وزن مالاویان تقریبی برای فرآیند واریانس گاما: تجزیه و تحلیل حساسیت از گزینه های سبک اروپا

The main objective of this work is to find an approximate Malliavin weight in order to calculate the delta of European style options where the underlying asset is modelled by a Variance Gamma process. We use a Malliavin integration by parts formula for a compound Poisson process. In order to apply this formula, a compound Poisson approximation of the Variance Gamma process is used. We calculate deltas using Monte Carlo simulations. An acceptance–rejection algorithm is used to generate random numbers for the approximated VG process.

In the past years, following the pioneering papers [1] and [2], Malliavin calculus (stochastic variational calculus) has been used extensively in the field of mathematical finance. Main topics discussed in terms of application of Malliavin calculus mostly include the computation of sensitivities and of conditional expectations. Earlier works of these applications focused on the lognormal type diffusion processes. Lately, jump type market models have been used by practitioners and researchers. Application of the Malliavin calculus in markets driven by Lévy processes, specifically by pure jump or jump diffusion processes are discussed in [3] and [4] and [5] and [6]. In [6], the Malliavin integration by parts formula is derived under the assumption that the law of random variables is absolutely continuous with respect to the Lebesque measure and smooth enough so that the first derivative exists and is continuous. The main purpose of this paper is to derive an integration by parts formula when the stock prices follow an exponential Variance Gamma (VG) process [7] in terms of the method introduced in [6]. However, increments of the VG process do not satisfy the necessary assumption of smoothness of the density. In order to overcome this problem, we approximate the VG process by using the compound Poisson approximation, [8]. Thus, we are able to derive an explicit formula for the deltas of European call and digital options. We also used fast Fourier transform (FFT) method in order to compute delta by the method appearing in [9]. We measured the performance of our results in Malliavin approach in terms of the FFT computations of the deltas. For more complex financial derivatives like Asian options we do not have the characteristic function information. Therefore, in this case the implementation of the inverse Fourier method is not possible and a method such as Malliavin approach is necessary. The paper is organized as follows: In Section 1 we give the analytic formulas that we obtain for the deltas in terms of the characteristic function of the log stock price. We show the explicit calculations in order to derive the pricing function for the digital option by using the inverse Fourier transform method. In the following section, approximation of the VG process and properties of the resulting compound Poisson process are discussed. In the subsection, an acceptance–rejection algorithm is introduced for the specific density of the jump size distribution. In Section 3, preliminaries of Malliavin calculus for simple functionals and the integration by parts formula of [6] are given. In Section 4, numerical schemes, formulas and the graphical results are presented. We use a finite difference method on the exact1 simulation of the VG process in order to measure the performance of the FFT method which we use as our benchmark. The explicit formula for delta and its derivation by using the Malliavin approach are presented in this section, as well. Results show that as the approximation parameter εε gets smaller, the approximation of delta gets better. However, this leads to a slower convergence of the Monte Carlo simulations.

The magnitude of the random numbers generated from the jump size distribution JiJi are necessarily larger than εε. The VG process has an infinite activity of smaller jumps. However, we obtain a finite activity process by truncating the original Lévy measure. Nevertheless, as a result of the inverse relation between U(ε)U(ε) and εε, as εε decreases, the number of the smaller jumps in the compound Poisson process for a given time period increases in probability. Thus, the approximation of the VG process jumps more frequently as we use smaller jumps. Thus, we conclude that choosing smaller values for εε improves the quality of the approximation. The disadvantage of the approximation of the VG process is the slowness of path generation. The acceptance–rejection method depends on εε. The smaller values of εε give a larger coefficient value of c++c−c++c−. Thus, the probability, View the MathML source1c++c−, of generating a random number from the algorithm decreases. However, when εε is at a level of 10−410−4, simulation of delta converges to the FFT result even though the number of paths generated for Monte Carlo is 104. Another important conclusion is that the FFT method gives accurate results in the calculation of delta which we verified by using the exact simulation of the VG process and by calculating the delta theoretically in the Black–Scholes model. This method can be extended in order to calculate the other Greeks if the characteristic function is continuously differentiable with respect to the parameter. However, when the FFT method is not applicable as in the Asian options’ case, the Malliavin approach can be used.