Numéraire

From Wikipedia, the free encyclopedia

1 Definition
2 Example
3 Change of Numeraire Technique
4 See Also

[edit] Definition

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.

Acting as the numéraire is one of the functions of money: to measure the worth of different goods and services relative to one another. The word is French and means counting/measuring.

When economic analysis refers to goods (g) as the numéraire, typically that analysis assumes that prices are normalized by g's price.

[edit] 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 with 5/6 of a can of soup, 6/5 times. In the latter case, the numeraire is the can of soup.

Note two things: firstly, the roles of buying and selling are 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.

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 packet 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.

[edit] 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 Black-Scholes formulae 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, $N (t)$ :

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