Numeraire and Change of Measure Explained

A dollar price uses currency as its visible unit, while asset pricing models often use a money-market account as the dynamic numeraire

If Nₜ is the chosen numeraire, the relative price of asset X is Xₜ/Nₜ. This is the amount of numeraire needed to buy X

Changing the unit does not create value. It reorganizes the same economic payoffs and changes the most convenient conditional expectation

The numeraire must be tradable and strictly positive over the relevant horizon so division is defined and the measure density remains valid

An asset that can hit zero cannot generally serve as a global numeraire without localization or a narrower state space

Funding, collateral, dividends, and reinvestment must be incorporated consistently when defining the numeraire's total-return value

Starting from money-market numeraire B and risk-neutral measure Qᴮ, another numeraire N defines a new measure Qᴺ through a density based on N/B

Under Qᴺ, every admissible tradable price divided by N is a martingale. Under Qᴮ, prices divided by B have that property instead

There is no measure-free claim that a raw asset price is a martingale. The numeraire and measure travel together

A payoff H at T can be valued as Nₜ times the Qᴺ conditional expectation of H/N_T

Using a different numeraire changes both the payoff units and probability weights so the currency value remains the same when assumptions are consistent

This identity is a pricing transformation, not a forecast that one measure's event probabilities are more physically likely

Choose a T-maturity zero-coupon bond P(t,T) as numeraire. Under the T-forward measure, eligible T-delivery forward prices become martingales

This can remove stochastic discounting from a payoff paid at T and simplify caps, swaptions, bond options, and other interest-rate claims

The maturity must match the payoff date or cash flows must be decomposed carefully. One forward measure does not simplify all dates at once

Choose a numeraire resembling a positive factor in the payoff. Dividing by it can turn a product into a simpler indicator or exchange option

In Monte Carlo, the transformed payoff may have lower variance or avoid simulating a separate discount factor, though improvement is not automatic

Numerical convenience cannot repair an inconsistent model. The transformed measure still needs correct drift, covariance, and density dynamics

For equity, the tradable numeraire must include reinvested dividends rather than only the quoted ex-dividend price

In foreign exchange, domestic and foreign money-market accounts naturally generate domestic and foreign risk-neutral measures

Quanto and cross-currency claims reveal the drift adjustments created by correlations between exchange rates, assets, rates, and the chosen numeraire

Price a benchmark under two numeraires and confirm agreement within analytic or numerical error after converting units back

Check density normalization, martingale tests for relative assets, payoff-date alignment, total-return conventions, and simulated drift adjustments

Document numeraire, associated measure, Radon–Nikodym derivative, settlement currency, and all cash-flow dates so the result is reproducible