《exact and explicit solution for the valuation of Amer》.pdf

Quantitative Finance, Vol. 6, No. 3, June 2006, 229–242 An exact and explicit solution for the valuation of American put options SONG-PING ZHU* School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia (Received 17 June 2005; in final form 11 March 2006) 2 In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the 1 0 valuation of American put options is presented for the first time. To the best of the author’s 2 y knowledge, a closed-form analytical formula has never been found for the valuation of l u American options of finite maturity, although there have been quite a few approximate J 5 solutions and numerical approaches proposed. The closed-form exact solution presented 0 8 here is written in the form of a Taylor’s series expansion, which contains infinitely many 2 terms. However, only about 30 terms are actually needed to generate a convergent numerical : 0 0 solution if the solution of the corresponding European option is taken as the initial guess of t the solution series. The optimal exercise boundary, which is the main difficulty of the problem, a ] is found as an explicit function of the risk-free interest rate, the volatility and the time to y t i expiration. A key feature of our