Hedging and Optimization Problems in Continuous-Time Financial ...

Hedging and Optimization Problems inContinuous-Time Financial Models ∗Huyên PHAMLaboratoire de Probabilités etModèles AléatoiresUMR 7599Université Paris 72 Place Jussieu75251 Paris Cedex 05e-mail : pham@gauss.math.jussieu.frand CREST, Laboratoire de Finance-Assurance.September 1999AbstractThis paper gives an overview of the results and developments inthe area of hedging contingent claims in an incomplete market. Westudy three hedging criteria. We first present the superhedging approach.We then study the mean-variance criterion and finally, we describethe shortfall risk minimization problem. From a mathematicalviewpoint, these optimization problems lead to nonstandard stochasticcontrol problems in PDE and new variants of decomposition theoremsin stochastic analysis.∗ Lecture presented at ISFMA Symposium on Mathematical Finance, Fudan University(August 1999).1

V T = H, a.s., or in other words, if H can be written as the sum of a constantand a stochastic integral with respect to S :H = H 0 +∫ T0θ H t dS t . (1.1)We say that the market is complete if every contingent claim is attainable.However, completeness of the market is a property largely denied byempirical studies on financial market, focusing attention of researchers onextensions of the Black-Scholes model : stochastic volatility models, jumpdiffusionmodels. In this case, given an arbitrary contingent claim H, representation(1.1) is no more possible and we say that the market is incomplete.We mention that there are many others modifications of the Black-Scholes model destroying the completeness property : These are portfolioconstraints, transaction costs on trading, etc ... More generally, we speakof imperfect markets. In this paper, we focus mainly on the incompletenesssituation. For a nonattainable contingent claim, it is then impossible to finda self-financing strategy with final value equal to H. The problem of pricingand hedging can then be formulated as follows : Approximate a contingentclaim H by the family of terminal wealth of self-financing strategies. Ofcourse, the approximation depends on the choice of the risk measure andleads to various stochastic optimization problems. A first approach is tolooking for trading strategies with terminal value V T larger than H : Inthis case, one (super)hedges the contingent claim. This idea, introducedby Bensaid, Lesne, Pagès and Scheinkman (1992), is behind the concept ofsuperreplication. An alternative approach is to introduce subjective criteriaaccording to which strategies are chosen. The measure of riskiness bya mean-variance criterion was first proposed by Föllmer and Sondermann(1986), and consists in minimizing the expected square of the replicationerror between the contingent claim and the terminal portfolio wealth. Onedrawback of this approach is the fact that one penalizes both situationswhere the terminal wealth is larger or smaller than H. Finally, a third measureof riskiness that circumvents this last point, is proposed by Föllmer andLeukert (1998). It consists in minimizing the expected shortfall (H − V T ) +weighted by some loss function.The paper is organized as follows. Section 2 introduces the model andformulates the hedging problems. Section 3 describes in detail the super-3

hedging approach. In Section 4, we study the mean-variance hedging criterionand the final Section 5 is devoted to the shortfall risk minimizationproblem.I would like to thank the organizers of the ISFMA symposium on mathematicalfinance at Fudan university, for their invitation, especially RamaCont, Jinhai Yan, Professors Li Ta-tsien and Jiongmin Yong.2 Hedging problems2.1 The modelWe consider a financial market that operates in uncertain conditions describedby a probability space (Ω, F, P ) equipped with a filtration IF ={F t } 0≤t≤T representing the flow of information on [0, T ]. For simplicity, weassume that F 0 is trivial and F T = F. There are d + 1 assets in the market.The 0-th asset is riskless, equal to 1 at any time. The last d assets couldbe risky (typically stocks), and their price process is given by an IR d -valuedsemimartingale S.An important notion in mathematical finance is the set of equivalentmartingale measures : We letP = {Q ∼ P : S is a Q − local martingale}denote the set of probability measures Q on F equivalent to P and suchthat S is a local martingale under Q. We shall assume that :P ≠ ∅. (2.1)This standing assumption is related to some kind of no-arbitrage conditionand we refer to Delbaen and Schachermayer (1994) for a general versionof this first fundamental theorem of asset pricing.A trading portfolio strategy is an IR d -valued predictable process θ, integrablewith respect to S. We denote θ ∈ L(S) and we refer to Jacod (1979)for vector stochastic integration with respect to a semimartingale. Here θ trepresents the number of shares invested in the assets of price S. Givenan initial capital x ∈ IR and a trading strategy θ ∈ L(S), the self-financed4

wealth process is governed by :∫ tV x,θt = x +0θ u dS u , 0 ≤ t ≤ T.2.2 Formulation of the problemAn (European) contingent claim is an F T -measurable random variable. Typicalexamples are call option of exercice price κ on the i-th asset S i , H =(ST i − κ)+ and put option of exercice price κ on the i-th asset S i , H =(κ − ST i )+ . More generally, H may depend on the whole history of S up totime T .Hedging of the contingent claim H in the financial market described inthe previous paragraph consists in approximating H by the terminal wealthvalues V x,θT. We say that the market is complete if every contingent claimH can be represented as :∫ TH = V H 0,θ HT= H 0 +0θu H dS u , (2.2)for some H 0 ∈ IR and θ ∈ L(S). Completeness property depends on the classof contingent claims and trading strategies considered and one has to precisethe integrability conditions on H and θ. Under the no-arbitrage assumption(2.1), a sufficient condition ensuring the completeness of the market is thatthe set of equivalent martingale measures P is reduced to a singleton. Thisresult follows from general results on martingale representation due to Jacod(1979) and is applied for the purpose of finance theory in Harrison and Pliska(1981). In this case, one can perfectly hedge (approximate) H by V H 0,θ HT.The initial capital H 0 is a fair price of H and θ H is called perfect replicatingstrategy of H. Typical example of complete market is the classical Black-Scholes model.In the general semimartingale model of Section 1, given an arbitraryH, representation (2.2) is no more possible. We say that the market isincomplete and in this case the set of equivalent martingale measures P isinfinite. Typical examples of incomplete markets are stochastic volatilitymodels and jump-diffusion models. In such a context, one can no moreperfectly hedge (approximate) H by V x,θTand one has to choose a hedgingcriteria. From a mathematical viewpoint, this leads to various stochastic5

optimization problems. In the sequel, we shall study the following threehedging criteria :- Superhedging approach- Mean-variance hedging criterion- Shortfall risk minimization3 The superhedging approachWe are given a nonnegative contingent claim of the form H = g(S T ), whereg is a continuous function from IR d into IR + , with linear growth condition.The integrability conditions on the trading strategies are defined as follows.Given x ≥ 0, we say that a trading strategy θ ∈ L(S) is admissible, and wenote θ ∈ A(x), if V x,θt ≥ 0, for all t in [0, T ]. The superhedging approachconsists in looking for an initial capital x ≥ 0 and an admissible tradingstrategy θ ∈ A(x), such that :∫ TV x,θT= x +0θ t dS t ≥ H = g(S T ), a.s.In this case, we say that the contingent claim H is superhedged (dominated).We define then the superreplication (superhedging) cost of H as the leastinitial capital that allows the superhedging of H :U 0 = inf{x ≥ 0 : ∃θ ∈ A(x), V x,θT≥ g(S T )}.This is a nonstandard stochastic control problem and we describe differentmethods for solving this problem, i.e. calculate U 0 and the associatedoptimal control.3.1 Dual approachThe starting point of the dual approach is to notice that for any Q ∈ P, x ≥0 and θ ∈ A(x), the wealth process V x,θ is a nonnegative local martingale,hence a supermartingale, under Q. It follows that :E Q [V x,θT] ≤ x. (3.1)6

Now, let x ≥ 0 such that there exists θ ∈ A(x) : X x,θT≥ g(S T ). By (3.1),we then have :E Q [g(S T )] ≤ x, ∀ Q ∈ P,and so by definition of the superreplication cost :V 0 := sup E Q [g(S T )] ≤ U 0Q∈PThe dual approach consists in studying and calculating V 0 and then U 0 .Remark 3.1 Actually, we have the equality V 0 = U 0 . The converse inequalityV 0 ≥ U 0 is proved by using optional decomposition for supermartingales,first proved by El Karoui and Quenez (1995) and then extended by Kramkov(1996). This theorem states that if V is a supermartingale under any Q ∈P, then V admits a decomposition of the form :∫ tV t = V 0 + θ u dS u − C t , 0 ≤ t ≤ T, (3.2)0where θ ∈ L(S) and C is an optional nondecreasing process. Applying thistheorem to the RCLL version of the process V t = esssup Q∈P E Q [g(S T )|F t ],we deduce from (3.2) for t = T , that g(S T ) ≤ V V 0,θT. By definition of thesuperreplication cost, this shows that V 0 ≥ U 0 . We mention that we don’tneed the equality in the dual approach but only the easy inequality V 0 ≤U 0 .Application : Stochastic volatility modelsWe use the dual approach to the computation of V 0 and of the superreplicationcost U 0 in the context of stochastic volatility models. This applicationis developed in Cvitanić, Pham and Touzi (1999a), see also Frey and Sin(1999).We consider a standard stochastic volatility diffusion model :)dS t = S t(µ(t, S t , Y t )dt + σ(t, S t , Y t )dWt1 (3.3)dY t = η(t, S t , Y t )dt + γ(t, S t , Y t )dW 2 t . (3.4)7

Here Y is an exogeneous factor influencing the volatility of the stockprice S and IF is the augmented filtration generated by the two-dimensionalBrownian motion (W 1 , W 2 ). In this model, we have a parametrization ofequivalent martingale measures : Consider the processZ ν t = exp(−(exp −∫ t0∫ t0µσ (u, S u, Y u )dWu 1 − 1 2ν u dWu 2 − 1 ∫ t )ν 22udu ,0∫ t0( µσ) 2(u, S u , Y u )du)where ν is an IF -adapted process satisfying ∫ T0 ν2 udu < ∞. Denote by D theset of processes ν such that E[ZT ν ] = 1. Then one can define a probabilitymeasure P ν equivalent to P , by dP ν /dP = ZT ν , and we have P = {P ν , ν ∈D}. It follows that :V 0 = V (0, S 0 , Y 0 ) := sup E P ν [g(S T )] (≤ U 0 ).ν∈DWe are then led to a more standard stochastic control problem, by studyingthe value function V (t, s, y). Indeed, by usual dynamic programmingprinciple, one shows that the value function V is a supersolution (in theviscosity sense) of the Bellman equation :− ∂V{∂t + inf −(η − νγ) ∂Vν∈IR∂y − 1 2 σ2 s 2 ∂2 V∂s 2 − 1 }2 γ2 ∂2 V∂y 2= 0,for all (t, s, y) ∈ [0, T ) × (0, ∞) × IR, and satisfies the terminal conditionV (T − , s, y) ≥ g(s).We show (formally by sending ν to ±∞ in Bellman equation) that :V does not depend on y : V (t, s, y) = V (t, s)so that V is supersolution of :.infy∈IR{− ∂V∂t − 1 }2 σ2 (t, s, y)s 2 ∂2 V∂s 2= 0, (3.5)8

At this point, the explicit calculation of V depends on the interval of variationof the volatility :[infyσ(t, s, y), sup σ(t, s, y)] = [σ(t, s), ¯σ(t, s)].We shall distinguish two cases.Case 1 : unbounded volatility modely¯σ(t, s) = ∞ and σ(t, s) = 0. (3.6)From the Bellman equation and the conditions (3.6), we show that :V is concave in s and V is nonincreasing in tUsing also the terminal condition V (T − , s, y) ≥ g(s), this shows that V (0, S 0 )≥ ĝ(S 0 ), where ĝ is the concave envelope of g, i.e. the least concave majorantfunction of g. Moreover, since ĝ is concave and is a majorant of g, wehave :ĝ(S 0 ) + ĝ ′ −(S 0 )(S T − S 0 ) ≥ ĝ(S T ) ≥ g(S T ),where ĝ ′ − is the left derivative of ĝ. This shows that g(S T ) can be superhedgedfrom an initial capital ĝ(S 0 ) and following the constant strategyĝ ′ −(S 0 ). We deduce that ĝ(S 0 ) ≥ U 0 ≥ V (0, S 0 ). In conclusion, we obtain :U 0 = V (0, S 0 ) = ĝ(S 0 )θ ∗ = ĝ ′ −(S 0 ) (constant : trivial buy-and-hold strategy).Case 2 :Paras)Model with bounded volatility (Avellaneda, Lévy and¯σ(t, s) < ∞ and σ(t, s) ≥ ε > 0.Then there exists an unique smooth solution W to the nonlinear PDE,called Black-Scholes-Barenblatt (BSB in short) PDE :9

− ∂W ∂t + 1 (∂2 ¯σ2 (t, s)s 2 2 )W∂s 2+− 1 (∂2 σ2 (t, s)s 2 2 )W∂s 2−= 0, (3.7)W (T, s) = g(s). (3.8)Actually, this Black-Scholes-Barenblatt PDE is the Bellman equation :{inf − ∂Vy∈IR ∂t − 1 }2 σ2 (t, s, y)s 2 ∂2 V∂s 2 = 0.By the maximum principle, we deduce that W ≤ V . Moreover, by Itô’sformula and using the fact that W solves the above Bellman equation, wehave :∫ Tg(S T ) = W (T, S T ) ≤ W (0, S 0 ) +0∂W∂s (t, S t)dS t ,and then by definition of the superreplication cost, W (0, S 0 ) ≥ U 0 ≥ V (0, S 0 ).In conclusion, we have that U 0 = W (0, S 0 ) (= V (0, S 0 )) is the unique smoothsolution of the BSB equation (3.7)-(3.8), and the optimal control is givenby :θ ∗ t = ∂W∂s (t, S t).Other applicationsBy the dual approach, one can also calculate the superreplication cost in :- models with jumps : see Eberlein and Jacod (1997) and Bellamy andJeanblanc (1998),- models with transaction costs : see Cvitanić, Pham and Touzi (1999b).3.2 Direct approachSoner and Touzi (1998) developed a dynamic programming principle directlyon the primal problem :U 0 = inf{x ≥ 0 : ∃θ ∈ A(x), V x,θT≥ g(S T )},10

and derived then an associated Bellman equation for U 0 . This Bellmanequation is actually the PDE (3.5) in the case of the stochastic volatilitymodel (3.3)-(3.4). Such a direct approach allows to study superreplicationin models with gamma constraints, i.e. constraints on the sensibility of θwith respect to stock price.3.3 BSDE approachThe superhedging problem can also be viewed as a problem of finding atriple (V, θ, C) of adapted processes, with C nonincreasing, solution of thebackward stochastic differential equation :dV t = θ t dS t − dC tV t = g(S T ).Such a method is studied in El Karoui, Peng and Quenez (1997), see alsoYong (1999).4 Mean-variance hedgingWe are looking for a strategy θ which minimizes the quadratic error ofreplication between the contingent claim H ∈ L 2 (P ) and the terminal wealthV x,θT= x + ∫ T0 θ tdS t :[minimize over θEH − x −∫ T0θ t dS t] 2. (4.1)This is problem of L 2 -projection of a random variable on a space of stochasticintegrals. In order to ensure the existence of a solution to this quadraticoptimization problem, we need to precise the class of admissible tradingstrategies θ so that the space of stochastic integrals is closed in L 2 (P ).Following Delbaen and Schachermayer (1996), we introduce the subset P 2of probability measure Q in P with square-integrable density : dQ/dP ∈L 2 (P ), and we assume that P 2 is nonempty. We then define the integrabilityconditions on the trading strategies :{Θ 2 =θ ∈ L(S) :∫ T011θ t dS t ∈ L 2 (P ) and

∫θdS is a Q − martingale, ∀Q ∈ P 2}.It is showed in Delbaen and Schachermayer (1996) that the set G T (Θ 2 ) ={ ∫ T0 θ tdS t : θ ∈ Θ 2 } is closed in L 2 (P ) so that for any H ∈ L 2 (P ), theproblemJ 2 (x) = minθ∈Θ 2E[H − x −∫ ] T 2θ t dS t ,0admits a solution. We now focus on the characterization of the solution.4.1 Case S martingale under PIn this paragraph, we assume that S is a continuous local martingale underP . This case was first considered in Föllmer and Sondermann (1986). Then,given H ∈ L 2 (P ), the Kunita-Watanabe projection theorem provides :∫ TH = E[H] + θt H dS t + R T ,0where θ H ∈ Θ 2 and R is a square-integrable martingale orthogonal to S. Itfollows immediately that the solution to J 2 (x) is given by :θ ∗ = θ H .ExampleConsider the stochastic volatility model :dS t = S t σ(t, S t , Y t )dW 1 tdY t = η(t, S t , Y t )dt + γ(t, S t , Y t )dW 2 t .Then, the integrand in the Kunita-Watanabe projection of H = g(S T ) is :θ H t= ∂V∂s (t, S t),where V (t, s, y) = E[g(S T )|S t = s, Y t = y]. More general applications andexamples are studied in Bouleau and Lamberton (1989).12

4.2 Case S semimartingaleWe now turn to the general situation where S is a continuous semimartingaleunder P . Recall that there exists a solution θ ∗ (x) ∈ Θ 2 to the problem J 2 (x).Characterization of the solution has been obtained by Duffie and Richardson(1991), Schweizer (1994), Hipp (1993) and Pham, Rheinländer and Schweizer(1998) under more and less restrictive assumptions. The most general resultsare obtained for the case where S is a continuous semimartingale,independently by Gouriéroux, Laurent and Pham (1998) (GLP in short)and Rheinländer and Schweizer (1997) (RS in short). We present here theapproach of the former authors. Their basic idea is to state an invarianceproperty of the space of stochastic integrals by a change of numéraire, andto combine this change of coordinates with an appropriate change of probabilitymeasure in order to transform J 2 (x) into an equivalent L 2 -projectionproblem corresponding to the martingale case.The suitable change of probability measure and change of coordinates usethe so-called variance-optimal martingale measure and hedging numéraire.Under the standing assumption (2.1) and the condition that S is continuous,Delbaen and Schachermayer (1996) prove that there exists an uniquesolution ˜P , called variance-optimal martingale measure, to the problemmin EQ∈P 2[ dQdP] 2. (4.2)Moreover, there exists ˜θ ∈ Θ 2 such that˜Z t := E ˜P [ d ˜P]dP ∣ F t] 2= V 1,˜θt ,EP a.s., 0 ≤ t ≤ T, (4.3)[ d ˜PdPand ˜θ, called hedging numéraire, is solution of the optimization problem :[ ]min E V 1,θ 2θ∈ΘT . (4.4)2It follows that the process ˜Z is a strictly positive Q-martingale for anyQ ∈ P 2 , with initial value 1. We can then associate to each Q ∈ P 2 theprobability measure ˜Q ∼ Q defined by :d ˜Q=dQ∣ ˜Z t , 0 ≤ t ≤ T. (4.5)Ft13

We denote then by ˜P 2 the set of all elements ˜Q ∼ Q defined by (4.5) when Qvaries in P 2 . In particular, we associate to ˜P ∈ P 2 the probability measure˜˜P ∈ ˜P 2 defined by the operator ∼ in (4.5). Notice that by definition of ˜Zin (4.3), the Radon-Nikodym density of ˜˜P with respect to P can be writtenas :d ˜˜P[ ] 2ddP = E ˜P˜Z2dP T . (4.6)We consider the IR d+1 -valued continuous process ˜X with ˜X 0 = 1/ ˜Z and˜X i = S i / ˜Z, i = 1, . . . , d. As a direct consequence of the fact that S isa continuous local martingale under any Q ∈ P 2 and Bayes formula, wededuce that the process ˜X is a continuous local martingale under any ˜Q∈ ˜P 2 . We denote by ˜Φ 2 the set of all IR d+1 -valued predictable processesφ ˜X-integrable, such that ∫ T0 φ td ˜X t ∈ L 2 ( ˜˜P ∫ ) and φd ˜X is a ˜Q-martingaleunder any ˜Q ∈ ˜P 2 .The following result is crucial in the method of resolution in GLP.Theorem 4.1 Assume that S is continuous. Let x ∈ IR. Then we have :{ ({}∫ ) }TV x,θT: θ ∈ Θ 2 = ˜Z T x + φ t d ˜X t : φ ∈ ˜Φ 2 . (4.7)Moreover, the relation between θ = (θ 1 , . . . , θ d ) ′ ∈ Θ 2 and φ = (φ 0 , . . . , φ d ) ′∈ ˜Φ 2 is given by :0φ 0 = V x,θ − θ ′ S and φ i = θ i , i = 1, . . . , d, (4.8)and∫θ i = φ i +(x ˜θ i +)φd ˜X − φ ′ ˜X , i = 1, . . . , d. (4.9)Proof. We follow arguments of GLP and RS. The proof is mainly based onItô’s product rule and for simplicity we omit the integrability questions.(1) By Itô’s product rule, we have :( )d S˜Z( )= Sd + 1˜Z 1˜Z dS + d < S, 1˜Z > . (4.10)14

Let θ ∈ Θ 2 . Then we have :∫ ( ) ∫ ( ) ∫θd = θSd +S˜Z1˜Zθ 1˜Z ∫dS +θd < S, 1˜Z > . (4.11)By Itô’s formula, the self-financing condition dV x,θ = θdS, and by (4.11) weobtain :( )Vx,θ( )d = V˜Zx,θ d + 1˜Z 1˜Z θdS + θ ′ d < S, 1˜Z >() ( ) ( )= V x,θ − θ ′ S d + θd 1˜ZS˜Z= φd ˜X, (4.12)with an ˜X-integrable process φ given by (4.8). Then relation (4.12) showsthat∫( ∫ )V x,θ = x + θdS = ˜Z x + φd ˜X . (4.13)Since ˜Z T and ∫ T0 θ tdS t ∈ L 2 (P ), we have ˜Z ∫ TT 0 φ td ˜X t ∈ L 2 (P ) or equivalentlyby (4.6) ∫ T0 φ td ˜X t ∈ L 2 ( ˜˜P ∫ ). Since θdS is a Q-martingale underany Q ∈ P 2 , we deduce by definition of ˜P 2 and (4.13) that ∫ φd ˜X is a ˜Qmartingaleunder any ˜Q ∈ ˜P 2 and so φ ∈ ˜Φ 2 . Therefore the inclusion ⊆ in(4.7) is proved.(2) The proof of the converse is very similar. By Itô’s product rule, wehave :d( ˜ZX) = ˜Zd ˜X + ˜Xd ˜Z + d < ˜Z, ˜X > . (4.14)Let φ ∈ ˜Φ 2 . By Itô’s formula, (4.14), (4.3) and definition of ˜X, we have :( ∫ ) ( ∫ )d ˜Z(x + φd ˜X) = x + φd ˜X d ˜Z + ˜Zφd ˜X + φ ′ d < ˜Z, ˜X >( ∫ )= x + φd ˜X d ˜Z + φd( ˜Z ˜X) − φ ˜Xd ˜Z= θdS,with the S-integrable process θ given by (4.9). We then obtain :( ∫ ) ∫˜Z x + φd ˜X = x + θdS. (4.15)15

Since ∫ T0 φ td ˜X t ∈ L 2 ( ˜˜P ∫ ), we see from (4.6) and (4.15) that T0 θ tdS t ∈ L 2 (P ).Moreover, ∫ φd ˜X is a ˜Q-martingale for any ˜Q ∈ ˜P 2 and so ∫ θdS is a Q-martingale for all Q ∈ P 2 . This shows that θ ∈ Θ 2 and so the inclusion ⊇in (4.7) is proved.✷By (4.6), we have H/ ˜Z T ∈ L 2 ( ˜˜P ) since H ∈ L 2 (P ). Moreover, theprocess ˜X is a continuous local martingale under ˜˜P ∈ ˜P 2 .apply the Kunita-Watanabe projection and obtain :[ ] ∫HT˜P= E + ˜φ˜Z TH˜ZT H t d ˜X t + ˜˜LH T , P a.s.0We can thenwhere ˜φ H ∈ ˜Φ 2 and ˜˜L is a square integrable martingale under ˜˜P orthogonalto ˜X. We have then the following characterization result of a solution tothe mean-variance hedging problem.Theorem 4.2 Assume that S is continuous. Then for all x ∈ IR, thereexists a unique solution θ ∗ (x) ∈ Θ 2 to problem J 2 (x) given by :∫)(θ ∗ (x)) i = ( ˜φ H ) i +(x ˜θ i + ˜φ H d ˜X − ˜φ H′ ˜X , i = 1, . . . , d.(4.16)The associated value function is given by :J 2 (x) =(E ˜P [H] − x) 2+ E˜PE[ d ˜PdPProof. In view of Theorem 4.1 and (4.6), we have :J 2 (x) =E1[ d ˜PdP] 2infφ∈˜Φ 2E˜P] 2[˜˜LH T] 2, x ∈ IR. (4.17)[ ∫ ] T 2− x − φ t d H˜ZT ˜X t . (4.18)0This is an optimization problem as in the martingale case (see Paragraph4.1), and therefore the unique solution of (4.18) is given by ˜φ H . The uniquesolution θ ∗ (x) to J 2 (x) is then obtained via (4.9) from ˜φ H . Moreover, wehave :J 2 (x) =(E˜P[ H˜ZT]− x) 2+ E˜PE[ d ˜PdP16] 2[˜˜LH T] 2, x ∈ IR. (4.19)

By definition of ˜˜P in function of ˜P , we then obtain (4.17).✷Remark 4.1 RS prove that H can be decomposed into :∫ TH = E ˜P [H] + ˜θ t H dS t + ˜L H T ,0where ˜θ H ∈ Θ 2 and ˜L H is a martingale under ˜P orthogonal to S. Theyobtain then a description of the solution to J 2 (x) in feedback form :θt ∗ (x) = ˜θ t H − ˜θ (∫ t )tx + E ˜P [H|F t −] − θu(x)dS ˜Z ∗ u . (4.20)t 0It is also checked in RS that the expression given in (4.20) coincide with theone given in (4.16).Description of the optimal hedging strategy requires finding the varianceoptimalmartingale measure and the hedging numéraire. Hipp (1993), Pham,Rheinländer and Schweizer (1998) studied the special case where the varianceoptimalmartingale measure coincide with the minimal martingale measureof Föllmer and Schweizer (1991). More general results have been obtained byLaurent and Pham (1999) in a multidimensional diffusion model by dynamicprogramming arguments, with applications to stochastic volatility models;see also in this direction the recent works of Heath, Platen and Schweizer(1998) and Biagini, Guasoni, Pratelli (1999).5 Shortfall risk minimizationThis is an alternative criterion to the extrem approach of superreplicationand to the symmetrical approach of the mean-variance hedging criterion.Here one penalizes only situations when :V x,θT≤ H,and we want to minimize the expected shortfall (H − V x,θT) + = max(H −, 0) weighted by some loss function l, i.e. l(0) = 0, l is nondecreasingV x,θTand convex on IR + .17

Given an initial capital x ≥ 0 and a nonnegative contingent claim H, weconsider the stochastic optimization problem :[]min E l(H − V x,θT ) + (P(x))θ∈A(x)Such a problem has been first studied by Föllmer and Leukert (1998a,b)in the context of incomplete semimartingale model. It is studied in thecontext of diffusion models by Cvitanić and Karatzas (1998), by Cvitanić(1998) for diffusion models with portfolio constraints. Pham (1999) extendedthis shortfall risk minimization problem to a general framework includingsemimartingale models with constrained portfolios, large investor models,labor income, reinsurance models.In a first step, notice that by the nondecreasing property of the lossfunction l, we can transform the original dynamic control problem into astatic one :[]min E l(H − V x,θT ) + (P(x))θ∈A(x)where= min E [l(H − X)] := J(x) (5.1)X∈C(x)C(x) = {X F T − measurable : 0 ≤ X ≤ H,}and ∃θ ∈ A(x), X ≤ V x,θT.Now, from the optional decomposition theorem for supermartingaleswhich gives a dual characterization of the superreplication cost (see Remark3.1), we have :∃θ ∈ A(x), X ≤ V x,θT⇐⇒ sup E Q [X] ≤ x. (5.2)Q∈PTherefore the set of constraints C(x) can be written as :C(x) = {X F T − measurable : 0 ≤ X ≤ H,}and sup E Q [X] ≤ xQ∈P. (5.3)We are then amounted to a static convex optimization J(x) with linearconstraints C(x) given by (5.3).18

The following result proves the existence of a solution to the dynamicproblem (P(x)) and relates it to the solution of the static problem J(x). Italso provides some qualitative properties of the associated value function.In the sequel, given a nonnegative contingent claim X, we denote by v 0 (X)= sup Q∈P E Q [X] its superreplication cost.Theorem 5.1 Assume that l(H) ∈ L 1 (P ).(1) For any x ≥ 0, there exists X ∗ (x) ∈ C(x) solution of J(x) and H issolution of J(x) for x ≥ v 0 (H). Moreover, if l is strictly convex, any twosuch solutions coincide P a.s.(2) The function J is nonincreasing and convex on [0, ∞), strictly decreasingon [0, v 0 (H)] and equal to zero on [v 0 (H), ∞). For any x ∈ [0, v 0 (H)], wehave :sup E Q [X ∗ (x)] = x. (5.4)Q∈PMoreover, if l is strictly convex, then J is strictly convex on [0, v 0 (H)].(3) For any x ≥ 0, there exists θ ∗ (x) ∈ A(x) such that X ∗ (x) ≤ V x,θ∗ (x)T,P a.s., and θ ∗ (x) is solution to the dynamic problem (P(x)).Proof. (1) Let x ≥ 0 and (X n ) n ∈ C(x) be a minimizing sequence for theproblem J(x), i.e.lim E[l(H − n→∞ Xn )] = inf E[l(H − X)].X∈C(x)Since X n ≥ 0, then by Lemma A.1.1 of Delbaen and Schachermayer (1994),there exists a sequence of F T -measurable random variables ˆX n ∈ conv(X n , X n+1 ,. . .) such that ˆX n converges almost surely to X ∗ (x) F T -measurable. Since0 ≤ ˆX n ≤ H, we deduce that 0 ≤ X ∗ (x) ≤ H. By Fatou’s lemma, we havefor all Q ∈ P :E Q [X ∗ (x)] ≤ lim infn→∞ EQ [ ˆXn ] ≤ x,hence X ∗ (x) ∈ C(x). Now, since l is convex and l(H) ∈ L 1 (P ), we have bythe dominated convergence theorem :inf E[l(H − X)] = lim E[l(H −X∈C(x) n→∞ Xn )]≥ lim E[l(H − ˆX n )]n→∞= E[l(H − X ∗ (x))],19

which proves that X ∗ (x) solves J(x). Now, suppose that x ≥ v 0 (H). ThenH ∈ C(x) and is obviously solution to J(x), and in this case J(x) = 0.Let X 1 and X 2 be two solutions of J(x) and ε ∈ (0, 1). Set X ε =(1 − ε)X 1 + εX 2 ∈ C(x). By convexity of function l, we have :[] []E [l(H − X ε )] ≤ (1 − ε)E l(H − X 1 ) + ɛE l(H − X 2 ) (5.5)= inf E[l(H − X)]. (5.6)X∈C(x)Suppose that P [X 1 ≠ X 2 ] > 0. Then by the strict convexity of l, we shouldhave strict inequality in (5.5), which is a contradiction with (5.6).(2) Let 0 ≤ x 1 ≤ x 2 . Since C(x 1 ) ⊂ C(x 2 ), we deduce that J(x 2 ) ≤ J(x 1 )and so J is nonincreasing on [0, ∞). Notice also that (X ∗ (x 1 ) + X ∗ (x 2 ))/2∈ C((x 1 + x 2 )/2). Then, by convexity of function l, we have :( ) x1 + x 2J2≤[ (E l H − X∗ (x 1 ) + X ∗ )](x 2 )2≤ 1 2 E [l(H − X∗ (x 1 ))] + 1 2 E [l(H − X∗ (x 2 ))]= 1 2 J(x 1) + 1 2 J(x 2),which proves the convexity of J on [0, ∞). We have already seen that J(x)= 0 for x ≥ v 0 (H). To end the proof of assertion (2), we now suppose thatv 0 (H) > 0 (otherwise there is nothing else to prove). First, notice that sincel is a nonnegative function, cancelling only on 0, it follows that J(x) = 0if and only if X ∗ (x) = H which implies that x ≥ v 0 (H). Therefore, for all0 ≤ x < v 0 (H), we have J(x) > 0. Let us check that J is strictly decreasingon [0, v 0 (H)]. On the contrary, there would exist 0 ≤ x 1 < x 2 < v 0 (H)such that J(x 1 ) = J(x 2 ). Then, there exists α ∈ (0, 1) such that x 2 =αx 1 + (1 − α)v 0 (H). By convexity of function l, we should have :J(x 2 ) ≤ αJ(x 1 ) + (1 − α)J(v 0 (H)) = αJ(x 1 ).Since J(x 1 ) = J(x 2 ) > 0, this would imply that α > 1, a contradiction. Letus now prove (5.4). On the contrary, we should have 0 ≤ ˜x := sup Q∈P E Q [X ∗ (x)]< x. Then X ∗ (x) ∈ C(˜x) and so J(˜x) ≤ E[l(H − X ∗ (x))] = J(x), a contradictionwith the fact that J is strictly decreasing on [0, v 0 (H)]. Let20

0 ≤ x 1 < x 2 ≤ v 0 (H). We have (X ∗ (x 1 ) + X ∗ (x 2 ))/2 ∈ C((x 1 + x 2 )/2).Moreover, since 0 < J(x 2 ) < J(x 1 ), we have X ∗ (x 1 ) ≠ X ∗ (x 2 ). Then, bythe strict convexity of function l, we obtain :( ) x1 + x 2J2≤E[l(H − X∗ (x 1 ) + X ∗ (x 2 )2)]< 1 2 E [l(H − X∗ (x 1 ))] + 1 2 E [l(H − X∗ (x 2 ))]= 1 2 J(x 1) + 1 2 J(x 2),which proves the strict convexity of J on [0, v 0 (H)].(3) The third assertion follows from (5.1) giving the relation between thedynamic problem (P(x)) and the static problem J(x).✷We provide a quantitative description of X ∗ (x) and of θ ∗ (x) solutions ofJ(x) and of (P(x)) by adopting a convex duality approach, which is now astandard tool in financial mathematics, see e.g. Karatzas (1998).We assume that the function l ∈ C 1 (0, ∞), the derivative l ′ is strictlyincreasing with l ′ (0 + ) = 0 and l ′ (∞) = ∞. We denote by I = (l ′ ) −1 theinverse function of l ′ . Starting from the state-dependent convex function 0 ≤x ≤ H ↦→ l(H − x), we consider its stochastic Fenchel-Legendre transform :˜L(y, ω) = max [−l(H − x) − xy] (5.7)0≤x≤H= −l (H ∧ I(y)) − y (H − I(y)) +, y ≥ 0.We now consider the dual control problem :[ ((D(y)) ˜J(y) = inf E ˜L y dQ )]Q∈P dP , ω , y ≥ 0.It is straightforward to see that ˜J is convex on [0, ∞).Our object is to provide a description of the solution to the problem(P(x)) in function of a solution to the problem (D(y)) when it exists. Thiscan be viewed as a verification theorem expressed in terms of the dual controlproblem. In a Markovian context, this is an alternative to the usualverification theorem of stochastic control problems expressed in terms ofthe value function of the primal problem. Notice that, due to the state constraints,the Bellman equation associated to the dynamic primal problem21

will involve non “classical” boundary conditions, which are delicate from atheoretical and numerical viewpoint (see e.g. Fleming and Soner 1993).In the sequel, we shall assume that the nonnegative contingent claim His not equal to zero a.s. and that its superreplication cost is finite. We thenassume that 0 < v 0 (H) < ∞.Theorem 5.2 Assume that l(H) ∈ L 1 (P ) and 0 < v 0 (H) < ∞. Supposethat for all y > 0, there exists a solution Q ∗ (y) ∈ P to problem (D(y)).Then :(1) ˜J is differentiable on (0, ∞) with derivative :for all y > 0.[ ( ( )) ]˜J ′ (y) = −E Q∗ (y)H − I y dQ∗ (y), (5.8)dP +(2) Let 0 < x < v 0 (H). Then, there exists ŷ > 0 that attains the infimumin inf y>0 { ˜J(y) + xy}, and we have :˜J ′ (ŷ) = −x. (5.9)The unique solution of J(x) is then given by :X ∗ (x) =( ( ))H − I ŷ dQ∗ (ŷ).dP +There exists θ ∗ (x) ∈ A(x) such that X ∗ (x) = V x,θ∗ (x)T, P a.s., and θ ∗ (x) issolution to (P(x)). Moreover, we have :V x,θ∗ (x)t = E Q∗ (ŷ) [X ∗ (x)| F t ] , 0 ≤ t ≤ T.(3) We have the duality relation :[J(x) = max − ˜J(y)]− xy , ∀x > 0.y≥0Proof. First notice that the maximum in (5.7) is attained for :χ(y, ω) = (H − I(y)) +, y ≥ 0. (5.10)22

The function ˜L(., ω) is convex, differentiable on (0, ∞) with derivative :˜L ′ (y, ω) = −χ(y, ω), y ≥ 0. (5.11)(1) Let y > 0. Then for all δ > 0, we have :˜J(y + δ) − ˜J(y)δ≤ 1 () ()][˜Lδ E (y + δ) dQ∗ (y)dP , ω − ˜L y dQ∗ (y)dP , ω [ ()]≤ − E Q∗ (y)χ (y + δ) dQ∗ (y)dP , ωwhere we used (5.11) and convexity of ˜L(., ω). By Fatou’s lemma, we deducethat :[ ()]˜J(y + δ) − ˜J(y)lim sup≤ −E Q∗ (y)χ y dQ∗ (y)δ↘0 + δdP , ω . (5.12)Similarly, for all δ < 0, y + δ > 0, we have :˜J(y + δ) − ˜J(y)[ ()]≥ −E Q∗ (y)χ (y + δ) dQ∗ (y)δdP , ω .Since |χ| is bounded by H and under the assumption that v 0 (H) < ∞, onecan apply the dominated convergence theorem to deduce that :lim infδ↗0 −˜J(y + δ) − ˜J(y)δ≥[ ()]−E Q∗ (y)χ y dQ∗ (y)dP , ω . (5.13)Relations (5.12)-(5.13) and convexity of the function ˜J imply the differentiabilityof ˜J and provide the expression (5.8) of ˜J ′ .(2) The function y ↦→ f x (y) = ˜J(y) + xy is convex on (0, ∞). Let uscheck that :lim f x(y) = ∞, ∀x > 0. (5.14)y→∞Indeed, by noting that ˜L(y, ω) ≥ −l(H), we have ˜J(y) ≥ −E[l(H)]. Wededuce that f x (y) ≥ −E[l(H)] + yx, which proves (5.14). We now checkthat for all 0 < x < v 0 (H), there exists y 0 > 0 such that f x (y 0 ) < 0. Onthe contrary, we should have :E[˜L(y dQdP , ω )]+ xy > 0, ∀y > 0, ∀Q ∈ P,23

and then[x > E − 1 (y ˜L y dQ )]dP , ω , ∀y > 0, ∀Q ∈ P.Since −˜L(ydQ/dP, ω)/y ≥ 0 and −˜L(ydQ/dP, ω)/y converges to HdQ/dPas y goes to infinity, we deduce by Fatou’s lemma that :x ≥ E Q [H] , ∀Q ∈ P,and then x ≥ v 0 (H), a contradiction. We can then deduce that for all 0 0 and since ˜J, andso f x , is differentiable on (0, ∞), we have f ′ x(ŷ) = 0, i.e. ˜J ′ (ŷ) = −x.Fix some y > 0 and let Q be an arbitrary element of P. Denote :Q ε = (1 − ε)Q ∗ (y) + εQ, ε ∈ (0, 1).Obviously, Q ε ∈ P so that by definition of ˜J, we have :⎡)0 ≤ 1 ˜L(y dQεε E dP⎣, ω − ˜L( )y dQ∗ (y)dP , ω ⎤⎦y⎡)≤1 ˜L(y dQεε E dP⎣, ω − ˜L( )y dQ∗ (y)dP , ω ⎤⎦y[ ( ))]dQ≤ E −dP − dQ∗ (y)χ(y dQεdP dP , ωwhere the third inequality from (5.11) and the convexity of ˜L. We obtainthen :)][)]E[χ(y Q dQεdP , ω ≤ E Q∗ (y)χ(y dQεdP , ω . (5.15)By the dominated convergence theorem and Fatou’s lemma applied respectivelyto the R.H.S. and the L.H.S. of (5.15), we have :()][ ()]E[χQ y dQ∗ (y)dP , ω ≤ E Q∗ (y)χ y dQ∗ (y)dP , ω .From (5.8) and (5.10), this can be written also as :()]sup E[χQ y dQ∗ (y)Q∈PdP , ω ≤ − ˜J ′ (y),24

for all y > 0. By choosing y = ŷ defined above, we get :()]sup E[χQ ŷ dQ∗ (ŷ)Q∈PdP , ω≤ x, (5.16)( )which proves that X ∗ (x) = χ ŷ dQ∗ (ŷ)dP , ω lies in C(x).Moreover, by definition (5.7) of ˜L and by definition of X ∗ (x), we havefor all X ∈ C(x) :()˜L ŷ dQ∗ (ŷ)dP , ω= −l(H − X ∗ (x)) − ŷ dQ∗ (ŷ)dP X∗ (x) (5.17)≥−l(H − X) − ŷ dQ∗ (ŷ)X. (5.18)dPTaking expectation under P in (5.17)-(5.18) and using the facts that :we obtain that :E Q∗ (ŷ) [X ∗ (x)] = − ˜J ′ (ŷ) = x, (5.19)E Q∗ (ŷ) [X] ≤ x, (5.20)E [l(H − X ∗ (x))] ≤ E [l(H − X)] ,which proves that X ∗ (x) is solution to problem (S(x)). Relations (5.16)and (5.19) show that Q ∗ (ŷ) attains the supremum in sup Q∈P E Q [X ∗ (x)],and by Theorem 5.1, this proves that there exists θ ∗ (x) ∈ A(x) such thatX ∗ (x) ≤ V x,θ∗ (x)T, a.s., and θ ∗ (x) is solution of the dynamic problem (P(x)).Moreover, since the associated wealth process V x,θ∗ (x) is a supermartingaleunder Q ∗ (ŷ), we have from (5.8)-(5.9) :x = E Q∗ (ŷ) [X ∗ (x)] ≤ E Q∗ (ŷ) [V x,θ∗ (x)T] ≤ x,which shows that X ∗ (x) = V x,θ∗ (x)T, a.s., and that the wealth process V x,θ∗ (x)is a martingale under Q ∗ (ŷ). The assertion (2) of Theorem 5.2 is then proved.(3) By definition (5.7) of the function ˜L, we have for all x ≥ 0, y ≥ 0,X ∈ C(x), Q ∈ P :−l(H − X) − y dQ (dP X ≤ ˜L y dQ )dP , ω ,25

hence by taking expectation under P :[ (−E [l(H − X)] − yx ≤ E ˜L y dQ )]dP , ω ,and therefore,[sup − ˜J(y)]− xyy≥0≤ J(x), ∀x ≥ 0. (5.21)For x ≥ v 0 (H), we have J(x) = 0 = − ˜J(0). Fix now 0 < x < v 0 (H). Fromrelations (5.17) and (5.19), we have :˜J(ŷ) =[ dQ−E [l(H − X ∗ ](ŷ)(x))] − ŷEdPX∗ (x)= −J(x) − xŷ,which proves that J(x) = − ˜J(ŷ) − xŷ. The proof is ended.✷The description of the optimal hedging strategy is proceeded in two steps.In the first step, one has to solve a dual problem. Notice that in a markovianframework, such as the stochastic volatility model described in the previoussections, one has a parametrization of the set P and so the dual problemleads to a classical stochastic control problem. The optimal hedging strategyis then obtained as the (super)replicating strategy of a modified contingentclaim, and can be computed via the superhedging approach described inSection 3.References[1] Avellaneda M., Lévy A. and A. Paras (1995) : “Pricing and Hedging DerivativeSecurities with Uncertain Volatilities”, Applied Mathematical Finance, 2, 73-88.[2] Bellamy N. and M. Jeanblanc (1998) : “Incompleteness of Markets driven bya Mixed Diffusion”, to appear in Finance and Stochastics.[3] Bensaid B., Lesne J.P., Pagès H. and J. Scheinkman (1992) : “DerivativeAsset Pricing with Transaction Costs”, Mathematical Finance, 2, 63-86.26

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