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
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
priced in terms of the numéraire (in this case, ), are
martingales with respect to a
risk-neutral measure, say
. That is:
: