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We show that the optimal exercise boundary for the American put option with non-dividend-paying asset is convex. With this convexity result, we then give a simple rigorous argument providing an accurate asymptotic behavior for the exercise boundary near expiry.KEY WORDS: American put option on a zero dividend asset, convexity of the early exercise boundary, free boundary problem, obstacle problem, near-expiry estimates.

The binomial tree method, first proposed by Cox, Ross, and Rubinstein [Journal of Financial Economics, 7 (1979), pp. 229-263], is one of the most popular approaches to pricing options. By introducing an additional path-dependent variable, such methods can be readily extended to the valuation of path-dependent options. In this paper, using numerical analysis and the notion of viscosity solutions, we present a unifying theoretical framework to show the uniform convergence of binomial tree methods for European/American path-dependent options, including arithmetic average options, geometric average options, and lookback options.

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