Change of Measure

Pricing a derivative under the real-world probabilities of stock moves is hard: you'd need to know investors' true risk preferences to discount payoffs correctly. But there is a shortcut. Reweight each outcome's probability — without changing which outcomes are possible, only how likely each one is treated as being — so that under the new weights every asset's expected return equals the risk-free rate. Once that reweighting is done, pricing collapses to a plain average: discount the expected payoff at the risk-free rate, full stop. That reweighting is a change of measure, and it is one of the most consequential tricks in quantitative finance.

The analogy: a loaded die that makes the arithmetic easy

Imagine a die whose true, real-world probabilities are unpleasant fractions: $0.12, 0.19, 0.15, 0.21, 0.14, 0.19$. Computing an expected payoff with these is a slog. Now suppose you're allowed to invent a different, cleverly chosen die — same six faces, same set of possible outcomes, but different probabilities — under which some target quantity (say, the average roll) comes out to a clean, convenient number. You haven't changed what the die can physically roll; you've only changed the weight you assign to each face for the purpose of a calculation. A change of measure is exactly this: same outcomes, different weights, chosen so that a specific expectation becomes easy — in finance, so that every discounted asset price becomes an average under the new weights.

Writing it down

Let $P$ be the real-world probability measure and $Q$ a new measure over the same set of outcomes (nothing that was impossible under $P$ becomes possible under $Q$, or vice versa). The Radon-Nikodym derivative $\dfrac{dQ}{dP}$ is the reweighting factor: for each outcome $\omega$, it tells you how much more or less weight $Q$ assigns compared to $P$. Expectations translate via

$$E^Q[X] = E^P\left[X \cdot \frac{dQ}{dP}\right] .$$

In words: to compute an expectation under the new measure $Q$, take the expectation under the old measure $P$ of the payoff multiplied by the reweighting factor — you never actually have to resample anything, you just weight each outcome differently inside the same average. In finance, $Q$ is chosen to be the risk-neutral measure: the unique reweighting under which every traded asset's expected return equals the risk-free rate $r$, so that a derivative's price is simply

$$V_0 = e^{-rT}\, E^Q[\text{payoff}] .$$

In words: discount the risk-neutral expected payoff at the risk-free rate — no risk premium needed anywhere in the formula, because the reweighting has already absorbed it.

Worked example 1: a one-period binomial reweighting

A stock at $100 either goes to $120 (up) or $90 (down) after one period, risk-free rate $r=0$ over the period for simplicity. Real-world probability of up is $P(\text{up}) = 0.7$. Find the risk-neutral probability $q$ of up such that $E^Q[\text{price}] = 100$ (matching the risk-free — here zero — growth): $q(120) + (1-q)(90) = 100$. Expand: $120q + 90 - 90q = 100$, so $30q = 10$, giving $q = 1/3$. Under $Q$, "up" has probability $1/3$, not the real-world $0.7$ — a big reweighting, but it makes the stock itself a fair bet under $Q$, which is exactly the property risk-neutral pricing needs.

Worked example 2: pricing a call with the reweighted probabilities

Same setup, strike $100, so the call pays $20 if up, $0 if down. Under real-world $P$, you'd need to know the market's risk premium to discount this correctly — a hard, subjective step. Under $Q$ from example 1, just average: $E^Q[\text{payoff}] = \tfrac13(20) + \tfrac23(0) = 6.67$. With $r=0$ the price is simply $6.67 — no discounting adjustment needed beyond the reweighting already baked into $q$. Compare: averaging under the real-world probability $0.7$ would (wrongly) give 0.7 \times 20 = \14$, badly overpriced, because it ignores that investors demand compensation for risk, which the$Q$-reweighting has already built in.

What this means in practice

Every Black-Scholes-style pricing formula is an expectation under a risk-neutral measure $Q$, not under real-world probabilities — this is why the drift term in the pricing PDE is the risk-free rate $r$ and not a stock's actual expected return. Girsanov's theorem is the continuous-time version of this same reweighting trick, used to shift the drift of a Brownian motion. Importance sampling in Monte Carlo pricing is the same idea again: reweight rare, important outcomes to sample them more often, then correct with the Radon-Nikodym factor.