Markov-Modulated and Shifted Wishart processes with applications in derivatives pricing

The popular Wishart (WI) processes, first introduced by Bru in 1991, exhibit convenient analytical properties for modeling asset prices, particularly a closed-form characteristic function, and the ability to jointly model stochastic volatility and correlation. These features tend to increase substantially during crisis periods, more than predicted by a Wishart dynamic. Moreover, the variance processes implied by the Wishart, similar to CIR models, have no buffer away from zero. In this paper, we introduced the Markov-Modulated Shifted Wishart processes (MMSW) and the embedded Shifted Wishart processes (SW) to address these shortcomings in the modeling of asset prices. We obtain analytical representations for several characteristic functions. We also estimate the parameters and evaluate the price of Spread options via the Fourier transform under the two new models compared to the standard Wishart. Our analyses demonstrate a significant impact of the MMSW process compared to the standard Wishart process of up to 7% in Spread option prices.