Mathematical and Statistical Methods for Actuarial Sciences and Finance

Exact and approximated option pricing in a stochastic

volatility jump-diffusion model

Fernanda D’Ippoliti, Enrico Moretto, Sara Pasquali, and Barbara Trivellato

Abstract.We propose a stochastic volatility jump-diffusion model for option pricing with

contemporaneous jumps in both spot return and volatilitydynamics. The model admits, in the

spirit of Heston, a closed-form solution for European-style options. To evaluate more complex

derivatives for which there is no explicit pricing expression, such as barrier options, a numerical

methodology, based on an “exact algorithm” proposed by Broadie and Kaya, is applied. This

technique is called exact as no discretisation of dynamics is required. We end up testing the

goodness of our methodology using, as real data, prices and implied volatilities from the DJ

Euro Stoxx 50 market and providing some numerical results for barrier options and their Greeks.

Key words:stochastic volatility jump-diffusion models, barrier option pricing, rejection sam-

pling

1 Introduction

In recent years, many authors have tried to overcome the Heston setting [11]. This

is due to the fact that the ability of stochastic volatility models to price short-time

options is limited [1, 14]. In [2], the author added (proportional) log-normal jumps to

the dynamics of spot returns in the Heston model (see [10] for log-uniform jumps) and

extended the Fourier inversion option pricing methodology of [11, 15] for European

and American options. This further improvement has not been sufficient to capture

the rapid increase of volatility experienced in financial markets. One documented

example of this feature is given by the market stress of Fall 1987, when the volatility

jumped up from roughly 20 % to over 50 %. To fill this gap, the introduction of jumps in

volatility has been considered the natural evolution of the existing diffusive stochastic

volatility models with jumps in returns. In [9], the authors recognised that “although

the motivation for jumps in volatility was to improve on the dynamics of volatility, the

results indicate that jumps in volatility also have an important cross-sectional impact

on option prices”.

In this context, we formulate a stochastic volatility jump-diffusion model that, in

the spirit of Heston, admits a closed-form solution for European-style options. The

evolution of the underlying asset is driven by a stochastic differential equation with

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010