Pricing American Parisian options under general time-inhomogeneous Markov models

This paper develops general approaches for pricing various types of American-style Parisian options (down-in/-out, perpetual/finite-maturity) with general payoff functions. These approaches are based on a continuous-time Markov chain (CTMC) approximation under general 1D time-inhomogeneous Markov models. For the down-in types, by conditioning on the Parisian stopping time, we reduce the pricing problem to that of a series of vanilla American options with different maturities; further integrating their prices against the distribution function of the Parisian stopping time then yields the American Parisian down-in option price. This facilitates an efficient application of CTMC approximation, in which the required quantities are calculated to obtain the approximate option price. For the perpetual down-in cases under time-homogeneous models, the computational cost can be substantially reduced. The down-out cases are more complicated: we use the state augmentation approach to record the excursion duration, then the approximate option price is obtained by recursively solving a series of variational inequalities using Lemke's pivoting method. We prove the convergence of CTMC approximation for all types of American Parisian options under general time-inhomogeneous Markov models, and the accuracy and efficiency of our algorithms are confirmed through extensive numerical experiments.