Stochastic Discount Factors and Global Dominance in Markets with ImperfectionsI Alejandro Balb´asa , Anna Downarowicz∗,b , Javier Gil-Bazoc a
University Carlos III of Madrid, Department of Business Administration b Instituto de Empresa Business School, Department of Finance c University Carlos III of Madrid, Department of Business Administration
Abstract In this paper, we address portfolio efficiency based on global dominance and its relationship to the existence of Stochastic Discount Factors in markets with frictions. In particular, we generalize some theoretical properties of the Stochastic Discount Factor to the imperfect market case and show that it is possible to construct a reachable payoff that yields a theoretical price process lying under the actual price process and matching the price of efficient portfolios. The implications of our results are far-reaching: (i) we provide a new asset and portfolio pricing technique not affected by the magnitude of market frictions, (ii) a new practical tool for testing for the existence of dominated portfolios in the market, (iii) a new method for quantification of the degree of efficiency of market prices. Keywords: Stochastic Discount Factor, Portfolio Choice Problem, Global Dominance, Portfolio Efficiency, Markets with Frictions JEL Classification: G11, G13
I
Partially funded by the Spanish Ministry of Education and Science (Refs.:SEJ200767448 and SEJ2006-15401-CO4). The usual caveat applies. ∗ Corresponding author at: Instituto de Empresa Business School, Calle Castell´ on de la Plana 8, 28006 Madrid, Spain. Tel.:+34 917821714; fax.: +34 917454762 Email addresses:
[email protected] (Alejandro Balb´ as),
[email protected] (Anna Downarowicz),
[email protected] (Javier Gil-Bazo)
Preprint submitted to Quantitative Finance
April 7, 2009
1. Introduction The absence of arbitrage (AA) is a basic assumption of Asset Pricing Models. In perfect markets, AA has been characterized by the existence of state prices or risk neutral probability measures under which the price process becomes a martingale (Harrison and Kreps (1979), Dalang et al. (1990), Balb´as et al. (2002)). Furthermore, it implies the fulfillment of the Law of One Price (LOP ) that leads to the existence of Stochastic Discount Factors (SDF ) providing us with pricing rules and, as long as risk levels are measured by standard deviations, also with optimal portfolios (see, for instance, Chamberlain and Rothschild (1983), or Hansen and Jagannathan (1997)). In particular, if one considers the classical Capital Asset Pricing Model (CAP M), there exists a close relationship between the Market Portfolio and the SDF . Finally, if the market is complete, the SDF may be understood as the density function between the risk neutral probability and the actual probability. Despite the convenience of the frictionless market assumption, market imperfections are receiving increasing attention from researchers since they capture better many properties of actual financial markets.1 In particular, the problem of replicating a contingent claim’s payoff in the presence of transaction costs or trading constraints has been addressed by Leland (1985), Boyle and Vorst (1992), and Edirisinghe et al. (1993), among others. These authors assume some dynamics for the underlying asset price and a specific functional form for transaction costs. In a more general setting, Jouini and Kallal (1995), Pham and Touzi (1999) and Schachermayer (2004) have shown that when frictions are considered, the martingale property of arbitrage-free markets is satisfied by a price process lying within the bid and the ask process.2 Besides, some type of SDF may be introduced in the sense that there exist theoretical (but not necessarily reachable) payoffs δ such that the expectation of every distorted reachable payoff X, i.e., the expectation of Xδ, must lay within the initial bid/ask spread of X (Luttmer 1996). Furthermore, for standard utility functions, optimal investments only purchase 1
The effect of market frictions on security prices has been studied among others by Garman and Ohlson (1981), He and Modest (1995), Luttmer (1996), Prisman (1986, 1997, 2003) and De Waegenaere and Wakker (2001). 2 More recently, Bizid and Jouini (2005) have investigated the constraint on the range of admisible prices imposed by market clearing conditions.
2
(sell) some securities whose bid (ask) price is obtained as the expectation of their distorted final payoff (He and Modest 1995). Jouini and Kallal (2001) extend previous results and characterize a general concept of efficiency based on stochastic dominance in imperfect markets. In this paper, we define a new weak and general form of efficiency based on global dominance. A portfolio is globally dominated if there is an alternative cheaper strategy that provides a higher pay-off in every state. The Jouini and Kallal efficiency implies ours but the converse does not hold in general. The main contribution of this paper is the introduction of a new kind of SDF that allows us to characterize and compute in practice those strategies that are not globally dominated. Moreover, this kind of SDF is also closely related to optimal portfolios in a mean-variance context. Thus, our SDF may be crucial when dealing with Portfolio Choice problems in the imperfect market case, which are becoming the object of growing concern to researchers and practitioners.3 Our paper differs from extant studies in important ways. First, with regard to Jouini and Kallal (1995) or Luttmer (1996), we are not only analyzing AA but also efficiency. Second, in comparison with He and Modest (1995) or Jouini and Kallal (2001), we are dealing with a much weaker concept of efficiency that does not draw on any sort of utility function. Consequently, those strategies that are not efficient under our point of view should not be considered by any Portfolio Choice Problem. This fact may be especially interesting when dealing with securities whose returns exhibit complex distributions such as derivatives. The findings of our paper have practical implication for brokers, portfolio managers and investors since they provide new practical pricing and investment methods that may outperform their clients’ portfolios. In particular, the pricing method proposed in Balb´as et al. (1999) for catastrophe-linked derivatives traded at CBOT ,4 can be extended to identify dominated securities. Further, since combining efficient securities may result in dominated portfolios, the method proposed in this paper potentially enables brokers to build and purchase dominating portfolios for their clients, rather than the 3
See, for instance, Sankaran and Patil (1999) or Alexander et al. (2006) for recent studies of the consequences of market imperfections on portfolio choice problem. 4 i.e., if the purchase (sale) of a given security or portfolio is dominated then they can ask (bid) this security/portfolio and simultaneously improve its market quote. If a new agent accepts and buys (sells) then the position may be hedged by an arbitrage.
3
initial ones, in order to generate additional earnings. Moreover, it is worth remarking that the proposed pricing technique will not be affected by significant frictions frequently reflected by financial markets, on the contrary, it allows traders to improve the quotes and provide liquidity. See for example Balb´s et al (2008) for the application of this methodology to pricing options in the oil-linked derivatives markets. Finally, the methodology proposed in this paper is general enough to apply to any financial market, including emerging markets, commodity markets, and any other possibly illiquid markets. It should be borne in mind that imperfections and other drawbacks may make it rather difficult to apply the classical pricing methods of frictionless markets. The outline of the paper is as follows. Section 2 introduces the general notation and definitions used to establish the main results of the paper. Section 3 states the main results and relates the existence of efficient portfolios to the existnce of SDF for imperfect markets. Finally, Section 4 concludes the article. 2. Notation and Preliminaries In this section we introduce the notation and theoretical notions necessary to establish the main results. Let (Ω, F , µ) be a probability space composed of the set Ω, the σ−algebra F and the probability measure µ. Suppose that [0, T ] represents a time interval and T ⊂ [0, T ] is a (finite or infinite) set of trading dates such that {0, T } ⊂ T . As usual, the arrival of information will be generated by the increasing family [Ft ]t∈T of σ−algebras of Ω (filtration) such that F0 = {∅, Ω} and FT = F . Prices, transaction costs and investment strategies will be introduced by following the approach of Jouini and Kallal (1995), since it incorporates those frictions created by the bid/ask spread and simultaneously generates a linear relationship between each portfolio and its current price.5 Thus we will consider n + 1 different securities, denoted by S0 , S1 , ..., Sn , whose bid and ask prices will be represented by two IRn+1−valued adapted processes {b(ω, t); ω ∈ Ω, t ∈ T } 5
Actually this approach enables us to represent many types of frictions, and not only those generated by the bid/ask spread. See Jouini and Kallal (1995) for a further discussion.
4
and {a(ω, t); ω ∈ Ω, t ∈ T } respectively. In order to simplify the notation, the previous pair of processes may be also denoted by (b, a). If b(ω, t) = (b0 (ω, t), b1(ω, t), ..., bn (ω, t)) where bj (ω, t) ∈ IR for ω ∈ Ω, t ∈ T and j = 0, 1, ..., n, then bj (ω, t) represents the bid price of Sj at t under the state ω. An analogous notation will be used for ask prices and, more generally, for any IRn+1−valued adapted stochastic process. The first asset S0 will play the role of a numeraire, and therefore we will impose the existence of v0 ∈ IR such that the inequalities 0 < v0 ≤ b0 (ω, t) ≤ a0 (ω, t)
(1)
hold for every ω ∈ Ω and t ∈ T . Obviously, we will require the assumption b(ω, t) ≤ a(ω, t) for every ω ∈ Ω and t ∈ T . For a fixed ω ∈ Ω the corresponding path or trajectory of the bid (ask) price will be denoted by b(ω, −) (a(ω, −)), while for any fixed trading date t ∈ T the symbol b(−, t) (a(−, t)) yields the random variable providing us with the bid (ask) price at t. A feasible portfolio (x, y) is a couple of IRn+1−valued adapted stochastic processes {x(ω, t); ω ∈ Ω, t ∈ T } and {y(ω, t); ω ∈ Ω, t ∈ T } such that its trajectories x(ω, −) and y(ω, −) are increasing functions of t ∈ T and there exist a finite number r ∈ IN, r ≥ 1, and a finite strictly increasing sequence {t0 = 0, t1 , t2 , ..., tr = T } ⊂ T (that depend on (x, y) but do not depend on ω) for which x(ω, −) and y(ω, −) remain constant on each T ∩ [ti−1 , ti ), i = 1, 2, ..., r. In addition, if [x(ω, ti ) − x(ω, ti−1 )] a(ω, ti ) − [y(ω, ti) − y(ω, ti−1)] b(ω, ti ) = 0 6
i = 1, 2, ..., r − 1, then (x, y) will be called self-financing. 6
Notice that products in (2) are scalar (or inner) products of IRn+1 .
5
(2)
The set of self-
financing portfolios will be denoted by S. It may be easily proved that α1 (x1 , y 1) + α2 (x2 , y 2) ∈ S if (xi , y i) ∈ S and αi ≥ 0 in IR, i = 1, 2. If (x, y) ∈ S then λ(x, y) = x(ω, 0)a(ω, 0) − y(ω, 0)b(ω, 0) ∈ IR
(3)
will be the price of (x, y). It immediately follows that λ (α1 (x1 , y 1 ) + α2 (x2 , y 2)) = α1 λ(x1 , y 1) + α2 λ(x2 , y 2) if (xi , y i) ∈ S and αi ≥ 0 in IR, i = 1, 2. Let (x, y) ∈ S. The FT −measurable random variable Λ(x, y)(ω) = [x(ω, tr−1 ) − y(ω, tr−1)]+ b(ω, T )−[y(ω, tr−1) − x(ω, tr−1)]+ a(ω, T ) (4) will be the final payoff of (x, y), and will be denoted by Λ(x, y) or, if necessary, by Λ(x, y)(ω).7 Once again, Λ (α1 (x1 , y 1 ) + α2 (x2 , y 2)) = α1 Λ(x1 , y 1 ) + α2 Λ(x2 , y 2) if (xi , y i) ∈ S and αi ≥ 0 in IR, i = 1, 2. We follow usual conventions in order to introduce the concept of arbitrage. Definition 2.1. Let (x, y) ∈ S. (x, y) is said to be an arbitrage if a) λ(x, y) ≤ 0 b) Λ(x, y)(ω) ≥ 0, µ − a.s. c) µ (Λ(x, y) − λ(x, y) > 0) > 0.
Hereafter we will assume that the market is arbitrage free. Definition 2.2. Let (x, y), (x , y ) ∈ S. We will say that (x, y) dominates or globally dominates (x , y ) if a) λ(x, y) ≤ λ(x , y ) b) Λ(x, y)(ω) ≥ Λ(x , y )(ω) , µ − a.s. c) µ [(Λ(x, y) − Λ(x , y )) + (λ(x , y ) − λ(x, y)) > 0] > 0. Henceforth non-dominated strategies will also be called efficient. Notice that Condition c) above is fulfilled if a) holds in terms of strict inequality. Arbitrage and dominance are closely related crucial notions. In 7
As usual, α+ = M ax{α, 0} and α− = M ax{−α, 0} if α ∈ IR, and, for every m ∈ IN + − − + − − and α ∈ IRm , α+ = (α+ 1 , α2 , ..., αm ) and α = (α1 , α2 , ..., αm ).
6
particular an arbitrage is a portfolio dominating the null strategy. It is well known that the AA is equivalent to the absence of dominated strategies in the frictionless case (b = a). However, under our present approach, the existence of transaction costs may lead to the existence of dominated portfolios in arbitrage free markets (see for example Balb´as et al. (2008)). As usual, the set M of marketed claims is composed of those FT −measurable IR−valued random variables m such that there exists (x, y) ∈ S with m ≤ Λ(x, y), µ − a.s. If so, (x, y) will be called a super-replication of m, and the set of super-replications of m will be represented by Sm . It may be very easily proved that M is a convex cone (i.e., α1 m1 + α2 m2 ∈ M if mi ∈ M and αi ≥ 0 in IR, i = 1, 2) and (1) shows that L∞ (FT ) ⊂ M, L∞ (FT ) being the space of IR−valued FT −measurable essentially bounded random variables. More generally, hereafter we will fix 1 ≤ p ≤ ∞ and a closed linear manifold V of Lp (FT ) such that V ⊂ M , where for p < ∞ the space Lp (FT ) is composed of the IR−valued FT − measurable random variables ξ such that the expectation of | ξ |p is bounded. We will define the function of minimum cost by M m → π(m) = Inf {λ(x, y); (x, y) ∈ Sm } ∈ IR ∪ {−∞}
(5)
It is easy to show that π is increasing and sublinear on M, i.e., π (m1 ) ≥ π (m2 )
(6)
whenever mi ∈ M, i = 1, 2, m1 ≥ m2 µ − a.s., π (m1 + m2 ) ≤ π (m1 ) + π (m2 )
(7)
if mi ∈ M, i = 1, 2, and π (αm) = απ (m)
(8)
if m ∈ M and α > 0 in IR. The positiveness of the bid and ask prices of S0 and the AA easily lead to π(0) = 0 and π(m) > −∞ for every m ∈ L∞ (FT ). We will also assume that π(m) > −∞ for every m ∈ V ⊂ Lp (FT ).8 8
Being V a vector space we have that a0 (−, T ) ∈ V is a sufficient condition. Indeed, suppose that v0 ∈ V and π(v0 ) = −∞. Then for every v ∈ V ⊂ M is v − v0 ∈ V ⊂ M and (7) gives π(v) ≤ π(v0 ) + π(v − v0 ) = −∞, which is a contradiction with the AA if one takes, for instance, v = a0 (−, T ).
7
If p < ∞ then q ∈ (1, ∞] will denote its conjugate value, i.e., (1/p) + (1/q) = 1 holds, and, according to the Riesz Representation Theorem, Lq (FT ) is the dual space of Lp (FT ). If p = ∞ then the dual space of L∞ (FT ) is composed of those finitely-additive measures FT A −→ δ(A) ∈ IR µ−continuous (i.e., µ(A) = 0 =⇒ δ(A) = 0) and with finite variation (see, for instance, Diestel and Uhl, 1977). This space will be denoted by M(FT ). 3. Generalized Stochastic Discount Factors Next we will establish the tie between the absence of dominance and the existence of SDF . Lemma 3.1. Let l ∈ IN and (x1 , y 1), (x2 , y 2 ), ..., (xl , y l) ∈ S. The following assertions l are equivalent: (i) i=1 βi (xi , y i) is non-dominated for every β1 , β2 , ..., βl ≥ 0. (ii) There exist α1 , α2 , ..., αl > 0 such that li=1 αi (xi , y i) is non-dominated. Before providing the proof of Lemma 3.1 we will present a simple lemma whose proof is immediate and therefore omitted. Lemma 3.2. Suppose that (x, y) and (x , y ) are self-financing portfolios such that (x, y) dominates (x , y ). Then θ(x, y) dominates θ(x , y ) for every θ > 0 and (x, y) + (x , y ) dominates (x , y ) + (x , y ) for every (x , y ) ∈ S.
Proof of Lemma 3.1. Suppose that li=1 αi (xi , y i) is non-dominated and ≥ 0. Consider θ > 0 with θβi < αi for i = 1, 2, ..., l. take β1 , β2 , ..., βl If we prove that li=1 θβi (xi , y i) is non-dominated then we can multiply by 1/θ > 0 and the efficiency of i∈I βi (xi , y i ) follows from the previous lemma. Suppose that (x, y) dominates li=1 θβi (xi , y i). Then (x, y) +
l
(αi − θβi )(xi , y i)
i=1
8
dominates l
θβi (xi , y i) +
i=1
l
(αi − θβi )(xi , y i) =
i=1
l
αi (xi , y i)
i=1
against the assumption.
Henceforth the operator E will be used to represent the expectation of any random variable. Theorems 3.3 and 3.5 below are established under the assumption that V satisfies the properties of a Banach Lattice. For instance, this assumption is fulfilled if V = Lp (FT ). The properties of a Banach Lattice may be found in Schaeffer (1974). Theorem 3.3. (Main Result) Let l ∈ IN and (xi , y i) ∈ S, i = 1, 2, ..., l. Suppose that (i) or (ii) of Lemma 3.1 hold. Suppose finally that Λ(xi , y i ) ∈ V , i = 1, 2, ..., l. Then: (i) If p < ∞ then there exists δ ∈ Lq (F ), δ ≥ 0 µ − a.s., such that E (mδ) ≤ π (m)
(9)
E Λ(xi , y i)δ = λ(xi , y i)
(10)
for every m ∈ V , and
i = 1, 2, ..., l. (ii)If p = ∞ then there exists a non-negative, finitely additive with finite variation µ− continuous measure δ ∈ M(F ) such that m(ω)dδ(ω) ≤ π (m) Ω
for every m ∈ V , and
Ω
Λ(xi , y i)(ω)dδ(ω) = λ(xi , y i)
i = 1, 2, ..., l.
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Before presenting the proof of the Theorem 3.3 we will provide without proof a lemma of Convex Analysis. Lemma 3.4. Let E be a vector space and Γ : E −→ IR an arbitrary function such that Γ(m1 + m2 ) ≤ Γ(m1 ) + Γ(m2 ) whenever m1 and m2 belong to E and Γ(αm) = αΓ(m) whenever α ≥ 0 and m belongs to E. If m0 ∈ E then there exists a linear function φ : E −→ IR such that φ(m) ≤ Γ(m) for every m ∈ E and φ(m0 ) = Γ(m0 ). If E is ordered and Γ is (strictly) increasing then φ is non-negative (positive). Proof of Theorem 3.3. Since li=1 (xi , y i) is efficient one has that l l l i i i i π Λ (x , y ) =λ (x , y ) = λ(xi , y i). i=1
i=1
i=1
Take E = V , Γ = π and m0 = li=1 Λ(xi , y i) and apply the previous lemma. There exists φ : V −→ IR such that φ(m) ≤ π(m) for every m ∈ V and φ
l
i
i
Λ(x , y )
i=1
=
l
λ(xi , y i).
i=1
The first inequality leads to φ (Λ(xi , y i)) ≤ λ(xi , y i), i = 1, 2, ..., l, and consequently, the equality gives φ (Λ(xi , y i)) = λ(xi , y i), i = 1, 2, ..., l. Furthermore, Sine π is increasing φ is non-negative. Hence the continuity of φ is guaranteed because every linear and real valued positive operator on a Banach Lattice is continuous (Schaeffer (1974)). According to the HanhBanach Theorem (Schaeffer (1974)), there exists a linear extension ϕ of φ to the whole space Lp (FT ). Whence the Riesz Representation Theorem implies the existence of δ. Expression (10) allows us to establish the link between prices and payoffs of efficient portfolios. Prices are given by the mathematical expectation 10
of distorted payoffs. This is in line with the usual conditions for the arbitrage absence in perfect markets, but there is a major difference since we are dealing with efficiency rather than arbitrage. As already mentioned, the absence of arbitrage in the presence of frictions is studied by the existence of price processes lying within the bid/ask spread and satisfying the martingale property with respect to some distorted probability measure. This is closely related to Expression (9),9 so we can conclude that the combination of (9) and (10) reflects the existence of linear expressions capturing both properties since they are lying under the price process and matching those prices of efficient portfolios. The distortion random variable δ is related to both risk neutral measures and SDF . In this paper we will focus on the second concept since it is more closely related to Portfolio Choice Problems. Therefore we will consider random variables with bounded expectation and variance (p = 2). Theorem 3.5. (Stochastic Discount Factors) Let l ∈ IN and (xi , y i) ∈ S, i = 1, 2, ..., l. Suppose that (i) or (ii) of Lemma 3.1 hold. Suppose finally that p = 2 and Λ(xi , y i) ∈ V , i = 1, 2, ..., l. Then there exists a unique δ ∈ V ⊂ L2 (FT ) such that (9) and (10) hold.
Proof. The existence of a continuous φ satisfying the conditions of the previous proof is absolutely similar. Being V a Hilbert Space the Riesz Representation Theorem leads to the existence of a unique δ ∈ V representing φ. There are two important differences with respect to Theorem 3.3 , since δ is unique and may be super-replicated (δ ∈ V ⊂ M). δ will be called SDF and is a genuine extension of the SDF of a perfect market (see, for instance, Chamberlain and Rothschild (1983) or Hansen and Jagannathan (1997)). As long as a riskless asset is available, if δ is a reachable payoff and is not risk-free it may be proved that every optimal portfolio in a mean/variance context is composed of the riskless asset minus kδ (k > 0), although we do not address this question that is beyond our scope. 9
Notice that if (x, y) ∈ S and Λ(x, y) ∈ V then (9) gives E (Λ(x, y)δ) ≤ λ(x, y).
11
An important particular case arises if we consider a static approach (T = {0, T }) and transaction costs are paid at the initial date, i.e., a(ω, T ) = b(ω, T ) for ω ∈ Ω. Both properties will be assumed in the rest of this section. We will also assume that aj (−, T ) has bounded variance (i.e., is in L2 (FT )) for j = 0, 1, ..., n. We will take as V the linear manifold of L2 (FT ) generated by {aj (−, T )}nj=0. It is obvious that δ is reachable in this framework and the discussion above applies. Consider partitions (J1 , J2 , J3 ) of J = {0, 1, ..., n}. For instance, if n = 3, a feasible partition is (J1 = {0, 2}, J2 = ∅, J3 = {1, 3}). Portfolio (x, y) is said to be associated with the partition (J1 , J2 , J3 ) if xj = 0 and yj > 0 whenever j ∈ J1 and xj > 0, yj = 0 for j ∈ J2 . Then Theorem 3.5 implies: Theorem 3.6. (Stochastic Discount Factors, static case) Consider an arbitrary partition (J1 , J2 , J3 ). If portfolios associated with this partition are efficient10 then there exists a unique reachable δ such that
for j = 0, 1, ..., n, for j ∈ J1 and
bj (−, 0) ≤ E (aj (−, T )δ) ≤ aj (−, 0)
(11)
bj (−, 0) = E (aj (−, T )δ)
(12)
E (aj (−, T )δ) = aj (−, 0)
(13)
for j ∈ J2 .
Proof. First of all take the set of strategies {((xJ = 0) , (yJ1 = (0, ..., 1, ...0), yJ2 = 0, yJ3 = 0))} and {((xJ1 = 0, xJ2 = (0, ..., 1, ...0), xJ3 = 0) , (yJ = 0))} 10
Recall that Lemma 3.1 implies that all the strategies associated with the partition are efficient if at least there exists one efficient associated portfolio.
12
As in the previous proof one can construct the function φ. Its continuity holds even if V is not a Banach Lattice because this space has a finite dimension. Thus, the existence of a unique δ ∈ V satisfying (9) and (10) is ensured. In particular, Expression (10) applied on the strategies above leads to (12) and (13), and Expression (9) applied on the whole set of available securities in long and short position, respectively, leads to (11). According to Chamberlain and Rothschild (1983) the SDF δ may be computed by fixing expected returns and minimizing variances. Moreover the SDF provides us with the family of optimal portfolios if investors measure the risk level by means of standard deviations.11 Thus Theorem 3.6 may be quite useful when testing for the existence of inefficient portfolios in imperfect markets. Indeed, in a first step one can check the existence of SDF for some feasible partitions in order to guarantee that their associated portfolios are non-dominated.12 This step is general and does not depend on the applied risk measurement methods, since every Portfolio Choice Problem rejects those strategies that are dominated in the sense of Definition 2.1 (See, for example, Balb´as et al. (2008) for the application of the Theorem 3.6 for testing price efficiency of oil-linked derivatives traded at NYMEX). The second step applies when the degree of risk is measured by means of standard deviations. In such a case the payoff δ of Theorem 6 indicates those portfolios that are optimal, since we can draw on the findings of Chamberlain and Rothschild (1983) for perfect markets. As mentioned above, the computation in practice of the SDF may be addressed by several optimization methods (see also Balb´as and Mayoral (2004)). 4. Conclusions In this paper, we have addressed global portfolio efficiency and its relationship to the existence of SDF in markets with frictions. We extend the notion of SDF to the case of imperfect markets and show that they provide a 11 It depends on their utility function and the cumulative distribution of each aj (−, T ). Although the standard deviation often applies as a risk measure, many alternative ways have been provided in order to compute the level of risk (see for instance Artzner et al (1999) for a general approach. 12 One could check the whole set of partitions but this set will be extremely large unless n is very small.
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useful link between payoffs and prices/quotes, since they yield a theoretical price process lying under the real price process and matching those quotes of the efficient portfolios. SDF can therefore be used to price securities and porfolios, analyze the existence of dominated portfolios in the market, compute optimal strategies in a mean/variance setting and to quantify the degree of efficiency of market prices. The implications of our results are far-reaching. First, we extend previous theoretical results for markets with frictions and portfolio efficiency. Second, we provide a new asset and portfolio pricing tool which is not affected by the magnitude of market frictions. This is of great practical importance to portfolio managers and risk managers, who may use this methodology to discard dominated investment or risk-management strategies. The method is also important for brokers and other intermediaries who may improve their clients’ portfolios without taking any additional risk. Additionaly, our results suggest a new method for evaluating the efficiency of a derivative market. Finally, the methodology is general enough to apply in any financial market, and therefore it may enable us to study emerging and maybe illiquid markets that are becoming very interesting for traders trying to correctly diversify their risk. It is worth to bear in mind that imperfections and other drawbacks may make it rather difficult to apply the classical pricing methods of frictionless markets. References Alexander, S., Coleman, T.F. Li, Y., 2006. Minimizing CVaR and VaR for a Portfolio of Derivatives. Journal of Banking & Finance 30, 583-605. Artzner, P., Delbaen, F., Eber, J.M., Heath, D., 1999. Coherent measures of risk. Mathematical Finance 9, 203-228. Balb´as A., Downarowicz, A., Gil-Bazo, J., 2008. Price Inefficiencies in Commodities-Linked Derivatives Markets: A Model-free Analysis. Available at SSRN: http://ssrn.com/abstract=1304085. Balb´as A., Longarela, I.R., Lucia, J., 1999. How Financial Theory Applies to Catastrophe-Linked Derivatives. An Empirical Test of Several Pricing Models. Journal of Risk and Insurance 66, 4, 551-582. Balb´as, A., Mayoral, S., 2004. Vector Optimization Approach for Pricing and Hedging in Imperfect Markets. INFOR 42, 3, 217-233. 14
Balb´as A., Mir´as, M., Mu´ noz-Bouzo, M.J., 2002. Projective system approach to the martingale characterization of the absence of arbitrage. Journal of Mathematical Economics 37, 311-323. Bizid, A., Jouini, E., 2005. Equilibrium Pricing in Incomplete Markets. Journal of Financial and Quantitative Analysis 40, 4, 833-849. Boyle, P., Vorst, T., 1992. Option Replication in Discrete-Time with Transaction Costs. Journal of Finance 47, 271-293. Chamberlain, G., Rothschild, M., 1983. Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Assets. Econometrica 51, 1281-1304. Dalang, R.C., Morton, A., Willinger, W., 1990. Equivalent Martingale Measures and No-Arbitrage in Stochastic Securities Market Models. Stochastics and Stochastic Reports 29, 185-201. De Waegenaere, A. Wakker, P.P., 2001. Nonmonotonic Choquet Integrals. Journal of Mathematical Economics 36, 45-60. Diestel, J. Uhl, J.J., 1977. Vector Measures. Math Surveys 15, American Mathematical Society, Providence R.I. Edirisinghe, C., V. Naik, Uppal, R., 1993. Optimal Replication Of Options With Transactions Costs And Trading Restrictions. Journal of Financial and Quantitative Analysis 28, 117-138. Hansen, L.P., Jagannathan, R., 1997. Assessing Specification Errors in Stochastic Discount Factor Models. Journal of Finance 52, 2, 567-590. Harrison, M., Kreps, D.M., 1979. Martingale and Arbitrage in Multiperiod Security Markets. Journal of Economic Theory 20, 381-408. He, H., Modest, D.M., 1995. Market Frictions and Consumption-Based Asset Pricing. Journal of Political Economy 103, 94-117. Jouini, E., Kallal, H., 1995. Martingales and Arbitrage in Securities Markets with Transaction Costs. Journal of Economic Theory 66, 178-197. Jouini, E., Kallal, H., 2001. Efficient Trading Strategies in Presence of Market Frictions. Review of Financial Studies 14, 343-369. 15
Leland, H., 1985. Option Pricing And Replication With Transactions Costs. Journal of Finance 40, 1283-1301. Luttmer, E., 1996. Asset Pricing in Economies with Frictions. Econometrica 64, 1439-1467. Muthuraman, K., Zha, H., 2008. Simulation-based Portfolio Optimization for Large Portfolios with Transaction Costs. Mathematical Finance 18, 1, 115-134. Pham, H., Touzi, N., 1999. The Fundamental Theorem of Asset Pricing with Cone Constraints. Journal of Mathematical Economics 31, 265-279. Rockafellar, R.T., Uryasev, S., 2002. Conditional Value-at-Risk for General Loss Distributions. Journal of Banking&Finance 26, 1443-1471. Schachermayer, W., 2004. The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time. Mathematical Finance 14, 1, 19-48. Sankaran, J.K., Patil, A.A., 1999. On the Optimal Selection of Portfolios under Limited Diversification. Journal of Banking & Finance 23, 16551666. Schaeffer, H.H., 1974. Banach Lattices and Positive Operators. Springer Verlag, Berlin.
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