Estimating Dynamics of S&P 500 Index and Volatility Using Option Market Data

Abstract

This paper studies the joint dynamics of the daily return on the S&P 500 index and its volatility using the option market data. We construct a measure of the stock return volatility based on the Black-Scholes implied volatilities of exchange traded options. The implied volatility measure is shown to be an efficient proxy of the daily stock return volatility as it dominates the conditional volatility estimate obtained from the GARCH models. Using the time series of the implied volatility, we examine the goodness-of-fit of several existing stochastic volatility models and find that a CEV specification with non-linear drift fits the data well. We further estimate a jump-diffusion model of stock return and volatility. We present evidence that both stock return and volatility processes contain jumps and the jump sizes in return and volatility are negatively correlated. We also investigate the information content of two implied volatility related variables: the slope of volatility smile and the term structure of implied volatility. We find evidence that the slope can predict future stock returns while both slope and term structure can predict future volatilities.