An Empirical Study on the Implied Volatility Function of S&P 500 Options

A better understanding of the empirical dynamics of Black-Scholes implied volatility surface has long been of considerable interest to both practitioners and academics. Basing on some findings about the ad hoc Black-Scholes valuation approach suggested in Dumas, Flemming and Whaley (1998), this essay studies the empirical performance of various volatility function forms that characterize the regularities underlying the Black-Scholes implied volatility skew or smile. Because the volatility function suggested in Dumas, Flemming and Whaley (1998) is unstable over time, we propose a new class of dynamics implied volatility function that separates a time-invariant implied volatility function from the random factors that drive changes in the individual implied volatilities. The random factors are incorporated through the at-the-money implied volatilities which are modelled as a function of lagged volatility and a non-linear function of the underlying asset return. This dynamic model is found to greatly improve the pricing performance.