# Application of Malliavin Calculus in Mean-Variance Hedging Strategy

Authors: Kefan Liu; Jingyao Chen; Jichao Zhang; Xili Tan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper considers an approach of Malliavin calculus to obtain the hedging ratio for mean-variance hedging (MVH) strategy under the stochastic volatility model with pure jump Lévy process (SVJ). Specifically speaking, there exists a correspondence between the martingale representation theorem and the Clark-Ocone formula for a square integrable contingent claim. Therefore, we can replace the diffusion term coefficients with the functions containing Malliavin derivatives to get a closed-form representation for the MVH strategy. By fast Fourier transform (FFT) algorithm, some numerical examples are performed to analyze the sensitivity of MVH strategy concerning strike price and current time.


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## Body

## 1. Introduction

Since the appearance of the Black-Scholes (B-S) model was proposed in 1973 [1], much research has been conducted on its application to the financial market. However, the different maturity dates and strike prices lead to the “smiles” and “incline” of volatility, which are inconsistent with the assumption that volatility is constant. Some classical models have been proposed to incorporate these phenomena in asset pricing, including two aspects. The first is presented by Heston [2], which assumes that volatility also follows a stochastic process, resulting in the volatility clustering effects can be captured. The second is proposed by Merton [3], which uses compound Poisson processes to describe the jump-diffusion phenomenon in the price process of the underlying asset. These models can be generally classified as Lévy processes for the above stochastic processes. Lévy processes have gained increasing importance in the option pricing literature due to the well-known flaws of the classical Black-Scholes geometric Brownian motion model [4]. They include the Brownian motion Poisson process and the compound Poisson process. Based on these characteristics, the model can well describe the leptokurtosis phenomenon and infinite small jump behaviours of asset return distribution which cannot be described by the Gaussian process. Therefore, it is useful and feasible to adopt Lévy process in option pricing research.In recent years, many researchers have yet to be satisfied with the pricing performance of the existing models and have developed various pricing models. He and Zhu [5] study the two-factor Heston-CIR model in which the interest rate process obeyed the CIR process and derived closed-form series solutions to variance and volatility swaps. Diffusion processes with double exponential jump features were examined by Kou and Wang [6]. A pricing model with stochastic volatility and stochastic interest rates with pure jumps can be seen in [7], in which the jump components of asset price follow the compound Poisson process, and volatility is stochastic. Carr and Wu [8] find that the compound Poisson process cannot describe the many small jumps in financial assets. However, the infinite activity Lévy process allows the jump components to have infinite activity and admits nearly an arbitrary distribution. Using an infinite activity Lévy process instead of a compound Poisson process is natural. Feng et al. [9] present a generalized European option-pricing model with stochastic volatility and stochastic interest rates and pure jumps under Lévy processes, and alternatively, the FFT algorithm is used to obtain the solution of option price. Bakshi et al. [10] exhibit empirical results of all those models above for the S&P500 call option data by the square of sum of pricing errors optimization and find the SVJ model has the least bias.Due to introducing a new random source, volatility, and pure jump, the market is usually incomplete such that the equivalent martingale measure is not unique. We cannot construct a portfolio to reduce intrinsic risk to zero in the incomplete market perfectly, so we focus on how to construct a hedging strategy to minimize risk. Gourieroux et al. [11] proposed the mean-variance hedging strategy, which minimizes the expected squared hedging cost in a global time step size. Another local risk-minimization strategy proposed by Föllmer and Sondermann [12] is minimizing the risk on each infinitesimal interval. Heath et al. [13, 14] compared these two optimization strategies and found that mean-variance hedging is far better than local risk-minimization in reducing risk. However, because of the complexity of the mean-variance hedging strategy, it is not easy to explicitly calculate the hedging ratio expression.In this paper, our work contribution is two-fold. Firstly, we build a mean-variance hedging strategy under a stochastic volatility model driven by exponential Lévy process. Secondly, we develop the closed-form expression for the hedging ratioϑtMVH by the correspondence between the Clark-Ocone formula and the martingale representation theorem under variance-optimal equivalent martingale measure Q. To demonstrate the sensitivity of MVH strategy concerning time and strike price, we adopt the FFT algorithm to calculate the conditional expectation contained in ϑtMVH. The underlying assets pricing and volatility processes are approximated with an involved recursive calculation, in which the discrete schemes for Gaussian process and pure jump are defined as [15].The structure of this paper is as follows. In Section1, we introduce the setting of the market and model. In Section 2, we describe the theoretical background of the mean-variance hedging strategy under SVJ model. Moreover, we derive the expression of Galtchouk-Kunita-Watanabe (GKW) decomposition under some conditions by using the martingale representation theorem. In Section 3, we introduce some fundamental conceptions of Malliavin calculus in Lévy space and obtain the explicit solution of mean-variance hedging ratio ϑtMVH by the Clark-Ocone formula under measure Q. Section 4 develops a numerical scheme with the help of FFT and a discrete scheme in [15].


## 2. Market and Model Settings

Consider a frictionless market with a probability spaceΩW×ΩL,F,Ρ. P is the probability measure of the actual market, and F=Ftt∈T, T∈0,∞ is the filtration generated by the process of stock price. ΩW and ΩL are the sample spaces composed of the one-dimensional Wiener process and pure jump Lévy process, respectively, ΩL=⋃n=0∞0,T×R0n, where R0=R\0. For simplicity, St denotes the single risky assets process, Bt is set as the numeraire, and then, the discount price of underlying assets St is Xt=St/Bt. We consider a jump-diffusion stochastic volatility model for this single risky asset, and its stochastic differential equation is given by(1)dXtXt=μt,Vtdt+βt,VtdWtx+∫R0γt,zN^dt,dz,dVt=gt,Xt,Vtdt+bt,Xt,VtdWtv,where Wtx and Wty are two standard Brownian motion under measure P.Vt is the volatility of the underlying asset. N is the Poisson random measure, and Nt,A=∑s≤t1AΔXs, A∈BR0, BR0 is a Borel field. Denoting the Levy measure of Nbyv, N^dt,dz=Ndt,dz−vdzdt is a compensated Poisson process of Ndt,dz. Functions μ, β, g, and b are assumed to be the predictable processes, γt,z is a stochastic process measurable with respect to the σ-algebra generated by A×s,u×B, A∈Fs, 0≤s<u<T and B∈BR0. For the application, we set(2)dWtx=ρdWt+1−ρ2dWt∗,dWtv=dWt,where dWtxdWtv=ρdt, ρ∈−1,1 is the correlation coefficient of Wtx and Wtv. Wt∗ and Wt are independent. Then, our model in (1) becomes(3)dXtXt=μt,Vtdt+ρβt,VtdWt+1−ρ2βt,VtdWt∗+∫R0γt,zN^dt,dz,dVt=gt,Vtdt+bt,VtdWt.


## 3. Mean-Variance Hedging Strategy

Considering a portfolioΘt=ϑt,ηt, ϑt and ηt are two R-valued predictable processes. ϑt denotes the number of risk assets (such as the stock). ηt represents the number of riskless (such as the bound). The value process of the portfolio Θt can be written as ΓtΘ=ϑtXt+ηt, and the cost process is(4)CtΘ=ΓtΘ−∫0tϑudXu,0≤t≤T.LetLX is the space of R-valued predictable X-integrable processes ϑ. Denote the space of all stochastic integral processes ∫0TϑudXu by Θ. Then, we call the Γ0MVH,ϑMVH is a mean-variance optimal strategy for a claim F if it minimizes(5)EF−ΓTΘ2=EF−Γ0MVH−∫0TϑtMVHdXt2=RMVH,where Γ0MVH is initial value of the value process, and ϑtMVH is called mean-variance hedging ratio. Due to the incompleteness of the market, the equivalent martingale measure is not unique, and we can only hope to minimize the global quadratic risk RMVH within the interval 0,T.In the absence of arbitrage opportunities, we assume that there is an equivalent martingale measureP1≈P, and then, XT under measure P1 is a martingale. Let P be the set of all such equivalent martingale measure P1, and if there is more than one element in P, the market is said to be incomplete. Therefore, finding an appropriate measure P1 to hedge a given contingent claim is particularly important.Gourieroux et al. [11] obtain the variance-optimal equivalent martingale measure Q by minimizing dP1/dPL2P=1+VardP1/dP, which is unique in P. Then, the density process of measure Q with respect to measure P can be denoted by ZtQ=EQdQ/dPFt.ZtQ is a martingale under Q, which can be represented by F¨llmer-Schweizer decomposition.(6)ZtQ=EQZTQFt=Z0Q+∫0tζuQdXu.For a contingent claimF∈L2P, the GKW decomposition of F under measure Q is(7)F=EQF+∫0tξtQdXu+LTQ,where ξtQ is a Rd-valued predictable process that coincides with the amount of the risky asset in the local risk-minimization strategy at time t. LTQ is a martingale under Q and is zero at the t=0, and LTQ is strongly Q-orthogonal to Xt. The form of hedging ratio ϑtMVH at time t is proposed by Pham with the following form:(8)ϑtMVH=ξtQ−ζtQZtQΓtQ−EQF−∫0tϑumvodXu=ξtQ−ζtQ∫0t−1/ZuQdLuQ,where F∈L2P is a contingent claim. From the above equation, it is not difficult to know that the key to solving ϑtMVH is to compute LtQ and ξtQ explicitly. For this purpose, we first give the forms of LtQ and ξtQ separately such that we can calculate them in the later sections.By the theorem given by Pham et al. [16], a martingale X can be decomposed into X=X0+Mt+At, where Mt is a real measurable martingale with the following form:(9)dMt=Xtρβt,VtdWt+1−ρ2βt,VtXtdWt∗+∫R0γt,zN^dt,dz,At is an adapted process with the form as(10)At=∫0tdMuλu=∫0tXuμu,Vudu,where(11)λt=dAtdMt=μ1t,VtXtβ2t,Vt+∫R0γt,z2vdz.The density processZtQ can be considered as the Radon–Nikodym derivative of Q. According to the concept of Schweizer [17], we can know ζtQ/ZtQ−λt, and the expression of ZtQ under model (3) can be written as(12)ZtQ=ε−∫0tλudMu−∫0taudWu=exp−∫0tρcu+audWt−∫0t1−ρ2cuWt∗−12∫0tcu2+au2dt+∫0t∫R0log1−θu,z+θu,zvdzdu+∫0t∫R0log1−θu,zN^du,dz,where εx represents the stochastic exponential of x, and the parameters are(13)ct=μt,Vtβt,Vtβ2t,Vt+∫R0γt,z2vdzat=∂Jt,Vt∂Vbt,Vt,θt,z=μt,Vtγt,zβ2t,Vt+∫R0γt,z2vdz,Jt,Vt=−logEexp−∫tTμs,Vs/β2s,Vs+∫R0γs,z2vdz2ds.Then, the stochastic differential equation ofXt under measure Q takes the form(14)dXt=Xtρβt,VtdW^t+1−ρ2βt,VtdW^t∗+∫R0γt,zN^Qdt,dz,dVt=gtQt,Vtdt+bt,VtdW^t,where gtQt,Vt=gtt,Vt−b2t,Vt∂Jt,Vt/∂V. The Girsanov’s theorem implies that W˜t=Wt+∫ρctdt and W˜t∗=Wt∗+∫1−ρ2ctdt are two standard Brownian motions under measure Q. N^Qdt,dz=θt,zvdzdt+N^dt,dz represents the compensated Poisson random measure of N^dt,dz.Theorem 1.
The expression of local risk-minimization strategyξtQ at time t is given by(15)ξtQ=ρβt,VtIt1+1−ρ2βt,VtIt2+∫R0γt,zIt3vdzXtVt+∫R0rt,z2vdz,where It1, It3, and It3 are unique adapted processes.Proof.
Due toLTQ is a martingale under measure Q, according to the martingale representation theorem, there exist two unique random processes r1, r2, such that(16)LTQ=L0Q+∫0Tr1ρdW˜u+∫0Tr11−ρ2dW˜u∗+∫0T∫R0r2N^Qdt,dz.
By the property ofLTQ strongly Q-orthogonal to Xt, we can obtain(17)βt,Vtr1Xt+∫R0r2γt,zvdz=0.
Substitute (16) into (7) to get(18)F=EQF+∫0Tr1+ξtβt,VtXtdW˜t+∫0T∫R0r2+ξtγt,zXtN^Qdt,dz.
SinceF is a martingale under measure Q, it can also be described by the martingale representation theorem with the following form:(19)F=EQF+∫0TIt1dW˜t+∫0TIt2dW˜t∗+∫R0It3N^Qdt,dz,where It1, It2, and It3 are unique adapted processes. Combine with (18) and (19), we can obtain the following equations:(20)ρr1+ρξtQβt,VtXt=It1,1−ρ2r1+1−ρ2ξtQβt,VtXt=It2,r2+ξtQγt,zXt=It3.
According to the condition given in (17), ξtQ can be obtained by solving equations (20).Remark 1.
The processesIt1, It2, and It3 are introduced by the martingale representation theorem, and it is hard to calculated them concretely. Next, we will introduce a method that we can obtain the explicit expression of It1, It2, and It3.


## 4. A Brief of Malliavin Calculus

In the current section, we overview the main concepts of the Malliavin calculus in the Wiener–Poisson space, and for more information, we can refer to [18].Assume thatZt=σWt+∫0t∫R0zN^dt,dz is a combination of Gaussian and pure jump lévy processes. Considering a set as follows:(21)Sn=a1k1,⋯,ankn∈∏i=1nAki:0≤t1,⋯,tn≤T,where Aki=0,T, k1,⋯,kn=0or1. There are some definitions as follows:(22)aiki=ti,ki=0,ti,z,ki=1,,Aki=0,T,ki=0,0,T×R,ki=1.LetdP0=dW, dP1∙=N˜∙,∙, and dP0=dt, dP0=vdzdt. Then, we have the following definitions.Definition 1.
(N-fold iteratedIto^ integrals). Letfk1,⋯,kn∈L2Sn, the n-fold iterated Ito^ integrals is defined as follows:(23)Jnk1,⋯,knfk1,⋯,kn=∫Snfk1,⋯,kna1k1,⋯,anknPk1da1k1⋯Pkndankn.
It can be known from definition of [19].(24)EInk1,⋯,knfk1,⋯,knImk1,⋯,kmgk1,⋯,km=0,n≠m,fk1,⋯,kn,gk1,⋯,km.n=m.
LetHn be a closed linear subspace on L2P consisting of random variables of the form Ink1,⋯,knfk1,⋯,kn. The inner product of Ink1,⋯,knfk1,⋯,kn and Imk1,⋯,kmgk1,⋯,km is 0 if n≠m, and it means that Hn and Hm are two orthonormal subspaces of L2P.Definition 2.
(Wiener chaos decomposition [20]). Hn:n>0 are dense subspaces in L2P, and then, L2P=⊕n=0∞Hn. If F∈L2P, its only expansion is as follows:(25)F=EF+∑n=1∞Fn,where EF∈H0 and Fi∈Hi, for i=1,⋯,n. Fi series converges in L2P.Definition 3.
(Malliavin derivative [21]). Assume(26)Dr=F∈L2P,F=EF+∑n=1∞∑ki=0,1i=0,⋯nJnk1,⋯,knfn:∑n=1∞∑ki=0,1i=0,⋯n∑j=1n1kj=r∫Akjfk1,⋯,kn⋯,ar,⋯L2Snj2dPrar<∞.
For anyF∈Dr⊂L2P, the Malliavin derivative of F can be defined by(27)DakiF=∑n=1∞nIn−1f=∑n=1∞∑ki=0,1i=0,⋯n∑j=1n1kj=rJk1,⋯,kj,⋯,knfk1,⋯,kn⋯,ar,⋯1Snjt,where(28)Snjt=a1k1,⋯,a^,⋯,ankn∈∏i=1nAki:0≤t1,⋯,tn≤T.Definition 4.
(Clark-Ocone formula for Lévy functionals [19]). Assume that F∈Dr,r=0,1. Then,(29)F=EF+∫0TEDsFFsdWs+∫0T∫R0EDs,zFFsN^ds,dz.


## 5. The Calculation of Hedging RatioϑtMVH

This section will use the related properties of the Malliavin derivative and the Clark-Ocone formula to get the expression of a contingent claim under the equivalent martingale measure. In addition, the correspondence between the martingale representation theorem and the Clark–Ocone formula lead to the closed-form solution of the MVH strategy. For this purpose, we need the following results:Theorem 2.
Suppose thatF,ZTQ∈D0,ZTQ is defined in (7). Then, the Malliavin derivative of ZTQF at time t is calculated as(30)DtZTQF=ZTQDtF+F−ρ+1−ρ2ct−at+H1t,where(31)H1t=−∫tTDtρcu+DtρaudW˜u−∫tTDt1−ρ2cudW˜u∗−∫tT∫R0Dtθu,z/1−θu,zN^du,dz.Proof.
Firstly, by the continuous case product rule of [19], we can deduce that(32)DtZTQF=ZTQDtF+FDtZTQ.
And according to the continuous case operation properties of the Malliavin derivative in [18], we have(33)DtZTQ=ZTQ−∫tTDtρcu+DtρaudW˜u−∫tTDt1−ρ2cudW˜u∗−∫tT∫R0Dtθu,z/1−θu,zN^du,dz−ρ+1−ρ2ct−at.
In order to simplify the notation, we denote(34)H1t=−∫tTDtρcu+DtρaudW˜u−∫tTDt1−ρ2cudW˜u∗−∫tT∫R0Dtθu,z/1−θu,zN^du,dz,
Then (33) can be expressed as(35)DtZTQ=ZTQH1t−ρ+1−ρ2ct−at,substitute (35) into (32), this theorem is proved. &square;Theorem 3.
Suppose thatF,ZTQ∈D1, let ZTQ=expΔ then the Malliavin derivative Dt,zZTQF at time t is(36)Dt,zZTQF=ZTQF1−θt,zH2t,z−1+1−θt,zH2t,zDt,zF,where(37)H2t,z=expDt,zΔ−log1−θt,z.Proof.
According to the discontinuous case product rule in [19], we get (38)Dt,zZTQF=FDt,zZTQ+ZTQDt,zF+Dt,zFDt,zZTQ.
By the chain rule of [12], we have(39)Dt,zZTQ=expΔ+Dt,zΔ−expF=ZTQexpDt,zΔ−1.
Substitute the above equation into (38) and get(40)Dt,zZTQF=ZTQFexpDt,zΔ−1+Dt,zFexpDt,zΔ.
Also, by the operation properties ofDt,zof [18], we can obtain(41)Dt,zΔ=−∫tTDt,zρcu+Dt,zaudWu−∫tTDt,z1−ρ2cudWu∗−12∫tTDt,zcu2du−12∫tTDt,zau2du+∫tT∫R0Dt,zlog1−θu,z+θu,zvdzdu+∫tT∫R0Dt,zlog1−θu,zN^du,dz+log1−θu,z=−∫tTDt,zρcu+Dt,zaudW˜u−∫tTDt,z1−ρ2cudW˜u∗+∫tT∫R0Dt,zθu,z+log1−Dt,zθu,z/1−θu,z1−θu,zvdzdu+∫tT∫R0log1−Dt,zθu,z/1−θu,zN^Qdu,dz+log1−θt,z.
In order to simplify this lengthy formula, we set(42)H2t,z=expDt,zΔ−log1−θt,z.
Then,Dt,zZTQ can be obtained as follows(43)Dt,zZTQ=ZTQ1−θt,zH2t,z.
By substituting the above equation into (40), we can obtain the expression of Dt,zZTQF.Theorem 4 (The Clark-Ocone formula under measureQ).
IfF is a Ft-measurable random variable, F∈Dr, and at,ct∈Dr. Suppose the following conditions are true(a)
EQF<∞,(b)
EQ1/2∫0TDtF2dt<∞,EQ1/2∫0TDt,zF2dt<∞,(c)
EZ2TF2∫0T−ρ+1−ρ2ct−at+H1t2dt<∞,(d)
EZ2T∫0TF1−θt,zHt,z−1+1−θt,zHt,zDt,zF2dt<∞.
Then, we get the Clark-Ocone formula under measureQ as follows:(44)F=EQF+∫0TEQρDtF+C1+FH1tFtdW˜t+∫0TEQ1−ρ2DtF+FH1t+C2FtdW˜t∗+∫0T∫R0EQFH2t,z−1+H2t,zDt,zFFtN^Qdt,dz,where(45)C1=−Fρ2+ρ1−ρ2−ρct−ρ−1at,and(46)C2=−Fρ+1−ρ2ct−Fat+F.Proof.
See Appendix A.Corollary 1.
Through the correspondence between (18) and (44), and assume that ct,at,θt,z are deterministic functions, we can get the following equations:(1)It1=EQρDtF+F−ρ2+ρ1−ρ2ct−ρat+H1tFt,(2)It2=EQ1−ρ2DtF+F−ρ+1−ρ2ct−at+1+H1tFt,(3)It3=EQFH2t,z−1+H2t,zDt,zFFt.Theorem 5.
Assume thatct,at, and θt,z are deterministic functions, then the closed-form expression of the hedge ratio ϑtMVH can be expressed as(47)ϑtMVH=ξtQ+ZtQλt∫0t−dFu−ξuQdSuZuQ.Proof.
Substitute the expressions ofIt1, It2, and It3 into (20) to get the following solution:(48)ξtQ=ρβIt1Xt+β1−ρ2It2Xt+γIt3Xt2β2+Xtγ.
Then, combining with (19), r1 and r2 can be solved as(49)r1=1ρIt1−ξtβXt,r2=It3−ξtγXt.
It means thatLtQ has the following form:(50)dLtQ=It1−ρξtQβt,VtXtdW˜t+1−ρ2ρIt1−ξtQβt,VtXtdW˜t∗+∫R0It3−ξtQγN^Qdt,dz=dFt−ξtQdXt.
By substituting (48) and (50) into (8), the closed-form expression of the hedging ratio ϑtMVH is obtained.


## 6. Numerical Analysis

In this section, we will propose a numerical example to compare across the one-dimensional Lévy model in [22]. We mainly discuss the sensitivity analysis for the hedging ratio ϑtMVH with respect to different strike prices and maturity times under these two models. We set F as a European Option with the payoff function F=XT−K+, K is the strike price, and T is maturity time.


### 6.1. Mathematical Preliminaries

For numerical simulation, we specify the SVJ model as the Heston model coupled with an exponential Lévy process(51)dXtXt=μ1dt+ρVtdWt+1−ρ2VtdWt∗+∫R0γzN^dt,dzdVt=ξ1η1−Vtdt+σ1VtdWt,where Vt and μ1 represent volatility and interest rates, ξ1 and η1 are the respective corresponding average mean reversion rates and long-run average volatility, σ1 is the volatility coefficient of the volatility process, γz=ez−1. The standard Brownian motion W,Wt∗ and the Poisson measure N^ are the same as Section 1. We have calculated the form of (51) under variance-optimal measure Q, as presented in Appendix B.Theorem 6.
Model (52) under variance-optimal measure Q satisfies the following form:(52)dXt=XtρVtdW^t+1−ρ2VtdW^t∗+∫R0γt,zN^Qdt,dz,dVt=ξ1η1−ξ1+σ12BtVtdt+σ1VtdW^t,with(53)Bt=1σ12ϖtan2ϖσ1T−t2,where W˜t=Wt+∫ρctdt, W˜t∗=Wt∗+∫1−ρ2ctdt are two Brownian motions, and N^Qdt,dz=θt,zvdzdt+N^dt,dz is a compensated Poisson random measure.Proof.
See Appendix B.
According to Theorem5, the discretization on 0,t for (47) is(54)ϑtMVH=ξtQ+ZtQλt∑i=1nΔFti−ξtiQΔXti/ZtiQ,#27,where(55)ξtQ=ρVtIt1+1−ρ2VtIt2+∫R0γt,zIt3vdzXtVt+∫R0rt,z2vdz.
To sum up, we need the following processes:(a)
Observing formulas (54) and (55), ΔFti, It1, It2, and It3 can be computed by the FFT algorithm(b)
ZtiQ and Xti can be obtained by iteration
We will deal with them separately.
For processing (a): by the example given by Arai and Imai [22] and properties of conditional expectation, the expressions for It1, It2, and It3 are(56)It1=VtEQXT1XT>kFt+−ρ2+ρ1−ρ2ct−ρatEQFFt,It2=1−ρ2ρIt1,It3=EQezXT−K+−XT−K+Ft.
The expectations contained in the results need to use the FFT algorithm to compute, and we will only takeEQXT1XT>kFt, for example.
Denote theYt=lnXt, and the characteristic function of YT can be expressed as(57)ΦQY,t;u=EQeiuYTFt.
The derivation for the expression of characteristic functionsΦQY,t;u has been shown in Appendix A, and this section only emphasizes the main framework of the FFT algorithm. Let Mtk=EQYT1YT>kFt, qT is the density of the process YT, and then,(58)ΦtQu=ΦQY,t;u=∫−∞∞eiuYqTYdY.
As Carr and Madan’s method in [23] introduces a modified function mT1=eαkMTk, α>0. By Fourier transform, we can get(59)ΨtQu=ΨQY,t;u=∫−∞∞eiukmTkdk.
We can representΦQY,t;u by Φtu as follows:(60)ΨtQu=∫−∞∞eiuk∫k∞e−αkeYTqTdsdk=ΦtQu−α+1iα−1+iu.
In turn, we apply inverse Fourier transform to get(61)Mtk=e−αkπ∫0∞e−iukΨtQudu=e−αkπ∫0∞e−iukΦtQu−α+1i/α−1+iudu.
From equations (58)–(61) and Simpson’s rule weightings, we consider the discrete version of the Mtk as(62)Mtkl=e−αklπ∑j=1Ne−i2π/Nj−1l−1eiujNω/2ΨTvη33+−1j−δj−1,where uj=j−N/2η, η denotes the integration steps, and δj is the Kronecker delta function that is unity for j=0 and zero otherwise. The FFT returns N values of kl=l−N/2ω and for a regular spacing size, l=1,⋯N.
For processing (b): according to the Euler scheme and method in [15], we get a discretization of the system (52) as(63)Xti+1=Xti+XtiρVtϵ1Δt+1−ρ2Vtiϵ2Δt+∑0≤τ1<⋯<τn<tKτi,Λτi,Vti+1=Vti+ξ1η1−ξ1+σ12FV,tiBtiVtiΔt+σ1Vtiϵ1Δt,where(64)∫0t∫R0Ks,zN^Qdt,dz=∑0≤τ1<⋯<τn<tKτi,Λτi.
Denote the countable set of definitions for stationary Poisson point processesΛ∙,ω by DΛω=τ1ω,⋯,τnω. The distance between τi+1 and τi follows exponential distribution with parameter λ1, and Λ∙,ω follows normal distribution N0,δ12. With the same schemes, other variables can also be discretized.


### 6.2. Numerical Results

We consider a European call option with the expiry timeT=1. The initial price of the underlying asset X0=1.2, and V0=0.12. For a mean reverting process with stochastic volatility and no jumps, μ1=0.05, ξ1 = 0.2, η1=0.5 , ρ=0.6. For Lévy process, γ=1, m=0, and δ=1. For parameters in a discrete scheme, δ1=0.1, λ1=0.5, Δt=0.01. We implement our FFT scheme with η=0.25 and N=256, which leads to a logarithm strike spacing of ω=8π/256. The modified coefficient is set to α = 1.5.As shown in Figure1, we learn that the MVH strategy of a European call option is sensitive to time t in the maturity time T. Both models L1 and L2 show a gradual increase in their overall trend. The only difference is that the hedge ratio of L1 changes sharply while that of L2 increases smoothly. Since there is no essential difference betweenL1 and L2 when time t is fixed, we only conducted a sensitivity analysis of MVH strategy on strike price K for L2. Ordinarily, we obtain t=0.5 and observe that the hedging ratio ϑtMVH decreases with the increase of strike price K as shown in Figure 2, which is consistent with the experimental situation in literature [13, 23].Figure 1
Hedging ratioϑtMVH for L1 and L2 with respect to time t.Figure 2
Hedging ratioϑtMVH for L2 with respect to strike price K.


## 6.1. Mathematical Preliminaries

For numerical simulation, we specify the SVJ model as the Heston model coupled with an exponential Lévy process(51)dXtXt=μ1dt+ρVtdWt+1−ρ2VtdWt∗+∫R0γzN^dt,dzdVt=ξ1η1−Vtdt+σ1VtdWt,where Vt and μ1 represent volatility and interest rates, ξ1 and η1 are the respective corresponding average mean reversion rates and long-run average volatility, σ1 is the volatility coefficient of the volatility process, γz=ez−1. The standard Brownian motion W,Wt∗ and the Poisson measure N^ are the same as Section 1. We have calculated the form of (51) under variance-optimal measure Q, as presented in Appendix B.Theorem 6.
Model (52) under variance-optimal measure Q satisfies the following form:(52)dXt=XtρVtdW^t+1−ρ2VtdW^t∗+∫R0γt,zN^Qdt,dz,dVt=ξ1η1−ξ1+σ12BtVtdt+σ1VtdW^t,with(53)Bt=1σ12ϖtan2ϖσ1T−t2,where W˜t=Wt+∫ρctdt, W˜t∗=Wt∗+∫1−ρ2ctdt are two Brownian motions, and N^Qdt,dz=θt,zvdzdt+N^dt,dz is a compensated Poisson random measure.Proof.
See Appendix B.
According to Theorem5, the discretization on 0,t for (47) is(54)ϑtMVH=ξtQ+ZtQλt∑i=1nΔFti−ξtiQΔXti/ZtiQ,#27,where(55)ξtQ=ρVtIt1+1−ρ2VtIt2+∫R0γt,zIt3vdzXtVt+∫R0rt,z2vdz.
To sum up, we need the following processes:(a)
Observing formulas (54) and (55), ΔFti, It1, It2, and It3 can be computed by the FFT algorithm(b)
ZtiQ and Xti can be obtained by iteration
We will deal with them separately.
For processing (a): by the example given by Arai and Imai [22] and properties of conditional expectation, the expressions for It1, It2, and It3 are(56)It1=VtEQXT1XT>kFt+−ρ2+ρ1−ρ2ct−ρatEQFFt,It2=1−ρ2ρIt1,It3=EQezXT−K+−XT−K+Ft.
The expectations contained in the results need to use the FFT algorithm to compute, and we will only takeEQXT1XT>kFt, for example.
Denote theYt=lnXt, and the characteristic function of YT can be expressed as(57)ΦQY,t;u=EQeiuYTFt.
The derivation for the expression of characteristic functionsΦQY,t;u has been shown in Appendix A, and this section only emphasizes the main framework of the FFT algorithm. Let Mtk=EQYT1YT>kFt, qT is the density of the process YT, and then,(58)ΦtQu=ΦQY,t;u=∫−∞∞eiuYqTYdY.
As Carr and Madan’s method in [23] introduces a modified function mT1=eαkMTk, α>0. By Fourier transform, we can get(59)ΨtQu=ΨQY,t;u=∫−∞∞eiukmTkdk.
We can representΦQY,t;u by Φtu as follows:(60)ΨtQu=∫−∞∞eiuk∫k∞e−αkeYTqTdsdk=ΦtQu−α+1iα−1+iu.
In turn, we apply inverse Fourier transform to get(61)Mtk=e−αkπ∫0∞e−iukΨtQudu=e−αkπ∫0∞e−iukΦtQu−α+1i/α−1+iudu.
From equations (58)–(61) and Simpson’s rule weightings, we consider the discrete version of the Mtk as(62)Mtkl=e−αklπ∑j=1Ne−i2π/Nj−1l−1eiujNω/2ΨTvη33+−1j−δj−1,where uj=j−N/2η, η denotes the integration steps, and δj is the Kronecker delta function that is unity for j=0 and zero otherwise. The FFT returns N values of kl=l−N/2ω and for a regular spacing size, l=1,⋯N.
For processing (b): according to the Euler scheme and method in [15], we get a discretization of the system (52) as(63)Xti+1=Xti+XtiρVtϵ1Δt+1−ρ2Vtiϵ2Δt+∑0≤τ1<⋯<τn<tKτi,Λτi,Vti+1=Vti+ξ1η1−ξ1+σ12FV,tiBtiVtiΔt+σ1Vtiϵ1Δt,where(64)∫0t∫R0Ks,zN^Qdt,dz=∑0≤τ1<⋯<τn<tKτi,Λτi.
Denote the countable set of definitions for stationary Poisson point processesΛ∙,ω by DΛω=τ1ω,⋯,τnω. The distance between τi+1 and τi follows exponential distribution with parameter λ1, and Λ∙,ω follows normal distribution N0,δ12. With the same schemes, other variables can also be discretized.


## 6.2. Numerical Results

We consider a European call option with the expiry timeT=1. The initial price of the underlying asset X0=1.2, and V0=0.12. For a mean reverting process with stochastic volatility and no jumps, μ1=0.05, ξ1 = 0.2, η1=0.5 , ρ=0.6. For Lévy process, γ=1, m=0, and δ=1. For parameters in a discrete scheme, δ1=0.1, λ1=0.5, Δt=0.01. We implement our FFT scheme with η=0.25 and N=256, which leads to a logarithm strike spacing of ω=8π/256. The modified coefficient is set to α = 1.5.As shown in Figure1, we learn that the MVH strategy of a European call option is sensitive to time t in the maturity time T. Both models L1 and L2 show a gradual increase in their overall trend. The only difference is that the hedge ratio of L1 changes sharply while that of L2 increases smoothly. Since there is no essential difference betweenL1 and L2 when time t is fixed, we only conducted a sensitivity analysis of MVH strategy on strike price K for L2. Ordinarily, we obtain t=0.5 and observe that the hedging ratio ϑtMVH decreases with the increase of strike price K as shown in Figure 2, which is consistent with the experimental situation in literature [13, 23].Figure 1
Hedging ratioϑtMVH for L1 and L2 with respect to time t.Figure 2
Hedging ratioϑtMVH for L2 with respect to strike price K.


## 7. Conclusion

In this article, we discussed a method of Malliavin calculus to obtain the explicit expression of the hedging ratioϑMVH of the mean-variance strategy. We calculated the Clark-Ocone formula under variance-optimal equivalent martingale measure Q. According to the correspondence of the Clark-Ocone formula and the martingale representation theorem, we express the diffusion term coefficients by Malliavin derivatives. By using the FFT algorithm and the discrete scheme for Lévy process, we conducted some numerical analyses for sensitivity analysis of MVH strategy with respect to time t and strike price K.However, two further extensions of the Malliavin calculus in the mean-variance hedging strategy would be interesting. One possible extension is incorporating stochastic interest into the SVJ model so that the hedging results are more consistent with the real market. Another extension would be to extend the study on the exotic options to make our conclusions more general.



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*Source: 3096866-2022-12-05.xml*