A General Equilibrium and Preference Free Model for Pricing Options Under Transformed Gamma Distribution

Abstract

The gamma class of distributions encompasses several important distributions either as special or limiting cases, or through simple transformations. In this paper, we established the link between the real and the risk neutral distributions, and provided a formal proof for the existence of the risk neutral valuation relationship between option price and the gamma distributed underlying asset. Closed form and preference free European option pricing formulae are derived for a variety of (transformed) gamma distributions, and Heston's (1993) model is obtained as a special case. The option pricing framework developed here significantly extends the Gaussian class distributions embedded in current option pricing theories. The gamma class of distributions is used historically in hydrology for modelling natural extreme events related to flood, rain and wind. Our models can be used to price European style derivatives associated with these natural phenomena, which will help to encourage greater risk sharing through financial securitization.

Keywords: Risk Neutral Valuation Relationship, General Equilibrium Framework, Transformed Gamma Distribution, Weather Derivatives

JEL Classification: G12, G13, G22