Estimating the Tail Shape Parameter from Option Prices

Abstract

In this paper, a method to estimate the tail shape parameter of the risk neutral density from option prices is developed. Closed form pricing formulas for out-of-the-money European style options are derived. The pricing formulas satisfy many well known model-free no-arbitrage properties for the options. The focus is only on the tails of the risk neutral density and not on the entire body of the density as many works have already done this. The method is quite general, and applies to a large class of risk neutral densities. Our method can be used without interpolating the implied volatility, or even the knowledge of the current index value or the dividend yield or the risk free rate. This is in contrast to every other method that attempts to estimate the risk neutral density. A case study using S&P 500 Index options is given. In particular, the estimation of the tail shape of S&P 500 index just prior to the market turmoil of the September 2008 shows a thickening of the left tail but a thinning in the midst of the turmoil.

Keywords: Options, Option Pricing, Risk Neutral Density, Generalized Pareto, S&P 500