Option Pricing under the Jump Diffusion and Multifactor Stochastic Processes

Shican Liu,Yanli Zhou,Yonghong Wu,Xiangyu Ge

doi:10.1155/2019/9754679

Abstract

In financial markets, there exists long-observed feature of the implied volatility surface such as volatility smile and skew. Stochastic volatility models are commonly used to model this financial phenomenon more accurately compared with the conventional Black-Scholes pricing models. However, one factor stochastic volatility model is not good enough to capture the term structure phenomenon of volatility smirk. In our paper, we extend the Heston model to be a hybrid option pricing model driven by multiscale stochastic volatility and jump diffusion process. In our model the correlation effects have been taken into consideration. For the reason that the combination of multiscale volatility processes and jump diffusion process results in a high dimensional differential equation (PIDE), an efficient finite element method is proposed and the integral term arising from the jump term is absorbed to simplify the problem. The numerical results show an efficient explanation for volatility smirks when we incorporate jumps into both the stock process and the volatility process.

Highlights

Due to the well-known phenomenon of volatility ‘smile’ and ‘smirk’ exhibited in option pricing processes, many attempts have been made to solve the problem by extending the classical Black-Scholes models [1] and relaxing the assumptions
The traditional Heston model [7] assumes that the underlying volatility process is a Cox Ingersoll Ross model (CIR) process with the power of 1/2; the 3/2 model assumes that the diffusion of volatility process is a flipped CIR process, raising the power of 3/2
The finite element method and the dimension reduction technique are applied to obtain the approximate solution for the classical European option price and the fair strike price of variance swaps under both multiscale stochastic volatility and jump diffusion process

Summary

Introduction

Due to the well-known phenomenon of volatility ‘smile’ and ‘smirk’ exhibited in option pricing processes, many attempts have been made to solve the problem by extending the classical Black-Scholes models [1] and relaxing the assumptions. In 2008, Fouque proposed a numerical algorithm based on asymptotic approximation and asymptotic homogenization to study the effect of the fast and the slow scale of the volatility OU process on option pricing (see [16]). Three simple applications of the FEM approach in option pricing are given in [26], including the standard Black-scholes equation, the stochastic volatility model, and the path-dependent Asian option. We develop an efficient FEM method to solve the problem numerically Inclusion of both of the two factors and the jump results in a high dimensional partial integral differential equation (PIDE).

Model Setup

Algorithm of FEM

Numerical Results and Discussion

Conclusions