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

Abstract

Crisis events such as the 1987 stock market crash, the Asian Crisis, and the collapse of Lehman Brothers have radically changed the view that extreme events in financial markets have negligible probability. In this article, Markose and Alentorn argue that the use of the generalized extreme value (GEV) distribution to model the implied riskneutral density (RND) function provides a flexible framework that captures the negative skewness and excess kurtosis of returns, and also delivers the market-implied tail index. The authors obtain an original analytical closed-form solution for the Harrison and Pliska [1981] no-arbitrage equilibrium price for the European option in the case of GEV asset returns. The GEV-based option pricing model successfully removes the in-sample pricing bias of the Black-Scholes model, and also shows greater out-of-sample pricing accuracy, while requiring the estimation of only two parameters. The authors explain how the implied tail index is efficacious at modeling the fat-tailed behavior and negative skewness of the implied RND functions, particularly around crisis events. Copyright © 2011 Norsk Regnesentral