Option Pricing When the Regime-Switching Risk is Priced

Recently, there has been considerable interest in investigating option valuation problem in the context of regime-switching models. However, most of the literature consider the case that the risk due to switching regimes is not priced. Relatively little attention has been paid to investigate the impact of switching regimes on the option price when this source of risk is priced. In this paper, we shall articulate this important problem and consider the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset switch over time according to the state of an economy, which is modeled by a continuous-time hidden Markov chain. We shall develop a two-stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. The latter is called a min-max entropy problem. We shall conduct numerical experiments to illustrate the effect of pricing regime-switching risk. The results of the numerical experiments reveal that the impact of pricing regime-switching risk on the option prices is significant.