Sigma approximation

Numéraire is a basic standard by which value is computed. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units. "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. In general equilibrium theory setting the price of one good to be 1 has the problem that this presumes (unwarrantedly) that this good will not be a free good in equilibrium. This is typically avoided by using the sum of the prices of all goods to be 1, that is, by restricting prices to the unit simplex.

Example

If a store sells 1 can of soup for $1.20, the numéraire is dollars. If the store would buy $1 for 5/6 of a can of soup, the numéraire is cans of soup. Trading a can of soup is simpler than trading fractional cans of soup, so most stores use a numéraire of money, which has fractional units.

The numéraire could be a third good, such as a packet of pasta. Suppose the store would sell the can of soup for $1.20 as above, and would also sell a packet of pasta for $2.80. The store should be willing to sell the can of soup for $1.20/$2.80 = 3/7 of a packet of pasta. In this case, the numéraire is packets of pasta. But the store probably does not want to handle broken packets of pasta, so the store demands 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 $M(t)=\exp (\underset{0}{\overset{t}{\int }}r(s)ds)$ is the price at time $t$ of $1 that was invested in the money market at time 0, then 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

$\frac{S(t)}{M(t)}=E_Q[\frac{S(T)}{M(T)}|\mathcal{ℱ}(t)]\mathrm{\forall }t\le T.$

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

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

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, $N(t)$:

$\begin{array}{rl}& E_{Q^N}[\frac{S(T)}{N(T)}|\mathcal{ℱ}(t)]\\ & =E_Q[\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}\frac{S(T)}{N(T)}|\mathcal{ℱ}(t)]/E_Q[\frac{M(0)}{M(T)}\frac{N(T)}{N(0)}|\mathcal{ℱ}(t)]\\ & =\frac{M(t)}{N(t)}{E}_Q[\frac{S(T)}{M(T)}|\mathcal{ℱ}(t)]=\frac{M(t)}{N(t)}\frac{S(t)}{M(t)}=\frac{S(t)}{N(t)}.\end{array}$

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

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