Changes of Numeraire for Pricing Futures, Forwards, and Options

Abstract

A change of numeraire argument is used to derive a general option parity, or equivalence, result relating American call and put prices, and to obtain new expressions for futures and forward prices. The general parity result unifies and extends a number of existing results. The new futures and forward pricing formulas are often simpler to compute in multifactor models than existing alternatives. We also extend previous work by deriving a general formula relating exchange options to ordinary call options. A number of applications to diffusion models, including stochastic volatility, stochastic interest rate, and stochastic dividend rate models, and jump-diffusion models are examined. A self-financing portfolio is called a numeraire if security prices, measured in units of this portfolio, admit an equivalent martingale measure. The most commonly used numeraire is the reinvested short-rate process; the corresponding equivalent martingale measure is the risk-neutral measure. Geman, El Karoui, and Rochet (1995) show that other numeraires can simplify many asset pricing problems. In this article, we build on their results and, using the reinvested asset price as the numeraire, unify and extend the literature on option parity, or equivalence, results relating American call and put prices for asset and futures options. The same numeraire change is used to obtain new pricing formulas for futures and forwards that are often simpler to compute in multifactor models. Finally, we use a numeraire change to simplify exchange option pricing, extending a similar result in Geman, El Karoui, and Rochet to dividend-paying assets. The change of numeraire method is most intuitive in the context of foreign currency derivative securities. As discussed by Grabbe (1983), an American call option to buy 1 DM, with dollar price process S, for K dollars is equivalent to an American put option to sell K dollars, with DM price process K=S, for a strike price of 1 DM. The dollar price of the call must therefore equal the product of the current exchange rate, S0, and the DM price of the put. The call price is computed using the dollar value of a U.S. I am grateful to Kerry Back (the editor) and Costis Skiadas for their many helpful suggestions. Thanks also to Peter DeMarzo, Matt Jackson, Naveen Khanna, David Marshall, Robert McDonald, Jim Moser, Phyllis Payette, and an anonymous referee for their comments. This article subsumes Schroder (1992).