The Measure We Choose: How Numéraires Simplify Pricing
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. ...