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ACCELERATING AMERICAN OPTION PRIcCING IN LATTICES Michael Curran is a quantitative analyst for Williams Energy Services Company in Tulsa, Oklahoma. This article describes a method of accelerating the pricing of American options in binomial lattices in a Black- Scholes environment. The standard backward induction method for solving an option valuation problem involves computations at every node of the binomial option price tree. ffi show that many of the intermediate calculations are actually unnecessary, and eliminating them leads to a dra- A ccelerating the pricing of American options in binomial and multinomial lattices in a Black-Scholes environment has been addressed in the literature (Kim and Byun [1994] (KB), Breen [1991]), but the method present- ed here (hereafter the diagonal method, or DM) is substantially more efficient and is applicable to a wider variety of cases. In particular, while the KB and Breen techniques yield an acceleration factor of three or four for the American put option, the DM results in acceleration factors in excess of an order of magnitude, while arriving at identical values as those obtained from the standard Cox, Ross, and Rubinstein [1979] (CRR) binomial method. The method is specific to CRR trees in that, starting at a fixed underlying price, an up move (u) followed by a down move (d) results in an underlying price equal to the original price, i.e., d = l/u. The DM also has potential for speeding the computation of American options on interest rate instruments and options involving multiple assets. 8 ACCELERATING AMERICAN OPTION PRICING IN LATTICES matic increase in computational ifficiency. Test cases demonstrate that valuing an American put option can be cu:celerated by at least an order of magnitude, while yielding the identical estimate given by the standard Cox, Ross, and Rubinstein binomial tree. In addition, we discuss how simi- lar techniques may be applied to pricing American options on interest rate derivatives and options involving multiple assets. I. THE DIAGONAL METHOD The DM method, as applied to the American put, is composed of two levels of acceleration. The first applies to arbitrary continuous dividend yields, while the second applies when r ;;:: y for puts and r ::;; y for call options, where r is the risk-free rate of interest and y is the continuous yield on the asset. These are precisely the cases when the early exercise premium is greatest, and include the important cases of options on assets paying no dividends and options on futures contracts. We confine ourselves to remarks on the American put, but, as McDonald and Schroder [1990] show, these results can be extended to American call options through the identity C (S, K, r, y, T, 0') = P (K, S, y, r, T, 0') where C (-) and P (-) are the American call and put valuation equations, whose arguments are the under- WINTER 1995 The Journal of Derivatives 1995.3.2:8-18. Downloaded from www.iijournals.com by COLUMBIA UNIVERSITY on 01/02/11. It is illegal to make unauthorized copies of this article, forward to an unauthorized user or to post electronically without Publisher permission.
