Numeraire pricing
V_t = N_t E_t^(Q^N)[X_T / N_T]Expressing the payoff in units of the numeraire makes the relative value a conditional expectation.
Research & Systems
Numeraire and Forward Measures
Choosing a numeraire (bank account, bond, annuity) induces a measure under which that numeraire-valued price is a martingale.
Mental model
Choosing a numeraire changes the convenient pricing measure so the numeraire-relative value becomes 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.
Explore the mechanism
Which numeraire makes the payoff or state variable easiest to value as a martingale?
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.
Numbers
r=2.00%, q=0.00%, σ=25.00%
P(0,T)=exp(-rT) ≈ 0.9802
Flat reference level ≈ 100.0000
Mean at T ≈ 97.0254 (deviation -2.9746)
Blue line is the mean over 300 paths.
Interpretation
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”.
UNDERSTAND
What to notice
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.
The quantitative structure
Notation
N_t- strictly positive numeraire value
Q^N- pricing measure associated with N
X_T- claim payoff at time T
This model assumes
- The numeraire is strictly positive and traded in the modeled market.
- Measures are equivalent and the required discounted processes are integrable.
Where it appears
- pricing bond and swap options
- deriving forward-measure expectations
- simplifying multi-currency valuation
Where the intuition breaks
What this view leaves out
- A numeraire must be strictly positive and tradable in the model; measure changes do not remove model, market, or calibration risk.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Numeraire and Forward Measures
Numeraire and Forward MeasuresYOU ARE HERE
ExtendsRisk-Neutral Measure (Intuition)Changing the numeraire generalises pricing from the money-market measure to other equivalent pricing measures.
Used byBlack-76 Pricing for Forwards and FuturesForward measures provide a natural framework for options whose underlying is a forward or futures price.
Supported byChange of Measure (Girsanov)Measure-change machinery underpins the transition between numeraires and associated martingale measures.
Extends · Risk-Neutral Measure (Intuition)
Changing the numeraire generalises pricing from the money-market measure to other equivalent pricing measures.
Explore concept →