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A binomial approximation to a diffusion is defined as "computationally simple" if the number of nodes grows at most linearly in the number of time intervals. It is shown how to construct computationally simple binomial processes that converge weakly to commonly employed diffusions in financial models. The convergence of the sequence of bond and European option prices from these processes to the corresponding values in the diffusion limit is also demonstrated. Numerical examples from the constant elasticity of variance stock price and the Cox, Ingersoll and Ross (1985) discount bond price are provided.The seminal work of Merton (1969) and Black and Scholes (1973) paved the way for the use of continuous-time models in finance. The usefulness of the underlying mathematical techniques has never been in doubt: the pricing of options and other contingent claims has relied heavily on these techniques. When Sharpe (1978) developed the binomial approach, the option pricing model became accessible to a much We thank

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Rational restrictions are derived for the values of American options on futures contracts. For these options, the optimal policy, in general, involves premature exercise. A model is developed for valuing options on futures contracts in a constant interest rate setting. Despite the fact that premature exercise may be optimal, the value of this American feature appears to be small and a European formula due to Black serves as a useful approximation. Finally, a model is developed to value these options in a world with stochastic interest rates. It is shown that the pricing errors caused by ignoring the location of the interest rate (relative to its long-run mean) range from -5% to 7%, when the current rate is ?200 basis points from its long-run value. The role of interest rate expectations is, therefore, crucial to the valuation. Optimal exercise policies are found from numerical methods for both models. * The Wharton School, University of Pennsylvania and Graduate School of Business, ColumbiaUniversity, respectively. We thank Mark B. Garman, M. Barry Goldman, and the referee for helpful comments, and In Joon Kim for comments and extensive computational assistance. We also thank the participants in the finance seminars at Duke University, New York University, Northwestern, M.I.T., and Wharton for their comments. The Center for the Study of Futures Markets at Columbia University supported this research.