While American calls on non-dividend-paying stocks may be valued as European, there is no completely explicit exact solution for the values of American puts. We use a technique called randomization to value American puts and calls on dividend-paying stocks. This technique yields a new semiexplicit approximation for American option values in the Black-Scholes model. Numerical results indicate that the approximation is both accurate and computationally efficient. Closed-form solutions for the

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... e of European-style options have been known since the seminal articles of Black and Scholes (1973) and Merton (1973) . Since American calls on non-dividend-paying stocks are not rationally exercised early, they can be valued in closed form. Unfortunately, the vast majority of listed options are American style and are subject to early exercise. Despite a profusion of research on the subject, no completely satisfactory analytic solution for the value of such options has been found. The principal difficulty in obtaining an analytic solution arises from the absence of a simple expression for I thank the participants