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We review the basic properties of American options and the difficulties of applying Monte Carlo valuation to American options. Recent progress on the Least Squares Monte Carlo (LSM) method is described, including the use of quasi-random sequences in LSM. A particle approach to evaluation of American options is formulated. Conclusions and prospects for future research are discussed.

This paper reviews the basic properties of American options and the difficulties of applying Monte Carlo valuation to American options. Asymptotic results by Keller and co-workers are described for the singularity in the early exercise boundary for time t near the final time T . Recent progress on application of Monte Carlo to American options is described including the following: Branching processes have been constructed to obtain upper and lower bounds on the American option price. A Martingale optimization formulation for the American option price can be used to obtain an upper bound on the price, which is complementary to the trivial lower bound. The Least Squares Monte Carlo (LSM) provides a direct method for pricing American options. Quasirandom sequences have been used to improve performance of LSM; a brief introduction to quasi-random sequences is presented. Conclusions and prospects for future research are discussed. In particular, we expect that the asymptotic results of Keller and co-workers could be useful for improving Monte Carlo methods.

This paper describes a fast, flexible numerical technique to price American options and generate their value surface through time. The method runs faster and more accurately than the standard CRR binomial method in practical cases and calculates options on a considerably broader family of new, useful underlying asset processes. The technique relies on the Fast Fourier Transform (FFT) to convolve a transition function for the underlying asset process. The method allows the underlying asset process to be quite general; the previously known standard geometric Brownian motion and the Variance Gamma process [8], and a novel, purely empirical transition function are compared by computing their respective American put value surface and the exercise boundaries.

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