Implied Volatility Functions

Abstract

The Black-Scholes model was derived under the assumption that the volatility of the underlying asset is constant and known, but decades of experience have consistently shown that reality is considerably more complicated. Volatilities implied in the prices of stock index options exhibit a persistent skewed smile pattern, with a clear linkage between the option9s strike price and its implied volatility. But the pattern seems to evolve over time. How should it best be modeled empirically? Is implied volatility a function of the exercise price itself, or is it the option9s “moneyness” that matters? Should we ignore the smile and simply fit a single volatility for all options and all time, or should we fit the current smile, whatever it is, and assume it will continue to have the same shape going forward? In this article, Rosenberg examines the empirical performance of the different choices for S&P 500 futures options and offers a composite approach that captures the main features of the implied volatility curve while allowing it to evolve over time. The dynamic implied volatility function model combines a time series model for the evolution of the at-the-money implied volatility, with a fixed cross-sectional smile pattern based on option moneyness relative to the at-the-money option.