CAPM and option pricing with elliptically contoured distributions

This paper offers an alternative proof of the Capital Asset Pricing Model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modelling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modelling the thickness of tails associated with the possibility that asset returns take extreme values with non-negligible probabilities. As summarized in this paper, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Stein’s lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is logelliptically distributed. The Black-Scholes call option price is a special case of this general result when the underlying is log-normally distributed.