Numéraire

The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradeable economic entity in terms of whose price the relative prices of all other tradeables are expressed. In a monetary economy, acting as the numéraire is one of the functions of money, to serve as a unit of account: to provide a common benchmark relative to which the worths of various goods and services are measured. Using a numeraire, whether monetary or some consumable good, facilitates value comparisons when only the relative prices are relevant, as in general equilibrium theory. When economic analysis refers to a particular good as the numéraire, one says that all other prices are normalized by the price of that good. For example, if a unit of good g has twice the market value of a unit of the numeraire, then the (relative) price of g is 2. Since the value of one unit of the numeraire relative to one unit of itself is 1, the price of the numeraire is always 1.

Change of numéraire

The notation in this section needs to be defined.

In a financial market with traded securities, one may use a change of numéraire 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 $N\left(t\right) >0$ 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 can be shown that $S(t)$ is a martingale under $Q^N$ when priced in terms of the new numéraire, $N(t)$:

$\begin{align} & {} \quad E_{Q^N}\left[\left.\frac{S(T)}{N(T)}\right| \mathcal{F}(t)\right] \\ & = E_{Q}\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}\right| \mathcal{F}(t)\right]/ E_Q\left[\left.\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\right| \mathcal{F}(t)\right] \\ & = \frac{M(t)}{N(t)}E_{Q}\left[\left.\frac{S(T)}{M(T)}\right| \mathcal{F}(t)\right]= \frac{M(t)}{N(t)}\frac{S(t)}{M(t)} = \frac{S(t)}{N(t)}. \end{align}$

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.

See also

• Price index
• Forward measure
• Unit of account

References

• Farshid Jamshidian (1989). "An Exact Bond Option Pricing Formula". The Journal of Finance. 44: 205–209. doi:10.1111/j.1540-6261.1989.tb02413.x.
• Helyette Geman; Nicole El Karoui; J.C. Rochet (1995). "Changes of Numeraire, Changes of Probability Measures and Pricing of Options". Journal of Applied Probability. 32: 443–458. doi:10.2307/3215299.
• Damiano Brigo; Fabio Mercurio (2006) [2001]. Interest Rate Models – Theory and Practice with Smile, Inflation and Credit (2 ed.). Springer Verlag. ISBN 978-3-540-22149-4.