Hedging and Optimization Problems in

Continuous-Time Financial Models∗

Huyen PHAM

Laboratoire de Probabilites et

Modeles Aleatoires

UMR 7599

Universite Paris 7

2 Place Jussieu

75251 Paris Cedex 05

e-mail : pham@gauss.math.jussieu.fr

and CREST, Laboratoire de Finance-Assurance.

September 1999

Abstract

This 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 ap-proach. We then study the mean-variance criterion and finally, we de-scribe the 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

1 Introduction

Option hedging is a prominent problem in finance and has become an im-portant field of theoretical and applied research in stochastic analysis andpartial differential equations. We start the introduction with some generalideas and financial motivation before turning to more precise mathematicaldefinitions.

We consider a financial market operating in continuous time and de-scribed by a probability space (Ω,F , P ), a time horizon T and a filtrationIF = Ft0≤t≤T representing the information available at time t. There ared+1 basic assets for trading. The 0-th asset is a bond with strictly positiveprice process and is used as numeraire. This means that we pass to dis-counted quantities so that asset 0 has discounted price equal to one at eachtime, and the i-th asset, i = 1, . . . , d, has a (discounted) price process Si.One central problem in financial mathematics is the pricing and hedging ofcontingent claims by means of dynamic trading strategies based on S. An(European) contingent claim is an FT -measurable random variable describ-ing the payoff at time T of some derivative security. A dynamic portfoliostrategy is pair (V, θ) where V is a real-valued IF -adapted process and θ

is an IRd-valued predictable process. In such a strategy, θit represents the

number of shares of asset i held at time t and Vt is the portfolio wealth.Notice that Vt− θ′tSt is the amount invested in asset 0 at time t. A strategy(V, θ) is called self-financing if :

Vt = V0 +∫ t

0θudSu,

which means that gain or losses on the portfolio wealth are uniquely due toprice fluctuations on S. Notice that a self-financing strategy is completelydetermined by the initial capital V0 and θ.

Suppose now that, given a contingent claim H, there exists a self-financingstrategy (V, θ) whose terminal value VT equals H almost surely. Under thebasic no-arbitrage assumption on the financial market, the price of H mustbe equal to V0 and θ provides a hedging strategy against H. This is the basicidea behind the seminal paper of Black and Scholes (1973) and it is mathe-matically clarified in the martingale theory by Harrison and Pliska (1981) :A contingent claim is attainable if there exists a self-financing strategy with

2

VT = 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 = H0 +∫ T

0θHt dSt. (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 by

empirical studies on financial market, focusing attention of researchers onextensions of the Black-Scholes model : stochastic volatility models, jump-diffusion models. In this case, given an arbitrary contingent claim H, rep-resentation (1.1) is no more possible and we say that the market is incom-plete. 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 VT larger than H : Inthis case, one (super)hedges the contingent claim. This idea, introducedby Bensaid, Lesne, Pages and Scheinkman (1992), is behind the concept ofsuperreplication. An alternative approach is to introduce subjective crite-ria according to which strategies are chosen. The measure of riskiness bya mean-variance criterion was first proposed by Follmer 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 mea-sure of riskiness that circumvents this last point, is proposed by Follmer andLeukert (1998). It consists in minimizing the expected shortfall (H − VT )+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 cri-terion and the final Section 5 is devoted to the shortfall risk minimizationproblem.

I would like to thank the organizers of the ISFMA symposium on math-ematical finance at Fudan university, for their invitation, especially RamaCont, Jinhai Yan, Professors Li Ta-tsien and Jiongmin Yong.

2 Hedging problems

2.1 The model

We consider a financial market that operates in uncertain conditions de-scribed by a probability space (Ω,F , P ) equipped with a filtration IF =Ft0≤t≤T representing the flow of information on [0, T ]. For simplicity, weassume that F0 is trivial and FT = 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 IRd-valuedsemimartingale S.

An important notion in mathematical finance is the set of equivalentmartingale measures : We let

P = 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 6= ∅. (2.1)

This standing assumption is related to some kind of no-arbitrage condi-tion and we refer to Delbaen and Schachermayer (1994) for a general versionof this first fundamental theorem of asset pricing.

A trading portfolio strategy is an IRd-valued predictable process θ, inte-grable with respect to S. We denote θ ∈ L(S) and we refer to Jacod (1979)for vector stochastic integration with respect to a semimartingale. Here θt

represents 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-financed

4

wealth process is governed by :

V x,θt = x +

∫ t

0θudSu, 0 ≤ t ≤ T.

2.2 Formulation of the problem

An (European) contingent claim is an FT -measurable random variable. Typ-ical examples are call option of exercice price κ on the i-th asset Si, H =(Si

T − κ)+ and put option of exercice price κ on the i-th asset Si, H =(κ− Si

T )+. 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,θ

T . We say that the market is complete if every contingent claimH can be represented as :

H = V H0,θH

T = H0 +∫ T

0θHu dSu, (2.2)

for some H0 ∈ 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 H0,θH

T .The initial capital H0 is a fair price of H and θH 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,θ

T and one has to choose a hedgingcriteria. From a mathematical viewpoint, this leads to various stochastic

5

optimization problems. In the sequel, we shall study the following threehedging criteria :

- Superhedging approach- Mean-variance hedging criterion- Shortfall risk minimization

3 The superhedging approach

We are given a nonnegative contingent claim of the form H = g(ST ), whereg is a continuous function from IRd 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,θ

t ≥ 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 :

V x,θT = x +

∫ T

0θtdSt ≥ H = g(ST ), 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 :

U0 = infx ≥ 0 : ∃θ ∈ A(x), V x,θT ≥ g(ST ).

This is a nonstandard stochastic control problem and we describe differ-ent methods for solving this problem, i.e. calculate U0 and the associatedoptimal control.

3.1 Dual approach

The 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 :

EQ[V x,θT ] ≤ x. (3.1)

6

Now, let x ≥ 0 such that there exists θ ∈ A(x) : Xx,θT ≥ g(ST ). By (3.1),

we then have :

EQ[g(ST )] ≤ x, ∀ Q ∈ P,

and so by definition of the superreplication cost :

V0 := supQ∈P

EQ[g(ST )] ≤ U0

The dual approach consists in studying and calculating V0 and then U0.

Remark 3.1 Actually, we have the equality V0 = U0. The converse inequal-ity V0 ≥ U0 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 :

Vt = V0 +∫ t

0θudSu − Ct, 0 ≤ t ≤ T, (3.2)

where θ ∈ L(S) and C is an optional nondecreasing process. Applying thistheorem to the RCLL version of the process Vt = esssupQ∈P EQ[g(ST )|Ft],we deduce from (3.2) for t = T , that g(ST ) ≤ V V0,θ

T . By definition of thesuperreplication cost, this shows that V0 ≥ U0. We mention that we don’tneed the equality in the dual approach but only the easy inequality V0 ≤U0.

Application : Stochastic volatility modelsWe use the dual approach to the computation of V0 and of the superreplica-tion cost U0 in the context of stochastic volatility models. This applicationis developed in Cvitanic, Pham and Touzi (1999a), see also Frey and Sin(1999).

We consider a standard stochastic volatility diffusion model :

dSt = St

(µ(t, St, Yt)dt + σ(t, St, Yt)dW 1

t

)(3.3)

dYt = η(t, St, Yt)dt + γ(t, St, Yt)dW 2t . (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 process

Zνt = exp

(−∫ t

0

µ

σ(u, Su, Yu)dW 1

u −12

∫ t

0

(µ

σ

)2

(u, Su, Yu)du

).

exp(−∫ t

0νudW 2

u −12

∫ t

0ν2

udu

),

where ν is an IF -adapted process satisfying∫ T0 ν2

udu < ∞. Denote by D theset of processes ν such that E[Zν

T ] = 1. Then one can define a probabilitymeasure P ν equivalent to P , by dP ν/dP = Zν

T , and we have P = P ν , ν ∈D. It follows that :

V0 = V (0, S0, Y0) := supν∈D

EP ν[g(ST )] (≤ U0).

We are then led to a more standard stochastic control problem, by study-ing the 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

ν∈IR

−(η − νγ)

∂V

∂y− 1

2σ2s2 ∂2V

∂s2− 1

2γ2 ∂2V

∂y2

= 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

2σ2(t, s, y)s2 ∂2V

∂s2

= 0, (3.5)

8

At this point, the explicit calculation of V depends on the interval of varia-tion of the volatility :

[infy

σ(t, s, y), supy

σ(t, s, y)] = [σ(t, s), σ(t, s)].

We shall distinguish two cases.

Case 1 : unbounded volatility model

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 t

Using also the terminal condition V (T−, s, y)≥ g(s), this shows that V (0, S0)≥ g(S0), where g is the concave envelope of g, i.e. the least concave majo-rant function of g. Moreover, since g is concave and is a majorant of g, wehave :

g(S0) + g′−(S0)(ST − S0) ≥ g(ST ) ≥ g(ST ),

where g′− is the left derivative of g. This shows that g(ST ) can be super-hedged from an initial capital g(S0) and following the constant strategyg′−(S0). We deduce that g(S0) ≥ U0 ≥ V (0, S0). In conclusion, we obtain :

U0 = V (0, S0) = g(S0)

θ∗ = g′−(S0) (constant : trivial buy-and-hold strategy).

Case 2 : Model with bounded volatility (Avellaneda, Levy andParas)

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+

12σ2(t, s)s2

(∂2W

∂s2

)+

− 12σ2(t, s)s2

(∂2W

∂s2

)−

= 0, (3.7)

W (T, s) = g(s). (3.8)

Actually, this Black-Scholes-Barenblatt PDE is the Bellman equation :

infy∈IR

−∂V

∂t− 1

2σ2(t, s, y)s2 ∂2V

∂s2

= 0.

By the maximum principle, we deduce that W ≤ V . Moreover, by Ito’sformula and using the fact that W solves the above Bellman equation, wehave :

g(ST ) = W (T, ST ) ≤ W (0, S0) +∫ T

0

∂W

∂s(t, St)dSt,

and then by definition of the superreplication cost, W (0, S0)≥ U0 ≥ V (0, S0).In conclusion, we have that U0 = W (0, S0) (= V (0, S0)) is the unique smoothsolution of the BSB equation (3.7)-(3.8), and the optimal control is givenby :

θ∗t =∂W

∂s(t, St).

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 Cvitanic, Pham and Touzi (1999b).

3.2 Direct approach

Soner and Touzi (1998) developed a dynamic programming principle directlyon the primal problem :

U0 = infx ≥ 0 : ∃θ ∈ A(x), V x,θT ≥ g(ST ),

10

and derived then an associated Bellman equation for U0. 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 θ

with respect to stock price.

3.3 BSDE approach

The 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 :

dVt = θtdSt − dCt

Vt = g(ST ).

Such a method is studied in El Karoui, Peng and Quenez (1997), see alsoYong (1999).

4 Mean-variance hedging

We are looking for a strategy θ which minimizes the quadratic error ofreplication between the contingent claim H ∈ L2(P ) and the terminal wealthV x,θ

T = x +∫ T0 θtdSt :

minimize over θ E

[H − x−

∫ T

0θtdSt

]2

. (4.1)

This is problem of L2-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 L2(P ).Following Delbaen and Schachermayer (1996), we introduce the subset P2

of probability measure Q in P with square-integrable density : dQ/dP ∈L2(P ), and we assume that P2 is nonempty. We then define the integrabilityconditions on the trading strategies :

Θ2 =

θ ∈ L(S) :

∫ T

0θtdSt ∈ L2(P ) and

11

∫θdS is a Q−martingale, ∀Q ∈ P2

.

It is showed in Delbaen and Schachermayer (1996) that the set GT (Θ2) =∫ T0 θtdSt : θ ∈ Θ2 is closed in L2(P ) so that for any H ∈ L2(P ), the

problem

J2(x) = minθ∈Θ2

E

[H − x−

∫ T

0θtdSt

]2

,

admits a solution. We now focus on the characterization of the solution.

4.1 Case S martingale under P

In this paragraph, we assume that S is a continuous local martingale underP . This case was first considered in Follmer and Sondermann (1986). Then,given H ∈ L2(P ), the Kunita-Watanabe projection theorem provides :

H = E[H] +∫ T

0θHt dSt + RT ,

where θH ∈ Θ2 and R is a square-integrable martingale orthogonal to S. Itfollows immediately that the solution to J2(x) is given by :

θ∗ = θH .

ExampleConsider the stochastic volatility model :

dSt = Stσ(t, St, Yt)dW 1t

dYt = η(t, St, Yt)dt + γ(t, St, Yt)dW 2t .

Then, the integrand in the Kunita-Watanabe projection of H = g(ST ) is :

θHt =

∂V

∂s(t, St),

where V (t, s, y) = E[g(ST )|St = s, Yt = y]. More general applications andexamples are studied in Bouleau and Lamberton (1989).

12

4.2 Case S semimartingale

We now turn to the general situation where S is a continuous semimartingaleunder P . Recall that there exists a solution θ∗(x) ∈ Θ2 to the problem J2(x).Characterization of the solution has been obtained by Duffie and Richardson(1991), Schweizer (1994), Hipp (1993) and Pham, Rheinlander and Schweizer(1998) under more and less restrictive assumptions. The most general re-sults are obtained for the case where S is a continuous semimartingale,independently by Gourieroux, Laurent and Pham (1998) (GLP in short)and Rheinlander 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 numeraire, andto combine this change of coordinates with an appropriate change of proba-bility measure in order to transform J2(x) into an equivalent L2-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 numeraire.Under the standing assumption (2.1) and the condition that S is continu-ous, Delbaen and Schachermayer (1996) prove that there exists an uniquesolution P , called variance-optimal martingale measure, to the problem

minQ∈P2

E

[dQ

dP

]2. (4.2)

Moreover, there exists θ ∈ Θ2 such that

Zt :=EP

[dPdP

∣∣∣Ft

]E[

dPdP

]2 = V 1,θt , P a.s., 0 ≤ t ≤ T, (4.3)

and θ, called hedging numeraire, is solution of the optimization problem :

minθ∈Θ2

E[V 1,θ

T

]2. (4.4)

It follows that the process Z is a strictly positive Q-martingale for anyQ ∈ P2, with initial value 1. We can then associate to each Q ∈ P2 theprobability measure Q ∼ Q defined by :

dQ

dQ

∣∣∣∣∣Ft

= Zt, 0 ≤ t ≤ T. (4.5)

13

We denote then by P2 the set of all elements Q ∼ Q defined by (4.5) when Q

varies in P2. In particular, we associate to P ∈ P2 the probability measure˜P ∈ P2 defined by the operator ∼ in (4.5). Notice that by definition of Z

in (4.3), the Radon-Nikodym density of ˜P with respect to P can be writtenas :

d˜P

dP= E

[dP

dP

]2

Z2T . (4.6)

We consider the IRd+1-valued continuous process X with X0 = 1/Z andXi = Si/Z, i = 1, . . . , d. As a direct consequence of the fact that S isa continuous local martingale under any Q ∈ P2 and Bayes formula, wededuce that the process X is a continuous local martingale under any Q

∈ P2. We denote by Φ2 the set of all IRd+1-valued predictable processesφ X-integrable, such that

∫ T0 φtdXt ∈ L2( ˜P ) and

∫φdX is a Q-martingale

under any Q ∈ P2.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 :V x,θ

T : θ ∈ Θ2

=

ZT

(x +

∫ T

0φtdXt

): φ ∈ Φ2

. (4.7)

Moreover, the relation between θ = (θ1, . . . , θd)′ ∈ Θ2 and φ = (φ0, . . . , φd)′

∈ Φ2 is given by :

φ0 = V x,θ − θ′S and φi = θi, i = 1, . . . , d, (4.8)

and

θi = φi + θi(

x +∫

φdX − φ′X

), i = 1, . . . , d. (4.9)

Proof. We follow arguments of GLP and RS. The proof is mainly based onIto’s product rule and for simplicity we omit the integrability questions.

(1) By Ito’s product rule, we have :

d

(S

Z

)= Sd

(1Z

)+

1Z

dS + d < S,1Z

> . (4.10)

14

Let θ ∈ Θ2. Then we have :∫θd

(S

Z

)=

∫θSd

(1Z

)+∫

θ1Z

dS +∫

θd < S,1Z

> . (4.11)

By Ito’s formula, the self-financing condition dV x,θ = θdS, and by (4.11) weobtain :

d

(V x,θ

Z

)= V x,θd

(1Z

)+

1Z

θdS + θ′d < S,1Z

>

=(V x,θ − θ

′S)

d

(1Z

)+ θd

(S

Z

)= φdX, (4.12)

with an X-integrable process φ given by (4.8). Then relation (4.12) showsthat

V x,θ = x +∫

θdS = Z

(x +

∫φdX

). (4.13)

Since ZT and∫ T0 θtdSt ∈ L2(P ), we have ZT

∫ T0 φtdXt ∈ L2(P ) or equiv-

alently by (4.6)∫ T0 φtdXt ∈ L2( ˜P ). Since

∫θdS is a Q-martingale under

any Q ∈ P2, we deduce by definition of P2 and (4.13) that∫

φdX is a Q-martingale under any Q ∈ P2 and so φ ∈ Φ2. Therefore the inclusion ⊆ in(4.7) is proved.

(2) The proof of the converse is very similar. By Ito’s product rule, wehave :

d(ZX) = ZdX + XdZ + d < Z, X > . (4.14)

Let φ ∈ Φ2. By Ito’s formula, (4.14), (4.3) and definition of X, we have :

d

(Z(x +

∫φdX)

)=

(x +

∫φdX

)dZ + ZφdX + φ

′d < Z, X >

=(

x +∫

φdX

)dZ + φd(ZX)− φXdZ

= θdS,

with the S-integrable process θ given by (4.9). We then obtain :

Z

(x +

∫φdX

)= x +

∫θdS. (4.15)

15

Since∫ T0 φtdXt ∈ L2( ˜P ), we see from (4.6) and (4.15) that

∫ T0 θtdSt ∈ L2(P ).

Moreover,∫

φdX is a Q-martingale for any Q ∈ P2 and so∫

θdS is a Q-martingale for all Q ∈ P2. This shows that θ ∈ Θ2 and so the inclusion ⊇in (4.7) is proved. 2

By (4.6), we have H/ZT ∈ L2( ˜P ) since H ∈ L2(P ). Moreover, the

process X is a continuous local martingale under ˜P ∈ P2. We can thenapply the Kunita-Watanabe projection and obtain :

H

ZT

= E˜P [ H

ZT

]+∫ T

0φH

t dXt + ˜L

H

T , P a.s.

where φH ∈ Φ2 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) ∈ Θ2 to problem J2(x) given by :

(θ∗(x))i = (φH)i + θi(

x +∫

φHdX − φH′X

), i = 1, . . . , d.(4.16)

The associated value function is given by :

J2(x) =

(EP [H]− x

)2+ E

˜P [˜LH

T

]2E[

dPdP

]2 , x ∈ IR. (4.17)

Proof. In view of Theorem 4.1 and (4.6), we have :

J2(x) =1

E[

dPdP

]2 infφ∈Φ2

E˜P [ H

ZT

− x−∫ T

0φtdXt

]2

. (4.18)

This is an optimization problem as in the martingale case (see Paragraph4.1), and therefore the unique solution of (4.18) is given by φH . The uniquesolution θ∗(x) to J2(x) is then obtained via (4.9) from φH . Moreover, wehave :

J2(x) =

(E˜P [ H

ZT

]− x

)2

+ E˜P [˜LH

T

]2E[

dPdP

]2 , x ∈ IR. (4.19)

16

By definition of ˜P in function of P , we then obtain (4.17). 2

Remark 4.1 RS prove that H can be decomposed into :

H = EP [H] +∫ T

0θHt dSt + LH

T ,

where θH ∈ Θ2 and LH is a martingale under P orthogonal to S. Theyobtain then a description of the solution to J2(x) in feedback form :

θ∗t (x) = θHt − θt

Zt

(x + EP [H|Ft− ]−

∫ t

0θ∗u(x)dSu

). (4.20)

It 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 variance-optimal martingale measure and the hedging numeraire. Hipp (1993), Pham,Rheinlander and Schweizer (1998) studied the special case where the variance-optimal martingale measure coincide with the minimal martingale measureof Follmer 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 minimization

This 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 −

V x,θT , 0) weighted by some loss function l, i.e. l(0) = 0, l is nondecreasing

and convex on IR+.

17

Given an initial capital x ≥ 0 and a nonnegative contingent claim H, weconsider the stochastic optimization problem :

minθ∈A(x)

E[l(H − V x,θ

T )+]

(P(x))

Such a problem has been first studied by Follmer and Leukert (1998a,b)in the context of incomplete semimartingale model. It is studied in thecontext of diffusion models by Cvitanic and Karatzas (1998), by Cvitanic(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 :

minθ∈A(x)

E[l(H − V x,θ

T )+]

(P(x))

= minX∈C(x)

E [l(H −X)] := J(x) (5.1)

where

C(x) = X FT −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

Q∈PEQ[X] ≤ x. (5.2)

Therefore the set of constraints C(x) can be written as :

C(x) = X FT −measurable : 0 ≤ X ≤ H,

and supQ∈P

EQ[X] ≤ x

. (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 v0(X)= supQ∈P EQ[X] its superreplication cost.

Theorem 5.1 Assume that l(H) ∈ L1(P ).(1) For any x ≥ 0, there exists X∗(x) ∈ C(x) solution of J(x) and H issolution of J(x) for x ≥ v0(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, v0(H)] and equal to zero on [v0(H),∞). For any x ∈ [0, v0(H)], wehave :

supQ∈P

EQ [X∗(x)] = x. (5.4)

Moreover, if l is strictly convex, then J is strictly convex on [0, v0(H)].

(3) For any x ≥ 0, there exists θ∗(x) ∈ A(x) such that X∗(x) ≤ Vx,θ∗(x)T ,

P a.s., and θ∗(x) is solution to the dynamic problem (P(x)).

Proof. (1) Let x ≥ 0 and (Xn)n ∈ C(x) be a minimizing sequence for theproblem J(x), i.e.

limn→∞

E[l(H −Xn)] = infX∈C(x)

E[l(H −X)].

Since Xn ≥ 0, then by Lemma A.1.1 of Delbaen and Schachermayer (1994),there exists a sequence of FT -measurable random variables Xn ∈ conv(Xn, Xn+1,

. . .) such that Xn converges almost surely to X∗(x) FT -measurable. Since0 ≤ Xn ≤ H, we deduce that 0 ≤ X∗(x) ≤ H. By Fatou’s lemma, we havefor all Q ∈ P :

EQ [X∗(x)] ≤ lim infn→∞

EQ[Xn]≤ x,

hence X∗(x) ∈ C(x). Now, since l is convex and l(H) ∈ L1(P ), we have bythe dominated convergence theorem :

infX∈C(x)

E[l(H −X)] = limn→∞

E[l(H −Xn)]

≥ limn→∞

E[l(H − Xn)]

= E[l(H −X∗(x))],

19

which proves that X∗(x) solves J(x). Now, suppose that x ≥ v0(H). ThenH ∈ C(x) and is obviously solution to J(x), and in this case J(x) = 0.

Let X1 and X2 be two solutions of J(x) and ε ∈ (0, 1). Set Xε =(1− ε)X1 + εX2 ∈ C(x). By convexity of function l, we have :

E [l(H −Xε)] ≤ (1− ε)E[l(H −X1)

]+ εE

[l(H −X2)

](5.5)

= infX∈C(x)

E[l(H −X)]. (5.6)

Suppose that P [X1 6= X2] > 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 ≤ x1 ≤ x2. Since C(x1) ⊂ C(x2), we deduce that J(x2) ≤ J(x1)and so J is nonincreasing on [0,∞). Notice also that (X∗(x1) + X∗(x2))/2∈ C((x1 + x2)/2). Then, by convexity of function l, we have :

J

(x1 + x2

2

)≤ E

[l

(H − X∗(x1) + X∗(x2)

2

)]≤ 1

2E [l(H −X∗(x1))] +

12E [l(H −X∗(x2))]

=12J(x1) +

12J(x2),

which proves the convexity of J on [0,∞). We have already seen that J(x)= 0 for x ≥ v0(H). To end the proof of assertion (2), we now suppose thatv0(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 ≥ v0(H). Therefore, for all0 ≤ x < v0(H), we have J(x) > 0. Let us check that J is strictly decreasingon [0, v0(H)]. On the contrary, there would exist 0 ≤ x1 < x2 < v0(H)such that J(x1) = J(x2). Then, there exists α ∈ (0, 1) such that x2 =αx1 + (1− α)v0(H). By convexity of function l, we should have :

J(x2) ≤ αJ(x1) + (1− α)J(v0(H)) = αJ(x1).

Since J(x1) = J(x2) > 0, this would imply that α > 1, a contradiction. Letus now prove (5.4). On the contrary, we should have 0≤ x := supQ∈P EQ[X∗(x)]< x. Then X∗(x) ∈ C(x) and so J(x) ≤ E[l(H − X∗(x))] = J(x), a con-tradiction with the fact that J is strictly decreasing on [0, v0(H)]. Let

20

0 ≤ x1 < x2 ≤ v0(H). We have (X∗(x1) + X∗(x2))/2 ∈ C((x1 + x2)/2).Moreover, since 0 < J(x2) < J(x1), we have X∗(x1) 6= X∗(x2). Then, bythe strict convexity of function l, we obtain :

J

(x1 + x2

2

)≤ E

[l

(H − X∗(x1) + X∗(x2)

2

)]<

12E [l(H −X∗(x1))] +

12E [l(H −X∗(x2))]

=12J(x1) +

12J(x2),

which proves the strict convexity of J on [0, v0(H)].

(3) The third assertion follows from (5.1) giving the relation between thedynamic problem (P(x)) and the static problem J(x). 2

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 ∈ C1(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 7→ l(H − x), we consider its stochastic Fenchel-Legendre transform :

L(y, ω) = max0≤x≤H

[−l(H − x)− xy] (5.7)

= −l (H ∧ I(y))− y (H − I(y))+ , y ≥ 0.

We now consider the dual control problem :

(D(y)) J(y) = infQ∈P

E

[L

(ydQ

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 con-trol problem. 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 con-straints, the Bellman equation associated to the dynamic primal problem

21

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 H

is not equal to zero a.s. and that its superreplication cost is finite. We thenassume that 0 < v0(H) < ∞.

Theorem 5.2 Assume that l(H) ∈ L1(P ) and 0 < v0(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 :

J ′(y) = −EQ∗(y)

[(H − I

(ydQ∗(y)

dP

))+

], (5.8)

for all y > 0.

(2) Let 0 < x < v0(H). Then, there exists y > 0 that attains the infimumin infy>0J(y) + xy, and we have :

J ′(y) = −x. (5.9)

The unique solution of J(x) is then given by :

X∗(x) =(

H − I

(ydQ∗(y)

dP

))+

.

There exists θ∗(x) ∈ A(x) such that X∗(x) = Vx,θ∗(x)T , P a.s., and θ∗(x) is

solution to (P(x)). Moreover, we have :

Vx,θ∗(x)t = EQ∗(y) [X∗(x)| Ft] , 0 ≤ t ≤ T.

(3) We have the duality relation :

J(x) = maxy≥0

[−J(y)− xy

], ∀x > 0.

Proof. 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

≤ 1δE

[L

((y + δ)

dQ∗(y)dP

, ω

)− L

(ydQ∗(y)

dP, ω

)]≤ − EQ∗(y)

[χ

((y + δ)

dQ∗(y)dP

, ω

)]where we used (5.11) and convexity of L(., ω). By Fatou’s lemma, we deducethat :

lim supδ0+

J(y + δ)− J(y)δ

≤ −EQ∗(y)[χ

(ydQ∗(y)

dP, ω

)]. (5.12)

Similarly, for all δ < 0, y + δ > 0, we have :

J(y + δ)− J(y)δ

≥ −EQ∗(y)[χ

((y + δ)

dQ∗(y)dP

, ω

)].

Since |χ| is bounded by H and under the assumption that v0(H) < ∞, onecan apply the dominated convergence theorem to deduce that :

lim infδ0−

J(y + δ)− J(y)δ

≥ −EQ∗(y)[χ

(ydQ∗(y)

dP, ω

)]. (5.13)

Relations (5.12)-(5.13) and convexity of the function J imply the differen-tiability of J and provide the expression (5.8) of J ′.

(2) The function y 7→ fx(y) = J(y) + xy is convex on (0,∞). Let uscheck that :

limy→∞

fx(y) = ∞, ∀x > 0. (5.14)

Indeed, by noting that L(y, ω) ≥ −l(H), we have J(y) ≥ −E[l(H)]. Wededuce that fx(y) ≥ −E[l(H)] + yx, which proves (5.14). We now checkthat for all 0 < x < v0(H), there exists y0 > 0 such that fx(y0) < 0. Onthe contrary, we should have :

E

[L

(ydQ

dP, ω

)]+ xy > 0, ∀y > 0,∀Q ∈ P,

23

and then

x > E

[−1

yL

(ydQ

dP, ω

)], ∀y > 0,∀Q ∈ P.

Since −L(ydQ/dP, ω)/y ≥ 0 and −L(ydQ/dP, ω)/y converges to HdQ/dP

as y goes to infinity, we deduce by Fatou’s lemma that :

x ≥ EQ [H] , ∀Q ∈ P,

and then x ≥ v0(H), a contradiction. We can then deduce that for all 0 <

x < v0(H), the function fx(.) attains an infimum for y > 0 and since J , andso fx, is differentiable on (0,∞), we have f ′x(y) = 0, i.e. J ′(y) = −x.

Fix some y > 0 and let Q be an arbitrary element of P. Denote :

Qε = (1− ε)Q∗(y) + εQ, ε ∈ (0, 1).

Obviously, Qε ∈ P so that by definition of J , we have :

0 ≤ 1εE

L(y dQε

dP , ω)− L

(y dQ∗(y)

dP , ω)

y

≤ 1

εE

L(y dQε

dP , ω)− L

(y dQ∗(y)

dP , ω)

y

≤ E

[−(

dQ

dP− dQ∗(y)

dP

)χ

(ydQε

dP, ω

)]where the third inequality from (5.11) and the convexity of L. We obtainthen :

EQ[χ

(ydQε

dP, ω

)]≤ EQ∗(y)

[χ

(ydQε

dP, ω

)]. (5.15)

By the dominated convergence theorem and Fatou’s lemma applied respec-tively to the R.H.S. and the L.H.S. of (5.15), we have :

EQ[χ

(ydQ∗(y)

dP, ω

)]≤ EQ∗(y)

[χ

(ydQ∗(y)

dP, ω

)].

From (5.8) and (5.10), this can be written also as :

supQ∈P

EQ[χ

(ydQ∗(y)

dP, ω

)]≤ −J ′(y),

24

for all y > 0. By choosing y = y defined above, we get :

supQ∈P

EQ[χ

(ydQ∗(y)

dP, ω

)]≤ x, (5.16)

which proves that X∗(x) = χ(y dQ∗(y)

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

(ydQ∗(y)

dP, ω

)= −l(H −X∗(x))− y

dQ∗(y)dP

X∗(x) (5.17)

≥ −l(H −X)− ydQ∗(y)

dPX. (5.18)

Taking expectation under P in (5.17)-(5.18) and using the facts that :

EQ∗(y) [X∗(x)] = −J ′(y) = x, (5.19)

EQ∗(y) [X] ≤ x, (5.20)

we obtain that :

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∗(y) attains the supremum in supQ∈P EQ[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∗(y), we have from (5.8)-(5.9) :

x = EQ∗(y)[X∗(x)] ≤ EQ∗(y)[V x,θ∗(x)T ] ≤ x,

which shows that X∗(x) = Vx,θ∗(x)T , a.s., and that the wealth process V x,θ∗(x)

is a martingale under Q∗(y). 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)− ydQ

dPX ≤ L

(ydQ

dP, ω

),

25

hence by taking expectation under P :

−E [l(H −X)]− yx ≤ E

[L

(ydQ

dP, ω

)],

and therefore,

supy≥0

[−J(y)− xy

]≤ J(x), ∀x ≥ 0. (5.21)

For x ≥ v0(H), we have J(x) = 0 = −J(0). Fix now 0 < x < v0(H). Fromrelations (5.17) and (5.19), we have :

J(y) = −E [l(H −X∗(x))]− yE

[dQ∗(y)

dPX∗(x)

]= −J(x)− xy,

which proves that J(x) = −J(y)− xy. The proof is ended. 2

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.

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