Pricing and Exercising American Options: an Asymptotic Expansion Approach

This paper proposes and implements a novel asymptotic expansion approach for pricing discretely monitored American options and approximating their optimal early exercise boundaries, under a generic class of multivariate derivative pricing models incorporating both stochastic volatility and Lévy-driven jumps in asset return. The price and the critical value can be expanded up to any arbitrary order around those under a simple constant-volatility jump-diffusion model. The expansion terms are then efficiently implemented by exactly solving some backward inductions via closed-form Fourier transforms, where a substantial extension of the Hilbert transform method of Feng and Linetsky (2008a, 2009) plays an important role. The efficiency of our method is illustrated through some representative examples. As applications, we analyze the impacts of various model parameters on the optimal early exercise boundary, compare the optimal early exercise boundaries under different models, and propose a derivative-proxy based method for practically exercising American options under models with stochastic volatility and jumps.