Volatility Cones and Their Sampling Properties

Abstract

Determining the volatility of the underlying asset is perhaps the single most important issue in practical option pricing. Many forecasting techniques exist, using historical returns data, implied volatility parameters from observed option prices, normal and non-normal probability distributions and more complex stochastic processes, non-market information, and more. But one of the most important aspects of the problem is seldom formally examined: estimation error. Even under ideal conditions, both measured volatility in a historical sample of returns and future realized volatility over an option9s lifetime are subject to sampling error. Shorter samples have larger estimation error. A “volatility cone” is a plot of the range of volatilities within a fixed probability band around the true parameter, as a function of sample length. In this article, Hodges and Tompkins examine volatility cones under different assumptions about the true returns process. One important contribution is a bias correction for estimation using an overlapping data sample that produces unbiased estimates and a substantial gain in efficiency. They then apply the analysis empirically to S&P 500 index futures returns and conclude that the observed volatility behavior is consistent with a stochastic volatility process that has fat-tailed innovations.