Numeraire and Forward Measures
Choosing a numeraire (bank account, bond, annuity) induces a measure under which that numeraire-valued price is a martingale.
A numeraire is the asset you measure value in; dividing prices by the numeraire and choosing the matching measure yields martingales by construction.
The money-market numeraire gives the standard risk-neutral measure, under which discounted prices are fair games and pricing uses discounted expectations.
Using a T-maturity zero-coupon bond as numeraire gives the T-forward measure, under which the corresponding forward price is a martingale.
In interest-rate and XVA engines, switching numeraires and measures simplifies dynamics and expectations for specific products or cash-flow structures.
Pick a numeraire. Divide by it. Under the matching measure, the resulting process is a martingale. Here we test that visually by plotting the cross-path mean of X_t = S_t / B_t (discounted price): it should stay near a flat level.
A numeraire is “what you measure value in”. When you divide by the chosen numeraire and use the matching pricing measure, the resulting process has zero drift in expectation.
Money-market numeraire gives the usual “discounted price” story. Bond numeraire gives the “forward measure” story: quantities naturally expressed in units of P(t,T) become martingales.
This toy model uses deterministic rates and a simple GBM. The goal is the invariance idea: change numeraire → change what is driftless, not “learn measure theory from sliders”.