# An Exact and Explicit Solution for the Valuation of American Put Options

*by Associate Professor Song-Ping Zhu*

*Institution:*School of Mathematics and Applied Statistics, University of Wollongong

*Date: Thu 23rd November 2006*

*Time: 3:15 PM*

*Location: Room 213, Richard Berry Building, The University of Melbourne*

*Abstract*: In this talk, an exact and explicit solution of the well-known Black-Scholes (1973) equation for the valuation of American put options is presented for the first time. To the author's best knowledge, never has a closed-form analytical formula 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 is constructed based on the homotopy-analysis method. The optimal exercise boundary, which forms the key difficulty of this highly nonlinear problem, has been elegantly and temporarily removed in the solution process, 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. The newly found analytical solution is also explicit as the optimal exercise boundary as well as the option price can be written as explicit functions of the risk-free interest rate, the volatility and the time to expiration.

*For More Information:* Professor Daniel Dufresne: dufresne@unimelb.edu.au