Numéraire
Numéraire
The numéraire (or numeraire) is a basic standard by which value is computed. In mathematical economics it is a tradable economic entity in terms of whose price the relative prices of all other tradables 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.
![\frac{S(t)}{M(t)} = E_Q\left[\left.\frac{S(T)}{M(T)} \right| \mathcal{F}(t)\right]\qquad \forall\, t \leq T.](https://wikimedia.org/api/rest_v1/media/math/render/svg/98c3913f8e32927f63f7e0ee6d34ef641c9b16ea)
![\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6751efdb216f354be70bcf5a15f11cd5e1ab642f)
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.
Numéraire in Financial Pricing
Determining an appropriate numéraire has foundation in several financial pricing models such as options and certain assets. Identifying a risky asset as numéraire has a correlation with the number of underlying assets to model. Underlying shifts are modeled by the following:
Where set 1 defines the new numéraire and can output risk.
See also
Price index
Forward measure
Unit of account