A Note on the Convergence of Binomial‐Pricing and Compound‐Option Models

Abstract

THE BINOMIAL-PRICING MODEL of Cox, Ross, and Rubinstein [2] and the compound-option model of Geske [3] are both elegant and self-contained frameworks for the valuation of American options and related contingent claims, each providing us with a distinctive valuation procedure flexible enough to be adapted to a wide variety of problems. For American-type options on assets with (a) prices following a lognormal diffusion process and (b) a continuous-time exercise policy, such as put options on individual stocks, calls and puts on foreign exchange, and calls and puts on futures contracts, the binomial-pricing model approximates both the stochastic process and the exercise-opportunity set, the latter by restricting exercise to a discrete set of dates. The compound-option model, on the other hand, approximates only the exercise-opportunity set. However, it pays for this with a high price in computational requirements; specifically, with N exercise dates allowed, multivariate normal probabilities of every order up to and including N must be evaluated. In their article on the valuation of put options, Geske and Johnson [4] provide an ingenious solution to this problem by applying acceleration techniques common in numerical analysis but new to the finance literature. Letting T be the time to expiration of the option and P, the value of a compound option with exercise permitted at the n uniformly spaced dates [T/n, 2T/n, ..., T], they generate the sequence of compound-option values [P1, P2, ... , PN]. Polynomial extrapolation is then used to estimate P = lim n -X oo Pn. They found this procedure to be very accurate over a limited sample of put options with low values of N. Though not discussed by Geske and Johnson, the reasons for the effectiveness of acceleration techniques in a compound-option model are apparent from a cursory examination. An acceleration technique will perform best when the approximating sequence converges uniformly on the true value of the contingent claim and worst when convergence includes complex and unpredictable oscilla