Option Pricing

Abstract

We have long understood that the lognormal distribution for stock returns does not adequately capture the way options are priced I the market. The existence of fat tails, a persistent implied volatility smile or skew, and other discrepancies is well-established empirically. One solution is simply to fit a general shape to the implied density function using non-parametric methods. This essentially allows any kind of irregularity in the distribution as dictated by the data, with little input from the user as to what “reasonable” shapes might be. Another approach, which puts more structure on the problem while allowing the distribution with different skewness, kurtosis, and higher moments than the lognormal. A variety of alternative distributions have also been explored in the literature. Li presents a horse race among models from a family that nests the common ones and contains a number of more general ones to see which one provides the best and the most reliable fit for options on the S&P 500 index. Looking at both in-sample fit and out-of-sample forecasting ability, the winners are skewed t distributions, with the most general formulation working best, except for longer-period out-of-sample prediction. Among the “also-rans,” Li finds that performance is improved more by fitting the non-lognormal skewness of the implied distribution than its kurtosis.