Stochastic Volatility Models

Abstract

Stochastic models for asset prices processes are now familiar to actuaries. Many of the models used in life office and pension fund valuation and asset-liability modelling studies assume deterministic volatility parameters. Empirical evidence however, suggests that volatility in asset prices varies with time. Further, volatilities implied by traded option prices show a term structure for implied volatility, as well as an apparent dependence on the "moneyness" of the option. These observations seem to be at odds with a constant volatility assumption. In this paper we present some empirical observations concerning volatility, and consider the impact of volatility on actuarial work. We then review some of the common models which incorporate stochastic volatility and consider issues related to parameterising such models. 1 We are indebted to Stuart Jones for his patience and for providing invaluable technical assistance. We would also like to thank Dylan Brooks and Carmela Calvosa for providing data and fruitful discussions.