Regime switching market models and applications
Pemy, Moustapha Njiahouo
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A regime switching model consists of a set of Black-Scholes models (geometric Brownian motions) coupled by a finite state Markov chain. This model is considered as one of the effective mathematical frameworks to study the valuation of stocks and their derivatives. Under this model the associated PDEs satisfied by the option price are quite involved. In the European option case we have a linear system of PDEs; and in the American option case the corresponding PDE is fully nonlinear. Both equations are difficult to solve, and they may not have classical solutions. In this work, we use the framework of viscosity solution to prove that in both cases the option price can be characterized as a unique viscosity solution of those PDEs. This enables us to construct a numerical scheme to approximate the option price. In addition, this framework is used to treat stock selling rule and search for an optimal selling strategy in order to maximize the reward resulted from a selling transaction.