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.
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 process allows the jump components to have infinite activity and admits nearly an arbitrary distribution. Using an infinite activity 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 ratio by the correspondence between the Clark-Ocone formula and the martingale representation theorem under variance-optimal equivalent martingale measure . To demonstrate the sensitivity of MVH strategy concerning time and strike price, we adopt the FFT algorithm to calculate the conditional expectation contained in . 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 Section 1, 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 by the Clark-Ocone formula under measure . 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 space . is the probability measure of the actual market, and , is the filtration generated by the process of stock price. and are the sample spaces composed of the one-dimensional Wiener process and pure jump Lévy process, respectively, , where . For simplicity, denotes the single risky assets process, is set as the numeraire, and then, the discount price of underlying assets is . We consider a jump-diffusion stochastic volatility model for this single risky asset, and its stochastic differential equation is given bywhere and are two standard Brownian motion under measure . is the volatility of the underlying asset. is the Poisson random measure, and , , is a Borel field. Denoting the measure of by , is a compensated Poisson process of . Functions , , , and are assumed to be the predictable processes, is a stochastic process measurable with respect to the -algebra generated by , , and . For the application, we setwhere , is the correlation coefficient of and . and are independent. Then, our model in (1) becomes
3. Mean-Variance Hedging Strategy
Considering a portfolio , and are two -valued predictable processes. denotes the number of risk assets (such as the stock). represents the number of riskless (such as the bound). The value process of the portfolio can be written as , and the cost process is
Let is the space of -valued predictable -integrable processes . Denote the space of all stochastic integral processes by . Then, we call the is a mean-variance optimal strategy for a claim if it minimizeswhere is initial value of the value process, and 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 within the interval .
In the absence of arbitrage opportunities, we assume that there is an equivalent martingale measure , and then, under measure is a martingale. Let be the set of all such equivalent martingale measure , and if there is more than one element in , the market is said to be incomplete. Therefore, finding an appropriate measure to hedge a given contingent claim is particularly important.
Gourieroux et al. [11] obtain the variance-optimal equivalent martingale measure by minimizing , which is unique in . Then, the density process of measure with respect to measure can be denoted by . is a martingale under , which can be represented by F¨llmer-Schweizer decomposition.
For a contingent claim , the GKW decomposition of under measure iswhere is a -valued predictable process that coincides with the amount of the risky asset in the local risk-minimization strategy at time . is a martingale under and is zero at the , and is strongly -orthogonal to . The form of hedging ratio at time is proposed by Pham with the following form:where is a contingent claim. From the above equation, it is not difficult to know that the key to solving is to compute and explicitly. For this purpose, we first give the forms of and separately such that we can calculate them in the later sections.
By the theorem given by Pham et al. [16], a martingale can be decomposed into , where is a real measurable martingale with the following form: is an adapted process with the form aswhere
The density process can be considered as the derivative of . According to the concept of Schweizer [17], we can know , and the expression of under model (3) can be written aswhere represents the stochastic exponential of , and the parameters are
Then, the stochastic differential equation of under measure takes the formwhere . The Girsanov’s theorem implies that and are two standard Brownian motions under measure . represents the compensated Poisson random measure of .
Theorem 1. The expression of local risk-minimization strategy at time is given bywhere , , and are unique adapted processes.
Proof. Due to is a martingale under measure , according to the martingale representation theorem, there exist two unique random processes , , such thatBy the property of strongly -orthogonal to , we can obtainSubstitute (16) into (7) to getSince is a martingale under measure , it can also be described by the martingale representation theorem with the following form:where , , and are unique adapted processes. Combine with (18) and (19), we can obtain the following equations:According to the condition given in (17), can be obtained by solving equations (20).
Remark 1. The processes , , and 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 , , and .
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 that is a combination of Gaussian and pure jump processes. Considering a set as follows:where , . There are some definitions as follows:
Let , , and , . Then, we have the following definitions.
Definition 1. (N-fold iterated integrals). Let, the n-fold iterated integrals is defined as follows:It can be known from definition of [19].Let be a closed linear subspace on consisting of random variables of the form . The inner product of and is 0 if , and it means that and are two orthonormal subspaces of .
Definition 2. (Wiener chaos decomposition [20]). are dense subspaces in , and then, . If , its only expansion is as follows:where and , for . series converges in .
Definition 3. (Malliavin derivative [21]). AssumeFor any , the Malliavin derivative of can be defined bywhere
Definition 4. (Clark-Ocone formula for Lévy functionals [19]). Assume that ,. Then,
5. The Calculation of Hedging Ratio
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 that , is defined in (7). Then, the Malliavin derivative of at time is calculated aswhere
Proof. Firstly, by the continuous case product rule of [19], we can deduce thatAnd according to the continuous case operation properties of the Malliavin derivative in [18], we haveIn order to simplify the notation, we denoteThen (33) can be expressed assubstitute (35) into (32), this theorem is proved. □
Theorem 3. Suppose that , let then the Malliavin derivative at time t iswhere
Proof. According to the discontinuous case product rule in [19], we get By the chain rule of [12], we haveSubstitute the above equation into (38) and getAlso, by the operation properties of of [18], we can obtainIn order to simplify this lengthy formula, we setThen, can be obtained as followsBy substituting the above equation into (40), we can obtain the expression of .
Theorem 4 (The Clark-Ocone formula under measure ). If is a -measurable random variable, , and . Suppose the following conditions are true(a)(b)(c)(d)Then, we get the Clark-Ocone formula under measure as follows:whereand
Proof. See Appendix A.
Corollary 1. Through the correspondence between (18) and (44), and assume that ,, are deterministic functions, we can get the following equations: (1) , (2) , (3) .
Theorem 5. Assume that ,, and are deterministic functions, then the closed-form expression of the hedge ratio can be expressed as
Proof. Substitute the expressions of , , and into (20) to get the following solution:Then, combining with (19), and can be solved asIt means that has the following form:By substituting (48) and (50) into (8), the closed-form expression of the hedging ratio 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 with respect to different strike prices and maturity times under these two models. We set as a European Option with the payoff function , is the strike price, and is maturity time.
6.1. Mathematical Preliminaries
For numerical simulation, we specify the SVJ model as the Heston model coupled with an exponential Lévy processwhere and represent volatility and interest rates, and are the respective corresponding average mean reversion rates and long-run average volatility, is the volatility coefficient of the volatility process, . The standard Brownian motion and the Poisson measure are the same as Section 1. We have calculated the form of (51) under variance-optimal measure , as presented in Appendix B.
Theorem 6. Model (52) under variance-optimal measure satisfies the following form:withwhere , are two Brownian motions, and is a compensated Poisson random measure.
Proof. See Appendix B.
According to Theorem 5, the discretization on for (47) iswhereTo sum up, we need the following processes:(a)Observing formulas (54) and (55), , , , and can be computed by the FFT algorithm(b) and can be obtained by iterationWe 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 , , and areThe expectations contained in the results need to use the FFT algorithm to compute, and we will only take , for example.
Denote the , and the characteristic function of can be expressed asThe derivation for the expression of characteristic functions has been shown in Appendix A, and this section only emphasizes the main framework of the FFT algorithm. Let , is the density of the process , and then,As Carr and Madan’s method in [23] introduces a modified function , . By Fourier transform, we can getWe can represent by as follows:In turn, we apply inverse Fourier transform to getFrom equations (58)–(61) and Simpson’s rule weightings, we consider the discrete version of the aswhere , denotes the integration steps, and is the Kronecker delta function that is unity for and zero otherwise. The FFT returns values of and for a regular spacing size, .
For processing (b): according to the Euler scheme and method in [15], we get a discretization of the system (52) aswhereDenote the countable set of definitions for stationary Poisson point processes by The distance between and follows exponential distribution with parameter , and follows normal distribution . With the same schemes, other variables can also be discretized.
6.2. Numerical Results
We consider a European call option with the expiry time . The initial price of the underlying asset , and . For a mean reverting process with stochastic volatility and no jumps, , = 0.2, , . For Lévy process, , , and . For parameters in a discrete scheme, , , . We implement our FFT scheme with and , which leads to a logarithm strike spacing of . The modified coefficient is set to α = 1.5.
As shown in Figure 1, we learn that the MVH strategy of a European call option is sensitive to time in the maturity time . Both models and show a gradual increase in their overall trend. The only difference is that the hedge ratio of changes sharply while that of increases smoothly. Since there is no essential difference between and when time is fixed, we only conducted a sensitivity analysis of MVH strategy on strike price for . Ordinarily, we obtain and observe that the hedging ratio decreases with the increase of strike price as shown in Figure 2, which is consistent with the experimental situation in literature [13, 23].
7. Conclusion
In this article, we discussed a method of Malliavin calculus to obtain the explicit expression of the hedging ratio of the mean-variance strategy. We calculated the Clark-Ocone formula under variance-optimal equivalent martingale measure . 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 and strike price .
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.
Appendix
A. Proof of Theorem 4
Proof. Let , . From thecorollary ofNunno [19], we can deduce that . With the help of the Clark-Ocone formula for Lévy functionals in Definition 2, we can write as followsApply It formula to getTherefore, by combining (A.1) and (A.2), we getBy (28) and (34), the above equation can be written aswhereandSince is a martingale under measure , we can know that and then the proof is completed.
B. Proof of Theorem 6
Proof. According the form of Radon-Nikodym derivative given in (12), (12), the parameters can be expressed asAssume that is the jumps intensity and the sizes of the jumps are distributed normally with mean and variance . Then we can calculate asDenote the result in (B.2) as . Let . From the Feynman-Kac formula, is the solution of the following backward Parabolic partial differential equation with the Cauchy problem:We set . Because of the affine structure of the volatility process in (B.2), we assume that has a solution as follows:We substitute (B.4) into (B.3) to obtainwith the boundary conditions . Assume that and set , and then, the ODE (B.5) can be solved asIt means that , following the stochastic differential equation in (14), and the form model (51) under measure can be obtained.
C. Derivation of the Characteristic Function
In Appendix C we will show the Solutions of characteristic functions (55). According to -It formula for in one-dimensional case, we have the following result:where , . The dynamic process can be written as , in which and . Because of the independence between and , we can obtain an expression owing to basic properties of the characteristic function as follows:
Theorem C.1. The characteristic function under variance-optimal measure iswhere parameters of the formula (37) are constrained as follows:in which
Proof. For , we set as the moment generating function of at time . The generalized Feynman–Kac formula implies the following partial differential difference equations:The exponential affine structure for characteristic function can be assumed as follows:Substitute (C.4) to (C.3) to getWith the boundary condition , we solve this ODE as (C.1) and (C.2).
By Lévy–Khintchine theorem, we can compute , and as the form ofCombining (C.4) and (C.5), we obtain the characteristic function .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported through the Education Department of Jilin Province (No. JJKH20200030KJ) and the Department of Science and Technology of Jilin Province (No. YDZJ202102ZYTS156).