Risk-neutral measure

In mathematical finance, a risk-neutral measure is the probability measure that results when one assumes that the future expected value of all financial assets are equal to the future payoff of the asset discounted at the risk-free rate. The concept is used in the pricing of derivatives.

Background
In an actual economy, the price of assets is affected by the amount investors are willing to pay to assume or eliminate risk. However, it is sometimes possible to calculate the prices of asset assuming that there was no risk. When the asset prices are corrected so that there is no risk, the probability that result are those of the risk-neutral measure.

The measure is so-called because, under that measure, all financial assets in the economy have the same expected rate of return, regardless of the 'riskiness' - i.e. the variability in the price - of the asset. This is in contrast to the physical measure - i.e. the actual probability distribution of prices where (almost universally ) more risky assets (those assets with a higher price volatility) have a greater expected rate of return than less risky assets.

Risk-neutral measures make it easy to express in a formula the value of a derivative. Suppose at some time T in the future a derivative (for example, a call option on a stock) pays off $$H_T$$ units, where $$H_T$$ is a random variable on the probability space describing the market. Further suppose that the discount factor from now (time zero) until time T is P(0, T), then today's fair value of the derivative is


 * $$H_0 = P(0,T) \operatorname{E}_Q(H_T).$$

where the risk-neutral measure is denoted by $$Q$$. This can be re-stated in terms of the physical measure P as


 * $$H_0 = \operatorname{E}_P\left(\frac{dQ}{dP}H_T\right)$$

where $$\frac{dQ}{dP}$$ is the Radon-Nikodym derivative of Q with respect to P.

Another name for the risk-neutral measure is the equivalent martingale measure. A particular financial market may have one or more risk-neutral measures. If there is just one then there is a unique arbitrage-free price for each asset in the market. This is the fundamental theorem of arbitrage-free pricing. If there is more than one such measure then there is an interval of prices in which no arbitrage is possible. In this case the equivalent martingale measure terminology is more commonly used.

Example 1 &mdash; Binomial model of stock prices
Suppose that we have a two state economy: the initial stock price $$S$$ can go either up to $$S^u$$ or down to $$S^d$$. If the interest rate is R-1>0, and we have the following relation $$S^d < RS < S^u$$, then the risk-neutral probability of an upward stock movement is given by the number


 * $$\pi = \frac{RS - S^d}{S^u - S^d}.$$

Given a derivative that has payoff $$X^u$$ when the stock price moves upward and $$X^d$$ when the stock price goes downward, we can price the derivative via


 * $$X = \frac{\pi X^u + (1- \pi)X^d}{R}.$$

Example 2 &mdash; Brownian motion model of stock prices
Suppose that our economy consists of one stock, one risk-free bond and that our model describing the evolution of the world is the Black-Scholes model. In the model the stock has dynamics


 * $$ dS_t = \mu S_t\, dt + \sigma S_t\, dW_t $$

where $$W_t$$ is a standard Brownian motion with respect to the physical measure. If we define


 * $$\tilde{W}_t = W_t + \frac{\mu -r}{\sigma}t,$$

then Girsanov's theorem states that there exists a measure $$Q$$ under which $$\tilde{W}_t$$ is a Brownian motion.

The market price of risk is recognizable as


 * $$ \frac{\mu -r}{\sigma}. $$

Substituting in we have


 * $$ dS_t = rS_t\,dt + \sigma S_t\, d\tilde{W}_t. $$

Q is the unique risk-neutral measure for the model. The (discounted) payoff process of a derivative on the stock $$H_t = \operatorname{E}_Q(H_T| F_t)$$ is a martingale under Q. Since S and H are Q-martingales we can invoke the martingale representation theorem to find a replicating strategy - a holding of stocks and bonds that pays off $$H_t$$ at all times $$t\leq T$$.