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.American put options, Series-expansion analytical solution, Homotopy-analysis method,

Pricing American Options under Stochastic Volatility and Stochastic Interest Rates

“…The only known analytical solution to this problem 2 is found by Zhu (2006) in the form of a Taylor series expansion. While the emphasis of that paper is to show the existence of an exact analytical solution, such a solution does not have a clear advantage over some fast numerical schemes from a computational point of view.…”

Some Notes On The Convergence Control Parameter In The Framework Of The Homotopy Analysis Method

“…Other than applied mathematicians, the method has found its way into different fields in engineering and sciences, for example chemists, physicians, biologists also take advantage of this powerful analytic technique for coping their own problems and equations. One can, also, find HAM footprints in other fields like finance problems [19].…”

A Survey on American Options: Old Approaches and New Trends

“…Section 5 reviews the analytic method of lines and its associated randomization method. Section 6 provides a critical review of Zhu's homotopy method [80], and Section 7 reviews analytic approximation of the early exercise boundary. Section 8 explains Monte Carlo methods, and Section 9 discusses recent topics: model uncertainty, backward stochastic differential equations, and real options.…”

“…In our opinion the following have been prominent ideas: early attempts to obtain approximate prices [58,9], the integral representation of the early exercise premium [51,53,19], the analytic method of lines and its associated randomization method [17], and a homotopy method [80], analytic approximations of the early exercise boundary [20], and Monte Calro methods for American options. We discuss these key ideas and identify problems that need to be answered for further development of their methods.…”