A numéraire is a strictly positive traded asset used as the unit in which all other asset prices are expressed. Changing the numéraire changes the associated martingale measure while preserving no-arbitrage prices.
If are strictly positive traded numeraires and is a mean-one martingale after normalizing its initial value under , this ratio defines . Conditional change of measure then makes every -martingale asset price into the corresponding -martingale , when the required integrability holds. In particular a martingale exchange rate under one currency's bank-account measure has a reciprocal martingale under the other currency's measure when their interest rates coincide.
For a non-dividend-paying stock in the Black-Scholes model, use density . Under , is Brownian motion, and . Thus . This change of numeraire is particularly useful for payments delivered in stock units.
In a one-period market, a numéraire portfolio has strictly positive value at both dates and can therefore be used to convert initial consumption into terminal wealth.
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Numéraire is a term used in economics, finance, and mathematics to refer to a unit of account or a standard numerical value that is used to measure the value of goods, services, or financial assets. In a broader sense, it acts as a common denominator that facilitates comparisons of value across different items. In the context of finance, numéraire can refer to a specific reference asset or currency used to express the value of other assets.