A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility
AA GENERAL MULTIDIMENSIONAL MONTE CARLO APPROACH FORDYNAMIC HEDGING UNDER STOCHASTIC VOLATILITY
DORIVAL LE ˜AO, ALBERTO OHASHI, AND VIN´ICIUS SIQUEIRA
Abstract.
In this work, we introduce a Monte Carlo method for the dynamic hedging of generalEuropean-type contingent claims in a multidimensional Brownian arbitrage-free market. Based onbounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, wepropose a feasible and constructive methodology which allows us to compute pure hedging strategies w.r.t arbitrary square-integrable claims in incomplete markets. In particular, the methodology canbe applied to quadratic hedging-type strategies for fully path-dependent options with stochasticvolatility and discontinuous payoffs. We illustrate the method with numerical examples based ongeneralized F¨ollmer-Schweizer decompositions, locally-risk minimizing and mean-variance hedgingstrategies for vanilla and path-dependent options written on local volatility and stochastic volatilitymodels. Introduction
Background and Motivation.
Let ( S, F , P ) be a financial market composed by a continuous F -semimartingale S which represents a discounted risky asset price process, F = { F t ; 0 ≤ t ≤ T } is afiltration which encodes the information flow in the market on a finite horizon [0 , T ], P is a physicalprobability measure and M e is the set of equivalent local martingale measures. Let H be an F T -measurable contingent claim describing the net payoff whose the trader is faced at time T . In orderto hedge this claim, the trader has to choose a dynamic portfolio strategy.Under the assumption of an arbitrage-free market, the classical Galtchouk-Kunita-Watanabe (hence-forth abbreviated by GKW) decomposition yields(1.1) H = E Q [ H ] + (cid:90) T θ H, Q (cid:96) dS (cid:96) + L H, Q T under Q ∈ M e , where L H, Q is a Q -local martingale which is strongly orthogonal to S and θ H, Q is an adapted process.The GKW decomposition plays a crucial role in determining optimal hedging strategies in a generalBrownian-based market model subject to stochastic volatility. For instance, if S is a one-dimensionalItˆo risky asset price process which is adapted to the information generated by a two-dimensionalBrownian motion W = ( W , W ), then there exists a two-dimensional adapted process φ H, Q :=( φ H, , φ H, ) such that H = E Q [ H ] + (cid:90) T φ H, Q t dW t , which also realizes(1.2) θ H, Q t = φ H, t [ S t σ t ] − , L H, Q t = (cid:90) t φ H, dW s ; 0 ≤ t ≤ T. Date : August 22, 2018.1991
Mathematics Subject Classification.
Primary: C02; Secondary: G12.
Key words and phrases.
Martingale representation, hedging contingent claims, path dependent options.We would like to thank Bruno Dupire and Francesco Russo for stimulating discussions and several suggestions aboutthe numerical algorithm proposed in this work. We also gratefully acknowledge the computational support from LNCC(Laborat´orio Nacional de Computa¸c˜ao Cient´ıfica - Brazil). The second author was supported by CNPq grant 308742. a r X i v : . [ q -f i n . P R ] A ug DORIVAL LE˜AO, ALBERTO OHASHI, AND VIN´ICIUS SIQUEIRA
In the complete market case, there exists a unique Q ∈ M e and in this case, L H, Q = 0, E Q [ H ] isthe unique fair price and the hedging replicating strategy is fully described by the process θ H, Q . Ina general stochastic volatility framework, there are infinitely many GKW orthogonal decompositionsparameterized by the set M e and hence one can ask if it is possible to determine the notion of non-self-financing optimal hedging strategies solely based on the quantities (1.2). This type of question wasfirstly answered by F¨ollmer and Sonderman [9] and later on extended by Schweizer [23] and F¨ollmerand Schweizer [8] through the existence of the so-called F¨ollmer-Schweizer decomposition which turnsout to be equivalent to the existence of locally-risk minimizing hedging strategies. The GKW decom-position under the so-called minimal martingale measure constitutes the starting point to get locallyrisk minimizing strategies provided one is able to check some square-integrability properties of thecomponents in (1.1) under the physical measure. See e.g [12] and [26] for details and other referencestherein. Orthogonal decompositions without square-integrability properties can also be defined interms of the the so-called generalized F¨ollmer-Schweizer decomposition (see e.g Schweizer [24]).In contrast to the local-risk minimization approach, one can insist in working with self-financinghedging strategies which give rise to the so-called mean-variance hedging methodology. In this ap-proach, the spirit is to minimize the expectation of the squared hedging error over all initial endow-ments x and all suitable admissible strategies ϕ ∈ Θ:(1.3) inf ϕ ∈ Θ ,x ∈ R E P (cid:12)(cid:12)(cid:12) H − x − (cid:90) T ϕ t dS t (cid:12)(cid:12)(cid:12) . The nature of the optimization problem (1.3) suggests to work with the subset M e := { Q ∈ M e ; d Q d P ∈ L ( P ) } . Rheinlander and Schweizer [22], Gourieroux, Laurent and Pham [10] and Schweizer [25] showthat if M e (cid:54) = ∅ and H ∈ L ( P ) then the optimal quadratic hedging strategy exists and it is given by (cid:0) E ˜ P [ H ] , η ˜ P (cid:1) , where(1.4) η ˜ P t := θ H, ˜ P t − ˜ ζ t ˜ Z t (cid:32) V H, ˜ P t − − E ˜ P [ H ] − (cid:90) t η ˜ P (cid:96) dS (cid:96) (cid:33) ; 0 ≤ t ≤ T. Here θ H, ˜ P is computed in terms of ˜ P , the so-called variance optimal martingale measure, ˜ ζ realizes(1.5) ˜ Z t := E ˜ P (cid:34) d ˜ P d P (cid:12)(cid:12)(cid:12) F t (cid:35) = ˜ Z + (cid:90) t ˜ ζ (cid:96) dS (cid:96) ; 0 ≤ t ≤ T, and V H, ˜ P := E ˜ P [ H | F · ] is the value option price process under ˜ P . See also Cern´y and Kallsen [4] for thegeneral semimartingale case and the works [16], [18] and [19] for other utility-based hedging strategiesbased on GKW decompositions.Concrete representations for the pure hedging strategies { θ H, Q ; Q = ˆ P , ˜ P } can in principle be ob-tained by computing cross-quadratic variations d [ V H, Q , S ] t /d [ S, S ] t for Q ∈ { ˜ P , ˆ P } . For instance, inthe classical vanilla case, pure hedging strategies can be computed by means of the Feynman-Kactheorem (see e.g Heath, Platen and Schweizer [12]). In the path-dependent case, the obtention ofconcrete computationally efficient representations for θ H, Q is a rather difficult problem. Feynman-Kac-type arguments for fully path-dependent options mixed with stochastic volatility typically facenot-well posed problems on the whole trading period as well as highly degenerate PDEs arise in thiscontext. Generically speaking, one has to work with non-Markovian versions of the Feynman-Kactheorem in order to get robust dynamic hedging strategies for fully path dependent options writtenon stochastic volatility risky asset price processes.In the mean variance case, the only quantity in (1.4) not related to GKW decomposition is ˜ Z whichcan in principle be expressed in terms of the so-called fundamental representation equations given byHobson [14] and Biagini, Guasoni and Pratelli [2] in the stochastic volatility case. For instance, YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 3
Hobson derives closed form expressions for ˜ ζ and also for any type of q -optimal measure in the Hestonmodel [13]. Recently, semi-explicit formulas for vanilla options based on general characterizationsof the variance-optimal hedge in Cern´y and Kallsen [4] have been also proposed in the literaturewhich allow for a feasible numerical implementation in affine models. See Kallsen and Vierthauer [17]and Cern´y and Kallsen [5] for some results in this direction. A different approach based on backwardstochastic differential equations can also be used in order to get useful characterizations for the optimalmean variance hedging strategies. See e.g Jeanblanc, Mania, Santacrose and Schweizer [15] and otherreferences therein.1.2. Contribution of the current paper.
In spite of deep characterizations of optimal quadratichedging strategies and concrete numerical schemes available for vanilla-type options, to our bestknowledge no feasible approach has been proposed to tackle the problem of obtaining dynamic optimalquadratic hedging strategies for fully path dependent options written on a generic multidimensionalItˆo risky asset price process. In this work, we attempt to solve this problem with a probabilisticapproach. The main difficulty in dealing with fully path dependent and/or discontinuous payoffs isthe non-Markovian nature of the option value and a priori lack of path smoothness of the pure hedgingstrategies. Usual numerical schemes based on PDE and martingale techniques do not trivially applyto this context.The main contribution of this paper is the obtention of flexible and computationally efficient multidi-mensional non-Markovian representations for generic option price processes which allow for a concretecomputation of the associated GKW decomposition (cid:0) θ H, Q , L H, Q (cid:1) for Q -square integrable payoffs H with Q ∈ M e . We provide a Monte Carlo methodology capable to compute optimal quadratic hedgingstrategies w.r.t general square-integrable claims in a multidimensional Brownian-based market model.This article provides a feasible and constructive method to compute generalized F¨ollmer-Schweizerdecompositions under full generality. As far as the mean variance hedging is concerned, we are ableto compute pure optimal hedging strategies θ H, ˜ P for arbitrary square-integrable payoffs. Hence, ourmethodology also applies to this case provided one is able to compute the fundamental representationequations in Hobson [14] and Biagini, Guasoni and Pratelli [2] which is the case for the classical Hestonmodel. In mathematical terms, we are able to compute Q -GKW decompositions under full generalityso that the results of this article can also be used to other non-quadratic hedging methodologieswhere orthogonal martingale representations play an important role in determining optimal hedgingstrategies.The starting point of this article is based on weak approximations developed by Le˜ao and Ohashi [20]for one-dimensional Brownian functionals. They introduced a one-dimensional space-filtration dis-cretization scheme constructed from suitable waiting times which measure the instants when theBrownian motion hits some a priori levels. In this work, we extend [20] to the multidimensional caseas follows: More general and stronger convergence results are obtained in order to recover incompletemarkets with stochastic volatility. Hitting times induced by multidimensional noises which drive thestochastic volatility are carefully analyzed in order to obtain Q -GKW decompositions under ratherweak integrability conditions for any Q ∈ M e . Moreover, a complete analysis is performed w.r.tweak approximations for gain processes by means of suitable non-antecipative discrete-time hedgingstrategies for square-integrable payoffs, including path-dependent ones.It is important to stress that the results of this article can be applied to both complete andincomplete markets written on a generic multidimensional Itˆo risky asset price process. One importantrestriction of our methodology is the assumption that the risky asset price process has continuouspaths. This is a limitation that we hope to overcome in a future work.Numerical results based on the standard Black-Scholes, local-volatility and Heston models are per-formed in order to illustrate the theoretical results and the methodology of this article. In particular,we briefly compare our results with other prominent methodologies based on Malliavin weights (com-plete market case) and PDE techniques (incomplete market case) employed by Bernis, Gobet andKohatsu-Higa [1] and Heath, Platen and Schweizer [12], respectively. The numerical experiments DORIVAL LE˜AO, ALBERTO OHASHI, AND VIN´ICIUS SIQUEIRA suggest that pure hedging strategies based on generalized F¨ollmer-Schweizer decompositions mitigatevery well the cost of hedging of a path-dependent option even if there is no guarantee of the exis-tence of locally-risk minimizing strategies. We also compare hedging errors arising from optimal meanvariance hedging strategies for one-touch options written on a Heston model with nonzero correlation.The remainder of this paper is structured as follows. In Section 2, we fix the notation and wedescribe the basic underlying market model. In Section 3, we provide the basic elements of theMonte Carlo methodology proposed in this article. In Section 4, we formulate dynamic hedgingstrategies starting from a given GKW decomposition and we translate our results to well-knownquadratic hedging strategies. The Monte Carlo algorithm and the numerical study are describedin Sections 5 and 6, respectively. The Appendix presents more refined approximations when themartingale representations admit additional hypotheses.2.
Preliminaries
Throughout this paper, we assume that we are in the usual Brownian market model with finite timehorizon 0 ≤ T < ∞ equipped with the stochastic basis (Ω , F , P ) generated by a standard p -dimensionalBrownian motion B = { ( B (1) t , . . . , B ( p ) t ); 0 ≤ t ≤ T } starting from 0. The filtration F := ( F t ) ≤ t ≤ T is the P -augmentation of the natural filtration generated by B . For a given m -dimensional vector J = ( J , . . . , J m ), we denote by diag( J ) the m × m diagonal matrix whose (cid:96) -th diagonal term is J (cid:96) . In this paper, for all unexplained terminology concerning general theory of processes, we refer toDellacherie and Meyer [6].In view of stochastic volatility models, let us split B into two multidimensional Brownian motionsas follows B S := ( B (1) , . . . , B ( d ) ) and B I := ( B ( d +1) , . . . , B ( p ) ). In this section, the market consists of d + 1 assets ( d ≤ p ): one riskless asset given by dS t = r t S t dt, S = 1; 0 ≤ t ≤ T, and a d -dimensional vector of risky assets ¯ S := ( ¯ S , . . . , ¯ S d ) which satisfies the following stochasticdifferential equation d ¯ S t = diag( ¯ S t ) (cid:0) b t dt + σ t dB St (cid:1) , ¯ S = ¯ x ∈ R d ; 0 ≤ t ≤ T. Here, the real-valued interest rate process r = { r t ; 0 ≤ t ≤ T } , the vector of mean rates of return b := { b t = ( b t , . . . , b dt ); 0 ≤ t ≤ T } and the volatility matrix σ := { σ t = ( σ ijt ); 1 ≤ i ≤ d, ≤ j ≤ d, ≤ t ≤ T } are assumed to be predictable and they satisfy the standard assumptions in such way thatboth S and ¯ S are well-defined positive semimartingales. We also assume that the volatility matrix σ is non-singular for almost all ( t, ω ) ∈ [0 , T ] × Ω. The discounted price S := { S i := ¯ S i /S ; i = 1 , . . . , d } follows dS t = diag( S t ) (cid:2) ( b t − r t d ) dt + σ t dB St (cid:3) ; S = x ∈ R d , ≤ t ≤ T, where d is a d-dimensional vector with every component equal to 1. The market price of risk is givenby ψ t := σ − t [ b t − r t d ] , ≤ t ≤ T, where we assume (cid:90) T (cid:107) ψ u (cid:107) R d du < ∞ a.s. In the sequel, M e denotes the set of P -equivalent probability measures Q such that the respectiveRadon-Nikodym derivative process is a P − martingale and the discounted price S is a Q -local mar-tingale. Throughout this paper, we assume that M e (cid:54) = ∅ . In our setup, it is well known that M e isgiven by the subset of probability measures with Radon-Nikodym derivatives of the form YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 5 d Q d P := exp (cid:34) − (cid:90) T ψ u dB Su − (cid:90) T ν u dB Iu − (cid:90) T (cid:8) (cid:107) ψ u (cid:107) R d + (cid:107) ν u (cid:107) R p − d (cid:9) du (cid:35) , for some R p − d -valued adapted process ν such that (cid:82) T (cid:107) ν t (cid:107) R p − d dt < ∞ a.s. Example : The typical example studied in the literature is the following one-dimensional stochasticvolatility model(2.1) (cid:40) dS t = S t µ ( t, S t , σ t ) dt + S t σ t dY (1) t dσ t = a ( t, S t , σ t ) dt + b ( t, S t , σ t ) dY (2) t ; 0 ≤ t ≤ T, where Y (1) and Y (2) are correlated Brownian motions with correlation ρ ∈ [ − , µ, a and b aresuitable functions such that ( S, σ ) is a well-defined two-dimensional Markov process. All continuousstochastic volatility model commonly used in practice fit into the specification (2.1). In this case, p = 2 > d = 1 and we recall that the market is incomplete where the set M e is infinity. The dynamichedging procedure turns out to be quite challenging due to extrinsic randomness generated by thenon-tradeable volatility, specially w.r.t to exotic options.2.1. GKW Decomposition.
In the sequel, we take Q ∈ M e and we set W S := ( W (1) , . . . , W ( d ) )and W I := ( W ( d +1) , . . . , W ( p ) ) where(2.2) W ( j ) t := B ( j ) t + (cid:90) t ψ ju du, j = 1 , . . . , dB ( j ) t + (cid:90) t ν ju du, j = d + 1 , . . . , p ; 0 ≤ t ≤ T, is a standard p -dimensional Brownian motion under the measure Q and filtration F := {F t ; 0 ≤ t ≤ T } generated by W = ( W (1) , . . . , W ( p ) ). In what follows, we fix a discounted contingent claim H . Recallthat the filtration F is contained in F , but it is not necessarily equal. In the remainder of this article,we assume the following hypothesis. (M) The contingent claim H is also F T -measurable. Remark 2.1.
Assumption (M) is essential for the approach taken in this work because the wholealgorithm is based on the information generated by the Brownian motion W (defined under the measure Q and filtration F ). As long as the numeraire is deterministic, this hypothesis is satisfied for anystochastic volatility model of the form (2.1) and a payoff Φ( S t ; 0 ≤ t ≤ T ) where Φ : C T → R is aBorel map and C T is the usual space of continuous paths on [0 , T ] . Hence, (M) holds for a very largeclass of examples founded in practice. For a given Q -square integrable claim H , the Brownian martingale representation (computed interms of ( F , Q )) yields H = E Q [ H ] + (cid:90) T φ H, Q u dW u , where φ H, Q := ( φ H, Q , , . . . , φ H, Q ,p ) is a p -dimensional F -predictable process. In what follows, we set φ H, Q ,S := ( φ H, Q , , . . . , φ H, Q ,d ), φ H, Q ,I := ( φ H, Q ,d +1 , . . . , φ H, Q ,p ) and(2.3) L H, Q t := (cid:90) t φ H, Q ,Iu dW Iu , ˆ V t := E Q [ H |F t ]; 0 ≤ t ≤ T. DORIVAL LE˜AO, ALBERTO OHASHI, AND VIN´ICIUS SIQUEIRA
The discounted stock price process has the following Q -dynamics dS t = diag( S t ) σ t dW St , S = x, ≤ t ≤ T, and therefore the Q -GKW decomposition for the pair of locally square integrable local martingales( ˆ V , S ) is given by ˆ V t = E Q [ H ] + (cid:90) t φ H, Q ,Su dW Su + L H, Q t = E Q [ H ] + (cid:90) t θ H, Q u dS u + L H, Q t ; 0 ≤ t ≤ T, (2.4)where(2.5) θ H, Q := φ H, Q ,S [diag( S ) σ ] − . The p -dimensional process φ H, Q which constitutes (2.3) and (2.5) plays a major role in several typesof hedging strategies in incomplete markets and it will be our main object of study. Remark 2.2.
If we set ν j = 0 for j = d +1 , . . . , p and the correspondent density process is a martingalethen the resulting minimal martingale measure ˆ P yields a GKW decomposition where L H, ˆ P is still a P -local martingale orthogonal to the martingale component of S under P . In this case, it is alsonatural to implement a pure hedging strategy based on θ H, ˆ P regardless the existence of the F¨ollmer-Schweizer decomposition. If this is the case, this hedging strategy can be based on the generalizedF¨ollmer-Schweizer decomposition (see e.g Th.9 in [24] ). The Random Skeleton and Weak Approximations for GKW Decompositions
In this section, we provide the fundamentals of the numerical algorithm of this article for theobtention of hedging strategies in complete and incomplete markets.3.1.
The Multidimensional Random Skeleton.
At first, we fix once and for all Q ∈ M e and a Q -square-integrable contingent claim H satisfying (M) . In the remainder of this section, we are goingto fix a Q -Brownian motion W and with a slight abuse of notation all Q -expectations will be denotedby E . The choice of Q ∈ M e is dictated by the pricing and hedging method used by the trader.In the sequel, [ · , · ] denotes the usual quadratic variation between semimartingales and the usualjump of a process is denoted by ∆ Y t = Y t − Y t − where Y t − is the left-hand limit of a cadlag process Y . For a pair ( a, b ) ∈ R , we denote a ∨ b := max { a, b } and a ∧ b := min { a, b } . Moreover, for anytwo stopping times S and J , we denote the stochastic intervals [[ S, J [[:= { ( ω, t ); S ( ω ) ≤ t < J ( ω ) } ,[[ S ]] := { ( ω, t ); S ( ω ) = t } and so on. Throughout this article, Leb denotes the Lebesgue measure onthe interval [0 , T ].For a fixed positive integer k and for each j = 1 , , . . . , p we define T k,j := 0 a.s. and(3.1) T k,jn := inf { T k,jn − < t < ∞ ; | W ( j ) t − W ( j ) T k,jn − | = 2 − k } , n ≥ , where W := ( W (1) , . . . , W ( p ) ) is the p -dimensional Q -Brownian motion as defined in (2.2).For each j ∈ { , . . . , p } , the family ( T k,jn ) n ≥ is a sequence of F -stopping times where the increments { T k,jn − T k,jn − ; n ≥ } is an i.i.d sequence with the same distribution as T k,j . In the sequel, we define A k := ( A k, , . . . , A k,p ) as the p -dimensional step process given componentwise by A k,jt := ∞ (cid:88) n =1 − k η k,jn { T k,jn ≤ t } ; 0 ≤ t ≤ T, where YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 7 (3.2) η k,jn :=
1; if W ( j ) T k,jn − W ( j ) T k,jn − = 2 − k and T k,jn < ∞−
1; if W ( j ) T k,jn − W ( j ) T k,jn − = − − k and T k,jn < ∞
0; if T k,jn = ∞ . for k, n ≥ j = 1 , . . . , p . We split A k into ( A S,k , A
I,k ) where A S,k is the d -dimensional processconstituted by the first d components of A k and A I,k the remainder p − d -dimensional process. Let F k,j := {F k,jt : 0 ≤ t ≤ T } be the natural filtration generated by { A k,jt ; 0 ≤ t ≤ T } . One should noticethat F k,j is a discrete-type filtration in the sense that F k,jt = ∞ (cid:95) (cid:96) =0 (cid:16) F k,jT k,j(cid:96) ∩ { T k,j(cid:96) ≤ t < T k,j(cid:96) +1 } (cid:17) , ≤ t ≤ T, where F k,j = { Ω , ∅} and F k,jT k,jm = σ ( T k,j , . . . , T k,jm , η k,j , . . . , η k,jm ) for m ≥ j = 1 , . . . , p . Here, (cid:87) denotes the smallest sigma-algebra generated by the union. One can easily check that F k,jT k,jm = σ ( A k,js ∧ T k,jm ; s ≥
0) and hence F k,jT k,jm = F k,jt a.s on (cid:8) T k,jm ≤ t < T k,jm +1 (cid:9) . With a slight abuse of notation we write F k,jt to denote its Q -augmentation satisfying the usualconditions.Let us now introduce the multidimensional filtration generated by A k . Let us consider F k := {F kt ; 0 ≤ t ≤ T } where F kt := F k, t ⊗ F k, t ⊗ · · · ⊗ F k,pt for 0 ≤ t ≤ T . Let T k := { T km ; m ≥ } be theorder statistics obtained from the family of random variables { T k,j(cid:96) ; (cid:96) ≥ j = 1 , . . . , p } . That is, weset T k := 0,(3.3) T k := inf ≤ j ≤ pm ≥ (cid:110) T k,jm (cid:111) , T kn := inf ≤ j ≤ pm ≥ (cid:110) T k,jm ; T k,jm ≥ T k ∨ . . . ∨ T kn − (cid:111) for n ≥
1. In this case, T k is the partition generated by all stopping times defined in (3.1). Thefinite-dimensional distribution of W ( j ) is absolutely continuous for each j = 1 , . . . , p and therefore theelements of T k are almost surely distinct for every k ≥
1. Moreover, the following result holds true.
Lemma 3.1.
For every k ≥ , the set T k is an exhaustive sequence of F k -stopping times such that sup n ≥ | T kn − T kn − | → in probability as k → ∞ .Proof. The following obvious estimate holdssup n ≥ | T kn − T kn − | ≤ max ≤ j ≤ p sup n ≥ | T k,jn − T k,jn − | → , in probability as k → ∞ and T kn → ∞ a.s as n → ∞ for each k ≥
1. Let us now prove that T k = { T kn ; n ≥ } is a sequence of F k -stopping times. In order to show this, we write ( T kn ) n ≥ in adifferent way. This sequence can be defined recursively as follows T k = 0 , T k = inf { t > (cid:107) A kt (cid:107) R p = 2 − k } , where (cid:107) · (cid:107) R p denotes the R p -maximum norm. Therefore, T k is an F k -stopping time. Next, let usdefine a family of F kT k -random variables related to the index j which realizes the hitting time T k asfollows DORIVAL LE˜AO, ALBERTO OHASHI, AND VIN´ICIUS SIQUEIRA (cid:96) k,j := , if | A k,jT k |(cid:54) = 2 − k , if | A k,jT k | = 2 − k , for any j = 1 , . . . , p . Then, we shift A k as follows˜ A k ( t ) := (cid:16) ˜ A k, ( t ) := A k, ( t + T k ) − A k, ( T k(cid:96) k, ); . . . ; ˜ A k,p ( t ) := A k,p ( t + T k ) − A k,p ( T k(cid:96) k,p ) (cid:17) , for t ≥
0. In this case, we conclude that ˜ A k is adapted to the filtration {F kt + T k ; t ≥ } , the hittingtime S k := inf { t > (cid:107) ˜ A k ( t ) (cid:107) R p = 2 − k } is a {F kt + T k : t ≥ } -stopping time and T k = T k + S k is a F k -stopping time. In the sequel, we definea family of F kT k -random variables related to the index j which realizes the hitting time T k as follows (cid:96) k,j := (cid:26) , if | ˜ A k,j ( S k ) |(cid:54) = 2 − k , if | ˜ A k,j ( S k ) | = 2 − k , for j = 1 , . . . , p . If we denote S k = 0, we shift ˜ A k as follows˜ A k ( t ) := (cid:16) ˜ A k, ( t ) := ˜ A k, ( t + S k ) − ˜ A k, ( S k(cid:96) k, ); . . . ; ˜ A k,p ( t ) = ˜ A k,p ( t + S k ) − ˜ A k,p ( S k(cid:96) k,p ) (cid:17) , for every t ≥
0. In this case, we conclude that ˜ A k is adapted to the filtration {F kt + T k ; t ≥ } , thehitting time S k = inf { t > (cid:107) ˜ A k ( t ) (cid:107) R p = 2 − k } is an {F kt + T k ; t ≥ } -stopping time and T k = T k + S k is a F k -stopping time. By induction, we concludethat ( T kn ) n ≥ is a sequence of F k -stopping times. (cid:3) With Lemma 3.1 at hand, we notice that the filtration F k is a discrete-type filtration in the sensethat F kT kn = F kt a.s on { T kn ≤ t < T kn +1 } , for k ≥ n ≥
0. Itˆo representation theorem yields E [ H |F t ] = E [ H ] + (cid:90) t φ Hu dW u ; 0 ≤ t ≤ T, where φ H is a p -dimensional F -predictable process such that E (cid:90) T (cid:107) φ Ht (cid:107) R p dt < ∞ . The payoff H induces the Q -square-integrable F -martingale X t := E [ H |F t ]; 0 ≤ t ≤ T . We nowembed the process X into the quasi left-continuous filtration F k by means of the following operator δ k X t := X + ∞ (cid:88) m =1 E (cid:2) X T km |F kT km (cid:3) { T km ≤ t Therefore, δ k X is indeed a Q -square-integrable F k -martingale and we shall write it as δ k X t = X + ∞ (cid:88) m =1 ∆ δ k X T km { T km ≤ t } = X + p (cid:88) j =1 ∞ (cid:88) n =1 ∆ δ k X T k,jn { T k,jn ≤ t } = X + p (cid:88) j =1 ∞ (cid:88) (cid:96) =1 ∆ δ k X T k,j(cid:96) ∆ A k,jT k,j(cid:96) ∆ A k,jT k,j(cid:96) { T k,j(cid:96) ≤ t } = X + p (cid:88) j =1 (cid:90) t D j δ k X u dA k,ju , (3.4)where D j δ k X := ∞ (cid:88) (cid:96) =1 ∆ δ k X T k,j(cid:96) ∆ A k,jT k,j(cid:96) [[ T k,j(cid:96) ,T k,j(cid:96) ]] , and the integral in (3.4) is computed in the Lebesgue-Stieltjes sense. Remark 3.1. Similar to the univariate case, one can easily check that F k → F weakly and since X has continuous paths then δ k X → X uniformly in probability as k → ∞ . See Remark 2.1 in [20] . Based on the Dirac process D j δ k X , we denote D k,j X := ∞ (cid:88) (cid:96) =1 D jT k,j(cid:96) δ k X [[ T k,j(cid:96) ,T k,j(cid:96) +1 [[ , k ≥ , j = 1 , . . . , p. In order to work with non-antecipative hedging strategies, let us now define a suitable F k -predictableversion of D k,j X as follows D k,j X := 011 [[0]] + ∞ (cid:88) n =1 E (cid:2) D k,j X T k,jn |F kT k,jn − (cid:3) ]] T k,jn − ,T k,jn ]] ; k ≥ , j = 1 , . . . , d. One can check that D k,j X is F k -predictable. See e.g [11], Ch.5 for details. Example : Let H be a contingent claim satisfying (M) . Then for a given j = 1 , . . . , p , we have(3.5) D k,j X t = E ∞ (cid:88) (cid:96) =1 (cid:34) E (cid:2) H (cid:12)(cid:12) F kT k(cid:96) (cid:3) − E (cid:2) H (cid:12)(cid:12) F kT k(cid:96) − (cid:3) W ( j ) T k,j − W ( j ) T k,j (cid:35) { T k,j = T k(cid:96) } , < t ≤ T k,j . One should notice that (3.5) is reminiscent from the usual delta-hedging strategy but the price isshifted on the level of the sigma-algebras jointly with the increments of the driving Brownian motioninstead of the pure spot price. For instance, in the one-dimensional case ( p = d = 1), we have D k, X t = E (cid:34) E (cid:2) H (cid:12)(cid:12) F kT k, (cid:3) − E [ H ] W (1) T k, − W (1) T k, (cid:35) , < t ≤ T k, , and hence a natural procedure to approximate pure hedging strategies is to look at D k, X T k, /S σ at time zero. In the incomplete market case, additional randomness from e.g stochastic volatilities areencoded by E [ H |F kT k ] where T k is determined not only by the hitting times coming from the riskyasset prices but also by possibly Brownian motion hitting times coming from stochastic volatility.In the next sections, we will construct feasible approximations for the gain and cost processes basedon the ratios (3.5). We will see that hedging ratios of the form (3.5) will be the key ingredient torecover the gain process in full generality. Weak approximation for the hedging process. Based on (2.3), (2.4) and (2.5), let us denote(3.6) θ Ht := φ H,St [diag( S t ) σ t ] − and L Ht := E [ H ] + (cid:90) t φ H,I(cid:96) dW I(cid:96) ; 0 ≤ t ≤ T. In order to shorten notation, we do not write ( φ H, Q ,S , φ H, Q ,I ) in (3.6). The main goal of this sectionis the obtention of bounded variation martingale weak approximations for both the gain and costprocesses, given respectively, by (cid:90) t θ Hu dS u , L Ht ; 0 ≤ t ≤ T. We assume the trader has some knowledge of the underlying volatility so that the obtention of φ H,S willbe sufficient to recover θ H . The typical example we have in mind are generalized F¨ollmer-Schweizerdecompositions, locally-risk minimizing and mean variance strategies as explained in the Introduction.The scheme will be very constructive in such way that all the elements of our approximation willbe amenable to a feasible numerical analysis. Under very mild integrability conditions, the weakapproximations for the gain process will be translated into the physical measure. The weak topology . In order to obtain approximation results under full generality, it is important toconsider a topology which is flexible to deal with nonsmooth hedging strategies θ H for possibly non-Markovian payoffs H and at the same time justifies Monte Carlo procedures. In the sequel, we makeuse of the weak topology σ (B p , M q ) of the Banach space B p ( F ) constituted by F -optional processes Y such that E | Y ∗ T | p < ∞ , where Y ∗ T := sup ≤ t ≤ T | Y t | and 1 ≤ p, q < ∞ such that p + q = 1. The subspace of the square-integrable F -martingales will be denoted by H ( F ). It will be also useful to work with σ (B , Λ ∞ )-topology given in [20]. For more details about these topologies, we refer to the works [6, 7, 20]. Itturns out that σ (B , M ) and σ (B , Λ ∞ ) are very natural notions to deal with generic square-integrablerandom variables as described in [20].In the sequel, we recall the following notion of covariation introduced in [20]. Definition 3.1. Let { Y k ; k ≥ } be a sequence of square-integrable F k -martingales. We say that { Y k ; k ≥ } has δ -covariation w.r.t jth component of A k if the limit lim k →∞ [ Y k , A k,j ] t exists weakly in L ( Q ) for every t ∈ [0 , T ] . Lemma 3.2. Let (cid:110) Y k,j = (cid:82) · H k,js dA k,j ; k ≥ , j = 1 , . . . , p (cid:111) be a sequence of stochastic integrals and Y k := (cid:80) pj =1 Y k,j . Assume that sup k ≥ E [ Y k , Y k ] T < ∞ . Then Y j := lim k →∞ Y k,j exists weakly in B ( F ) for each j = 1 , . . . , p with Y j ∈ H ( F ) if, and only if, { Y k ; k ≥ } admits δ -covariation w.r.t jth component of A k . In this case, lim k →∞ [ Y k , A k,j ] t = lim k →∞ [ Y k,j , A k,j ] t = [ Y j , W ( j ) ] t weakly in L ; t ∈ [0 , T ] for j = 1 , . . . , p .Proof. The proof follows easily from the arguments given in the proof of Prop. 3.2 in [20] by using thefact that { W ( j ) ; 1 ≤ j ≤ p } is an independent family of Brownian motions, so we omit the details. (cid:3) YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 11 In the sequel, we present a key asymptotic result for the numerical algorithm of this article. Theorem 3.1. Let H be a Q -square integrable contingent claim satisfying (M) . Then (3.7) lim k →∞ d (cid:88) j =1 (cid:90) · D k,js XdA k,js = d (cid:88) j =1 (cid:90) · φ H,ju dW ( j ) u = (cid:90) · θ Hu dS u , and (3.8) L H = lim k →∞ p (cid:88) j = d +1 (cid:90) · D k,js XdA k,js weakly in B ( F ) . In particular, (3.9) lim k →∞ D k,j X = φ H,j , weakly in L ( Leb × Q ) for each j = 1 , . . . , p. Proof. We divide the proof into three steps. Throughout this proof C is a generic constant which maydefer from line to line. STEP1. We claim that(3.10) lim k →∞ (cid:90) · D k,j X s dA k,js = (cid:90) · φ H,ju dW ( j ) u weakly in B ( F )for each j = 1 , . . . , p . In order to prove (3.10), we begin by noticing that Lemma 3.1 states thatthe elements of T k are F -stopping times. By assumption, X is Q -square integrable martingale andhence one may use similar arguments given in the proof of Lemma 3.1 in [20] to safely state that thefollowing estimate holds(3.11)sup k ≥ E [ δ k X, δ k X ] T = sup k ≥ E p (cid:88) j =1 (cid:90) T | D k,j X s | d [ A k,j , A k,j ] s ≤ sup k ≥ E ∞ (cid:88) m =1 ( X T km − X T km − ) { T km ≤ T } < ∞ . Now, we notice that the sequence F k converges weakly to F , X is continuous and therefore δ k X → X uniformly in probability (see Remark 3.1). Since X ∈ B ( F ), then a simple application of Burkh¨olderinequality allows us to state that δ k X converges strongly in B ( F ) and a routine argument based onthe definition of the B -weak topology yields(3.12) lim k →∞ δ k X = X weakly in B ( F ) . Now under (3.12) and (3.11), we shall prove in the same way as in Prop.3.2 in [20] that(3.13) lim k →∞ [ δ k X, A k,j ] t = [ X, W ( j ) ] t = (cid:90) t φ H,ju du ; 0 ≤ t ≤ T, holds weakly in L for each t ∈ [0 , T ] and j = 1 , . . . , p due to the pairwise independence of { W ( j ) ; 1 ≤ j ≤ p } . Summing up (3.11) and (3.13), we shall apply Lemma 3.2 to get (3.10). STEP 2. In the sequel, let ( · ) o,k and ( · ) p,k be the optional and predictable projections w.r.t F k ,respectively. Let us consider the F k -martingales given by M kt := p (cid:88) j =1 M k,jt ; 0 ≤ t ≤ T, where M k,jt := (cid:90) t D k,j X s dA k,js ; 0 ≤ t ≤ T, j = 1 , . . . , p. We claim that sup k ≥ E [ M k , M k ] T < ∞ . One can check that D k,j X T k,jn = (cid:16) D k,j X (cid:17) p,kT k,jn a.s for each n, k ≥ j = 1 . . . , p (see e.g chap.5, section 5 in [11]). Moreover, by the very definition(3.14) { ( t, ω ) ∈ [0 , T ] × Ω; ∆[ A k,j , A k,j ] t ( ω ) (cid:54) = 0 } = ∞ (cid:91) n =1 [[ T k,jn , T k,jn ]] . Therefore, Jensen inequality yields E [ M k , M k ] T = E p (cid:88) j =1 (cid:90) T | D k,j X s | d [ A k,j , A k,j ] s = E p (cid:88) j =1 (cid:90) T (cid:12)(cid:12)(cid:12)(cid:16) D k,j X (cid:17) p,ks (cid:12)(cid:12)(cid:12) d [ A k,j , A k,j ] s ≤ E p (cid:88) j =1 (cid:90) T (cid:16) ( D k,j X s ) (cid:17) p,ks d [ A k,j , A k,j ] s = p (cid:88) j =1 E ∞ (cid:88) n =1 E (cid:2) ( D k,j X T k,jn ) |F kT k,jn − (cid:3) − k { T k,jn ≤ T } := J k , (3.15)where in (3.15) we have used (3.14) and the fact that (cid:16) ( D k,j X ) (cid:17) p,kT k,jn = E (cid:2) ( D k,j X T k,jn ) |F kT k,jn − (cid:3) a.sfor each n, k ≥ j = 1 . . . , p . We shall write J k in a slightly different manner as follows(3.16) J k = p (cid:88) j =1 E ∞ (cid:88) n =1 E (cid:104) ( D k,j X T k,jn ) |F kT k,jn − (cid:105) − k { T k,jn − ≤ T } − p (cid:88) j =1 E (cid:104) E (cid:2) ( D k,j X T k,jq ) |F kT k,jq − (cid:3)(cid:105) − k { T k,jq − ≤ T STEP 3 . We claim that for a given g ∈ L ∞ , t ∈ [0 , T ] and j = 1 . . . , p we have(3.18) lim k →∞ E g [ M k − δ k X, A k,j ] t = 0 . By using the fact that D k,j X is F k -optional and D k,j X is F k -predictable, we shall use duality of the F k -optional projection to write E g [ M k − δ k X, A k,j ] t = E (cid:90) t ( g ) o,ks (cid:16) D k,j X s − D k,j X s (cid:17) d [ A k,j , A k,j ] s . In order to prove (3.18), let us check that(3.19) lim k →∞ E (cid:90) t ( g ) p,ks (cid:16) D k,j X s − D k,j X s (cid:17) d [ A k,j , A k,j ] s = 0 , and(3.20) lim k →∞ E (cid:90) t (cid:0) ( g ) o,ks − ( g ) p,ks (cid:1)(cid:16) D k,j X s − D k,j X s (cid:17) d [ A k,j , A k,j ] s = 0 . The same trick we did in (3.16) together with (3.14) yield E (cid:90) t ( g ) p,ks (cid:16) D k,j X s − D k,j X s (cid:17) d [ A k,j , A k,j ] s = E (cid:104) ( g ) p,kT k,jq D k,j X T k,jq (cid:105) − k { T k,jq − ≤ t In this section, we apply Theorem 3.1 for the formulation of a dynamic hedging strategy startingwith a given GKW decomposition(4.1) H = E [ H ] + (cid:90) T θ Ht dS t + L HT , where H is a Q -square integrable European-type option satisfying (M) for a given Q ∈ M e . The typi-cal examples we have in mind are quadratic hedging strategies w.r.t a fully path-dependent option. Werecall that when Q is the minimal martingale measure then (4.1) is the generalized F¨ollmer-Schweizerdecomposition so that under some P -square integrability conditions on the components of (4.1), θ H isthe locally risk minimizing hedging strategy (see e.g [12], [24]). In fact, GKW and F¨ollmer-Schweizerdecompositions are essentially equivalent for the market model assumed in Section 2. We recall thatdecomposition (4.1) is not sufficient to fully describe mean variance hedging strategies but the addi-tional component rests on the fundamental representation equations as described in Introduction. Seealso expression (6.4) in Section 6.For simplicity of exposition, we consider a financial market (Ω , F , P ) driven by a two-dimensionalBrownian motion B and a one-dimensional risky asset price process S as described in Section 2. Westress that all results in this section hold for a general multidimensional setting with the obviousmodifications.In the sequel, we denote θ k,H := ∞ (cid:88) n =1 D k, X T k, n σ T k, n − S T k, n − [[ T k, n − ,T k, n [[ where D k, X T k, n = E (cid:104) D k, X T k, n |F kT k, n − (cid:105) for k, n ≥ Corollary 4.1. For a given Q ∈ M e , let H be a Q -square integrable claim satisfying (M) . Let H = E [ H ] + (cid:90) T θ Ht dS t + L HT be the correspondent GKW decomposition under Q . If d P d Q ∈ L ( P ) and (4.2) E P sup ≤ t ≤ T (cid:12)(cid:12)(cid:12) (cid:90) t θ Hu dS u (cid:12)(cid:12)(cid:12) < ∞ , then ∞ (cid:88) n =1 θ k,HT k, n − ( S T k, n − S T k, n − )11 { T k, n ≤·} → (cid:90) · θ Ht dS t as k → ∞ , in the σ ( B , Λ ∞ ) -topology under P .Proof. We have E | d P d Q | = E P | d P d Q | d Q d P = E P d P d Q < ∞ . To shorten notation, let Y kt := (cid:82) t D k, s XdA k, s and Y t := (cid:82) t θ H(cid:96) dS (cid:96) for 0 ≤ t ≤ T. Let G be an arbitrary F -stopping time bounded by T and let g ∈ L ∞ ( P ) be an essentially P -bounded random variable and F G -measurable. Let J ∈ M be acontinuous linear functional given by the purely discontinuous F -optional bounded variation process J t := g E (cid:104) d P d Q (cid:12)(cid:12) F G (cid:105) { G ≤ t } ; 0 ≤ t ≤ T, where the duality action (cid:0) · , · (cid:1) is given by (cid:0) J, N (cid:1) = E (cid:82) T N s dJ s ; N ∈ B ( F ). See [20] for more details.Then Theorem 3.1 and the fact d P d Q ∈ L ( Q ) yield YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 15 E P gY kG = E Y kG g d P d Q = (cid:0) J, Y k (cid:1) → (cid:0) J, Y (cid:1) = E Y G g d P d Q = E P gY G as k → ∞ . By the very definition, (cid:90) t D k, s XdA k, s = ∞ (cid:88) n =1 E (cid:104) D k, X T k, n (cid:12)(cid:12) F kT k, n − (cid:105) ∆ A k, T k, n { T k, n ≤ t } (4.3) = ∞ (cid:88) n =1 θ k,HT k, n − σ T k, n − S T k, n − ( W (1) T k, n − W (1) T k, n − )11 { T k, n ≤ t } = ∞ (cid:88) n =1 θ k,HT k, n − ( S T k, n − S T k, n − )11 { T k, n ≤ t } ; 0 ≤ t ≤ T. Then from the definition of the σ (B , Λ ∞ )-topology based on the physical measure P , we shall concludethe proof. (cid:3) Remark 4.1. Corollary 4.1 provides a non-antecipative Riemman-sum approximation for the gainprocess (cid:82) · θ Ht dS t in a multi-dimensional filtration setting where none path regularity of the pure hedgingstrategy θ H is imposed. The price we pay is a weak-type convergence instead of uniform convergencein probability. However, from the financial point of view this type of convergence is sufficient for theimplementation of Monte Carlo methods in hedging. More importantly, we will see that θ k,H can befairly simulated and hence the resulting Monte Carlo hedging strategy can be calibrated from marketdata. Remark 4.2. If one is interested only at convergence at the terminal time < T < ∞ , then assump-tion (4.2) can be weakened to E P | (cid:82) T θ Ht dS t | < ∞ . Assumption E P d P d Q < ∞ is essential to change the Q -convergence into the physical measure P . One should notice that the associated density process isno longer a P -local-martingale and in general such integrability assumption must be checked case bycase. Such assumption holds locally for every underlying Itˆo risky asset price process. Our numericalresults suggest that this property behaves well for a variety of spot price models. Of course, in practice both the spot prices and trading dates are not observable at the stoppingtimes so we need to translate our results to a given deterministic set of rebalancing hedging dates.4.1. Hedging Strategies. In this section, we provide a dynamic hedging strategy based on a refinedset of hedging dates Π := 0 = s < . . . < s p − < s p = T . For this, we need to introduce some objects.For a given s i ∈ Π, we set W ( j ) s i ,t := W ( j ) s i + t − W ( j ) s i ; 0 ≤ t ≤ T − s i for j = 1 , 2. Of course, by thestrong Markov property of the Brownian motion, we know that W ( j ) s i , · is an ( F js i ,t ) ≤ t ≤ T − s i -Brownianmotion for each j = 1 , F js i , where F js i ,t := F js i + t for 0 ≤ t ≤ T − s i . Similarto Section 3.1, we set T k, s i , := 0 and T k,js i ,n := inf { t > T k,js i ,n − ; | W ( j ) s i ,t − W ( j ) s i ,T k,jsi,n − | = 2 − k } ; n ≥ , j = 1 , . For a given k ≥ j = 1 , 2, we define H k,js i ,n as the sigma-algebra generated by { T k,js i ,(cid:96) ; 1 ≤ (cid:96) ≤ n } and W ( j ) s i ,T k,jsi,(cid:96) − W ( j ) s i ,T k,jsi,(cid:96) − ; 1 ≤ (cid:96) ≤ n . We then define the following discrete jumping filtration F k,js i ,t := H k,js i ,n a.s on { T k,js i ,n ≤ t < T k,js i ,n +1 } . In order to deal with fully path dependent options, it is convenient to introduce the following aug-mented filtration G k,js i ,t := F js i ∨ F k,js i ,t ; 0 ≤ t ≤ T − s i , for j = 1 , 2. The bidimensional information flows are defined by F s i ,t := F s i ,t ⊗ F s i ,t and G ks i ,t := G k, s i ,t ⊗ G k, s i ,t for 0 ≤ t ≤ T − s i . We set G ks i := {G ks i ,t ; 0 ≤ t ≤ T − s i } . We shall assume that they satisfythe usual conditions. The piecewise constant martingale projection A k,js i based on W ( j ) s i is given by A k,js i ,t := E [ W ( j ) s i ,T − s i |G k,js i ,t ]; 0 ≤ t ≤ T − s i . We set { T ks i ,n ; n ≥ } as the order statistic generated by the stopping times { T k,js i ,n ; j = 1 , , n ≥ } similar to (3.3).If H ∈ L ( Q ) and X t = E [ H |F t ]; 0 ≤ t ≤ T , then we define δ ks i X t := E [ H |G ks i ,t ]; 0 ≤ t ≤ T − s i , so that the related derivative operators are given by D k,js i X := ∞ (cid:88) n =1 D jT k,jsi,n δ ks i X [[ T k,jsi,n ,T k,jsi,n +1 [[ , where D j δ ks i X := ∞ (cid:88) n =1 ∆ δ ks i X T k,jsi,n ∆ A k,jT k,jsi,n [[ T k,jsi,n ,T k,jsi,n ]] ; j = 1 , , k ≥ . An G ks i -predictable version of D k,js i X is given by D k,js i X := 011 [[0]] + ∞ (cid:88) n =1 E (cid:2) D k,js i X T k,jsi,n |G ks i ,T k,jsi,n − (cid:3) ]] T k,jsi,n − ,T k,jsi,n ]] ; j = 1 , . In the sequel, we denote(4.4) θ k,Hs i := ∞ (cid:88) n =1 D k, s i X T k, si,n σ s i ,T k, si,n − S s i ,T k, si,n − [[ T k, si,n − ,T k,jsi,n [[ ; s i ∈ Π , where σ s i , · is the volatility process driven by the shifted filtration {F s i ,t ; 0 ≤ t ≤ T − s i } and S s i , · isthe risky asset price process driven by the shifted Brownian W (1) s i .We are now able to present the main result of this section. Corollary 4.2. For a given Q ∈ M e , let H be a Q -square integrable claim satisfying (M) . Let H = E [ H ] + (cid:90) T θ Ht dS t + L HT be the correspondent GKW decomposition under Q . If d P d Q ∈ L ( P ) and E P (cid:12)(cid:12)(cid:12) (cid:90) T θ Hu dS u (cid:12)(cid:12)(cid:12) < ∞ , then for any set of trading dates Π = { ( s i ) pi =0 } , we have (4.5) lim k →∞ (cid:88) s i ∈ Π ∞ (cid:88) n =1 θ k,Hs i ,T k, si,n − (cid:0) S s i ,T k, si,n − S s i ,T k, si,n − (cid:1) { T k, si,n ≤ s i +1 − s i } = (cid:90) T θ Ht dS t weakly in L under P . YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 17 Proof. Let Π = { ( s i ) pi =0 } be any set of trading dates. To shorten notation, let us define(4.6) R ( θ k,H , Π , k ) := (cid:88) s i ∈ Π ∞ (cid:88) n =1 θ k,Hs i ,T k, si,n − (cid:0) S s i ,T k, si,n − S s i ,T k, si,n − (cid:1) { T k, si,n ≤ s i +1 − s i } for k ≥ { T k, s i ,n − T k, s i ,n − ; n ≥ , s i ∈ Π } is an i.i.d sequence withabsolutely continuous distribution. In this case, the probability of the set { T k, s i ,n ≤ s i +1 − s i } is alwaysstrictly positive for every Π and k, n ≥ 1. Hence, R ( θ k,H , Π , k ) is a non-degenerate subset of randomvariables. By making a change of variable on the Itˆo integral, we shall write (cid:90) T θ Ht dS t = (cid:90) T φ H, t dW (1) t = (cid:88) s i ∈ Π (cid:90) s i +1 s i φ H, t dW (1) t =(4.7) (cid:88) s i ∈ Π (cid:90) s i +1 − s i φ H, s i + t dW (1) s i ,t . Let us fix Q ∈ M e . By the very definition, R ( θ k,H , Π , k ) = (cid:88) s i ∈ Π (cid:90) s i +1 − s i D k, s i X (cid:96) dA k, s i ,(cid:96) under Q Now we notice that Theorem 3.1 holds for the two-dimensional Brownian motion (cid:0) W (1) s i , W (2) s i (cid:1) , foreach s i ∈ Π with the discretization of the Brownian motion given by A k, s i . Moreover, using the factthat E | d P d Q | < ∞ and repeating the argument given by (4.3) restricted to the interval [ s i , s i +1 ), wehave lim k →∞ R ( θ k,H , Π , k ) = (cid:88) s i ∈ Π lim k →∞ (cid:90) s i +1 − s i D k, s i X (cid:96) dA k, s i ,(cid:96) = (cid:90) T θ Ht dS t , (4.8)weakly in L ( P ) for each Π. This concludes the proof. (cid:3) Remark 4.3. In practice, one may approximate the gain process by a non-antecipative strategy asfollows: Let Π be a given set of trading dates on the interval [0 , T ] so that | Π | = max ≤ i ≤ p | s i − s i − | is small. We take a large k and we perform a non-antecipative buy and hold-type strategy among thetrading dates [ s i , s i +1 ); s i ∈ Π in the full approximation (4.6) which results (4.9) (cid:88) s i ∈ Π θ k,Hs i , (cid:0) S s i ,s i +1 − s i − S s i , (cid:1) where θ k,Hs i , = E (cid:104) D k, s i X T k, si, (cid:12)(cid:12) F s i (cid:105) σ s i , S s i , ; s i ∈ Π . Convergence (4.5) implies that the approximation (4.9) results in unavoidable hedging errors w.r.tthe gain process due to the discretization of the dynamic hedging, but we do not expect large hedgingerrors provided k is large and | Π | small. Hedging errors arising from discrete hedging in completemarkets are widely studied in literature. We do not know optimal rebalancing dates in this incompletemarket setting, but simulation results presented in Section 6 suggest that homogeneous hedging dateswork very well for a variety of models with and without stochastic volatility. A more detailed study isneeded in order to get more precise relations between Π and the stopping times, a topic which will befurther explored in a forthcoming paper. Let us now briefly explain how the results of this section can be applied to well-known quadratichedging methodologies. Generalized F¨ollmer-Schweizer : If one takes the minimal martingale measure ˆ P , then L H in (4.1)is a P -local martingale and orthogonal to the martingale component of S . Due this orthogonalityand the zero mean behavior of the cost L H , it is still reasonable to work with generalized F¨ollmer-Schweizer decompositions under P without knowing a priori the existence of locally-risk minimizinghedging strategies. Local Risk Minimization : One should notice that if (cid:82) θ H dS ∈ B ( F ), L H ∈ B ( F ) under P and d ˆ P d P ∈ L ( P ), then θ H is the locally risk minimizing trading strategy and (4.1) is the F¨ollmer-Schweizerdecomposition under P . Mean Variance hedging : If one takes ˜ P , then the mean variance hedging strategy is not completelydetermined by the GKW decomposition under ˜ P . Nevertheless, Corollary 4.2 still can be used toapproximate the optimal hedging strategy by computing the density process ˜ Z based on the so-calledfundamental equations derived by Hobson [14]. See (1.4) and (1.5) for details. For instance, in theclassical Heston model, Hobson derives analytical formulas for ˜ ζ . See (6.4) in Section 6. Hedging of fully path-dependent options : The most interesting application of our results is the hedgingof fully path-dependent options under stochastic volatility. For instance, if H = Φ( { S t ; 0 ≤ t ≤ T } ) then Corollary 4.2 and Remark 4.3 jointly with the above hedging methodologies allow us todynamically hedge the payoff H based on (4.9). The conditioning on the information flow {F s i ; s i ∈ Π } in the hedging strategy θ k,Hhedg := { θ k,Hs i ; s i ∈ Π } encodes the continuous monitoring of a path-dependentoption. For each hedging date s i , one has to incorporate the whole history of the price and volatilityuntil such date in order to get an accurate description of the hedging. If H is not path-dependentthen the information encoded by {F s i ; s i ∈ Π } in θ k,Hhedg is only crucial at time s i .Next, we provide the details of the Monte Carlo algorithm for the approximating pure hedgingstrategy θ k,Hhedg = { θ k,Hs i , ; s i ∈ Π } . 5. The algorithm In this section we present the basic algorithm to evaluate the hedging strategy for a given European-type contingent claim H ∈ L ( Q ) satisfying assumption (M) for a fixed Q ∈ M e at a terminal time0 < T < ∞ . The structure of the algorithm is based on the space-filtration discretization schemeinduced by the stopping times { T k,jm ; k ≥ , m ≥ , j = 1 , . . . , p } . From the Markov property, the keypoint is the simulation of the first passage time T k,j for each j = 1 . . . , p for which we refer the workof Burq and Jones [3] for details. (Step 1) Simulation of { A k,j ; k ≥ , j = 1 , . . . , p } .(1) One chooses k ≥ { T k,j(cid:96) − T k,j(cid:96) − ; (cid:96) ≥ } according to the algorithm described byBurq and Jones [3].(3) One simulates the family { η k,j(cid:96) ; (cid:96) ≥ } independently from { T k,j(cid:96) − T k,j(cid:96) − ; (cid:96) ≥ } . This i.i.d family { η k,j(cid:96) ; (cid:96) ≥ } must be simulated according to the Bernoulli random variable η k,j withparameter 1 / i = − , 1. This simulates the jump process A k,j for j = 1 , . . . , p .The next step is the simulation of D k,j X T k,j where the conditional expectations in (3.5) play a keyrole. For this, we need to simulate H based on { S t ; 0 ≤ t ≤ T } as follows. YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 19 (Step 2) Simulation of the risky asset price process { S i ; i = 1 , . . . , d } .(1) Generate a sample of A k,i according to Step 1 for a fixed k ≥ T k at hand, we can apply some appropriate approximation method toevaluate the discounted price. Generally speaking, we work with some Itˆo-Taylor expansionmethod.The multidimensional setup requires an additional notation as follows. In the sequel, t k,j(cid:96) denotesthe realization of the T k,j(cid:96) by means of Step 1, t k(cid:96) denotes the realization of T k(cid:96) based on the finestrandom partition T k . Moreover, any sequence ( t k < t k < . . . < t k,j ) encodes the information generatedby the realization of T k until the first hitting time of the j -th partition. In addition, we denote t k,j − as the last time in the finest partition previous to t k,j . Let ν (cid:96)k = ( ν (cid:96) ,k , ν (cid:96) ,k ) be the pair which realizes t k(cid:96) = t k,ν (cid:96) ,k ν (cid:96) ,k , k, (cid:96) ≥ η kt k(cid:96) as the realization of the random variable η k,ν (cid:96) ,k ν (cid:96) ,k . Recallexpression (3.2). (Step 3) Simulation of the stochastic derivative D k,j X T k,j .Based on Steps 1 and 2, for each j = 1 , . . . , p one simulates D k,j X T k,j as follows. In the sequel, ˆ E denotes the conditional expectation computed in terms of the Monte Carlo method:(5.1) ˆ D k,jt k,j X := 12 − k η k,j (cid:110) ˆ E (cid:104) H (cid:12)(cid:12)(cid:12) (cid:16) t k , η kt k (cid:17) , . . . , (cid:16) t k,j , η kt k,j (cid:17)(cid:105) − ˆ E (cid:104) H (cid:12)(cid:12)(cid:12) (cid:16) t k , η kt k (cid:17) , . . . , (cid:16) t k,j − , η kt k,j − (cid:17)(cid:105) (cid:111) , where with a slight abuse of notation, η k,j in (5.1) denotes the realization of the Bernoulli variable η k,j . Then we define(5.2) ˆ φ H,S,k := (cid:16) ˆ D k, t k, X, . . . , ˆ D k,dt k,d X (cid:17) , The correspondent simulated pure hedging strategy is given by(5.3) ¯ θ k,H , := ( ˆ φ H,S,k ) (cid:62) [diag( S ) σ ] − . (Step 4) Simulation of θ k,Hhedg .Repeat these steps several times and(5.4) ˆ θ k,H , := mean of ¯ θ k,H , . The quantity (5.4) is a Monte Carlo estimate of θ k,H , . Remark 5.1. In order to compute the hedging strategy θ k,Hhedg over a trading period { s i ; i = 0 , . . . , q } ,one perform the algorithm described above but based on the shifted filtration and the Brownian motions W ( j ) s i for j = 1 , . . . , p as described in Section 4.1. Remark 5.2. In practice, one has to calibrate the parameters of a given stochastic volatility modelbased on liquid instruments such as vanilla options and volatility surfaces. With those parameters athand, the trader must follow the steps (5.1) and (5.4). The hedging strategy is then given by calibrationand the computation of the quantity (5.4) over a trading period. Numerical Analysis and Discussion of the Methods In this section, we provide a detailed analysis of the numerical scheme proposed in this work.6.1. Multidimensional Black-Scholes model. At first, we consider the classical multidimensionalBlack-Scholes model with as many risky stocks as underlying independent random factors to be hedged( d = p ). In this case, there is only one equivalent local martingale measure, the hedging strategy θ H is given by (3.6) and the cost is just the option price. To illustrate our method, we study a veryspecial type of exotic option: a BLAC (Basket Lock Active Coupon) down and out barrier optionwhose payoff is given by H = (cid:89) i (cid:54) = j { min s ∈ [0 ,T ] S is ∨ min s ∈ [0 ,T ] S js >L } . It is well-known that for this type of option, there exists a closed formula for the hedging strategy.Moreover, it satisfies the assumptions of Theorem 7.2. See e.g Bernis, Gobet and Kohatsu-Higa [1]for some formulas.For comparison purposes with Bernis, Gobet and Kohatsu-Higa [1], we consider d = 5 underlyingassets, r = 0% for the interest rate and T = 1 year for the maturity time. For each asset, we set initialvalues S i = 100; 1 ≤ i ≤ S with discretization level k = 3 , , , (cid:107) σ (cid:107) = 35%, (cid:107) σ (cid:107) = 35%, (cid:107) σ (cid:107) = 38%, (cid:107) σ (cid:107) = 35% and (cid:107) σ (cid:107) = 40%, the correlation matrix defined by ρ ij = 0 , i (cid:54) = j ,where σ i = ( σ i , · · · , σ i ) (cid:62) and we use the barrier level L = 76. Table 1 provides the numericalresults based on the algorithm described in Section 5 for the pointwise hedging strategy θ H . Due toTheorem 7.2, we expect that when the discretization level k increases, we obtain results closer to thetrue value and this is what we find in our Monte Carlo experiments. The standard deviation andpercentage % error in Table 1 are related to the average of the hedging strategies calculated by MonteCarlo and the difference between the true and the estimated hedging value, respectively. k Result St. error True value Diference % error . . × − . . . . . × − . . . . . × − . . . . . × − . . . Table 1. Monte Carlo hedging strategy of a BLAC down and out option for a 5-dimensionalBlack-Scholes model. In Figure 1, we plot the average hedging estimates with respect to the number of simulations. Oneshould notice that when k increases, the standard error also increases, which suggests more simulationsfor higher values of k .6.2. Hedging Errors. Next, we present some hedging error results for two well-known non-constantvolatility models: The constant elasticity of variance (CEV) model and the classical Heston stochasticvolatility model [13]. The typical examples we have in mind are the generalized F¨ollmer-Schweizer,local risk minimization and mean variance hedging strategies, where the optimal hedging strategies arecomputed by means of the minimal martingale measure and the variance optimal martingale measure,respectively. We analyze digital and one-touch one-dimensional European-type contingent claims asfollows YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 21 Figure 1. Monte Carlo hedging strategy of a BLAC down and out option for a 5-dimensionalBlack-Scholes model. Digital option: H = 11 { S T < } , One-touch option: H = 11 { max t ∈ [0 ,T ] S t > } . By using the algorithm described in Section 5, we compute the error committed by approximatingthe payoff H by (cid:99) E Q [ H ] + (cid:80) n − i =0 ˆ θ k,Ht i , ( S t i ,t i +1 − t i − S t i , ). This error will be called hedging error . Thecomputation of this error is summarized in the following steps:(1) We first simulate paths under the physical measure and compute the payoff H .(2) Then, we consider some deterministic partition of the interval [0,T] into n points t , t , . . . , t n − such that t i +1 − t i = Tn , for i = 0 , . . . , n − t = 0 the option price (cid:99) E Q [ H ] and the initial hedging estimate ˆ θ k,H , from (5.2), (5.3) and (5.4) under a fixed Q ∈ M e following the algorithm described in Section5.(4) We simulate ˆ θ k,Ht i , by means of the shifting argument based on the strong Markov property ofthe Brownian motion as described in Section 4.1.(5) We compute ˆ H by(6.1) ˆ H := (cid:99) E Q [ H ] + n − (cid:88) i =0 ˆ θ k,Ht i , ( S t i ,t i +1 − t i − S t i , ) . (6) Finally, the hedging error estimate γ and the percentual error e γ are given by γ := H − ˆ H and e γ := 100 × γ/ (cid:99) E Q [ H ], respectively. Remark 6.1. When no locally-risk minimizing strategy is available, we also expect to obtain lowhedging errors when dealing with generalized F¨ollmer-Schweizer decompositions due to the orthogonalmartingale decomposition. In the mean variance hedging case, two terms appear in the optimal hedgingstrategy: the pure hedging component θ H, ˜ P of the GKW decomposition under the optimal variancemartingale measure ˜ P and ˜ ζ as described by (1.4) and (1.5). For the Heston model, ˜ ζ was explicitly calculated by Hobson [14] . We have used his formula in our numerical simulations jointly with ˆ θ k,H under ˜ P in the calculation of the mean variance hedging errors. See expression (6.4) for details. Constant Elasticity of Variance (CEV) model. The discounted risky asset price process de-scribed by the CEV model under the physical measure is given by(6.2) dS t = S t (cid:104) ( b t − r t ) dt + σS ( β − / t dB t (cid:105) , S = s, where B is a P -Brownian motion. The instantaneous sharpe ratio is ψ t = b t − r t σS ( β − / t such that themodel can be rewritten as(6.3) dS t = σ t S β/ t dW t where W is a Q -Brownian motion and Q is the equivalent local martingale measure. For both thedigital and one-touch options, we consider the parameters r = 0 for the interest rate, T = 1 (month)for the maturity time, σ = 0 . S = 100 and β = 1 . − . 4. Wesimulate the hedging error along [0 , T ] considering discretization levels k = 3 , 4, and 1 and 2 hedgingstrategies per day, which means approximately 22 and 44 hedging strategies, respectively, along theinterval [0 , T ]. From Corollary 4.2, we know that this procedure is consistent. For the digital option,we also recall that the hedging strategy has continuous paths up to some stopping time (see Zhang [27])so that Theorem 7.2 and Remark 7.2 apply accordingly. The hedging error results for the digital andone-touch options are summarized in Tables 2 and 3, respectively. The standard deviations are relatedto the hedging errors. Simulations k Hedges/day Hedging error γ St. dev. Price % Error e γ 200 3 1 − . . . . − . . . . . . . . − . . . . Table 2. Hedging error of a digital option for the CEV model. Simulations k Hedges/day Hedging error γ St. dev. Price % Error e γ 600 3 1 0 . . . . . . . . . . . . . . . . Table 3. Hedging error of one-touch option for the CEV model.6.2.2. Heston’s Stochastic Volatility Model. Here we consider two types of hedging methodologies:Local-risk minimization and mean variance hedging strategies as described in the Introduction andRemark 6.1. The Heston dynamics of the discounted price under the physical measure is given by (cid:26) dS t = S t ( b t − r t )Σ t dt + S t √ Σ t dB (1) t d Σ t = 2 κ ( θ − Σ t ) dt + 2 σ √ Σ t dZ t , ≤ t ≤ T, where Z = ρB (1) + ¯ ρB (2) t , ¯ ρ = (cid:112) − ρ , with ( B (1) B (2) ) two independent P -Brownian motions and κ, m, β , µ are suitable constants in order to have a well-defined Markov process (see e.g Heston [13]).Alternatively, we can rewrite the dynamics as YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 23 dS t = S t Y t ( b t − r t ) dt + S t Y t dB (1) t dY t = κ (cid:32) mY t − Y t (cid:33) dt + σdZ t , ≤ t ≤ T, where Y = √ Σ t and m = θ − σ κ . Local-Risk Minimization . For comparison purposes with Heath, Platen and Schweizer [12], weconsider the hedging of a European put option H written on a Heston model with correlation parameter ρ = 0. We set S = 100, strike price K = 100, T = 1 (month) and we use discretization levels k = 3 , κ = 2 . θ = 0 . ρ = 0, σ = 0 . r = 0 and Y = 0 . 02. In thiscase, the hedging strategy θ H, ˆ P based on the local-risk-minimization methodology is bounded withcontinuous paths so that Theorem 7.2 applies to this case. Moreover, as described by Heath, Platenand Schweizer [12], θ H, ˆ P can be obtained by a PDE numerical analysis.Table 4 presents the results of the hedging strategy ˆ θ k,H , by using the algorithm described in Sec-tion 5. Figure 2 provides the Monte Carlo hedging strategy with respect to the number of simulationsof order 10000. We notice that our results agree with the results obtained by Heath, Platen andSchweizer [12] by PDE methods. In this case, the true value of the hedging at time t = 0 is ap-proximately − . 44. The standard errors in Table 4 are related to the hedging and prices computed,respectively, from the Monte Carlo method described in Section 5. k Hedging Standard error Monte Carlo price Standard error − . . × − . 417 5 . × − − . . × − . 422 3 . × − − . . × − . 409 2 . × − Table 4. Monte Carlo local-risk minimization hedging strategy of a European put option withHeston model. Hedging with generalized F¨ollmer-Schweizer decomposition for one-touch option. Basedon Corollary 4.2, we also present the hedging error associated to one-touch options for a Heston modelwith non-zero correlation. We simulate the hedging error along the interval [0 , 1] using k = 3 , κ = 3 . θ = 0 . ρ = − . σ = 0 . r = 0, b = 0 . Y = 0 . S = 100 where the barrier is 105. The hedgingerror result for the one-touch option is summarized in Table 5. The standard deviations in Table 5are related to the hedging error.To our best knowledge, there is no result about the existence of locally-risk minimizing hedgingstrategies for one-touch options written on a Heston model with nonzero correlation. As pointed outin Remark 6.1, it is expected that pure hedging strategies based on the generalized F¨ollmer-Schweizerdecomposition mitigate very-well the hedging error. This is what we get in the simulation results. Mean variance hedging strategy . Here we present the hedging errors associated to one-touchoptions written on a Heston model with non-zero correlation under the mean variance methodology.Again, we simulate the hedging error along the interval [0 , 1] using k = 3 , r = 0, b = 0 . κ = 3 . θ = 0 . ρ = − . σ = 0 . Y = 0 . S = 100 with barrier 105. The computation of the optimal hedging strategy Figure 2. Monte Carlo local-risk minimization hedging strategy of a European put option withHeston model. Simulations k Hedges/day Hedging error γ St. dev. Price % error e γ 600 3 1 0 . . . . . . . . . . . . . . . . Table 5. Hedging error with generalized F¨ollmer-Schweizer decomposition: One-touch option with Heston model.follows from Remark 6.1. The quantity ˜ ζ is not related to the GKW decomposition but it is describedby Theorem 1.1 in Hobson [14] as follows. The process ˜ ζ appearing in (1.4) and (1.5) is given by(6.4) ˜ ζ t = ˜ Z ρσF ( T − t ) − ˜ Z b ; 0 ≤ t ≤ T, where F is given by (see case 2 of Prop. 5.1 in Hobson [14]) F ( t ) = CA tanh (cid:18) ACt + tanh − (cid:18) ABC (cid:19)(cid:19) − B ; 0 ≤ t ≤ T, with A = (cid:112) | − ρ | σ , B = κ +2 ρσbσ | − ρ | and C = (cid:112) | D | where D = 2 b + ( κ +2 ρσb ) ) σ (1 − ρ ) . The initialcondition ˜ Z is given by ˜ Z = Y F ( T ) + κθ (cid:90) T F ( s ) ds. YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 25 The hedging error results are summarized in Table 6 where the standard deviations are relatedto the hedging error. In comparison with the local-risk minimization methodology, the results showsmaller percentual errors when k increases. Also, in all the cases, we had smaller values of the standarddeviation which suggests the mean variance methodology provides more accurate values of the hedgingstrategy. Simulations k Hedges/day Hedging error γ St. dev. Price % error e γ 600 3 1 0 . . . . . . . . . . . . . . . . Table 6. Hedging error in the mean variance hedging methodology for one-touchoption with Heston model. 7. Appendix This appendix provides a deeper understanding of the Monte Carlo algorithm proposed in this workwhen the representation ( φ H,S , φ H,I ) in (3.6) admits additional integrability and path smoothnessassumptions. We present stronger approximations which complement the asymptotic result given inTheorem 3.1. Uniform-type weak and strong pointwise approximations for θ H are presented and theyvalidate the numerical experiments in Tables 1 and 4 in Section 6. At first, we need of some technicallemmas. Lemma 7.1. Suppose that φ H = ( φ H, , . . . , φ H,p ) is a p -dimensional progressive process such that E sup ≤ t ≤ T (cid:107) φ Ht (cid:107) R p < ∞ . Then, the following identity holds (7.1) ∆ δ k X T k,j = E (cid:34)(cid:90) T k,j φ H,js dW ( j ) s | F kT k,j (cid:35) a.s ; j = 1 , . . . , p ; k ≥ . Proof. It is sufficient to prove for p = 2 since the argument for p > H be the linear space constituted by the bounded R -valued F -progressive processes φ = ( φ , φ )such that (7.1) holds with X = X + (cid:82) · φ s dW (1) s + (cid:82) · φ s dW (2) s where X ∈ F . Let U be theclass of stochastic intervals of the form [[ S, + ∞ [[ where S is a F -stopping time. We claim that φ = (cid:0) [[ S, + ∞ [[ , [[ J, + ∞ [[ (cid:1) ∈ H for every F -stopping times S and J . In order to check (7.1) for such φ , we only need to show for j = 1 since the argument for j = 2 is the same. With a slight abuse ofnotation, any sub-sigma algebra of F T of the form Ω ∗ ⊗ G will be denoted by G where Ω ∗ is the trivialsigma-algebra on the first copy Ω .At first, we split Ω = (cid:83) ∞ n =1 { T kn = T k, } and we make the argument on the sets { T kn = T k, } ; n ≥ F kT k, = F k, T k, ⊗ F k, T k, n − a.s and∆ δ k X T k,j = ∆ δ k (cid:16) W (1) T k, − W (1) S (cid:17) { S Let B be a one-dimensional Brownian motion and S kn := inf { t > S kn − ; | B t − B S kn − | =2 − k } with S k = 0 a.s , n ≥ . If ϕ is an absolutely continuous and non-negative adapted process thenthere exists a deterministic constant C which does not depend on m, k ≥ such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) S km S km − ϕ t dB t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { S km ≤ T } ≤ C sup ≤ t ≤ T | ϕ t | − k a.s ; k, m ≥ . Proof. For given m, k ≥ 1, Young inequality and integration by parts yield (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) S km S km − ϕ t dB t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:110) | ϕ S km | | B S km | + | ϕ S km − | | B S km − | + (cid:12)(cid:12) (cid:90) S km S km − B t dϕ t (cid:12)(cid:12) (cid:111) YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 27 ≤ C − k sup ≤ t ≤ T | ϕ t | + C sup S km − ≤ t ≤ S km | B t | | V ar ( ϕ ) S km − V ar ( ϕ ) S km − | ≤ C − k sup ≤ t ≤ T | ϕ t | + C − k | V ar ( ϕ ) S km − V ar ( ϕ ) S km − | = C − k sup ≤ t ≤ T | ϕ t | + C − k | ϕ S km − ϕ S km − | ≤ C − k sup ≤ t ≤ T | ϕ t | a.s on { S km ≤ T } , for some constant C which does not depend on m, k ≥ (cid:3) Lemma 7.3. Assume that φ H,j ∈ B ( F ) for some j = 1 , . . . , p . Then there exists a constant C suchthat sup k ≥ E sup ≤ t ≤ T | D k,j X t | ≤ C E sup ≤ t ≤ T | φ H,j | . Proof. By repeating the argument employed in Lemma 7.1 for k ≥ n > j ∈ { , . . . , p } , weshall write D k,j X t = E (cid:34) A k,jT k,jn (cid:90) T k,jn T k,jn − φ H,jt dW ( j ) t (cid:12)(cid:12) F kT k,jn − (cid:35) a.s on { T k,jn − < t ≤ T k,jn } . Doob maximal inequalities combined with Jensen inequality yield(7.2) E sup ≤ t ≤ T | D k,j X t | ≤ C k E sup n ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T k,jn T k,jn − φ H,jt dW ( j ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { T k,jn ≤ T } , for k ≥ C . Now, we need a path-wise argument in order to estimatethe right-hand side of (7.2). For this, let us define ϕ (cid:96),jt := (cid:96) (cid:90) tt − (cid:96) φ H,js ds ; (cid:96) ≥ 1; 0 ≤ t ≤ T. Lemma 7.2 and the fact that sup ≤ t ≤ T | ϕ (cid:96),jt | ≤ sup ≤ t ≤ T | φ H,jt | ∀ (cid:96) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T k,jn T k,jn − ϕ (cid:96),jt dW ( j ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { T k,jn ≤ T } ≤ C sup ≤ t ≤ T | φ H,jt | − k ; (cid:96), n, k ≥ , where C is the constant in Lemma 7.2. Now, by applying Lemma 2.4 in Nutz [21], the estimate (7.3)and a routine localization procedure, the following estimate holds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T k,jn T k,jn − φ H,jt dW ( j ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { T k,jn ≤ T } ≤ C sup ≤ t ≤ T | φ H,jt | − k ; k ≥ , and therefore(7.4) E sup n ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T k,jn T k,jn − φ H,jt dW ( j ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { T k,jn ≤ T } ≤ C E sup ≤ t ≤ T | φ H,jt | − k ∀ k ≥ . The estimate (7.2) combined with (7.4) allow us to conclude the proof if φ H,j ≥ a.s ( Leb × Q ). Bysplitting φ H,j = φ H,j, + − φ H,j, − into the negative and positive parts, we may conclude the proof ofthe lemma. (cid:3) The following result allows us to get a uniform-type weak convergence of D k,j X under very mildintegrability assumption. Theorem 7.1. Let H be a Q -square integrable contingent claim satisfying assumption (M) andassume that H admits a representation φ H such that φ H,j ∈ B ( F ) for some j ∈ { , . . . , p } . Then lim k →∞ D k,j X = φ H,j weakly in B ( F ) .Proof. Let us fix j = 1 , . . . , p . From Lemma 7.3, we know that { D k,j X ; k ≥ } is bounded in B ( F )and therefore this set is weakly relatively compact in B ( F ). By Eberlein Theorem, we also know thatit is B ( F )-weakly relatively sequentially compact. From Theorem 3.1,lim k →∞ D k,j X = φ H,j weakly in L ( Leb × Q ) and since (cid:107) · (cid:107) B is stronger than (cid:107) · (cid:107) L ( Leb × Q ) , we necessarily have the fullconvergence lim k →∞ D k,j X = φ H,j in σ (B , M ). (cid:3) Next, we analyze the pointwise strong convergence for our approximation scheme.7.1. Strong Convergence under Mild Regularity. In this section, we provide a pointwise strongconvergence result for GKW projectors under rather weak path regularity conditions. Let us considerthe stopping times τ j := inf (cid:8) t > | W ( j ) t | = 1 (cid:9) ; j = 1 , . . . , p, and we set ψ H,j ( u ) := E | φ H,jτ j u − φ H,j | , for u ≥ , j = 1 . . . , p. Here, if u satisfies τ j u ≥ T we set φ H,jτ j u := φ H,jT and for simplicity we assume that ψ H,j (0 − ) = 0. Theorem 7.2. If H is a Q -square integrable contingent claim satisfying (M) and there exists arepresentation φ H = ( φ H, , . . . , φ H,p ) of H such that φ H,j ∈ B ( F ) for some j ∈ { , . . . , p } and theinitial time t = 0 is a Lebesgue point of u (cid:55)→ ψ H,j ( u ) , then (7.5) D k,j X T k,j → φ H,j as k → ∞ . Proof. In the sequel, C will be a constant which may differ from line to line and let us fix j = 1 , . . . , p .For a given k ≥ 1, it follows from Lemma 7.1 that D k,j X T k,j = E (cid:104)(cid:82) T k,j φ H,js dW ( j ) s | F kT k,j (cid:105) ∆ A k,jT k,j = E (cid:104)(cid:82) T k,j (cid:16) φ H,js − φ H,j + φ H,j (cid:17) dW ( j ) s | F kT k,j (cid:105) ∆ A k,jT k,j = E (cid:104)(cid:82) T k,j (cid:16) φ H,js − φ H,j (cid:17) dW ( j ) s | F kT k,j (cid:105) ∆ A k,jT k,j + E (cid:104) φ H,j | F kT k,j (cid:105) . (7.6) YNAMIC HEDGING UNDER STOCHASTIC VOLATILITY 29 We recall that T k,j law = 2 − k τ j so that we shall apply the Burkholder-Davis-Gundy and Cauchy-Schwartz inequalities together with a simple time change argument on the Brownian motion to getthe following estimate E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:104)(cid:82) T k,j (cid:16) φ H,js − φ H,j (cid:17) dW ( j ) s | F kT k,j (cid:105) ∆ A k,jT k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) T k,j (cid:16) φ H,js − φ H,j (cid:17) dW ( j ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 k E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) − k (cid:16) φ H,jτ j s − φ H,j (cid:17) dW ( j ) τ j s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) − k (cid:16) φ H,jτ j s − φ H,j (cid:17) τ j ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / ≤ C E / τ j E / − k (cid:90) − k (cid:16) φ H,jτ j u − φ H,j (cid:17) du = C E / − k (cid:90) − k (cid:16) φ H,juτ j − φ H,j (cid:17) du. (7.7)Therefore, the right-hand side of (7.7) vanishes if, and only if, t = 0 is a Lebesgue point of u (cid:55)→ ψ H,j ( u ),i.e.,(7.8) 12 − k (cid:90) − k E | φ H,juτ j − φ H,j | du → k → ∞ . The estimate (7.7), the limit (7.8) and the weak convergence of F kT k,j to the initial sigma-algebra F yield lim k →∞ D k,j X T k,j = lim k →∞ E (cid:104) φ H,j | F kT k,j (cid:105) = φ H,j strongly in L . Since D k,j X T k,j = E (cid:2) D k,j X T k,j (cid:3) ; k ≥ (cid:3) Remark 7.1. At first glance, the limit (7.5) stated in Theorem 7.2 seems to be rather weak sinceit is not defined in terms of convergence of processes. However, from the purely computational pointof view, we shall construct a pointwise Monte Carlo simulation method of the GKW projectors interms of D k,j X T k,j given by (3.5). This substantially simplifies the algorithm introduced by Le˜ao andOhashi [20] for the unidimensional case under rather weak path regularity. Remark 7.2. For each j = 1 , . . . , p , let us define ψ H,j ( t , u ) := E | φ H,jt + τ j u − φ H,jt | , for t ∈ [0 , T ] , u ≥ . One can show by a standard shifting argument based on the Brownian motion strong Markov propertythat if there exists a representation φ H such that u (cid:55)→ ψ H,j ( t , u ) is cadlag for a given t then onecan recover pointwise in L -strong sense the j -th GKW projector for that t . We notice that if φ H,j belongs to B ( F ) and it has cadlag paths then u (cid:55)→ ψ H,j ( t , u ) is cadlag for each t , but the conversedoes not hold. Hence the assumption in Theorem 7.2 is rather weak in the sense that it does not implythe existence of a cadlag version of φ H,j . References [1] G. Bernis, E. Gobet, and A. Kohatsu-Higa. Monte carlo evaluation of greeks for multidimensional barrier andlookback options. Mathematical Finance , 13(1):99–113, 2003. [2] F. Biagini, P. Guasoni, and M. Pratelli. Mean-variance hedging for stochastic volatility models. 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Universidade de S˜ao Paulo, 13560-970, S˜ao Carlos- SP, Brazil E-mail address : [email protected] E-mail address : [email protected] Departamento de Matem´atica, Universidade Federal da Para´ıba, 13560-970, Jo˜ao Pessoa - Para´ıba, Brazil E-mail address ::