Option Pricing and the Implied Tail Index with the Generalized Extreme Value (GEV) Distribution
The 1987 stock market crash, the LTCM debacle, the Asian Crisis, the bursting of the high technology Dot-Com bubble of 2001-2 with 30% losses of equity values, events such as 9/11 and sudden corporate collapses of the magnitude of Enron - have radically changed the view that extreme events have negligible probability. The well known drawback of the Black-Scholes model is that it cannot account for the negative skewness and the excess kurtosis of asset returns. Since the work of Jackwerth and Rubinstein (1996) which demonstrated the discontinuity in the implied skewness and kurtosis across the divide of the 1987 stock market crash - a large literature has developed, which aims to extract the risk neutral probability density function from traded option prices so that the skewness and fat tail properties of the distribution are better captured than in the case of lognormal models. This paper argues that the use of the Generalized Extreme Value Distribution (GEV) for asset returns provides not just a flexible framework that subsumes as special cases a number of classes of distributions that have been assumed to date in more restrictive settings â€“ but also delivers the market implied tail index for the assets returns. Under the postulation of the GEV distribution in the Risk Neutral Density (RND) function for the asset returns, we obtain an original analytical closed form solution for the Harrison and Pliska (1981) no arbitrage equilibrium price for the European call option. The implied GEV parameters and RND are estimated from traded option prices for the period from 1997 to 2003. The pricing performance of the GEV option pricing model is compared to the benchmark Black-Scholes model and found to be superior at all time horizons and at all levels of moneyness. We explain how the implied tail index extracted from traded put prices are efficacious at identifying the fat tailed behaviour of losses or negative returns and hence of the skew in the left tail of the RND function for the underlying price. The GEV implied RNDs before and after special events such as the Asian Crisis, the LTCM crisis and 9/11 are also analyzed
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