The Pricing of Contingent Claims in a Multivariate Gamma Distributed Economy

Abstract

In this paper we establish a Risk Neutral Valuation Relationship and develop a framework for pricing multivariate European-style contingent claims in a discrete-time economy using a multivariate gamma distribution. In our framework, risk neutrality is obtained by using market equilibrium conditions, leading to preference-free contingent claim pricing equations. Multivariate contingent claim pricing models are of particular interest when payo¤s depend on two or more stochastic variables, such as options to exchange one asset for another, options on mutual funds, and options with a stochastic strike price in general. In our model each underlying stochastic variable depends on a systematic gamma distributed term and on an idiosyncratic one, where the former has a direct impact on the correlation structure of the underlying variables. To illustrate the applicability of our framework, we present multivariate gamma distributed versions of well-known multivariate normally/lognormally distributed contingent claim pricing formulae. The gamma distribution is particularly suitable to price stochastic variables that present implied volatilities that are an increasing function of the strike price.