Abstract
This paper is concerned with the numerical solution of partial integrodifferential equation for option pricing models under a tempered stable process known as CGMY model. A double discretization finite difference scheme is used for the treatment of the unbounded nonlocal integral term. We also introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions. Integration formula of open type is used in order to improve the accuracy of the approximation of the integral part. Stability and consistency are also studied. Illustrative examples are included.
Highlights
In Black-Scholes model, it is assumed that the probability distribution of the stock price is lognormal and the instantaneous log return is a geometric Brownian motion
We introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions
Several models have been proposed to overcome these shortcomings such as models where the volatility follows a stochastic process like Heston model [2] and models incorporating jumps in the underlying asset following Levy processes [3] and [4, chap. 14, 15]
Summary
In Black-Scholes model, it is assumed that the probability distribution of the stock price is lognormal and the instantaneous log return is a geometric Brownian motion. The singularity of the integral kernel and the nonsmoothness of initial conditions are treated using the viscosity solutions and applying this technique to Merton and Variance Gamma models. The option pricing for jump diffusion models with finite jump intensity has been treated using ADI finite difference method, accelerated by the fast Fourier transformation [13]. In [20], a three-time-level finite difference method is proposed showing a second order convergence rate in the numerical experiments for infinite activity models. An explicit scheme has been used in [15], applying the trapezoidal rule to approximate the integral term after removing the singularity of the kernel and including the unbounded domain using a double discretization technique [14]. Our objective is to construct a stable and conditionally consistent numerical scheme that guarantees positive solutions for the PIDE governing the CGMY model.
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