An exact and explicit solution for the valuation of American put options

Abstract

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.

Keywords:

• American put options
• Series-expansion analytical solution
• Homotopy-analysis method

Acknowledgements

The author wishes to gratefully acknowledge the financial support from the City University of Hong Kong for a visiting professorship during April and May of 2001; the fundamental concept presented in this paper was conceived there and some preliminary work that leads to the final completion of this paper was carried out there as well. Dr Xiaoping Lu's help in proof-reading the manuscript is also gratefully acknowledged.