The MEMMs for Markov-Modulated GBMs

In this paper, we consider the option pricing problem when the risky underlying assets are driven by Markov-modulated geometric Brownian motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the risky asset, depend on unobservable states of the economy which are modeled by a continuous-time hidden Markov chain. The market described by the Markov-modulated GBM model is incomplete in general, and, hence, the martingale measure is not unique. We adopt the minimal relative entropy martingale measure (MEMM) for the Markov-modulated GBM model as the suitable martingale measure and we obtain the MEMM for the market in general sense.