American option pricing and filtering with a hidden regime-switching jump diffusion

The valuation of an American-style contingent claim is discussed in a hidden Markov regime-switching jump-diffusion market, where the evolution of a hidden economic state process over time is described by a continuous-time, finite-state, hidden Markov chain. Filtering theory is applied to introduce a filtered market where the valuation problem is discussed. A probabilistic approach to American option pricing is considered, where a decomposition formula for the price of an American put option is given as the sum of its European counterpart and an early exercise premium. Then the valuation of a perpetual American put option is considered. A (semi-)analytical approximation to the perpetual American put price is obtained. Numerical results for the perpetual American put prices and critical values are provided to illustrate the approximation and to examine the impacts of probability beliefs on hidden economic regimes and jumps on the put prices and critical values.