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Introduction: credit risk under incomplete information. 2. Pricing and hedging credit derivatives via nonlinear filtering: the. [Frey et al., 2007] model. Main ideas:.
Pricing and Hedging of Credit Derivatives via Nonlinear Filtering R¨ udiger Frey Universit¨at Leipzig May 2008 [email protected] www.math.uni-leipzig.de/~frey based on work with T. Schmidt, W. Runggaldier, H. M¨ uhlichen and A. Gabih

Overview 1. Introduction: credit risk under incomplete information 2. Pricing and hedging credit derivatives via nonlinear filtering: the [Frey et al., 2007] model. Main ideas: • We model evolution of investors believes about credit quality, as those are driving credit spreads. • We use innovations approach to nonlinear filtering for deriving dynamics of traded credit derivatives.

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1.0

Attainable Correlations

-1.0

-0.5

correlation 0.0

0.5

min. correlation max. correlation

0

1

2

3

4

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sigma

Attainable correlations for two lognormal variables, X1 ∼ Ln(0, 1), X2 ∼ Ln(0, σ 2); (from McNeil, Frey, Embrechts, Quantitative Risk Management, Princeton University Press 2005) 2

1. Credit Risk and Incomplete Information Basically we have two classes of dynamic credit risk models. • Structural models: Default occurs if the asset value Vi of firm i falls below some threshold Ki, interpreted as liability, so that default time is τi := inf{t ≥ 0 : Vt,i ≤ Ki}. τi is (typically) predictable; dependence between defaults via dependence of the Vi. • Reduced form models: Default occurs at the first jump of some point process, typically with stochastic intensity λt,i. (τi is totally inaccessible.) Usually λt,i = λi(t, Xt), where X is a common state variable process introducing dependence between default times. 3

Incomplete information In both model-classes it makes sense to assume that investors have only limited information about state variables of the model • Asset value Vi is hard to observe precisely ⇒ consider firm-value models with noisy information about V (see for instance [Duffie and Lando, 2001], [Jarrow and Protter, 2004], [Coculescu et al., 2006] or [Frey and Schmidt, 2006]). • In reduced-form models state variable process X is usually not associated with observable economic quantities and needs to be backed out from observables such as prices.

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Implications of incomplete information • Under incomplete informations τi typically admits an intensity. • Natural two-step-procedure for pricing: prices are first computed under full information (using Markov property) and then projected on the investor filtration ⇒ Pricing and model calibration naturally lead to nonlinear filtering problems. • Information-driven default contagion. In real markets one frequently observes contagion effects, i.e. spreads of non-defaulted firms jump(upward) in reaction to default events. Models with incomplete information mimic this effect: given that firm i defaults, conditional distribution of the state-variable is updated, ⇒ default intensity of surviving firms increases ([Sch¨ onbucher, 2004], [Collin-Dufresne et al., 2003], . . . .) 5

Some literature (mainly reduced-form models) • Simple doubly-stochastic models with incomplete information such as [Sch¨onbucher, 2004], [Duffie et al., 2006], extensions in recent work by Giesecke. • [Frey and Runggaldier, 2007]. Relation between credit risk and nonlinear filtering and analysis of filtering problems in very general reduced-form model; dynamics of credit risky securities not studied. • Default-free term-structure models: [Landen, 2001]: construction of short-rate model via nonlinear filtering; [Gombani et al., 2005]: calibration of bond prices via filtering. • [Frey and Runggaldier, 2008] A general overview over nonlinear filtering in term-structure and credit risk models. 6

2. Our information-based model Overview. Three layers of information: 1. Underlying default model (full information) Default times τi are conditionally independent doubly-stochastic random times; intensities are driven by a finite-state Markov chain X. 2. Market information. Prices of traded credit derivatives are determined by informed market-participants who observe default history and some (abstract) process Z giving X in additive Gaussian noise (market information FM := FY ∨ FZ ); Filtering results wrt FM are used to obtain asset price dynamics. 3. Investor information. Z represents abstract form of ‘insider information’ and is not directly observable. ⇒ study pricing and hedging of credit derivatives for secondary-market investors with investor information FI ⊂ FM . 7

Advantages • Prices are weighted averages of full-information values (the theoretical price wrt FX ∨ FY ), so that most computations are done in the underlying Markov model. Since the latter has a simple structure, computations become relatively easy. • Rich credit-spread dynamics with spread risk (spreads fluctuate in response to fluctuations in Z) and default contagion (as defaults lead to an update of the conditional distribution of Xt given FtM ). • Model has has a natural factor structure with factors given by the conditional probabilities πtk = Q(Xt = k | FM ) , 1 ≤ k ≤ K. • Great flexibility for calibration. In particular, we may view observed prices as noisy observation of the state Xt and apply calibration via filtering. 8

Notation • We work on probability space (Ω, F, Q), Q the risk-neutral measure, with filtration F. All processes will be F adapted. • We consider portfolio of m firms with default state Y t = (Yt,1, . . . , Yt,m) for Yt,i = 1{τi≤t}. Yti is obtained from Yt by flipping ith coordinate. Ordered default times denoted by T0 < T1 < . . . < Tm; ξn ∈ {1, . . . , m} gives identity of the firm defaulting at Tn. • Default-free interest rate r(t), t ≥ 0, deterministic. Here r(t) ≡ 0.

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The underlying full-information model Consider a finite-state Markov chain X with S X := {1, . . . , K} and generator QX . A1 The default times are conditionally independent, doubly stochastic random times with (Q, F)-default intensity (λi(Xt)). Implications. • The processes Yt,j − martingales.

R t∧τj 0

λj (Xs−)ds, 1 ≤ j ≤ m, are F-

X ; in particular no • τ1, . . . , τm are conditionally independent given F∞ joint defaults.

• The pair process (X, Y) is Markov wrt F . 10

Examples 1. Homogeneous model (default intensities of all firms are identical). Default intensities are modelled by some increasing function λ : {1, . . . , K} → (0, ∞) of the states of the economy. Elements of S X thus represent different states of the economy (1 is the best state and K the worst state). Various possibilities for generator QX ; a very simple model takes X to be constant (Bayesian analysis instead of filtering). 2. Global- and industry factors. Assume that we have r¯ different industry groups. Let S X = {1, . . . , κ} × {0, 1}r ; write X 0,. . . , X r¯ for the components of X, modelled as independent Markov chains. X r is the state of industry r which is good (X r = 0) or bad (X r = 1); X 0 represents the global factor. Default intensity of firm i from industry group r takes the form λi(x) = gi(x0) + fi(xr ) for increasing functions fi and gi. 11

Full-information-values Define the full-information value of a FTY -measurable claim H (a typical credit derivative) by   EQ H | Ft =: h(t, Xt, Yt) ; (1) the last definition makes sense since (X, Y ) is Markov w.r.t. F. Computation of full-information values. Many possibilities: • Bond prices or legs of a CDS can be computed via Feynman-Kac • For portfolio products such as CDOs we can use conditional independence and compute Laplace transform of portfolio loss, (as in [Graziano and Rogers, 2006]) or use Poisson- and normal approximations, combined with Monte Carlo. 12

Market information Recall that the informational advantage of informed market participants is modelled via observations of a process Z. Formally, A2 FM = FY ∨ FZ , where the l-dim. process Z solves the SDE dZt = a(Xt)dt + dBt. Here, B is an l-dim standard F-Brownian motion independent of X and Y , and a(·) is a function from S X to Rl. b the Notation. Given a generic RCLL process U , we denote by U optional projection of U w.r.t. the market filtration FM ; recall that b is a right continuous process with U bt = E(Ut|FtM ) for all t ≥ 0. U

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3. Dynamics of Security Prices Traded securities. We consider N liquidly traded credit derivatives (eg. corporate bonds) with maturity T and FTY -measurable payoff PT,1, . . . , PT,N . We use martingale modelling:  M Q A3 Prices of traded securities are given by pbt,i := E PT,i|Ft . Market-pricing. Denote by pi(t, Xt, Yt) the full-information value of security i. We get from iterated conditional expectations   M M pbt,i = E E(PT,i|Ft) | Ft = E pi(t, Xt, Yt)|Ft . (2) Note that this is solved if we know the conditional distribution of Xt given FtM (a nonlinear filtering problem). Goal. Study the dynamics of traded security prices pbt,i; this is a prerequisite for hedging and risk management. 14

Innovations processes As a first towards determining the dynamics of the traded security prices step we introduce the innovations processes: t∧τj

Z

λj\ (Xs−)ds ,

Mt,j := Yt,j −

j = 1, · · · , m

Z0 t a\ i(Xs) ds ,

µt,i := Zt,i −

i = 1, · · · , l.

0

Properties. • Mj is an FM -martingale and µ is FM -Brownian motion. • Every FM -martingale can be represented as stochastic integral wrt M and µ.

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General filtering equations Proposition 1 (General filtering equations). Consider a FRt semimartingale of the form Jt = J0 + 0 Asds + MtJ , M J an FR t J,i J martingale with [M , B] = 0. Suppose that [J, Yi]t = 0 Rs−dYs,i. Then Jb has the representation Z t Z t Z t > bsds + Jbt = Jb0 + A γ> dM + α (3) s s s dµs; 0

0

0

γ and α are given by b\ αt = J\ ta(Xt) − Jta(Xt),  1  d \ J,iλ ) (Jλi)t− + Jbt−(λbi)t− + (R γt,i = i t− . (λbi)t−

(4) (5)

Proof based on innovations approach to nonlinear filtering. 16

Security-price dynamics Theorem 2. Under A1 - A3 the (discounted) price process of the traded securities has the martingale representation Z t Z t pbt,i = pb0,i + γ psbi,>dMs + αpsbi,>dµs, with 0

0

αptbi = p\ bt,i abt t,i · at − p

p bi γt,j = as in (5) with Rtpi,j = pi(t, Xt, Yti) − p(t, Xt, Yt).

The predictable quadratic variations of the asset prices with respect to the market information FM satisfy dhb pi, pbj iMt = vtij dt with vtij

=

m X n=1

p bj b p bi γt,n γt,n λt−,n

+

l X

p bj p bi αt−,nαt−,n.

(6)

n=1 17

Filtering Define the conditional probability vector π t = (πt1, . . . , πtK )> with πtk := Q(Xt = k|FtM ). π t is the natural state variable; under market information FM all quantities of interest are functions of π t. Kushner-Stratonovich equation. (K-dim SDE-system for π) Let q(ι, k), 1 ≤ ι, k ≤ K denote generator matrix of X. Then dπtk =

K X

q(ι, k)πtιdt + (γ k (π t−))> dMt + (αk (π t))> dµt , with

ι=1

(7) 

γjk (π) = πk PK 

λj (k)

n=1 λj (n)πn K X

 −1 ,

 πna(n) .

αk (π) = πk a(k) −

1 ≤ j ≤ m,

(8)

(9)

n=1 18

Default contagion • Updating at the default time τj . 

∆πτkj = πτkj − PK

λj (k)



n λ (n)π j τ n=1 j−

−1 .

• Default contagion. At τj default intensity of firm i jumps: bτ ,i − λ bτ −,i = λ j j

K X k=1

λi(k) ·

πτkj −



λj (k) PK

l l=1 λj (l)πτj −

 −1

=

cov

π τj −

λi , λ j

π E τj − (λj )

 .

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The filter in action

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4. Secondary market investors Recall that secondary market investors do not observe Z. Their information set is given by FI ⊂ FM ; typically FI contains default history and noisy price information. Pricing. Consider non-traded FTY -measurable claim H. Define its secondary-market value as E(H|FtI ). Let ht(Xt) = E(H | Ft) (full-information value of H). We get from iterated conditional expectations K X  E(H|FtI ) = E E(H|FtM ) FtI = E(πtk | FtI ) ht(k), k=1

i.e. pricing wrt FI reduces to finding E(πtk | FtI ).

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Hedging. We look for risk-minimizing strategies under restricted information in the sense of [Schweizer, 1994]. • Quadratic criterion combines well with incomplete information • On credit markets it is natural to minimize risk wrt martingale measure Q as historical default intensities are hard to determine. The risk-minimizing strategy θH can be computed by suitably projecting the FM -risk-minimizing hedging strategy ξtH on the set of FI -predictable strategies. For instance we get with only one traded asset that θt is left-continuous version of E(vtξtH

|

FtI )



E(vt | FtI ) .

Recall that vt and ξt are nonlinear functions of π t. ⇒ We need to determine νt(dπ), the conditional distribution of π given FtI . 22

Modelling FI and Calibration Strategies Pragmatic calibration. Here prices of traded securities are PK observable). Recall that pbt,i = k=1 πtk pi(t, k, Yt). If N ≥ K (more securities than states) and if the matrix p(t, Yt) := (pi(t, k, Yt)) of fundamental values has full rank, the vector π t could be implied by standard calibration: π t = argmin{π ≥0,PK

k=1 πk =1}

N X n=1



wn pbt,n −

K X

pn(t, k, Yt)πk

2

,

k=1

for suitable weights w1, . . . , wN . In that case pricing and hedging for secondary market investors and informed market participants coincides.

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Preliminary numerical results

Left: itraxx spreads from last winter for different maturities; Right: homogeneous model with 3 states and state probabilities calibrated to itraxx; note that probability of worst state increases over time. 24

Calibration via filtering Alternatively, assume that FI = FY ∨ FU where the N -dim process U solves the SDE dUt = pbtdt + dWt = p(t, Yt)π tdt + dWt for a Brownian motion W independent of X, Y, Z. U can be viewed as cumulative noisy price information of the traded assets pb1, . . . , pbN ; noise reflects observation errors and model errors. Recall that π solves the KS-equation (7). Hence computation of the conditional distribution of π t given FtI is a nonlinear filtering problem with signal process π and observation process U and Y .

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Filtering problem for secondary-market investors Challenging problem: • Observations of mixed type; Joint jumps of state process π and observation Y at defaults (see for instance [Frey and Runggaldier, 2007]) • Typically high-dimensional problem ⇒ use particle filtering as in [Crisan and Lyons, 1999] • Numerical analysis work in progress.

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References [Coculescu et al., 2006] Coculescu, D., Geman, H., , and Jeanblanc, M. (2006). Valuation of default sensitive claims under imperfect information. working paper, Universit´e d’ Evry. [Collin-Dufresne et al., 2003] Collin-Dufresne, P., Goldstein, R., and Helwege, J. (2003). Is credit event risk priced? modeling contagion via the updating of beliefs. Preprint, Carnegie Mellon University. [Crisan and Lyons, 1999] Crisan, D. and Lyons, T. (1999). A particle approximation of the solution of the Kushner-Stratonovich equation. Probability Theory and Related Fields, 115:549–578. [Duffie et al., 2006] Duffie, D., Eckner, A., Horel, G., and Saita, L. (2006). Frailty correlated defaullt. preprint, Stanford University. [Duffie and Lando, 2001] Duffie, D. and Lando, D. (2001). Term structure of credit risk with incomplete accounting observations. Econometrica, 69:633–664. 27

[Frey and Runggaldier, 2007] Frey, R. and Runggaldier, W. (2007). Credit risk and incomplete information: a nonlinear filtering approach. preprint, Universit¨at Leipzig,submitted. [Frey and Runggaldier, 2008] Frey, R. and Runggaldier, W. (2008). Nonlinear filtering in models for interest-rate and credit risk. preprint, submitted to Handbook of Nonlinear Filtering. [Frey and Schmidt, 2006] Frey, R. and Schmidt, T. (2006). Pricing corporate securities under noisy asset information. preprint, Universit¨at Leipzig,forthcoming in Mathematical Finance. [Frey et al., 2007] Frey, R., Schmidt, T., and Gabih, A. (2007). Pricing and hedging of credit derivatives via nonlinear filtering. preprint, Universit¨at Leipzig. available from www.math.uni-leipzig.de/%7Efrey/publications-frey.html. [Gombani et al., 2005] Gombani, A., Jaschke, S., and Runggaldier, W. (2005). A filtered no arbitrage model for term structures with noisy data. Stochastic Processes and Applications, 115:381–400. 28

[Graziano and Rogers, 2006] Graziano, G. and Rogers, C. (2006). A dynamic approach to the modelling of correlation credit derivatives using Markov chains. working paper, Statistical Laboratory, University of Cambridge. [Jarrow and Protter, 2004] Jarrow, R. and Protter, P. (2004). Structural versus reduced-form models: a new information based perspective. Journal of Investment management, 2:1–10. [Landen, 2001] Landen, C. (2001). Bond pricing in a hidden markov model of the short rate. Finance and Stochastics, 4:371–389. [Sch¨ onbucher, 2004] Sch¨ onbucher, P. (2004). Information-driven contagion. Preprint, Department of Mathematics, ETH Z¨ urich.

default

[Schweizer, 1994] Schweizer, M. (1994). Risk minimizing hedging strategies under restricted information. Math. Finance, 4:327–342.

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