A New Analytical-Approximation Formula for the Optimal Exercise Boundary of American Put Options
This article was originally published as Zhu, SP, A New Analytical-Approximation Formula for the Optimal Exercise Boundary of American Put Options, International Journal of Theoretical and Applied Finance, 9(7), 2006, 1141-1177. Copyright World Scientific Publishing Company. Original journal available here.
In this paper, a new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counter- parts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to sys- tems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.