On the Use of Numeraires in Option Pricing SIMON BENNINGA, TOMAS BJÖRK, AND ZVI WIENER SIMON BENNINGA Significant computational simplification is achieved •Pricing options whose strike price is cor- is a professor of finance when option pricing is approached through the change related with the short-term interest rate. at Tel-Aviv University of numeraire technique. By pricing an asset in terms in Israel. [email protected] of another traded asset (the numeraire), this technique The standard Black-Scholes (BS) formula reduces the number of sources of risk that need to be prices a European option on an asset that fol- TOMAS BJÖRK accounted for. The technique is useful in pricing com- lows geometric Brownian motion. The asset’s is a professor of mathe- plicated derivatives. uncertainty is the only risk factor in the model. matical finance at the A more general approach developed by Black- Stockholm School of This article discusses the underlying theory of the Merton-Scholes leads to a partial differential Economics in Sweden. [email protected] numeraire technique, and illustrates it with five equation. The most general method devel- pricing problems: pricing savings plans that offer a oped so far for the pricing of contingent claims ZVI WIENER choice of interest rates; pricing convertible bonds; is the martingale approach to arbitrage theory is a senior lecturer in pricing employee stock ownership plans; pricing developed by Harrison and Kreps [1981], Har- finance at the School options whose strike price is in a currency different rison and Pliska [1981], and others. of Business at the Hebrew from the stock price; and pricing options whose strike Whether one uses the PDE or the stan- University of Jerusalem in Israel. price is correlated with the short-term interest rate. dard risk-neutral valuation formulas of the [email protected] martingale method, it is in most cases very hile the numeraire method is hard to obtain analytic pricing formulas. Thus, well-known in the theoretical for many important cases, special formulas literature, it appears to be (typically modifications of the original BS for- infrequently used in more mula) have been developed. See Haug [1997] Wapplied research, and many practitioners seem for extensive examples. unaware of how to use it as well as when it is One of the most typical cases with mul- profitable (or not) to use it. To illustrate the uses tiple risk factors occurs when an option involves (and possible misuses) of the method, we dis- a choice between two assets with stochastic cuss in some detail five concrete applied prob- prices. In this case, it is often of considerable lems in option pricing: advantage to use a change of numeraire in the pricing of the option. We show where the •Pricing employee stock ownership plans. numeraire approach leads to significant sim- •Pricing options whose strike price is in plifications, but also where the numeraire change a currency different from the stock price. is trivial, or where an obvious numeraire change •Pricing convertible bonds. really does not simplify the computations. • Pricing savings plans that provide a choice of indexing. ITW INTERIS 2002ILLEGAL TO REPRODUCE THIS ARTICLE TINHE JOURNAL ANY OF D ERIVATIVESFORMAT43 Copyright © 2002 Institutional Investor, Inc. All Rights Reserved • Express all prices in the market, including that of the THE HISTORY OF THE NUMERAIRE APPROACH option, in terms of the chosen numeraire. In other IN OPTION PRICING words, perform all the computations in a relative The idea of using a numeraire to simplify option price system. pricing seems to have a history almost as long as the • Since the numeraire asset in the new price system is Black-Scholes formula—a formula that might itself be riskless (by definition), we have reduced the number interpreted as using the dollar as a numeraire. In 1973, of risk factors by one, from n to n – 1. If, for example, the year the Black-Scholes paper was published, we start out with two sources of risk, eliminating Merton [1973] used a change of numeraire (though one may allow us to apply standard one-risk factor without using the name) to derive the value of a Euro- option pricing formulas (such as Black-Scholes). pean call (in units of zero coupon bond) with a •We thus derive the option price in terms of the stochastic yield on the zero. Margrabe’s 1978 paper in numeraire. A simple translation from the numeraire the Journal of Finance on exchange options was the first back to the local currency will then give the price to give the numeraire idea wide press. Margrabe appears of the option in monetary terms. also to have primacy in using the “numeraire” nomen- clature. In his paper Margrabe acknowledges a sug- The standard numeraire reference in an abstract set- gestion from Steve Ross, who had suggested that using ting is Geman, El Karoui, and Rochet [1995]. We first one of the assets as a numeraire would reduce the consider a Markovian framework that is simpler than problem to the Black-Scholes solution and obviate any theirs, but that is still reasonably general. All details and further mathematics. proofs can be found in Björk [1999].1 In the same year that Margrabe’s paper was pub- lished, two other papers, Brenner-Galai [1978] and Assumption 1. Given a priori are: Fischer [1978], made use of an approach which would now be called the numeraire approach. In the fol- • An empirically observable (k + 1)-dimensional lowing year Harrison-Kreps [1979, p. 401] used the stochastic process: price of a security with a strictly positive price as numeraire. Hence, their numeraire asset has no market X = (X1, …, Xk + 1) risk and pays an interest rate that equals zero, which is convenient for their analysis. In 1989 papers by Geman with the notational convention that the process k + 1 is [1989] and Jamshidian [1989] formalized the mathe- the riskless rate: matics behind the numeraire approach. Xk + 1(t) = r(t) The main message is that in many cases the change •We assume that under a fixed risk-neutral martin- of numeraire approach leads to a drastic simplification in gale measure Q the factor dynamics have the form: the computations. For each of five different option pricing problems, we present the possible choices of numeraire, µ δ dXi(t) = i(t, X(t))dt + i(t, X(t))dW(t) discuss the pros and cons of the various numeraires, and i = 1, …, k + 1 compute the option prices. where W = (W1, …, Wd)´ is a standard d-dimensional δ δ δ δ I. CHANGE OF NUMERAIRE APPROACH Q-Wiener process and i = ( i1, i2, …, id) is a row vector. The superscript ´ denotes transpose. The basic idea of the numeraire approach can be described as follows. Suppose that an option’s price •A risk-free asset (money account) with the dynamics: depends on several (say, n) sources of risk. We may then compute the price of the option according to this scheme: dB(t) = r(t)B(t)dt • Pick a security that embodies one of the sources of The interpretation is that the components of the risk, and choose this security as the numeraire. vector process X are the underlying factors in the economy. 44 ON THE USE OF NUMERAIRES IN OPTION PRICING WINTER 2002 It is illegal to reproduce this article in any format. Email [email protected] for Reprints or Permissions. We make no a priori market assumptions, so whether a particular component is the price process of a traded asset (2) in the market will depend on the particular application. Assumption 2 introduces asset prices, driven by the underlying factors in the economy. where ΠZ denotes the arbitrage-free price in the Z economy. ~ ~ Assumption 2 •For any T-claim Y ( Y = Y/S0(T ), for example) its arbitrage-free price process ΠZ in the Z economy •We consider a fixed set of price processes S0(t), …, is given by: Sn(t), each assumed to be the arbitrage-free price process for some traded asset without dividends. • Under the risk-neutral measure Q, the S dynamics (3) have the form where E0 denotes expectations with regard to Q0. The pricing formula (2) can be written (1) (4) for i = 0, …, n – 1. • The n-th asset price is always given by • The Q0 dynamics of the Z processes are given by S (t) = B(t) n (5) σ and thus (1) also holds for i = n with nj = 0 for j = 1, σ σ σ σ σ …, d. where i = ( i1, i2, …, id), and 0 is defined similarly. We now fix an arbitrary asset as the numeraire, and • The Q0 dynamics of the price processes are given by for notational convenience we assume that it is S0. We may then express all other asset prices in terms of the (6) numeraire S0, thus obtaining the normalized price vector 0 0 Z = (Z0, Z1, …, Zn), defined by where W is a Q -Wiener process. • The Q0 dynamics of the X processes are given by (7) We now have two formal economies: the S economy where prices are measured in the local currency (such as dollars), and the Z economy, where prices are measured • The measure Q0 depends upon the choice of in terms of the numeraire S0. numeraire asset S0, but the same measure is used for The main result is a theorem that shows how to all claims, regardless of their exercise dates. price an arbitrary contingent claim in terms of the chosen numeraire. For brevity, we henceforth refer to a contin- In passing, note that if we use the money account gent claim with exercise date T as a T-claim.