Option pricing under multiscale stochastic volatility

Abstract

The stochastic volatility model proposed by Fouque, Papanicolaou, and Sircar (2000) explores a fast and a slow time-scale fluctuation of the volatility process to end up with a parsimonious way of capturing the volatility smile implied by close to the money options. In this paper, we test three different models of these authors using options on the S&P 500. First, we use model independent statistical tools to demonstrate the presence of a short time-scale, on the order of days, and a long time-scale, on the order of months, in the S&P 500 volatility. Our analysis of market data shows that both time-scales are statistically significant. We also provide a calibration method using observed option prices as represented by the so-called term structure of implied volatility. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the implied volatility surface. In addition, to test the model’s ability to price options, we simulate options prices using four different specifications for the Data generating Process. As an illustration, we price an exotic option.