The MEMMs for Markov-Modulated GBMs

In this paper, we consider the optionpricing problem when the risky underlying assets are driven byMarkov-modulated geometric Brownian motion (GBM). That is, themarket parameters, for instance, the market interest rate, theappreciation rate and the volatility of the risky asset, depend onunobservable states of the economy which are modeled by acontinuous-time hidden Markov chain. The market described by theMarkov-modulated GBM model is incomplete in general, and, hence, themartingale measure is not unique. We adopt the minimal relativeentropy martingale measure (MEMM) for the Markov-modulated GBM modelas the suitable martingale measure and we obtain the MEMM for themarket in general sense.