A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options

Abstract

Abstract I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the Black– Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems. Many plaudits have been aptly used to describe Black and Scholes’s (1973) contribution to option pricing theory. Despite subsequent development of option theory, the original Black–Scholes formula for a European call option remains the most successful and widely used application. This formula is particularly useful because it relates the distribution of spot returns to the cross-sectional properties of option prices. In this article, I generalize the model while retaining this feature.