Change of numeraire
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Girsanov-based transformation between equivalent martingale measures induced by switching the unit-of-account asset. If N and M are two strictly positive numeraires, the Radon-Nikodým derivative is left.dfracdmathbbQ^NdmathbbQ^Mright|_mathcalF_T = (N_T / N_0)/(M_T / M_0). Switching to the T-forward measure (numeraire P(t,T)) lets one pull the discount factor out of the expectation — the standard trick for pricing swaptions and FRAs. Switching to the annuity measure makes the swap rate a martingale (LMM/SABR habitat).
Change of numeraire in plain English
When you price something complicated, the math is much easier in some units than others — the same way a recipe is easier in grams than in 'pinches,' or a road trip is simpler measured in hours than in footsteps. 'Change of numeraire' is the trick of re-measuring all values against a cleverly chosen yardstick (the numeraire, meaning the 'unit of account' you divide every price by) so that a messy formula collapses into a tidy one. There's a precise recipe for switching yardsticks correctly without changing the true answer, and quants use it constantly: pick the right unit and an awkward average suddenly simplifies, letting them price tricky interest-rate options in a few clean steps.
Practice Change of numeraire
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