The Measure We Choose: How Numéraires Simplify Pricing

Tue, 12 May 2026 00:00:00 +0000

Why This Matters

In the article on Girsanov’s Theorem, we studied how the real-world measure $\mathbb{P}$ and the risk-neutral measure $\mathbb{Q}$ relate, and showed that switching between them amounts to reweighting paths via the Girsanov exponential. Throughout, the risk-free bond was the numéraire: the asset against which all prices were expressed. But this is a convenient choice, not a fundamental one.

Any strictly positive self-financing wealth process can serve as a numéraire, and each choice gives a different probability measure under which asset prices, expressed in units of that numéraire, become martingales. The price of a derivative is invariant; what changes is how the problem is represented. So instead of viewing pricing as a fixed-measure expectation problem, it is often more natural to think of it as choosing the numéraire that best matches the structure of the payoff.

Annuity Measure on Inflection Quant Lab

The Measure We Choose: How Numéraires Simplify Pricing

Why This Matters