On the Use of Numeraires in Option Pricing

Abstract

It is natural to think of asset prices in terms of dollars per unit. The original Black-Scholes equation does so, giving the price of an option in dollars as a function of the stock price and strike price in dollars. But as models become more complicated, allowing stochastic interest rates, for example, each new source of risk makes solving the model significantly harder. Changing the numeraire can often simplify matters substantially. By recasting the problem using a different unit of account, one can capture the impact of some of the risk in terms of the value of a traded asset, whose price can simply be observed in the market. For example, under stochastic interest rates, the current market price of a zero coupon bond that matures on the option expiration date impounds the interest rate process over that period, stochastically time-varying though it may be. Changing numeraire and writing the option pricing problem in terms of zero coupon bonds per share, rather than dollars, eliminates the need to model the interest rate process at all. The problem is solved in terms of bonds, and the answer is then converted to dollars by multiplying the bond-denominated option value by the current bond price in the market. In this article, Benninga, Björk, and Wiener explain the change of numeraire technique fully and show how useful it can be, and how creatively it can be used, in simplifying option pricing problems.