Consequences for option sricing of a long memory in volatility

Abstract

Conditionally heteroscedastic time series models are used to describe the volatility of stock index returns. Volatility has a long memory property in the most general models and then the autocorrelations of volatility decay at a hyperbolic rate; contrasts are made with popular, short memory specifications whose autocorrelations decay more rapidly at a geometric rate.Options are valued for ARCH volatility models by calculating the discounted expectations of option payoffs for an appropriate risk-neutral measure. Monte Carlo methods provide the expectations. The speed and accuracy of the calculations is enhanced by two variance reduction methods, which use antithetic and control variables. The economic consequences of a long memory assumption about volatility are documented, by comparing implied volatilities for option prices obtained from short and long memory volatility processes. Results are given for options on the S & P 100-share index, with lives up to 2 years. The long memory assumption is found to have a significant impact upon the term structure of implied volatilities and a relatively minor impact upon smile shapes. These conclusions are important because evidence for long memory in volatility has been found in the prices of many assets.