Numéraire

Numéraire
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Numéraire".

content

This article does not cite any references or sources.
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed. (April 2008)

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 why 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 numeraire technique

In a financial market with traded securities, one may use a change of numeraire to price assets. For instance, if $M(t) = \exp\left(\int_0^t r(s) ds\right)$ 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

$\frac{S(t)}{M(t)} = E_Q\left[\left.\frac{S(T)}{M(T)} \right| \mathcal{F}(t)\right]\qquad \forall\, t \leq T.$

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 $Q N$ by the Radon-Nikodym derivative

$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(0)}.$

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 numeraire, $N (t)$ :

$E_{Q^N}\left[\frac{S(T)}{N(T)}| \mathcal{F}(t)\right]$

$= E_{Q}\left[\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}| \mathcal{F}(t)\right]/ E_Q\left[\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}| \mathcal{F}(t)\right]$

$= \frac{M(t)}{N(t)}E_{Q}\left[\frac{S(T)}{M(T)}| \mathcal{F}(t)\right]= \frac{M(t)}{N(t)}\frac{S(t)}{M(t)} = \frac{S(t)}{N(t)}$

This technique has many important applications in LIBOR and swap market models, as well as commodity markets.

Example

Change of numeraire technique

See also