Convexity of the Exercise Boundary of the American Put Option on a Zero Dividend Asset

The Black–Scholes model is widely used to value options. An important advantage of the model is that European options can be valued analytically by the Black–Scholes formula (Merton 1992; Hull 1997). The situation is quite different, however, for American put options with optimal early exercise. While considerable progress has been made, no completely satisfactory analytic solution has been found. As a result, people resort routinely either to numerical methods or to analytic approximations. There is a considerable literature in these fields; see, for example, McKean (1965), Van Moerbeke (1976), MacMillan (1986), Barone-Adesi and Whaley (1987), Barone-Adesi and Elliott (1991), Carr (1992), Barle (1995), Broadie and Detemple (1996), Hull (1997), Kuske and Keller (1998), and Stamicar (1999). A recent list of references can be found in Aitsahlia and Lai (2001) and Chen and Chadam (2006). With the hypothesis of log-normal underlying asset pricing S and based on standard arbitrage–free arguments, the price P(S, T) (T-current time) for the American put option on a non-dividend-paying asset can be formulated as an