Change of Numeraire
Also known as: Numeraire Switching, Measure Change
Change of numeraire is a mathematical technique for simplifying option pricing by changing the choice of discount factor (numeraire). By selecting a numeraire aligned with the payoff structure, complex problems become simple. The technique is essential for LIBOR market models and multi-currency derivatives.
Key highlights
- Mathematical elegance: allows choice of simplest problem formulation via numeraire selection
- Computational efficiency: choosing the right numeraire can reduce dimensionality and speed pricing
- Flexibility: for any product with payoff dependent on future rates or prices, there exists a numeraire that makes pricing simple
- No-arbitrage by construction: the technique preserves arbitrage-freeness regardless of numeraire choice
Intuition
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How it works
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When to use it
Use change of numeraire when the natural numeraire for your problem is not the money market account. Essential for swaption pricing (use the bond maturing at the option's maturity date as numeraire). Useful for FX derivatives (use a foreign bond as numeraire). Less useful if the payoff depends on multiple numeraires.
Strengths & limitations
- Mathematical elegance: allows choice of simplest problem formulation via numeraire selection
- Computational efficiency: choosing the right numeraire can reduce dimensionality and speed pricing
- Flexibility: for any product with payoff dependent on future rates or prices, there exists a numeraire that makes pricing simple
- No-arbitrage by construction: the technique preserves arbitrage-freeness regardless of numeraire choice
- Measure change complexity: changing numeraires requires Radon-Nikodym derivatives; errors lead to drift mistakes
- Limited applicability: some payoffs don't have a natural simplifying numeraire
- Path-dependent sensitivity: for path-dependent derivatives, the numeraire choice doesn't simplify as much
- Multi-currency complications: multiple numeraires in different currencies require careful attention to exchange rates
Common pitfalls
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Applications
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Frequently asked
What is a numeraire?
A numeraire is any positive traded asset. Prices measured in units of the numeraire must form martingales under an appropriate measure. The money market account is the standard numeraire; bond prices measured in units of the money market are martingales under the risk-neutral measure.
How do I compute the Radon-Nikodym derivative?
If you change from numeraire N1 to N2, the Radon-Nikodym derivative is dQ2/dQ1|_T = (N2(0) / N2(T)) * (N1(T) / N1(0)). This appears as a correction to the drift in the SDE when you switch measures.
Why is the terminal bond the natural numeraire for swaptions?
A European swaption payoff depends on the fixed rate of a swap at the exercise date. If you use the bond (zero-coupon bond maturing at the exercise date) as numeraire, the payoff simplifies: LIBOR-like rates become martingales under the bond measure, making the pricing problem one-dimensional.
Can I use any asset as a numeraire?
Any positive asset can serve as numeraire mathematically. But not all choices simplify the problem. Choose a numeraire that (1) is aligned with the payoff structure, (2) has volatility that is simple or zero, or (3) naturally appears in the product.
Sources
- 1.Geman, H., El Karoui, N., & Rochet, J. C. (1995). Changes of numeraire, changes of probability measure and option pricing. Journal of Applied Probability, 32(2), 443-458.
- 2.Brigo, D., & Mercurio, F. (2006). Interest Rate Models: Theory and Practice (2nd ed.). Springer-Verlag.
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Cite this page
ScholarGate. (2026, June 3). Change of Numeraire. ScholarGate. https://scholargate.app/quantitative-finance/change-of-numeraire