Numéraire is a basic standard by which values are measured, such as gold in a monetary system. Acting as the numéraire is one of the functions of money: to measure the worth of different goods and services relative to one another. "Numéraire goods" are goods with a fixed price of 1 used to facilitate calculations when only the relative prices are relevant, as in general equilibrium theory or in effect for base-year dollars. When economic analysis refers to goods (g) as the numéraire, typically that analysis assumes that prices are normalized by g's price.
Example
In a supermarket, Adam can buy 1 can of soup for $1.20. In this case, the numéraire is the currency—dollars. The same trade could be analyzed differently: Adam could also sell $1 for 5/6 of a can of soup. In the latter case, the numeraire is the can of soup.
Note two things: firstly, the focus on buying or selling is reversed when the numeraire changes. Secondly, it is natural to talk about one can of soup rather than 5/6 cans of soup, which is one of the reasons everyone thinks in cash which has fractional monetary units.
Next, we could change numeraires to a third good: for instance a packet of pasta. Suppose now that 1 packet of pasta costs $2.80. If Adam had 3/7 (= 1.20/2.80) of a packet of pasta, he could purchase one can of soup. In the latter case, the numeraire is the packet of pasta. Again, because of the difficulty of breaking a packet of pasta into fractions, it is significantly easier to use cash as the numéraire.
Change of numéraire technique
In a financial market with traded securities, one may use a change of numéraire to price assets. For instance, if 
is the price at time t of $1 that was invested in the money market at time 0, then the Black-Scholes formula says that all assets (say S(t)), priced in terms of the money market, are martingales with respect to the risk-neutral measure, (say Q). That is
![\frac{S(t)}{M(t)} = E_Q\left[\left.\frac{S(T)}{M(T)} \right| \mathcal{F}(t)\right]\qquad \forall\, t \leq T.](http://wpcontent.answers.com/math/7/d/1/7d1e28a7bf565abb190f47de97d3c3f5.png)
Now, suppose that N(t) > 0 is another strictly positive traded asset (and hence a martingale when priced in terms of the money market). Then, we can define a new probability measure QN by the Radon–Nikodym derivative
Then, by using the abstract Bayes' Rule it is not hard to show that S(t) is a martingale when priced in terms of the new numéraire, N(t):
![\begin{align}
& {} \quad E_{Q^N}\left[\left.\frac{S(T)}{N(T)}\right| \mathcal{F}(t)\right] \\
& = E_{Q}\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}\right| \mathcal{F}(t)\right]/ E_Q\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\right| \mathcal{F}(t)\right] \\
& = \frac{M(t)}{N(t)}E_{Q}\left[\left.\frac{S(T)}{M(T)}\right| \mathcal{F}(t)\right]= \frac{M(t)}{N(t)}\frac{S(t)}{M(t)} = \frac{S(t)}{N(t)}.
\end{align}](http://wpcontent.answers.com/math/5/d/e/5de434e6cdae29ea60da4f18c1dd2c31.png)
This technique has many important applications in LIBOR and swap market models, as well as commodity markets. Jamshidian (1989) first used it in the context of the Vasicek model for interest rates in order to calculate bond options prices. Geman, El Karoui and Rochet (1995) introduced the general formal framework for the change of numéraire technique. See for example Brigo and Mercurio (2001) for a change of numéraire toolkit.
References
- Farshid Jamshidian (1989). An Exact Bond Option Pricing Formula. The Journal of Finance 44, 205-209.
- Helyette Geman, Nicole El Karoui and J.C. Rochet (1995) Changes of Numeraire, Changes of Probability Measures and Pricing of Options. Journal of Applied Probability 32, 443-458.
- Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models - Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag. ISBN 978-3-540-22149-4.
- Notes on Quantitative Analysis in Finance
See also
