The Generalized Extreme Value Distribution, Implied Tail Index, and Option Pricing

Abstract

Black and Scholes assumed the returns for an option’s underlying stock followed a lognormal diffusion, which produces a lognormal probability density for the stock price at option expiration. This was mathematically very convenient but, unfortunately, not very satisfactory empirically. Observed return distributions for actual stocks, stock indexes, and most other kinds of assets exhibit distinctly non-lognormal shapes, with negative skewness and fat tails. Numerous non- Black–Scholes pricing models have been developed to try to capture these important features of real-world returns. Here, Markose and Alentorn propose the use of the generalized extreme value (GEV) distribution in place of the lognormal. Tail shape in the GEV distribution is governed by a single parameter, and it can be proven that the extreme tails of any plausible return density resemble those from a GEV. The authors develop a pricing model under GEV returns, whose form is very like Black–Scholes, along with equations for its Greek letter risks. They then demonstrate its superior ability to fit FTSE option prices from 1997 to 2009 and explore its behavior around critical market events, such as the 1997 Asian crisis, the 9/11 attack in 2001, and the collapse of Lehman Brothers in 2008.