FINITE-DIFFERENCE BISECTION ALGORITHMS FOR FREE BOUNDARIES OF AMERICAN OPTIONS

This paper presents two algorithms: one for American call options and the other for American put options. Formulated on the implicit system of option price and free boundary of an American option, each algorithm computes numerically both the option price and the free boundary. For the option price, an upwind flnite-difierence scheme solves numerically Jamshidian equation, which is a nonhomogeneous Black-Scholes equation. For the free boundary, a bisection scheme determines a sequence of subintervals bracketing the free boundary in connection with the computed option price. The algorithms are accurate, e-cient, and applicable to most cases such as r > d, rd, and for long time and short time. Using the theory of M-matrix, we proved the necessarytheoremsfordesigningouralgorithmsandshowedthatthealgorithmsareunconditionally stable. Also, we applied our algorithms to a series of numerical experiments to compare them with Binomial Treealgorithms.