General Equilibrium and Risk Neutral Framework for Option Pricing with a Mixture of Distributions

The continuous-time framework for option pricing leads to the very desirable property that a continuous hedging strategy allows us to price options as if investors were risk neutral. But the real world doesn't permit continuous rebalancing with no transactions costs, and discrete-time models cannot get away from risk preferences. However, it has been shown that risk neutral valuation still holds for certain specific combinations of risk preference and returns process. In this article, Vitiello and Poon substantially broaden the class of such models to returns distributions that can be expressed as a mixture of g distributions. The g distribution is obtained through a transformation of a Gaussian variable. The authors derive the pricing kernel that pairs with a given g distribution to produce risk neutral valuation, and then derive option valuation models for calls and puts. As special cases, their formulas produce the known results for lognormal and normal returns. Extending the model further, they obtain pricing formulae for mixtures of two g distributions. Finally, they illustrate the use of the new models to value options on the S&P 500 index and on Heating Degree Days (HDD). One of the key virtues of this approach is its ability to price options on underlyings like HDD, that are not susceptible to delta hedging because they are illiquid or not traded at all.