riskneutral

Riskneutral describes two related ideas used in economics and finance. Broadly, a risk-neutral person is someone who is indifferent to risk, valuing outcomes solely by their expected value. In financial modeling, however, risk-neutral refers to a probability measure used for pricing assets, known as the risk-neutral or equivalent martingale measure.

Under a risk-neutral measure Q, the prices of traded assets evolve so that discounted prices are martingales. If there is a risk-free rate r, the process S_t e^{-rt} is a martingale under Q. This means that the present value of any future payoff can be computed as an expected value under Q, rather than under the real-world probability P.

Pricing with the risk-neutral measure works as follows: for a contingent claim with payoff X_T at time T, its current price is the discounted expected payoff under Q, P0 = E_Q[e^{-rT} X_T]. The change from P to Q is governed by a Radon-Nikodym derivative and, in continuous models, often described by Girsanov’s theorem, which adjusts the drift of asset dynamics to reflect risk premia.

In complete markets there is a unique risk-neutral measure, yielding a single fair price via replication. In incomplete markets, multiple risk-neutral measures may exist, producing a range of arbitrage-free prices. The concept is a mathematical tool for pricing, not a statement about actual investor beliefs; a risk-neutral agent, in everyday terms, is not necessarily the same as a risk-neutral probability measure used for pricing.