ACCELERATING AMERICANOPTION PRIcCING IN LATTICESMichael Curranis a quantitative analyst for Williams Energy Services Company in Tulsa,Oklahoma.
This article describes a method of accelerating thepricing ofAmerican options in binomial lattices in a BlackScholes environment. The standard backward inductionmethod for solving an option valuation problem involvescomputations at every node of the binomial option price tree.ffi show that many of the intermediate calculations areactually unnecessary, and eliminating them leads to a dra-
Accelerating the pricing of American optionsin binomial and multinomial lattices in aBlack-Scholes environment has beenaddressed in the literature (Kim and Byun
[1994] (KB), Breen [1991]), but the method presented here (hereafter the diagonal method, or DM) issubstantially more efficient and is applicable to awider variety of cases. In particular, while the KBand Breen techniques yield an acceleration factor ofthree or four for the American put option, the DMresults in acceleration factors in excess of an order ofmagnitude, while arriving at identical values as thoseobtained from the standard Cox, Ross, andRubinstein [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 underlyingprice equal to the original price, i.e., d = l/u. TheDM also has potential for speeding the computationof American options on interest rate instruments andoptions 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 ofmagnitude, whileyielding the identical estimate given by the standard Cox, Ross,and Rubinstein binomial tree. In addition, we discuss how similar techniques may be applied to pricing American options oninterest rate derivatives and options involving multiple assets.
I. THE DIAGONAL METHOD
The DM method, as applied to the Americanput, is composed of two levels of acceleration. Thefirst applies to arbitrary continuous dividend yields,while the second applies when r ;;:: y for puts andr ::;; y for call options, where r is the risk-free rate ofinterest and y is the continuous yield on the asset.These are precisely the cases when the early exercisepremium is greatest, and include the important casesof options on assets paying no dividends and optionson futures contracts.
We confine ourselves to remarks on theAmerican put, but, as McDonald and Schroder[1990] show, these results can be extended toAmerican 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 putvaluation equations, whose arguments are the under-
WINTER 1995
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