risk-neutral pricing

Risk-neutral pricing is the central computational principle of mathematical finance: the arbitrage-free price of a derivative is the expected value of its discounted payoff, computed not under the real-world probabilities but under a special, artificial 'risk-neutral' probability measure. This is initially startling — why would the right price ignore investors' actual beliefs about how likely the stock is to go up? The answer is that the price is fixed by NO-ARBITRAGE (the impossibility of riskless profit) and by the cost of REPLICATING the payoff with a hedging portfolio, not by anyone's view of the odds; the risk-neutral measure is exactly the bookkeeping device that makes replication look like a simple expectation.

Here is the precise content. A measure Q equivalent to the real-world measure P is an equivalent martingale measure (risk-neutral measure) if, under Q, every traded asset's price DISCOUNTED by the risk-free bank account is a martingale: S_t / B_t is a Q-martingale, where dB_t = r B_t dt. Under such a Q every asset earns, in expectation, exactly the risk-free rate r — that is what 'risk-neutral' means: it is the fictional world in which no one demands extra return for bearing risk. The pricing formula is then V_t = B_t E^Q[ V_T / B_T given F_t ] = E^Q[ e^(-r(T-t)) V_T given F_t ]: the value today is the Q-expected, discounted future payoff. The change from P to Q is performed by Girsanov's theorem, which re-tilts the Brownian drift; the market price of risk (the Sharpe-ratio-like quantity (mu - r)/sigma) is exactly the drift adjustment Girsanov applies to turn the real drift mu into the risk-free rate r. The deep reason this works is the fundamental theorem of asset pricing: no-arbitrage is EQUIVALENT to the existence of such a Q.

Risk-neutral pricing matters because it converts every pricing problem into a (conditional) expectation under Q, computable by Feynman-Kac (a PDE), by Monte Carlo, or by trees, and because it cleanly separates pricing (which needs only Q) from forecasting (which needs P). Honest caveats that must not be dropped: the existence of Q encodes no-arbitrage, but UNIQUENESS of Q encodes market COMPLETENESS (every payoff is replicable). When the market is incomplete (stochastic volatility, jumps, transaction costs) there are MANY equivalent martingale measures, the no-arbitrage price is only an INTERVAL, and selecting one Q requires an extra economic input (a utility, a calibration, a choice of risk premium). Also Q is NOT the real-world measure — it must never be used to forecast actual returns or to estimate real-world probabilities of events; using risk-neutral probabilities as if they were physical probabilities is a classic and costly error.