K 1,P − K 2,P is nondecreasing for all P ∈ PH , where
n
K i,P , P ∈ PH
o
are the non
(F i , X i ),
decreasing processes associated to the 2BSDEs i = 1, 2. Then the minimum condition 1,P 2,P 1 holds for the family K − K , P ∈ PH , and Yt ≥ Yt2 , t ∈ [0, T ], P a.s. for all P ∈ PH . Proof. Let δY = Y 1 − Y 2 , δZ = Z 1 − Z 2 and δK = K 1 − K 2 . Then using Assumption 4.6, iv) on F 1 and the classical linearization technique, one can construct λ, η two bounded, Fprogressively measurable processes valued in R and Rn respectively such that for all t ∈ [0, T ] it holds P a.s for any P ∈ PH that δYt = (X 1 − X 2 ) +
Z
T
b1/2 δ2 Fbs + λs δYs + ηstr a s δZs ds −
t
Z
T
t
δZstr dBs + δKTP − δKtP ,
where δ2 Fbt = Fbt2 (Yt2 , Zt2 ) − Fbt1 (Yt2 , Zt2 ) ≥ 0, t ∈ [0, T ], P a.s. By assumption, the process n o P 1,P 2,P P δK := K −K is nondecreasing and starts at 0. Moreover, δK , P ∈ PH also satisfies the minimum condition (4.8). Indeed let P ∈ PH and t ∈ [0, T ], then for all P 0 ∈ PH (t+ , P ) holds 0
0
0
0
0
0
0
δKtP = δKtP ≤ EtP [δKTP ] = EtP [KT1,P ] − EtP [KT2,P ], P a.s.. Taking the essential infimum over all P 0 ∈ PH (t+ , P ) on both sides yields δKtP ≤
ess inf
P
P 0 ∈PH (t+ ,P )
0
0
EtP [δKTP ] = ≤
ess inf
P
P 0 ∈PH (t+ ,P )
ess inf
P
P 0 ∈PH (t+ ,P )
0
0
0
0
EtP [KT1,P ] − EtP [KT2,P ] , 0
0
EtP [KT1,P ] −
ess inf
P
P 0 ∈PH (t+ ,P )
P a.s.. 0
0
EtP [KT2,P ],
P a.s.,
Section 4.1. Mathematical framework and preliminaries
Page 154 n
which by the minimum condition on K 1,P , P ∈ PH and K 2,P , P ∈ PH
δKtP ≤
ess inf
P 0 ∈P
H
P
0
(t+ ,P )
0
EtP [δKTP ] ≤ Kt1,P − Kt2,P = δKtP
o
yields
P a.s..
This implies that δK P , P ∈ PH satisfies the minimum condition. By assumptions on F 1 , F 2 it clearly holds δ2 Fb ∈ H2H . Now since δ2 Fb ≥ 0 and X1 − X2 ≥ 0, then Lemma 4.11 implies δY ≥ 0.
The following are direct consequences of Proposition 4.12 and Theorem 4.13, that could be used to describe in terms of 2BSDEs the solution to optimization problems that are stated with respect to mutually singular measures in PH . Corollary 4.14. Let X, X ϑ ∈ L2H and F, F ϑ associated to nonlinear functions H, H ϑ satisfying (4.5) and Assumption 4.6, (i),(ii),(iv), for ϑ in some index set Θ. Let (Y, Z), (Y ϑ , Z ϑ ) in D2H × H2H be solutions to the 2BSDEs with parameters (F, X), (F ϑ , X ϑ ) and having the representation (4.9). Suppose there exists ϑ¯ ∈ Θ such that P
¯
X = ess inf X ϑ = X ϑ , ϑ∈Θ
P a.s., P ∈ PH
P ¯ Fbt (YtP , ZtP ) = ess sup Fbtϑ (YtP , ZtP ) = Fbtϑ (YtP , ZtP ),
and P a.s., t ∈ [0, T ], P ∈ PH ,
ϑ∈Θ
where (Y P , Z P ), (Y ϑ,P , Z ϑ,P ) denote the solutions under P to the standard BSDEs with P
¯
parameters (−Fb , X), (−Fb ϑ , X ϑ ) respectively. Then Yt = ess inf Ytϑ = Ytϑ P a.s., for all ϑ∈Θ
t ∈ [0, T ], P ∈ PH . ¯
Proof. By the hypotheses on the generators, follows Y P = Y ϑ,P , P a.s., P ∈ PH . This ¯ ¯ implies by the representation (4.9) that Y = Y ϑ . Now Proposition 4.12 yields Ytϑ = Yt ≤ Ytϑ , P a.s. for all t ∈ [0, T ], P ∈ PH , for all ϑ ∈ Θ. After taking the essential infimum over all ϑ ∈ Θ, this yields P
P
¯
ess inf Ytϑ ≤ Ytϑ = Yt ≤ ess inf Ytϑ P a.s., for all t ∈ [0, T ], P ∈ PH , ϑ∈Θ
ϑ∈Θ
which is the required result. Corollary 4.15. Let X, X ϑ ∈ L2H and F, F ϑ associated to nonlinear functions H, H ϑ satisfying Assumption 4.6, for ϑ in some index set Θ. Let (Y, Z), (Y ϑ , Z ϑ ) ∈ D2H × H2H be solutions to the 2BSDEs with parameters (F, X), (F ϑ , X ϑ ). Suppose there exists ϑ¯ ∈ Θ such that P
¯
X = ess inf X ϑ = X ϑ P a.s., for all P ∈ PH , ϑ∈Θ
P ¯ Fbt (Yt , Zt ) = ess sup Fbtϑ (Yt , Zt ) = Fbtϑ (Yt , Zt ), P a.s., for all t ∈ [0, T ], P ∈ PH , ϑ∈Θ
Section 4.2. Market model and gooddeal constraint under volatility uncertainty and K ϑ,P − K P is nondecreasing for all ϑ ∈ Θ, P ∈ PH , where n
K ϑ,P , P ∈ PH
o
Page 155
n
K P , P ∈ PH
o
and
are the nondecreasing components of the solutions (Y, Z) and (Y ϑ , Z ϑ )
for the 2BSDEs with parameters (F, X) and (F ϑ , X ϑ ). Then for all t ∈ [0, T ], P ∈ PH , one has P
P
¯
Yt = ess inf Ytϑ = Ytϑ ϑ∈Θ
¯
KtP = ess inf Ktϑ,P = Ktϑ,P , P a.s..
and
ϑ∈Θ
Proof. By the hypotheses on the generators it follows for all P ∈ PH and t ∈ [0, T ] that KtP = Y0 − Yt + = Y0 − Yt +
t
Z 0
0 t
Z 0
¯
Since X = X ϑ and
n
t
Z
Fbs (Ys , Zs )ds +
K P , P ∈ PH
o
¯
Fbsϑ (Ys , Zs )ds +
Zstr dBs , P a.s.
t
Z 0
(4.15)
Zstr dBs , P a.s..
satisfies the minimum condition, then by uniqueness ¯
¯
¯
of solutions to 2BSDEs, (4.15) implies (Y, Z) = (Y ϑ , Z ϑ ). This yields KtP = Ktϑ,P , P a.s., t ∈ [0, T ], for all P ∈ P. Moreover by Theorem 4.13 one obtains from the hypotheses on the generators and the associated nondecreasing processes that Yt ≤ Ytϑ holds P a.s., for all t ∈ [0, T ], P ∈ PH , for all ϑ ∈ Θ. Now taking the essential infimum over all ϑ ∈ Θ yields P
P
¯
ess inf Ytϑ ≤ Ytϑ = Yt ≤ ess inf Ytϑ ϑ∈Θ
ϑ∈Θ
P a.s., for all t ∈ [0, T ], P ∈ PH .
Furthermore, observe that since for all ϑ ∈ Θ the process K ϑ,P − K P is nondecreasing P a.s., then the process
P
ess inf K ϑ,P − K P is also P a.s. nondecreasing and starts at 0 for all ϑ∈Θ
¯
P ∈ PH . In addition, since K P = K ϑ,P , for all P ∈ PH , then the inequalities P
P
¯
0 ≤ ess inf Ktϑ,P − KtP ≤ ess inf Ktϑ,P − K ϑ,P ≤ 0. ϑ∈Θ
ϑ∈Θ
hold.
4.2
Market model and gooddeal constraint under volatility uncertainty
We apply the preceding 2BSDE theory to gooddeal valuation and hedging of contingent claims in incomplete financial markets under volatility uncertainty. Recall (cf. [DM06, DK13a, NS12, EJ13, EJ14, Vor14]) that in the framework of volatility uncertainty, the reference probability measures interpreted as generalized scenarios in the market (cf. [ADE+ 07]) are no longer dominated and may actually be mutually singular. In comparison to standard BSDEs which are
Section 4.2. Market model and gooddeal constraint under volatility uncertainty
Page 156
used in Chapters 3 and 2 in the presence of drift uncertainty or absence of uncertainty at all, 2BSDEs seem to be an appropriate tool for describing worstcase valuations in the presence of volatility uncertainty (see also [MPZ15]). We will characterize worstcase gooddeal bounds and associated robust hedging strategies via solutions to 2BSDEs. As in [CR00, BS06], we consider gooddeal constraints imposed as bounds on the Sharpe ratios (equivalently bounds on the optimal growth rates as in [Bec09]) in the financial market extended by additional wealth processes. First let us specify the model for the market with uncertainty about the volatility.
4.2.1
Financial market with volatility uncertainty
The financial market consists of d tradeable stocks (d ≤ n) with discounted price processes (S i )di=1 = S modelled by dSt = diag(St )σt dBt , t ∈ [0, T ], PH q.s.,
S0 ∈ (0, ∞)d ,
(4.16)
where σ is a Rd×n valued Fpredictable process, each σt being uniformly continuous in ω with respect to k · k∞ . We assume that σσ tr is uniformly bounded and uniformly elliptic, i.e. there exists K, L > 0 such that
K Id ≤ σσ tr ≤ L Id ,
PH ⊗ dtq.s.,
(4.17)
b1/2 is PH ⊗ dtq.s. of maximal rank where Id denotes the d × d identity matrix. In particular σ a tr bt σt is uniformly elliptic and bounded (using (4.1) and (4.17)). d ≤ n, since σt a
Remark 4.16.
1. From (4.17) and (4.1) holds supP ∈PH E P
RT σt a1/2 2 dt < ∞, and 0 n R o (P )
σs dBs , P ∈ PH
hence by [DM06, Lemma 2.4 and Theorem 2.8] the family
of
R·
stochastic integrals can be aggregated into a single process 0 σs dBs that is defined PH quasisurely. In fact under additional assumptions (e.g. c`adl`ag integrands as in [Kar95], or continuum hypothesis as in [Nut12b]) they can even be defined pathwise without exception of a nullset. 2. The market model captures uncertainty about the volatility in the sense that under each 1/2 bt dWtP , where W P is a P Brownian motion. In measure P ∈ PH , one has dBt = a fact, substituting this in the dynamics of S in (4.16) one sees that under the reference b1/2 plays the role of the instantaneous volatility matrix measure P ∈ PH , the process σ a for the stock prices S. In this sense, Knightian uncertainty (ambiguity) about future volatility scenarios is captured by the local martingale laws P ∈ PH for S. 3. The bounds a, a ¯ and the uniform bounds on σσ tr can be viewed as setting a confidence region for future volatility values, calibrated e.g. from extreme implied (or historical) volatilities in the market.
Section 4.2. Market model and gooddeal constraint under volatility uncertainty
Page 157
4. The financial market described is incomplete under any scenario P ∈ PH for the volatility b1/2 if d < n, since σ a b1/2 is of full rank PH ⊗ dtq.s.. σa Let Me (P ) be the set of equivalent local martingale measures of S under each model P , for P ∈ PH . Denoting (P )E(M ) := exp M − M0 − 12 hM iP the stochastic exponential of the local P martingale M under P , we have the following Lemma 4.17. For P ∈ PH , the set Me (P ) consists of the equivalent measures Q ∼ P such 1/2 bt ), t ∈ [0, T ]. that dQ = (P )E(η · W P )dP with η Fprogressively measurable and ηt ∈ Ker (σt a Proof. Let P ∈ PH . By the martingale representation theorem under P (see Lemma 4.2), any Q ∈ Me (P ) satisfies dQ = (P )E(η · W P )dP for a Fprogressive measurable process η such R that 0T ηs 2 ds < ∞, P a.s. holds. By Girsanov theorem, one can rewrite the dynamics of S R 1/2 1/2 bt dWtQ , t ∈ [0, T ], where W Q = W P − 0· ηs ds bt ηt dt + σt a under Q as dSt = diag(St ) σt a is a QBrownian motion by Levy’s characterization. Now Q is a local martingale measure for S 1/2 1/2 bt ηt = 0 for all t ∈ [0, T ], i.e. if and only if ηt ∈ Ker (σt a bt ), t ∈ [0, T ]. if and only if σt a Remark 4.18. 1. Note from (4.16) that the measures P ∈ PH are also local martingale measures for S. This implies that P ∈ Me (P ) 6= ∅, for any P ∈ PH . As a consequence, the market satisfies the nofree lunch with vanishing risk condition (see [DS94]) under each P ∈ PH . This is equivalent to a robust notion for noarbitrage under uncertainty (see [BBKN14]). 2. Modeling the stock price process directly under localmartingale measures (i.e. setting its drift to zero) is a technical assumption rather than financially justified. In our case, this will ensure convexity (in a ∈ S>0 n ) of the generators F of the upcoming pricing and hedging 2BSDEs. This convexity is essential in 2BSDE theory since F is defined by (4.3) as the convex conjugate of a function H. Confer part 1 of Remark 4.21 for further notes about the possible limitations for the applicability of 2BSDE theory if one includes a nonzero drift in (4.16) . We parametrize trading strategies ϕ = (ϕi )di=1 in terms of the amount ϕi of wealth invested in the stock with price process S i , with ϕ being a F+ progressively measurable process with suitable integrability properties. In this respect, the wealth process V ϕ associated to a trading strategy ϕ with initial capital V0 (so that (V0 , ϕ) quasisurely satisfies the selffinancing requirement) has the dynamics Vtϕ = V0 +
Z 0
t
ϕtr s σs dBs ,
t ∈ [0, T ], PH q.s..
Section 4.2. Market model and gooddeal constraint under volatility uncertainty
Page 158
Reparameterizing trading strategies in terms of integrands φ := σ tr ϕ ∈ Im σ tr with respect to B, the dynamics of the wealth process V φ := V ϕ rewrites Vtφ = V0 +
Z 0
t
φtr s dBs = V0 +
t
Z
(P )
0
1
bs2 dWsP , P a.s., t ∈ [0, T ], P ∈ PH . φtr sa
(4.18)
We denote Φ(P ), P ∈ PH the set of trading strategies that are permitted under P (referred to as P permitted), defined as n
Φ(P ) := φ : φ is F prog. meas., E +
P
Z
T
0
o
tr 2 tr b1/2 a , s φs  ds < ∞, and φ ∈ Im σ
with “prog. meas.” abbreviating progressively measurable. We use the following definition of the set of permitted trading strategies. Definition 4.19. The set Φ of permitted trading strategies under volatility uncertainty consists of all F+ progressively measurable processes φ ∈ Im σ tr satisfying sup E P
Z
P ∈PH
0
T
tr 2 b1/2 a s φs  ds < ∞
and such that the family of stochastic integrals single process
n
R (P ) · φtr dB , s 0 s
P ∈ PH
o
aggregates into a
R·
tr 0 φs dBs .
· tr By its quasisure definition, the integral 0· φtr s dBs , for a strategy φ ∈ Φ, satisfies 0 φs dBs = R · (P ) φtr dB , P a.s. for all P ∈ P . The trading strategies in Φ are termed as P permitted s H H 0 s T (or simply permitted). Clearly V φ is a P martingale for any φ ∈ Φ ⊆ P ∈PH Φ(P ) and b1/2 of the P ∈ PH , hence excluding existence of arbitrage strategies in Φ for any scenario σ a volatility.
R
4.2.2
R
Nogooddeal constraint
In the absence of uncertainty, we consider a nogooddeal constraint defined as a bound on the instantaneous Sharpe ratios, for any market extension by additional derivative price processes obtained from the nogooddeal pricing measures (cf. [CR00, BS06] and references therein). This nogooddeal constraint is equivalent to a bound on the optimal expected growth rates of returns, again in any market extension (see [Bec09]). Classically, such can be ensured (using the HansenJagannathan inequality) by imposing a bound on the norm of Girsanov kernels for riskneutral pricing measures. In the presence of drift (rather than volatility) uncertainty, results about gooddeal valuation and robust hedging are provided in Chapter 3. Our aim here is to derive analogs of these results in the presence of volatility uncertainty. The nogooddeal constraint under volatility uncertainty consists of imposing the same bound h on the Girsanov
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 159
kernels of pricing measures in every model P ∈ PH separately. By doing this, we obtain for each P a set of nogooddeal measures Qngd (P ) ⊆ Me (P ). Following a worstcase approach to gooddeal valuation under uncertainty (as in (3.49) in Chapter 3, but taking into account here the possible singularity of the priors P ∈ PH ), this will yield a larger gooddeal bound obtained as the supremum of prices taken over all nogooddeal measures for all reference measures P ∈ PH . To be more precise let h be a fixed positive bounded Fprogressively measurable process that is uniformly continuous in ω with respect to k · k∞ . We consider the set Qngd (P ) of nogooddeal measures in the model P ∈ PH as the subset of Me (P ) consisting of equivalent local martingale measures Q, whose Girsanov kernels η with respect to the P Brownian motion W P are bounded by h, i.e. ηt (ω) ≤ ht (ω) for all (t, ω) ∈ [0, T ] × Ω. In other words, using Lemma 4.17, we define n
Qngd (P ) := Q ∼ P dQ/dP = (P )E η · W P , with Fprog. meas. η o
b1/2 ) and η ≤ h . satisfying η ∈ Ker (σ a
Clearly, for all P ∈ PH holds P ∈ Qngd (P ) 6= ∅. Note that uniform continuity of h and σ will ensure that the forthcoming 2BSDE generators satisfy Assumption 4.6, iii), needed for wellposedness of the associated 2BSDEs (see Theorem 4.9). As in part b) of Lemma 3.1 in Chapter 3, one can show that for P ∈ PH the set Qngd (P ) is convex and multiplicatively stable (in short mstable). Mstability of a set of priors is usually key for obtaining timeconsistency of the corresponding process dynamically defined as essential supremum over conditional expectations over the priors; see [Del06] for the definition and a general study of mstability when the priors are dominated. Mstability is also referred to as rectangularity in the economic literature [CE02].
4.3
Gooddeal bounds and hedging under volatility uncertainty
Using 2BSDEs, we describe gooddeal bounds in the market model of Section 4.2.1 and study an associated notion of robust hedging in the framework of volatility uncertainty. We first define the gooddeal valuation bounds whose financial motivation comes from the nogooddeal restriction mentioned previously. Then we characterize the corresponding gooddeal bounds in terms of solutions to Lipschitz 2BSDEs. After that, we derive hedging strategies as minimizers of some dynamic coherent apriori risk measure ρ under volatility uncertainty (e.g. as in [NS12]), so that the gooddeal bound arises as the market consistent risk measure associated to ρ, in the spirit of [BE09]. Our definition of the gooddeal bounds and hedging strategies will take into account the dependence of the nogooddeal restriction on the prior, and the aversion of investors to volatility uncertainty.
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
4.3.1
Page 160
Gooddeal bounds under volatility uncertainty
As in Chapter 3, Section 3.3, the main idea behind gooddeal valuation under uncertainty is to view aversion to model uncertainty as a penalization of the nogooddeal restriction yielding a larger gooddeal bound than in the absence of uncertainty. We use a worstcase approach to uncertainty aversion in the spirit of [GS89, HS01, CE02]. This approach has been used for example in [ALP95, Lyo95, NS12, Vor14] to study robust arbitrage bounds and superhedging strategies in a financial market with volatility uncertainty or in [SW05, Sch07, Que04, DK13a, MPZ15] for robust utility maximization under model uncertainty. Intuitively, an uncertaintyaverse investor faced with insufficient knowledge about the actual financial market volatility, would opt for a worstcase approach to valuation in order to compensate for eventual losses due to the wrong choice of the volatility. Acting this way, she would sell (resp. buy) financial risks at the largest (resp. smallest) gooddeal bounds over all possible scenarios in her confidence set of volatility values, corresponding to the set PH of reference priors. Acknowledging that mutual singularity of the reference measures in PH brings additional technical difficulties in making rigorous sense of essential suprema, we define the (robust) worstcase gooddeal bound π·u (X) in our dynamic framework for a financial risk X ∈ L2H as the unique process π·u (X) ∈ D2H (if it exists) that satisfies πtu (X) =
P
ess sup
P0
ess sup EtQ [X], t ∈ [0, T ], P a.s., for all P ∈ PH .
P 0 ∈PH (t+ ,P ) Q∈Qngd (P 0 )
(4.19)
The definition of the lower gooddeal bound π·l (X) = −π·u (−X) is analogous, replacing the essential suprema in (4.19) by essential infima; for this reason we focus only on studying the upper bound. For X ∈ L2H , the gooddeal bound π·u (X) will be shown to be a single universal process corresponding to the Y component of the solution of a 2BSDE. Before proceeding, let us introduce some notations that will be used throughout the sequel. a,⊥ a For a ∈ S>0 n , we denote by Πt (·) and Πt (·) respectively the orthogonal projections onto the subspaces Im (σt a1/2 )tr and Ker (σt a1/2 ) of Rn , t ∈ [0, T ]. More precisely for each a ∈ S>0 n n and t ∈ [0, T ], we define the projections of z ∈ R as Πat (z) = (σt a1/2 )tr (σt aσttr )−1 (σt a1/2 )z
a and Πa,⊥ t (z) = z − Πt (z).
(4.20)
b at ,⊥ at b t (·) := Πb b⊥ In particular we define (in a pathwise sense) Π (·). For each t (·) and Πt (·) := Πt 0 + P ∈ PH , t ∈ [0, T ] and P ∈ PH (t , P ), the standard gooddeal bound in the model P 0 is 0 0 given as usual by πtu,P (X) := ess supPQ∈Qngd (P 0 ) EtQ [X], P a.s., so that by (4.19) one has
πtu (X) =
P
ess sup
P 0 ∈PH (t+ ,P )
0
πtu,P (X), P a.s., t ∈ [0, T ], P ∈ PH , for X ∈ L2H . 0
(4.21)
Note from Theorem 3.15 in Chapter 3 that the gooddeal bound π·u,P (X) for P 0 ∈ PH (t+ , P ) and P ∈ PH is the value process of the standard BSDE under P with generator −Fbt (·) =
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
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bt ), t ∈ [0, T ], and terminal condition X, with F given for z ∈ Rn , a ∈ Rn×n by −Ft (·, a −h Πa,⊥ a 12 z t t F (t, z, a) = +∞
if a ∈ S>0 n and a ≤ a ≤ a,
(4.22)
otherwise.
For X ∈ L2H , we consider the 2BSDE Yt = X −
Z t
T
Zstr dBs
−
T
Z
Fbs (Zs )ds + KT − Kt ,
t ∈ [0, T ],
t
PH q.s.,
(4.23)
bt ) for F given by (4.22). Using (4.19) and the representation formula in where Fbt (·) := Ft (·, a Proposition 4.9, we show the following
Theorem 4.20. 1. If X ∈ L2H and (Y, Z) ∈ D2H × H2H is a solution to the 2BSDE (4.23), then the gooddeal bound is uniquely given by πtu (X) = Yt , t ∈ [0, T ], PH q.s. and satisfies πtu (X) = Yt =
P
ess sup
P0
ess sup EtQ [X],
P 0 ∈PH (t+ ,P ) Q∈Qngd (P 0 )
t ∈ [0, T ], P a.s. for all P ∈ PH .
2. For X ∈ L2H , there exists a unique solution (Y, Z) ∈ D2H × H2H to the 2BSDE (4.23). Proof. For z ∈ Rn , t ∈ [0, T ], the generator F (t, z, a) writes explicitly for a ∈ S>0 n ∩ [a, a] as 1/2
F (t, z, a) = −ht z tr a − aσttr (σt aσttr )−1 σt a) z
.
First we need to show that the function F (t, z, ·) : Rn×n → R is convex on its domain DFt = S>0 n ∩ [a, a], from which the FenchelMoreau theorem would imply that F (t, z, ·) is the convex conjugate of a nonlinear function H such that (4.3) holds. For this purpose, it suffices to show that the function Gt : a 7→ aσttr (σt aσttr )−1 σt a is S>0 n convex. Let then µ ∈ [0, 1] >0 and a, a ˜ ∈ Sn . Using the Schur complement condition for positive semidefiniteness [HJ12, Theorem 7.7.7 or Theorem 7.7.16], convexity of Gt is equivalent to positive semidefiniteness of the matrix At ∈ R(n+d)×(n+d) given by
At =
tr
µ aσttr (σt aσttr )−1 σt a + (1 − µ) a ˜σttr (σt a ˜σttr )−1 σt a ˜
σt (µa + (1 − µ)˜ a)
=µ
aσttr (σt aσttr )−1 σt a σt a
=: µA1t + (1 − µ)A2t .
σt (µa + (1 − µ)˜ a)σttr
σt (µa + (1 − µ)˜ a) tr
σt a
σt aσttr
+ (1 − µ)
a ˜σttr (σt a ˜σttr )−1 σt a ˜ ˜ σt a
tr
σt a ˜
σt a ˜σttr
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Now since σt aσttr and σt a ˜σttr are positive definite and the set of positive semidefinite matrices is a convex cone, then the Schur complement condition applied to A1t and A2t implies that At is positive semidefinite. For existence and uniqueness of the solution to the 2BSDE (4.23) we aim to apply part 2 of Proposition 4.9. To this end, we show that F satisfies parts (i)(v) of Assumption 4.6. Part (i) is clear by definition of F in (4.22) and the fact that DFt = S>0 n ∩ [a, a]. As for part (ii), it holds from the progressive measurability of the processes σ and h. To show that part (iii) about uniform continuity of F holds, recall that the pointwise product of two bounded uniformly continuous functions is uniformly continuous, and that the composition of two uniformly continuous functions is also uniformly continuous. With this it follows that F is uniformly continuous in ω with respect to k · k∞ for fixed (t, z, a) ∈ [0, T ] × Rn × Rn×n , since h and σ are uniformly continuous and bounded, σσ tr and σaσ tr are uniformly elliptic and bounded in the matrix norm, and the square root function is uniformly continuous. Since Fb 0 = 0, then part (v) obviously holds. It remains to show part (iv) about the Lipschitz continuity of Fb in z. By the Minkowski inequality one has PH q.s. for all t ∈ [0, T ], that
1
1
1
1
b ⊥ 2 b ⊥ 2 0 0 2 b⊥ a b 2 (z − z 0 ) b t z − Πt a bt z ≤ ht Π Fbt (z) − Fbt (z 0 ) = ht Π t a t bt (z − z ) ≤ khk∞ a t
holds. Hence part (iv) follows and this concludes that F satisfies Assumption 4.6. 0
Part 1: Recall from the discussion preceding the statement of the theorem that π·u,P (X) for bt ), t ∈ P 0 ∈ PH (t+ , P ) and P ∈ PH solves the standard BSDE with generator Fbt (·) = Ft (·, a [0, T ], under P , for F given by (4.22). Part 1 is now a direct consequence of part 1 of Proposition 4.9, and the definition (4.19) of the gooddeal bound π·u (X). Part 2: Direct application of part 2 of Proposition 4.9 gives the claim. Remark 4.21. 1. Were the dynamics (4.16) of the stock price processes rather given by the SDE dSt = diag(St )(bt dt + σt dBt ), with nonzero drift b, a candidate for the generator of the 2BSDE (4.23) would have been by Theorem 3.15 in Chapter 3 given for t ∈ [0, T ], z ∈ Rn , a ∈ Rn×n as F (t, z, a) =
( 2 1/2 a,⊥ 1 1 Πt a2 z ξta tr Πat a 2 z − h2t − ξta +∞
if a ∈ S>0 n ∩ [a, a] otherwise,
(4.24)
with ξta := (σt a1/2 )tr (σt aσttr )−1 bt ∈ Im (σt a1/2 )tr being the market price of risk in a model with volatility σa1/2 . Clearly this involves an additional dependence of F in a ∈ S>0 n ∩ [a, a], for which it becomes very difficult to see whether F is convex or not in a ∈ S>0 n ∩ [a, a]. Indeed a sufficient condition for the convexity of F (t, z, ·) given by (4.24) is that each summand is convex in a ∈ S>0 n ∩ [a, a]. However, the second summand of F is a product of two functions, and would be convex if the two components of the product are convex and either monotone increasing or monotone decreasing functions in
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
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a ∈ S>0 n (cf. [BV04, Exercise 3.32]). But we have not been able to verify these properties and also the Schur complement condition is no longer enough to show convexity of the product. For these technical reasons we have modelled S directly as local martingale measures under P ∈ PH , i.e. with zero drift b = 0 (cf. (4.16) and Remark 4.18). 2. Theorem 4.20 shows in particular that the family of essential supremums in (4.19) indexed by the measures P ∈ PH effectively aggregates into a single process π·u (X). In fact using r.c.p.d. π·u (X) can be constructed without exception of a nullset, for X ∈ UCb (Ω) and then extended by density to X ∈ L2H (see [STZ13]). Moreover [STZ13, Proposition 4.11] implies that π·u (X) is actually Fprogressively measurable, for X ∈ L2H . Hence by the Blumenthal ZeroOne law (cf. Lemma 4.2) π0u (X) is constant and given by π0u (X) = supP ∈PH π0u,P (X). 3. By [STZ13, Proposition 4.7], the gooddeal bound π·u (·) satisfies a dynamic programing principle (recursiveness): for all s ≤ t ≤ T , X ∈ L2H , holds P a.s. for all P ∈ PH that πsu (X) =
P
ess sup
P 0 ∈PH (s+,P )
0
πsu,P (πtu (X)) =
P
ess sup
P0
ess sup EsQ [πtu (X)] = πsu (πtu (X)).
P 0 ∈PH (s+,P ) Q∈Qngd (P 0 )
This is equivalent to a time consistency property of the process π·u (X), for X ∈ L2H . 4. Using Proposition 4.12, it holds analogously to Lemma 3.1 in Chapter 3 (see also [KS07b, Theorem 2.7] or [Bec09, Proposition 2.6]) that the gooddeal bound π·u (X) satisfies the properties of dynamic coherent risk measures (with generalized scenarios consisting of measures that can be associated to volatility uncertainty). In addition by part 2. it is timeconsistent. These facts will be used to define gooddeal hedging in terms of minimization of a risk measure of the type of π·u (·). We refer to [NS12] for a general study of dynamic risk measures under volatility uncertainty. Note that our subsequent results on hedging are, differently from [NS12], not on superhedging. Remark 4.22. We are not able to give more general examples of elements in L2H than those provided in part 3 of Remark 4.10. This is restrictive for financial applications where one would typically be interested in X being contingent claims that have some exponential dependence in + BT and hBiT , e.g. X = K − exp BT − hBiT /2 ∈ L2H in dimension n = 1 modeling a put option with strike K > 0 on a BlackScholes risky asset with uncertain volatility. Clearly, this Markovian claim does not fit into the examples given in part 3 of Remark 4.10. Fortunately for some 2BSDE generators one can sometimes identify the solution to the 2BSDE via PDE arguments, even if X ∈ L2H does not belong to L2H ; cf. e.g. Section 4.3.3.
4.3.2
Robust gooddeal hedging under volatility uncertainty
Our aim now is to define and characterize the gooddeal hedging strategy using solutions to 2BSDEs. Here the objective of the investor is to find a PH permitted trading strategy that
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
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minimizes her residual risk (measured under some risk measure ρ) from any time onward when holding a liability X and trading dynamically in the market. Since the investor (say the seller) requires the premium π·u (X) for X, then she would like the gooddeal valuation to be the minimal capital requirement to make her position acceptable. In this sense, the gooddeal bound would be the market consistent risk measure associated to gooddeal hedging via ρ; cf. [BE09]. The risk to be minimized is measured in terms of a dynamic risk measure compatible with the nogooddeal constraint in the market and the uncertaintyaversion of the investor. The second objective of the investor should be towards robustness (of hedges and valuations) with respect to volatility uncertainty. As in Proposition 3.25 of Chapter 3 we show robustness of the gooddeal hedging strategy as a supermartingale property of its tracking (hedging) error with respect to a class of apriori valuation measures P ngd ⊇ ∪P ∈PH Qngd (P ), i.e. uniformly over all reference models P ∈ PH . Recalling the definition of π·u (X) (for X ∈ L2H ) in (4.19) and previous results on gooddeal valuation and hedging in the absence of model uncertainty (cf. e.g. [Bec09, Theorem 5.4] or Theorem 3.17 in Chapter 3), one has for all P ∈ PH , and P 0 ∈ PH (t+ , P ) P
0
0
πtu,P (X) = ess inf0 ρPt X −
T
Z
φ∈Φ(P )
t
P a.s., t ∈ [0, T ],
φtr s dBs ,
(4.25)
P
where for P ∈ PH we define ρPt (X) := ess sup EtQ [X], P a.s., t ∈ [0, T ] with Q∈P ngd (P )
n
o
P ngd (P ) := Q ∼ P  dQ/dP = (P )E(λ · W P ), λ progressively measurable, λ ≤ h . Here P ngd (P ) is the set of apriori valuation measures equivalent to P which satisfy the nogooddeal restriction under P , but might fail to be local martingale measures for the stock price process S (yet they are with respect to the trivial market with only the riskless asset S 0 ≡ 1). In particular for each P ∈ PH , the set P ngd (P ) is also mstable and convex. This implies that the dynamic coherent risk measure ρP : L2 (P ) → L2 (P, Ft ) is timeconsistent (see e.g. Lemma 3.1 in Chapter 3) satisfying ρP· (X) ≥ π·u,P (X) since P ngd (P ) ⊇ Qngd (P ). Furthermore from (4.21) and (4.25), we have for all t ∈ [0, T ] and P ∈ PH that πtu (X) =
P
P
ess sup
0
ess inf0 ρPt X −
P 0 ∈PH (t+ ,P ) φ∈Φ(P )
Z t
T
φtr s dBs , P a.s..
(4.26)
In addition for X ∈ L2H it can be inferred n from [Bec09, Theorem o 5.4] (see also Theorem 3.17 in P ¯ Chapter 3) that there exists a family φ ∈ Φ(P ), P ∈ PH of trading strategies satisfying πtu (X)
=
P
ess sup
P 0 ∈PH (t+ ,P )
0 ρPt
X−
Z t
T
0
(φ¯Ps )tr dBs , P a.s., for all t ∈ [0, T ], P ∈ PH . (4.27)
Moreover φ¯P is given for P ∈ PH by 1/2 ¯P b b1/2 Z P,X , t ∈ [0, T ], P a.s., bt φ a t = Πt a t t
(4.28)
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 165
where (Y P,X , Z P,X ), P ∈ PH , is the solution to the standard BSDE under P with terminal bt ), t ∈ [0, T ], for F defined in (4.22), satisfying condition X and generator −Fbt (·) = −Ft (·, a u,P P,X Y = π· (X). If PH were a singleton PH = {P }, then for X ∈ L2H = L2 (P ) the strategy P φ¯ would be PH permitted and hence already the solution to the gooddeal hedging problem with the valuation π·u (X) = π·u,P (X) associated to the risk measure ρP . In the present nondominated framework however, the situation is more subtle because the strategies φ¯P and risk measures ρP may be defined only up to a nullset of the associated probability measure P ∈ PH . Since we are looking for a PH permitted hedging strategy, one way is to investigate appropriate conditions under which the family {φ¯P , P ∈ PH } can be aggregated into a single strategy φ¯ ∈ Φ, i.e. φ¯ = φ¯P P ⊗ dta.s., for any P ∈ PH . If this were possible, then (4.27) would write πtu (X) =
P
0
ess sup
P 0 ∈PH (t+ ,P )
= ρt X −
Z t
T
ρPt X −
Z t
T
φ¯tr s dBs , t ∈ [0, T ], P a.s. for all P ∈ PH
φ¯tr s dBs , t ∈ [0, T ], P a.s. for all P ∈ PH ,
where ρ· (X) ∈ D2H is defined for X ∈ L2H as the unique process (if it exists) that satisfies ρt (X) =
P
ess sup
P 0 ∈PH (t+ ,P )
0
ρPt (X), t ∈ [0, T ], P a.s. for all P ∈ PH .
(4.29)
As general conditions for aggregation (see e.g. [STZ11]) can be somewhat restrictive and technical, we will express the hedging strategy in terms of the control component Z of the unique solution (Y, Z) to the 2BSDE (4.23). Note that even in case there exists a worstcase ¯ measure P¯ ∈ PH such that ρ = ρP , it is not not clear at all whether a hedging strategy in the model P¯ is robust with respect to all measures in P ngd (P ) for any P ∈ PH , in the sense that the supermartingale property of tracking errors holds uniformly under any Q ∈ ∪P ∈PH P ngd (P ). An analogous issue was already noticed in Subsection 3.3.4 of Chapter 3 under drift uncertainty. The issue was addressed there by first considering a larger valuation bound for which a robust hedging strategy uniformly with respect to all priors exists, i.e. a strategy that satisfies a supermartingale property of tracking error under all measures apriori valuation measure uniformly over all priors. A subsequent step was then to identify this larger bound with the standard gooddeal valuation bound. Here relying on the intuition from Theorem 3.28 and Theorem 3.30 in Chapter 3, we can write down what a candidate hedging strategy (cf. (4.34)) in our setup in terms of the solution to the 2BSDE (4.23). From this we can then proceed in a more straightforward manner to show directly that this candidate strategy is indeed a gooddeal hedging strategy and that it satisfies the required robustness property with respect to uncertainty. Clearly ρ is a dynamic coherent risk measure analogous to π·u (X). The gooddeal hedging problem under volatility uncertainty consists in minimizing over PH permitted trading strategies
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 166
the dynamic residual risk measured under ρ. This is done in such a way that at every time the minimal capital required for acceptability coincides with the gooddeal valuation bound. More precisely for a contingent claim X ∈ L2H , we aim to find φ¯ ∈ Φ such that for all t ∈ [0, T ] and P ∈ PH holds πtu (X)
P
= ess inf ρt X −
Z
φ∈Φ
t
T
φtr s dBs
= ρt X −
Z
T
t
φ¯tr s dBs , P a.s..
(4.30)
To introduce the notion of robustness with respect to volatility uncertainty, recall the definition of the tracking error Rφ (X) of a permitted strategy φ ∈ Φ for a claim X ∈ L2H : Rtφ = πtu (X) − π0u (X) −
t
Z 0
φtr s dBs ,
t ∈ [0, T ], P a.s. for all P ∈ PH .
(4.31)
In other words, the tracking error is the difference between the dynamic variations in the capital requirement and the profit or loss from trading. As in Subsection 3.3.3 of Chapter 3, ¯ we will say that a gooddeal hedging strategy φ(X) for a claim X is robust with respect to ¯ φ uncertainty if R (X) is a supermartingale under every measure Q ∈ P ngd (P ) uniformly for all P ∈ PH . Again as in Chapter 3, this means that a robust hedging strategy φ¯ is at least meanselffinancing uniformly over all Q ∈ ∪P ∈PH P ngd (P ). Let us make a short transit and provide a 2BSDE description of the risk measure ρ· (X), for X ∈ L2H . As in part 4. of Remark 4.21, this yields in particular timeconsistency of the dynamic risk measure ρ over contingent claims X in L2H . For this purpose, define the function F 0 : Ω × [0, T ] × Rn × Rn×n → R by −h a1/2 z, t F 0 (t, z, a) = +∞
if a ∈ S>0 n ∩ [a, a], otherwise.
(4.32)
Consider the 2BSDE Yt0 = X −
Z t
T
tr
Z 0 s dBs −
Z
T
t
Fbs0 (Zs0 )ds + KT0 − Kt0 ,
t ∈ [0, T ], PH q.s.,
(4.33)
with generator F 0 defined in (4.32). Proposition 4.23. 1. If X ∈ L2H and (Y 0 , Z 0 ) ∈ D2H × H2H is a solution to the 2BSDE (4.33), then ρ· (X) is uniquely given by ρt (X) = Yt0 , t ∈ [0, T ], PH q.s. and satisfies ρt (X) = Yt0 =
P
ess sup
P0
ess sup EtQ [X],
P 0 ∈PH (t+ ,P ) Q∈P ngd (P 0 )
t ∈ [0, T ], P a.s. for all P ∈ PH .
2. For X ∈ L2H , there exists a unique solution (Y 0 , Z 0 ) ∈ D2H × H2H to the 2BSDE (4.33).
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 167
0 tr 1/2 and using again the Proof. Rewriting F 0 on a ∈ S>0 n ∩ [a, a] as F (t, z, a) = −ht (z az) Schur complement condition (see proof of Theorem 4.20), one also proves that that F 0 is a convex function of a for fixed (t, z) ∈ [0, T ] × Rn . This by FenchelMoreau theorem implies that F 0 is the convex conjugate of a nonlinear function H 0 such that an analog of (4.3) holds. In addition, it is easy to verify as in the proof of Theorem 4.20 that F 0 satisfies Assumption 4.6.
Part 1: By [Bec09], it is known that Y˜tP,X = ρPt (X), P a.s., t ∈ [0, T ], where (Y˜ P,X , Z˜P,X ) denotes the unique solution to the standard Lipschitz BSDE under P with generator −Fb 0 and terminal condition X, for P ∈ PH . Hence part 1 is also a direct consequence of part 1 of Proposition 4.9 and the definition of ρ in (4.29). Part 2: As a consequence of part 2 of Proposition 4.9, for X ∈ L2H the 2BSDE (4.33) admits a unique solution (Y 0 , Z 0 ) ∈ D2H × H2H . We characterize φ¯ in terms of the unique solution (Y, Z) of the 2BSDE (4.23) and show that it is robust with respect to volatility uncertainty. Using the intuition from robust hedging in the presence of drift uncertainty (see Theorem 3.28 and Theorem 3.30 in Chapter 3), a candidate ¯ gooddeal hedging strategy for X ∈ L2H is φ¯ := φ(X) defined by 1/2 ¯ b b1/2 Zt ), bt φ a t := Πt (a t
t ∈ [0, T ], PH q.s.,
(4.34)
where (Y, Z) is a solution to the 2BSDE (4.23). Since Z is already defined PH quasisurely ¯ in (4.34) is also defined PH quasisurely and it b is defined pathwise, then the strategy φ and a can be shown it is indeed a robust gooddeal hedging strategy if the “gain/loss” family of n that o R · tr (P ) processes 0 Zt dBt , P ∈ PH aggregates. The precise result is the following 2 is a solution to the 2BSDE (4.23) Theorem 4.24. Assume X ∈ L2H and that (Y, Z) ∈ D2H ×H nH o
with generator F given by (4.22) such that the integrals into a single process
R·
tr 0 Zt dBt
R (P ) · Z tr dB , t 0 t
P ∈ PH , aggregate
(equivalently 2BSDE (4.23) admits a solution (Y, Z, K)). Then:
¯ 1. The strategy φ¯ = φ(X) given by (4.34) is in Φ and solves the gooddeal hedging problem under uncertainty (4.30). ¯
¯ 2. The tracking error process Rφ (X) of the hedging strategy φ¯ = φ(X) is a supermartingale ngd under any Q in ∪P ∈PH P (P ). Proof. We first prove part 2., since the proof of part 1. will use it. By Theorem 4.20, we know that π·u (X) = Y for (Y, Z) solution to the 2BSDE (4.23). Let P ∈ PH and Q ∈ P ngd (P ). Then ¯ ¯ Q is equivalent to P and dQ = (P )E(λ · W P )dP for λ ≤ h. The dynamics of Rφ := Rφ (X)
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 168
is then given under P by ¯ P −dRtφ = −Fbt (Zt )dt − Zttr dBt + φ¯tr t dBt + dKt ,
P a.s.,
1 2
bt dWtP + dKtP , = −Fbt (Zt )dt − (Zt − φ¯t )tr a
P a.s.,
for all t ∈ [0, T ], with {K P , P ∈ PH } the nondecreasing adapted processes defined as in R (4.7). Changing measures to Q for the QBrownian motion W Q = W P − 0· λt dt gives for t ∈ [0, T ] that ¯
1
1
¯t ))dt − (Zt − φ¯t )tr a bt2 dWtQ + dKtP , bt2 (Zt − φ −dRtφ = (−Fbt (Zt ) − λtr t a
P a.s.
holds. Now with (4.22) and the expression of φ¯ in (4.34) one rewrites P a.s. for t ∈ [0, T ] ¯ −dRtφ
1 1 ⊥ 1 tr b ⊥ 2 2 b = ht Πt (b at Zt ) − λt Πt (b at Zt ) dt − (Zt − φ¯t )tr b at2 dWtQ + dKtP . 1
1
2 b ⊥ b 2 Zt ) = ht Π b ⊥ (a Since K P is nondecreasing and maxλt ≤ht λtr t Πt (a t bt Zt ) , then the finite t ¯ ¯ variation part of the Qsemimartingale Rφ is nonincreasing. Note that Rφ ∈ S 2 (P ) since p π·u (X) ∈ D2H ⊂ S 2 (P ) and φ¯ ∈ Φ(P ). Finally, since λ is bounded, dQ dP is in L (P ) for ¯ any p < ∞ and by H¨older’s inequality it follows that Rφ ∈ S 2−ε (Q) (ε > 0) holds. As a ¯ consequence Rφ is clearly a Qsupermartingale.
Now to prove part 1. note first that by the condition on the integral Z, the strategy φ¯ given by (4.34) belongs to Φ. Now to show that φ¯ solves the hedging problem (4.30), let P ∈ PH , and P 0 ∈ PH (t+ , P ). Then for any φ ∈ Φ it holds φ · B is a Qmartingale in S 1 (Q) for any Q ∈ Qngd (P 0 ) since the Girsanov kernels of measures Q with respect to P 0 are all uniformly R 0 0 bounded. Because Qngd (P 0 ) ⊆ P ngd (P 0 ), this implies that πtu,P (X) = πtu,P (X− tT φtr s dBs ) ≤ R T tr 0 P 0 0 + ρt (X − t φs dBs ), P a.s.. Taking the essential supremum over P ∈ PH (t , P ) first and R then the essential infimum over φ ∈ Φ yields πtu (X) ≤ ess infPφ∈Φ ρt (X − tT φtr s dBs ), P a.s.. ¯ Hence to show that φ is a gooddeal hedging strategy satisfying (4.30), it suffices to show R ngd (P 0 ) and P 0 ∈ P (t+ , P ). that πtu (X) ≥ EtQ X − tT φ¯tr H s dBs , P a.s. holds for all Q ∈ P 0 + ngd 0 To this end, let P ∈ PH (t , P ) and Q ∈ P (P ). From part 1. of the theorem, the ¯ ¯ supermartingale property of the tracking error Rφ := R·φ (X) of φ¯ under Q implies that R R Q T ¯tr u πtu (X) − π0u (X) − 0t φ¯tr s dBs ≥ Et X − π0 (X) − 0 φs dBs . Reorganizing the last inequality yields the claim. In general, the Itˆo’s stochastic integrals of the form 0· Zttr dBt are only defined P almost surely under each P ∈ PH . As already mentioned before, sufficient conditions for aggregation of R processes can be quite restrictive. By a result of [Kar95] it is possible to define 0· Zttr dBt pathwise, and in particular such that it satisfies the hypothesis of Theorem 4.24, if the process Z is c`adl`ag. Note that in our setup the Zcomponent of a 2BSDE solution (Y, Z) does not have R
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 169
to be a c`adl`ag process in general. We emphasize however that this does not make Theorem 4.24 totally inapplicable. Indeed under some Markovian assumptions one may sometimes be able to use PDE arguments to show that the Zcomponent is c`adl`ag. An example of such a situation is provided in Section 4.3.3 below, where we obtain explicit solutions to the 2BSDE (4.23), for some contingent claim satisfying X ∈ L2H and probably not X ∈ L2H . In a general context, a R result of [Nut12b] shows that the stochastic integral 0· Zttr dBt can be defined pathwise for any predictable process Z if one complements the ZermeloFraenkel axioms of set theory (which are by now wellaccepted) with the combination “continuum hypothesis plus the axiom of choice” or the softer one “negation of the continuum hypothesis plus the socalled Martin’s axiom” (see [DM78, Chapter II, Sections 2729]). Note under the conditions of [Nut12b] that for a solution (Y, Z) of a 2BSDE for which Y is PH quasisurely defined, the family {K P , P ∈ PH } will automatically aggregate into a single process K such that (Y, Z, K) becomes a solution to the 2BSDE. As for the 2BSDE (4.23) of interest in Theorem 4.24, we already know by Theorem 4.20 that this would be the case if X ∈ L2H . As a further remark, note that Part 2. of Theorem 4.24 can be interpreted as a robustness property of the gooddeal hedging strategy φ¯ with respect to volatility uncertainty. Finally, a direct consequence of Theorem 4.24 (when its conditions are satisfied) is the following minmax identity: for all t ∈ [0, T ] , P ∈ PH one has by (4.26) and (4.30) that π ¯tu (X)
P
:= ess inf φ∈Φ
=
4.3.3
P
ess sup
P 0 ∈PH (t+ ,P ) P
ess sup
P
0 ρPt
0
X−
ess inf0 ρPt X −
P 0 ∈PH (t+ ,P ) φ∈Φ(P )
T
Z t
Z t
T
φtr s dBs
u φtr s dBs = πt (X), P a.s.,
Example for options on nontraded assets
We provide an example for robust gooddeal valuation and hedging of European put options on a nontraded asset under volatility uncertainty. The financial market consists of a traded stock of BlackScholes’ type with (discounted) price process S and a nontraded asset with value process L. Hence d = 1 and n = 2 for the framework of Section 4.2. For the canonical process B = (B 1 , B 2 ), the set PH of local martingale measures is defined as in (4.1) via constant diagonal matrices a, a ∈ S>0 2 given by a = diag(a1 , a2 ) and a = diag(a1 , a2 ), such b ≤ a, PH ⊗ dtq.s.. We model (S, L) as that a ≤ a dSt = St σ S dBt1
and dLt = Lt γdt + β(ρdBt1 +
q
1 − ρ2 dBt2 ) ,
PH q.s.,
with S0 , L0 > 0, a volatility matrix σ := (σ S , 0) ∈ R1×2 of maximal rank 1 = d < n = 2, σ S , β ∈ (0, ∞), γ ∈ R, and P 0 correlation coefficient ρ ∈ [−1, 1] for returns of S and L. For a constant bound h ∈ [0, ∞) on the instantaneous Sharpe ratios, we derive explicit formulas
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 170
for the worstcase gooddeal valuation bound and robust hedging strategy of European put options X = (K − LT )+ , with strike K ∈ R+ and maturity T . We denote !
b11 a b12 a b12 a b22 a
b= a
1 2
b = and a
b b11
b b12
b b12
b b22
!
,
b being the S>0 bt dt pathwise and a ≤ a b ≤ a, PH ⊗dtq.s. for a 2 valued process satisfying dhBit = a as in (4.1). One has b11 = (b a b11 )2 + (bb12 )2 ,
b12 = b a b12 (bb11 + bb22 ),
b22 = (b a b22 )2 + (bb12 )2 ,
(4.35)
2 b11 a b22 − (a b12 )2 = b and a b11bb22 − (bb12 )2 . 1
b 2 = σ S (b Since σ a b11 , bb12 ), then from their respective definitions hold 1 b 2 )tr = z ∈ R2 : b Im (σ a b12 z1 − bb11 z2 = 0
1 b 2 ) = z ∈ R2 : b and Ker (σ a b11 z1 + bb12 z2 = 0 ,
which imply for z ∈ R2 that !
(bb11 )2 z1 + bb11bb12 z2 b b11bb12 z1 + (bb12 )2 z2
1 b Π(z) = 11 b a
(bb12 )2 z1 − bb11bb12 z2 (bb11 )2 z2 − bb11bb12 z1
b ⊥ (z) = 1 and Π b11 a
!
.
(4.36)
˜ there satisfies Clearly LT ∈ L2H follows from the estimate (4.44) below, since the process L P ˜ Et [L] ≤ 1 for any t ∈ [0, T ], P ∈ PH . In addition since the put option payoff function x 7→ (K − x)+ is bounded and Lipschitz continuous, it follows that X = (K − LT )+ ∈ L2H . Recall from (4.21) that the worstcase gooddeal bound πtu (X) (if it exists) for X for t ∈ [0, T ] satisfies for any P ∈ PH πtu (X) =
P
ess sup
P 0 ∈PH (t+ ,P )
0
πtu,P (X) =
P
ess sup
P
ess sup EtQ [X], P a.s..
P 0 ∈PH (t+ ,P ) Q∈Qngd (P 0 )
b⊥ a b11 a b22 − (a b1/2 z = a b12 )2 From (4.36) and using (4.35), follows Π solution to the 2BSDE (4.23) which rewrites here as
Yt = X −
Z
T
Fb (s, Zs )ds −
t
Z t
T
1/2
b11 a
−1/2 z2 . is a
Zstr dBs + KT − Kt , t ∈ [0, T ], PH q.s.,
(4.37)
1/2
−1/2
z2 , with generator given by F (t, z, a) = −h Πa,⊥ a1/2 z = −h a11 a22 − (a12 )2 a11 >0 tr 2 for a ∈ S2 ∩ [a, a] and F (t, z, a) = +∞ otherwise, for z = (z1 , z2 ) ∈ R . We show in Lemma 4.25 below that the solution to the BSDE (4.37) is given by
Yt = v(t, Lt )
and Zt = βLt
q tr ∂v (t, Lt ) ρ, 1 − ρ2 , PH q.s. ∂x
hold for every t ∈ [0, T ], where v ∈ C 1,2 [0, T ) × (0, ∞), R is the classical solution to the BlackScholes’ type PDE ( p √ ∂v 1 2 2 ∂v ∂2v + γ − hβ 1 − ρ2 a2 x ∂x + 2 β ρ a1 + (1 − ρ2 )a2 ) x2 ∂x 2 = 0 ∂t (4.38) v(T, LT ) = (K − LT )+ .
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 171
We need some preparations towards this result. Let P a = P 0 ◦ (a1/2 B)−1 ∈ PH be the local martingale measure satisfying hBit = at, P a a.s., for all t ∈ [0, T ]. The dynamics of the process L under P a is the geometric Brownian motion q √ √ a a dLt = Lt γdt + β ρ a1 dWt1,P + 1 − ρ2 a2 dWt2,P , t ∈ [0, T ], a which can be rewritten as dLt = Lt γdt + β¯ ρ¯dWt1,P +
β¯ := β ρ2 a1 + (1 − ρ2 )a2
1/2
>0
1 − ρ¯2 dWt2,P
p
a
, t ∈ [0, T ], for
−1/2 √ and ρ¯ := ρ a1 ρ2 a1 + (1 − ρ2 )a2 ∈ [−1, 1],
where W P = (W 1,P , W 2,P ) = (a) −1/2 B is a P a Brownian motion. The BlackScholes formula applied for the dynamics of L under P a provides a closedform expression for v(t, Lt ), for v solution to the PDE (4.38). Using arguments analogous to the ones in the derivations of (3.31) in Section 3.2.2 of Chapter 3 it can be shown that v(t, Lt ) coincides with the gooddeal a
a
a
a
valuation bound πtu,P (K − LT )+ in the model under P a . Furthermore, an explicit formula for both is given by K
a
v(t, Lt ) = πtu,P (X) (4.39)
= KN (−d− ) − Lt em(T −t) N (−d+ ) =em(T −t) ∗ B/Sputprice time: t, spot price: Lt , strike: Ke−m(T −t) , vol: β¯ ,
with “ B/Sputprice” being the standard BlackScholes formula for interest rate being zero, with “vol” being the argument for volatility in the BlackScholes model, where q q √ 2 ¯ m := γ − hβ 1 − ρ¯ = γ − hβ 1 − ρ2 a2 , −1 d± := ln Lt /K + m ± 12 β¯2 (T − t) β¯ (T − t) and N is the cumulative distribution function of the standard normal law. The following lemma identifies the solution to the 2BSDE (4.37) via the solution v of the PDE (4.38). p
Lemma 4.25. The solution (Y, Z, K) of the 2BSDE (4.37) is given by Yt = v(t, Lt ), Zt = p tr ∂v βLt ∂x (t, Lt ) ρ, 1 − ρ2 and K given by (4.41), for t ∈ [0, T ], with (Y, Z) ∈ D2H × H2H R such that the stochastic integral 0· Zttr dBt is pathwise defined. Proof. For any P ∈ PH , applying Itˆo’s formula and using (4.38) yields for t ∈ [0, T ] v(t, Lt ) = X −
Z t
T
Zstr dBs
Z
+h t
T
2 b11 b22 b12 a s a s − (a s )
1/2
b11 a s
−1/2 2 Z ds + KT − Kt , P a.s., s
with Z = (Z 1 , Z 2 )tr given from (4.39) by q q tr tr ∂v m(T −t) 2 Zt = βLt (t, Lt ) ρ, 1 − ρ = −βe N (−d+ )Lt ρ, 1 − ρ2 ∂x
(4.40)
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 172
and Z t
Kt :=
√ ∂v 22 12 2 1/2 11 −1/2 b11 b b b (4.41) a2 (s, Ls ) a a − ( a ) a − s s s s ∂x p 1 2 2 ∂2v 2 11 2 22 12 bs ) + (1 − ρ )(a2 − a bs ) − 2ρ 1 − ρa bs + β Ls 2 (s, Ls ) ρ (a1 − a ds. 2 ∂x q
hβ 1 − ρ2 Ls
0
b ≤ a P ⊗ dta.s. yields a b1/2 ≤ To show that K is a nondecreasing process, note that a a1/2 P ⊗ dta.s. and both inequalities imply that b11 ) + (1 − ρ2 )(a2 − a b22 ) − 2ρ 1 − ρ a b12 ρ2 (a1 − a p
= ρ,
q
1 − ρ2
tr
a ρ,
q
1 − ρ2 − ρ,
and b11 a b22 − (a b12 )2 a
1/2
b11 a
q
−1/2
1 − ρ2
tr
b22 ≤ a
b ρ, a
1/2
≤
q
1 − ρ2 ≥ 0
√ a2
(4.42)
(4.43)
hold P ⊗ dtalmost surely. Thus the process K is P a.s. nondecreasing, because the delta of the put option in the BlackScholes model is nonpositive and the gamma is nonnegative, ∂v ∂2v i.e. ∂x (t, Lt ) ≤ 0 and ∂x 2 (t, Lt ) ≥ 0 for all t ∈ [0, T ] using (4.39). Moreover the process K satisfies the minimum condition (4.8). This can be shown following arguments analogous to those in the proof of [STZ12, Theorem 5.3]; we reproduce the arguments for the convenience of the reader. Indeed, let us define l : (0, ∞) × R2 → R basically as the generator function (minus the γterm) of the PDE (4.38) defined for (x, p, q) ∈ R+ × R2 by q √ 1 l(x, p, q) := −hβ 1 − ρ2 a2 xp + β 2 ρ2 a1 + (1 − ρ2 )a2 ) x2 q, 2
so that K =
R·
kt = l Lt ,
0 ks ds
holds with
p 1 ∂v 2 22 12 b b b11 b + (1 − ρ ) a + 2ρ (t, Lt ), Γt − β 2 L2t ρ2 a 1 − ρ a t t t Γt + F (t, Zt ), ∂x 2
∂v ∂2v (t, Lt ) and Zt given by (4.40). Since l Lt , ∂x (t, Lt ), Γt is by (4.42) and (4.43) the ∂x2 √ 1 2 2 2 11 2 22 12 supremum of 2 β Lt ρ at + (1 − ρ )at + 2ρ 1 − ρat Γt − Fb (t, Zt ) over a ∈ DF := [a, a], then by measurable selection arguments there exists for every > 0 a predictable process a
for Γt =
valued in DF such that l Lt ,
p 1 ∂v + (1 − ρ2 )a,22 + 2ρ 1 − ρa,12 Γt (t, Lt ), Γt ≤ β 2 L2t ρ2 a,11 t t t ∂x 2 − F (t, Zt , at ) + .
Now let P α ∈ PH and t0 ∈ [0, T ] be fixed, and define recursively the sequence (τn )n of random times τ0 := inf{t ≥ t0 : Kt ≥ Kt0 + } ∧ T , and n
τn+1 := inf t ≥ τn :
l Lt ,
∂v (t, Lt ), Γt + F (t, Zt , aτn ) ≥ ∂x
o p 1 2 2 2 ,11 ,12 β Lt ρ aτn + (1 − ρ2 )a,22 + 2ρ 1 − ρa Γ + 2 ∧ T. t τn τn 2
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 173
Since K, L, Z, Γ are continuous, then τn is a Fstopping time for any n, and τ0 > t0 . Furthermore since l is continuous and F (·, a) also is for fixed a in DF , then for PH quasi all ω the function l Lt ,
p 1 ∂v ,12 2 ,22 + 2ρ (t, Lt ), Γt − β 2 L2t ρ2 a,11 + (1 − ρ )a 1 − ρa Γt + F (t, Zt , aτn ) τn τn τn ∂x 2
is uniformly continuous in t on the compact interval [τn , T ]. Hence uniformly over n holds (ω) − τ (ω) ≥ δ(, ω) > 0 whenever {τ (ω) < T }, which implies τ (ω) = T for large τn+1 n n n enough n. Now from the arguments in [STZ11, Example 4.5] applied to the interval [τ0 , T ], there exists a Fprogressively measurable process α valued in DF such that α = α on [0, τ0 ]
b= and a
∞ X
α aτn 1[τn ,τn+1 ⊗ dta.s. on Ω × [τ0 , T ]. ), P
n=0
It follows that k ≤ 2, P α ⊗ dta.s. on Ω × [τ0 , T ], which implies for P := P α ∈ PH that 0≤
P
α
0
ess inf
P 0 ∈PH (t0 +,P )
EtP0 [KT − Kt0 ] ≤ + EtP0 [KT − Kτ0 ] ≤ + 2(T − t0 ), P a.s.,
since P α ∈ P(t0 +, P ) because τ0 > t0 . Taking the limit as tends to zero yields that K satisfies the minimum condition (4.8). It remains to show that v(·, L· ) ∈ D2H and Z ∈ H2H . This will conclude by uniqueness of the solution to the 2BSDE (4.23) (see Theorem 4.20) that (v(·, L· ), Z) for Z given in (4.40) is the unique solution to the 2BSDE (4.37). Since v is of class C 1,2 and L is PH q.s. continuous, then v(·, L· ) and Z are F+ progressively measurable. That v(·, L· ) is in D2H now follows from (4.39) which indeed implies that 0 ≤ v(t, Lt ) ≤ K holds pathwise. From (4.40) and since 1/2 2 b ≤ a holds P a.s. for any P ∈ PH , one has a bt Zt ≤ max(a1 , a2 )β 2 e2mT L2t for a ≤ a all t ∈ [0, T ], P a.s. for any P ∈ PH . Hence to show Z ∈ H2H it suffices to show that R supP ∈PH E P 0T L2t dt < ∞. For this purpose, note that for any P ∈ PH it holds that T
Z 0
L2t dt
≤β
−2
−1
(min(a1 , a2 ))
hLiT
and
L2T
≤
2 ˜T L20 e 2γ+β max(a1 ,a2 ) T L
(4.44)
˜ satisfying L ˜ = 1 + · 2L ˜ s β ρBs1 + 1 − ρ2 Bs2 , PH q.s.. Clearly P almost surely, for L 0 ˜ T ] ≤ 1 holds for every P ∈ PH , and thus taking expectations in (4.44) gives E P [L R
E
P
Z 0
T
L2t dt
≤β
−2
p
2 (min(a1 , a2 ))−1 L20 e 2γ+β max(a1 ,a2 ) T ,
for all P ∈ PH .
Now taking the supremum over P ∈ PH implies the result. So (v(·, L· ), Z) is the unique R solution to the 2BSDE (4.37) in D2H × H2H . Finally that 0· Zttr dBt is pathwise defined follows from [Kar95] since Z is continuous and F+ adapted.
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 174
Lemma 4.25 implies that πtu (K − LT )+ = v(t, Lt )
and Zt = Zt1 , Zt2
tr
= βLt
tr q ∂v (t, Lt ) ρ, 1 − ρ2 , t ∈ [0, T ]. ∂x
Hence the robust gooddeal bound πtu (K − LT )+ is attained for the largest “volatility matrix” a, and can be computed as in the absence of uncertainty, but under a worstcase a measure P a ∈ PH for which hBit = at P a a.s., yielding πtu (K − LT )+ = πtu,P (K − LT )+ for t ∈ [0, T ]. In addition, πtu (K − LT )+ is given explicitly by the BlackScholes type formula (4.39), for modified strike price and volatility corresponding to K exp(−m(T − t)) and 1/2 respectively. Similarly, one can show that the lower gooddeal β¯ = β ρ2 a1 + (1 − ρ2 )a2 l + bound πt (K − LT ) can be computed as in the absence of uncertainty, but under the a worstcase measure P ∈ PH corresponding to the lowest “volatility matrix” a. Furthermore, ¯ the robust gooddeal hedging strategy φ¯ := φ(X) for the put option X = (K − LT )+ is given 1/2 ¯ b b1/2 Zt ), for Z = Z 1 , Z 2 tr given by (4.40), i.e. bt φ by a t = Πt (a t
−1/2 b 1/2 bt bt ρ, φ¯t = −βem(T −t) N (−d+ )Lt a Πt a
q
1 − ρ2
tr
,
for all t ∈ [0, T ], PH q.s..
Now for a vector z = (z1 , z2 )tr ∈ R2 , straightforward calculations using (4.36) and (4.35) imply b a
−1/2 b
1/2
b Π a
1
b b22
b b11bb22 − (bb12 )2
−bb12
z =
· = =
! b b12 (bb11 )2 + bb11bb12bb22 z2 b b11 (bb12 )2 + bb22 (bb12 )2 z2 11 ! b b z1 + a b12 z2 b11bb22 − (bb12 )2 a
(bb11 )2 + (bb12 )2 1 b b11bb22 − (bb12 )2 ! b a12 z1 + b z 2 11 a .
!
(bb11 )3 + bb11 (bb12 )2 z1 + b b12 (bb11 )2 + (bb12 )3 z1 +
1
b11 a
−bb12 b b11
0
0
Hence an explicit formula for φ¯t is q tr b12 a t m(T −t) ¯ φt = −βe N (−d+ )Lt ρ + 11 1 − ρ2 , 0 , for all t ∈ [0, T ], PH q.s.. bt a
(4.45)
As the optimal growth rate bound h tends to infinity, the gooddeal bound π·u (X) increases towards the robust upper noarbitrage bound under volatility as studied in [ALP95, Lyo95, DM06, NS12, Vor14]. The put option X = (K − LT )+ being a claim with convex payoff function, our result agrees with those of [ALP95, Lyo95, EJPS98, Vor14] according to which in the presence of volatility uncertainty, noarbitrage valuation of put options under maximal (resp. minimal) volatility corresponds to the worstcase for the seller (resp. buyer). The latter works focus on the robust superreplication problem under volatility uncertainty for valuation
Section 4.3. Gooddeal bounds and hedging under volatility uncertainty
Page 175
with respect to the worstcase noarbitrage bound. Here we instead study the robust gooddeal hedging problem under volatility uncertainty for valuation with respect to the worstcase gooddeal bound. Let us also mention that [ALP95, EJPS98, Vor14] work in a onedimensional model with a single risky asset and obtain as superreplicating strategy the delta of the option under the worstcase measure. This is included in our case study as a special case for ρ = 1. In a generalization towards a twodimensional model, we consider possibly nonperfectly correlated (traded and nontraded) risky assets and derive a robust gooddeal hedging strategy for the worstcase gooddeal valuation in a market that is possibly incomplete under each fixed prior P ∈ PH . Furthermore the robust gooddeal hedging strategy φ¯ here is not (the risky asset component of) the superreplicating strategy, in particular, when 0 < ρ < 1. Indeed since 0 ≤ X ≤ K holds pathwise, the noarbitrage bound process Vb (X) under P a defined by Pa
Vbt (X) := ess sup EtQ [X], t ∈ [0, T ], P a a.s. Q∈Me (P a )
a
a
a
satisfies πtu,P (X; h) ≤ Vbt (X) ≤ K, P a a.s. for πtu,P (X; h) given by πtu,P (X) in (4.39). In a
addition if ρ < 1 then πtu,P (X; h) % K as h % +∞ (since then m → −∞, d± → −∞). These imply that if ρ < 1 then Vbt (X) = K1{t