Mixture Dynamics and Option Pricing: a Regime Switching Model

The need of considering price dynamics alternative to the classical Black-Scholes model for derivatives pricing is widely known. The stochastic variability of market parameters and in particular the empirical evidence of non constant surfaces of implied volatility in real markets require more realistic models for the assets dynamics. Many approaches are available to obtain a better fitting of market data, each aiming to relax some of the restrictive assumption implied by the geometric Browian motion dynamic of the Black Scholes model. These modifications typically consider explicit models for the local volatility, the addition of jump components and/or the introduction of stochastic volatility in the underlying diffusive dynamic. In a series of papers Brigo and Mercurio (2000, 2001) proposed a class of one-dimensional diffusion models characterized by a local volatility function which induces a mixture marginal distribution for the asset price. In particular, the case of log-normal mixtures has been deeply developed. The main advantages of their model are: