A General Formula for Option Prices in a Stochastic Volatility Model

Abstract

We consider the pricing of European derivatives in a Black–Scholes model with stochastic volatility. We show how Parseval's theorem may be used to express those prices as Fourier integrals. This is a significant improvement over Monte Carlo simulation. The main ingredient in our method is the Laplace transform of the ordinary (constant volatility) price of a put or call in the Black–Scholes model, where the transform is taken with respect to maturity (T); this does not appear to have been used before in pricing options under stochastic volatility. We derive these formulas and then apply them to the case where volatility is modelled as a continuous-time Markov chain, the so-called Markov regime-switching model. This model has been used previously in stochastic volatility modelling, but mostly with only states. We show how to use states without difficulty, and how larger number of states can be handled. Numerical illustrations are given, including the implied volatility curve in two- and three-state models. The curves have the ‘smile’ shape observed in practice.

Key Words:

• Laplace transform
• Black–Scholes formula
• Parseval's theorem
• Markov regime switching
• marked point process

Acknowledgement

We thank one of the reviewers for pointing out unanswered questions, which led to improvements to the paper.