Estimating the tail shape parameter from option prices

In this paper, a method to estimate the tail shape parameter of the risk-neutral density from option prices is developed and 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. Our focus is only on the tails of the risk-neutral density and not on the entire body of the density. Our method is quite general, and applies to a large class of risk-neutral densities. Unlike all other methods of estimating the risk-neutral density, it can be used without interpolating the implied volatility, or even without the knowledge of the current index value, the dividend yield or the risk-free rate. A case study using Standard & Poor's 500 (S&P 500) index options is given. In particular, the estimation of the tail shape of the S&P 500 index shows a thickening of the left tail just prior to the market turmoil of September 2008, but a thinning of the left tail in the midst of the turmoil.