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.

Drift Lives in the Measure: An Intuitive Look at Girsanov's Theorem

Wed, 06 May 2026 00:00:00 +0000

Why This Matters

We want to price a derivative. Under the real world measure $\mathbb{P}$, we face two problems. First, we do not know the true drift $\mu$ of the underlying, and historical estimates are notoriously unreliable. Second, even if we knew $\mu$, taking the expected payoff under $\mathbb{P}$ would still not give the market price. Risky cash flows must be discounted more heavily than guaranteed ones because investors are risk averse. Pricing under $\mathbb{P}$ requires both the true probabilities of outcomes and a model for how the market prices risk. Both are fundamentally unobservable. So what can we do?

Change of Measure on Inflection Quant Lab

The Measure We Choose: How Numéraires Simplify Pricing

Why This Matters

Drift Lives in the Measure: An Intuitive Look at Girsanov's Theorem

Why This Matters