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The numéraire (or numeraire) is a basic standard by which value is computed. In
mathematical economics Mathematical economics is the application of Mathematics, mathematical methods to represent theories and analyze problems in economics. Often, these Applied mathematics#Economics, applied methods are beyond simple geometry, and may include diff ...
it is a tradable economic entity in terms of whose price the relative prices of all other tradables are expressed. In a monetary economy, one of the functions of
money Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic context. The primary functions which distinguish money are: m ...
is to act as the numéraire, i.e. to serve as a
unit of account In economics, unit of account is one of the functions of money. A unit of account is a standard numerical monetary unit of measurement of the market value of goods, services, and other transactions. Also known as a "measure" or "standard" of ...
and therefore provide a common benchmark relative to which the value of various
goods In economics, goods are anything that is good, usually in the sense that it provides welfare or utility to someone. Alan V. Deardorff, 2006. ''Terms Of Trade: Glossary of International Economics'', World Scientific. Online version: Deardorffs ...
and services can be measured against. Using a numeraire, whether monetary or some consumable good, facilitates value comparisons when only the relative prices are relevant, as in
general equilibrium theory In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
. 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

In a financial market with traded securities, one may use a numéraire to price assets. For instance, let M (t) be the price at time of $1 that was invested in the money market at time . The
fundamental theorem of asset pricing The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical finance, provide necessary and sufficient conditions for a market to be arbitrage-free, and for a market to be complete. An a ...
says that all assets S(t) priced in terms of the numéraire (in this case, ), are martingales with respect to a risk-neutral measure, say Q. That is: :\frac = E_Q\left frac\right/math> 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 = \frac\frac = \frac Then it can be shown that S(t) is a martingale under Q^N when priced in terms of the new numéraire N(t): : \begin & \quad E_\left frac\right\\ & = E_\left frac\frac\right E_Q\left frac\right\\ & = \fracE_\left frac\right\ &= \frac\frac\\ & = \frac \end This technique has many important applications in
LIBOR The London Inter-Bank Offered Rate (Libor ) was an interest rate average calculated from estimates submitted by the leading Bank, banks in London. Each bank estimated what it would be charged were it to borrow from other banks. It was the prim ...
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 modelled by the following: : Z_i:= \frac : X = (X_0, X_1, ..., X_n) \to Z = (1, Z_1, ..., Z_n) where ''1'' defines the numéraire.


See also

* Forward measure *
Price index A price index (''plural'': "price indices" or "price indexes") is a normalized average (typically a weighted average) of price relatives for a given class of goods or services in a specific region over a defined time period. It is a statistic ...
*
Unit of account In economics, unit of account is one of the functions of money. A unit of account is a standard numerical monetary unit of measurement of the market value of goods, services, and other transactions. Also known as a "measure" or "standard" of ...


References


Sources

* * * * {{DEFAULTSORT:Numeraire General equilibrium theory Mathematical finance Stock market Finance theories Stochastic processes