Recovering the Tail Shape Parameter of the Risk Neutral Density from Option Prices

• Activity Number: 107
• Type: Invited
• Date/Time: Monday, August 1, 2011 : 8:30 AM to 10:20 AM
• Sponsor: Business and Economic Statistics Section
• Author(s): Kamal Hamidieh*+
• Companies: Rice University
• Keywords: Risk Neutral Density ; Options ; Tail Shape Parameter ; Generalized Pareto Distribution ; Extreme Value Theory ; Asset Pricing

Abstract

In this paper closed form pricing formulas for the out of the money European style options are derived, and a method to recover the tail shape of the risk neutral density from the observed option prices is developed. The pricing formulas satisfy many well known model-free no-arbitrage bounds for the European style options. The method is quite general, and applies to a large class of risk neutral densities which also includes the lognormal density. The method is original and unique in the sense that 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. A case study using the S&P 500 option prices is given. In particular, the estimation of the tail shape of the S&P 500 index just prior to the market turmoil of the September 2008 shows a "thickening" of the left tail of the risk neutral density but "thinning" in midst of the turmoil. Information gained from this method would be useful to the risk managers, researchers, and practitioners interested in assessing and quantifying future extreme market conditions based on the observed option prices.