Risk-Neutral Measure (Intuition)
A probability tilt that replaces real-world drift with the risk-free rate so discounted prices become martingales.
Under the physical measure P, asset prices drift at the real-world rate μ, which reflects risk premia.
Under the risk-neutral measure Q, we tilt probabilities so that discounted prices have zero drift; effectively μ is replaced by r.
Pricing by arbitrage uses expectations under Q, not P: the fair price is the discounted risk-neutral expectation of future payoffs.
This diagram compares physical (P) and risk-neutral (Q) distributions for a GBM-style asset. Under Q the drift is r, so the discounted price becomes a martingale while volatility stays the same.
The blue density is the physical distribution under P, where the drift μ includes risk premia. The green dashed density is the risk-neutral distribution under Q, where the drift is pinned to the risk-free rate r with the same volatility σ.
Pricing by arbitrage uses expectations under Q: discounting EQ[S_T] at r gives back S₀, so the discounted price has zero drift (a martingale). Under P the asset may drift faster or slower, but that affects forecasting, not fair pricing. The key mental model is: P for risk and inference, Q for pricing.