The General Mixture Diffusion Sde and its Relationship with an Uncertain-Volatility Option Model with Volatility-Asset Decorrelation

Abstract

In the present paper, given an evolving mixture of probability densities, we define a candidate diffusion process whose marginal law follows the same evolution. We derive as a particular case a stochastic differential equation (SDE) admitting a unique strong solution and whose density evolves as a mixture of Gaussian densities. We present an interesting result on the comparison between the instantaneous and the terminal correlation between the obtained process and its squared diffusion coefficient. As an application to mathematical finance, we construct diffusion processes whose marginal densities are mixtures of lognormal densities. We explain how such processes can be used to model the market smile phenomenon. We show that the lognormal mixture dynamics is the one-dimensional diffusion version of a suitable uncertain volatility model, and suitably reinterpret the earlier correlation result. We explore numerically the relationship between the future smile structures of both the diffusion and the uncertain volatility versions.

Keywords: Stochastic Differential Equations, Mixtures of Densities, Mixtures of Gaussians, Mixtures of Lognormals, Risk-Neutral Valuation, Option Pricing, Volatility-Underlying Correlation, Smile Modeling

JEL Classification: G13