Option Pricings in an Incomplete Market with Regime Switching

We discuss a model of an incomplete market by adjoining the Black–Scholes exponential Brownian motion model for stock fluctuations to a hidden Markov process which represents the state of information in the investors' community. We investigate option pricing procedures for this incomplete model. Under an additional economic assumption, we provide an arbitrage-free pricing framework. In addition to European call options, we study optimal stopping problems related to some nonstandard types of options, such as perpetual American put and lookback options. We obtain explicit solutions by extending the technique of smooth fit to allow jump discontinuities. The optimal strategy involves jumping over the optimal boundary. In the end, we discuss the corresponding discrete CRR-type model and the first passage time problem for this regime-switching model.