General approximation schemes for option prices in stochastic volatility models

Abstract

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.

Keywords:

• Applied mathematical finance
• Stochastic volatility
• Option pricing
• Stochastic applications

Acknowledgements

Financial support from the Jan Wallander and Tom Hedelius Foundation and the Swedish Institute for Banking Research is gratefully acknowledged.