Pricing American Options: A Duality Approach

Abstract

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.

Monte Carlo Methods in Financial Engineering

“…Moreover, if aM is deterministic then LM(t) has lognormal distribution LN( -ai.r(t)/2, ai.r(t)) with aM(t) = ~ rt IlaM(u)112 duo t 10 (3.115) In fact, the choice of M is arbitrary: each Ln is a martingale (lognormal if an is deterministic) under the forward measure Pn+1 associated with Tn+1. 172 3 Generating Sample Paths…”

“…Haugh and Kogan [172] and Rogers [308] have established dual formulations in which the price is represented through a minimization problem. The dual minimizes over a class of supermartingales or martingales.…”

“…A general strategy is to construct a martingale from an approximate value function or stopping rule. This idea is developed and shown to be effective in Andersen and Broadie [15] and Haugh and Kogan [172]. A suboptimal exercise policy provides a lower bound on the price of an American option; the dual value defined by extracting a martingale from the suboptimal policy complements the lower bound with an upper bound.…”

“…This term is conditionally unbiased (given Xi-d, so the conditional expectation of Ai given X il is zero, as required by (8.61). It follows that using these Ai in (8.65) is guaranteed to produce a high-biased estimator, even for finite n. Haugh and Kogan [172] propose and test other methods for constructing upper bounds from approximate value functions. Their formulation of the dual problem takes an infimum over supermartingales rather than martingales, and they therefore use ~ to construct supermartingales.…”

Simulation in financial engineering

“…Haugh and Kogan (2001) and Rogers (2001) offer methods of producing an estimator biased high by considering the dual of the American optimal stopping problem (3). Rogers shows that this dual approach is related to the American option seller's hedging strategy and does not depend on finding the American option buyer's exercise policy and the related low-biased estimator.…”