Risk-neutral measure

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In mathematical finance, a risk-neutral measure is a probability measure in which today's fair (i.e. arbitrage-free) price of a derivative is equal to the expected value (under the measure) of the future payoff of the derivative discounted at the risk-free rate.

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.^[1] This is in contrast to the physical measure - i.e. the actual probability distribution of prices where (almost universally^[2]) 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.

[edit] Example

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.

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

is recognizable as the market price of risk.

Substituting in we have

$dS_t = rS_tdt + \sigma S_t\, d\tilde{W}_t.$

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 $t\leq T$ .

[edit] Notes

^ In fact the rate of return is equal to the short rate in the measure.
^ This is true in all risk-averse markets, which consists of all large financial markets. Example of risk-seeking markets are casinos and lotteries. A player could choose to play no casino games (zero risk, expected return zero) or play some games (significant risk, expected negative return). The player pays a premium to have the entertainment of taking a risk.