Searching for a closed-form exact solution for American put options under the Black-Scholes framework has been a long standing problem in the past; many researchers believe that it is impossible to find such a solution. In this paper, a closed-form exact solution, in the form of a Taylor's series expansion, of the well-known Black-Scholes equation is presented for the first time. As a result of this analytic solution, 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.
Zhu, S. (2005). A closed-form exact solution for the value of American put and its optimal exercise boundary. SPIE International Symposium (pp. 186-199). Online: SPIE. Conference Proceedings SPIE 5848, Noise and Fluctuations in Econophysics and Finance, 186 (June 01, 2005).