Pricing and Inference with Mixtures of Conditionally Normal Processes

Abstract

The main objective of this paper is to exploit important properties of the mixtures of normal distributions and of conditionally normal processes in order to propose flexible and tractable discrete-time option pricing models, based on the exponential-affine stochastic discount factor (SDF) modeling principle. These models deal simultaneously and coherently with the historical and risk-neutral worlds, they can be static or dynamic and, moreover, they can be parametric or semi-parametric. In the static setting our approach provide explicit historical and risk-neutral stock returns distributions and closed-form formulas for European option prices of any residual maturity. These pricing formulas turn out to be convex combinations of Black and Scholes formulas. The previous approach is extended to the dynamic case in several ways. First, by assuming that a Mixed Normal distribution is valid for the standardized error terms. Second, by considering mixtures of conditionally Gaussian processes and, third, by adopting a semi-parametric approach which estimates the distribution of standardized error terms by mixtures of normals based on a kernel method. Some numerical exercise shows the ability of our European option pricing models to replicate implied volatility smiles, volatility skews and volatility surfaces coherently with observations.