Numéraire is a basic standard by which values are measured, such as gold in a monetary system. Acting as the numéraire is one of the functions of
moneyMoney is anything that is generally accepted as payment for goods and services and repayment of debts. The main functions of money are distinguished as: a medium of exchange, a unit of account, a store of value, and occasionally, a standard of deferred payment...
: to measure the worth of different goods and services relative to one another. "Numéraire goods" are goods with a fixed price of 1 used to facilitate calculations when only the
relative priceRelative price is the price of a commodity such as a good or service in terms of another; ie, the ratio of two prices. A relative price may be expressed in terms of a ratio between any two prices or the ratio between the price of one particular good and a weighted average of all other goods...
s are relevant, as in
general equilibriumGeneral equilibrium theory is a branch of theoretical neoclassical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets, by seeking to prove that equilibrium prices for goods exist and that all prices are at equilibrium, hence...
theory or in effect for base-year dollars.
Numéraire is a basic standard by which values are measured, such as gold in a monetary system. Acting as the numéraire is one of the functions of
moneyMoney is anything that is generally accepted as payment for goods and services and repayment of debts. The main functions of money are distinguished as: a medium of exchange, a unit of account, a store of value, and occasionally, a standard of deferred payment...
: to measure the worth of different goods and services relative to one another. "Numéraire goods" are goods with a fixed price of 1 used to facilitate calculations when only the
relative priceRelative price is the price of a commodity such as a good or service in terms of another; ie, the ratio of two prices. A relative price may be expressed in terms of a ratio between any two prices or the ratio between the price of one particular good and a weighted average of all other goods...
s are relevant, as in
general equilibriumGeneral equilibrium theory is a branch of theoretical neoclassical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many markets, by seeking to prove that equilibrium prices for goods exist and that all prices are at equilibrium, hence...
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.
Example
In a supermarket, Adam can buy 1 can of soup for $1.20. In this case, the numéraire is the currency—dollars. The same trade could be analyzed differently: Adam could also sell $1 for 5/6 of a can of soup. In the latter case, the numeraire is the can of soup.
Note two things: firstly, the focus on buying or selling is reversed when the numeraire changes. Secondly, it is natural to talk about one can of soup rather than 5/6 cans of soup, which is one of the reasons everyone thinks in cash which has fractional monetary units.
Next, we could change numeraires to a third good: for instance a packet of pasta. Suppose now that 1 packet of pasta costs $2.80. If Adam had 3/7 (= 1.20/2.80) of a packet of pasta, he could purchase one can of soup. In the latter case, the numeraire is the packet of pasta. Again, because of the difficulty of breaking a packet of pasta into fractions, it is significantly easier to use 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 is the price at time of $1 that was invested in the money market at time 0, then the
Black-ScholesThe term Black–Scholes refers to three closely related concepts:* The Black–Scholes model is a mathematical model of the market for an equity, in which the equity's price is a stochastic process....
formula says that all assets (say ),
priced in terms of the money market, are
martingalesIn probability theory, a martingale is a stochastic process such that the conditional expected value of an observation at some time t, given all the observations up to some earlier time s, is equal to the observation at that earlier time s...
with respect to the
risk-neutral measureIn mathematical finance, a risk-neutral measure, equivalent martingale measure, or Q-measure is a probability measure that results when one assumes that the current value of all financial assets is equal to the expected value of the future payoff of the asset discounted at the risk-free rate...
, (say ). That is
Now, suppose that 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 by the Radon–Nikodym derivative
Then, by using the abstract Bayes' Rule it is not hard to show that is a martingale when priced in terms of the new numéraire, :
This technique has many important applications in LIBOR and
swapA swap generally refers to the bartering of one thing for another, but it may also refer to:Finance* Swap , a derivative in which two parties agree to exchange one stream of cash flows against anotherTechnology...
market models, as well as commodity markets. Jamshidian (1989) first used it in the context of the
Vasicek modelIn finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of "one-factor model" as it describes interest rate movements as driven by only one source of market risk...
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.