Risk-neutral measure
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Probability measure mathbbQ — equivalent to the physical measure mathbbP but reweighted — under which the discounted price of every traded asset is a martingale: S_t / B_t = mathbbE^mathbbQ[S_T / B_T mid mathcalF_t], where B_t is the money-market numeraire. By FTAP-I, mathbbQ exists iff the market is arbitrage-free; by FTAP-II it is unique iff the market is complete. Constructed via Girsanov by shifting GBM drift from μ to r with Radon-Nikodým density involving the market price of risk θ = (μ - r)/σ. Not the 'true' measure — a pricing artefact under which risk premia are absorbed into the numeraire.
Risk-neutral measure in plain English
When you price an option, you're not betting on how likely things REALLY are — you're finding a price that leaves no free money on the table, and the risk-neutral measure is the clever set of 'pretend probabilities' that makes that math work. The trick: under these adjusted odds, every asset is expected to grow at just the safe interest rate (as if nobody demanded extra reward for taking risk), which makes today's price simply the average future payoff discounted back — a 'fair' price with no arbitrage (no risk-free profit) possible. It exists exactly when the market has no free-lunch loopholes, and it's the ONLY such set of pretend-odds when the market is 'complete' (every payoff can be built from tradable pieces). Crucially it is NOT the true real-world probability — it's a pricing tool where the reward-for-risk gets quietly baked into the interest-rate discounting instead of the odds.
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