Optimal Stopping and Early Exercise: An Eigenfunction Expansion Approach

This paper proposes a new approach to solve finite-horizon optimal stopping problems for a class of Markov processes that includes one-dimensional diffusions, birth--death processes, and jump diffusions and continuous-time Markov chains obtained by time-changing diffusions and birth-and-death processes with Lévy subordinators. When the expectation operator has a purely discrete spectrum in the Hilbert space of square-integrable payoffs, the value function of a discrete optimal stopping problem has an expansion in the eigenfunctions of the expectation operator. The Bellman's dynamic programming for the value function then reduces to an explicit recursion for the expansion coefficients. The value function of the continuous optimal stopping problem is then obtained by extrapolating the value function of the discrete problem to the limit via Richardson extrapolation. To illustrate the method, the paper develops two applications: American-style commodity futures options and Bermudan-style abandonment and capacity expansion options in commodity extraction projects under the subordinate Ornstein-Uhlenbeck model with mean-reverting jumps with the value function given by an expansion in Hermite polynomials.