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

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

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 $0.7 \times 20 = \textdollar 14$, 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.