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Pricing American Options: A Duality Approach

Department of IE and OR, Columbia University, New York, New York 10027
Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142
martin.haugh{at}columbia.edu
lkogan{at}mit.edu

We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worst-case performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest.

Subject classifications: finance: asset pricing: American options; duality; Monte Carlo simulation; dynamic programming: approximate dynamic programming; optimal stopping.
History: Received June 2001; revision received February 2002; accepted December 2002.