Analytical approximations for American option pricing under regime-switching models

Regime-switching models have been widely used to describe the market dynamics, due to their characterization of the evolution of financial conditions into different states (regimes). However, the incorporation of regime-switching feature into the modeling brings analytical challenges in quantitative finance problems (e.g. American option pricing of our interest), such as long computational time for the solution (e.g. pricing formula; if any); the pain points are more pronounced when the number of regimes is large. To address this concern, this paper presents two analytical approximation frameworks for both risk-neutral American option prices and the associated optimal exercise strategies, perturbed around a Black–Scholes (BS) or a short-tenor solution. The short-tenor solution is relevant in its own right as we observed a promising trend of increasing short-lived options. Our approximated solutions are computationally efficient and accurate as the approximation terms are obtained explicitly while our error analyses provide error bounds for the approximation. Our numerical examples verify our theoretical claims and from which, we can observe that when it is near the expiry, the optimal exercise prices in all regimes are identical as with the underlying asset adopting a constant volatility (i.e. BS model), even though the RS model does not generally degenerate to the BS model.