Risk-neutral measure

Risk-neutral measure

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

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 it to express in a formula the value of a derivative. Suppose at some time T years into the future a derivative (for example, a call option on a stock) pays off units, where is a function from the probability space describing the market to the real line. Further suppose that the discount factor from now (time zero) until time T is P(t,T), then today's fair value of the derivative is

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

where 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 meaure 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

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

where is a standard Brownian motion with respect to the physical measure. If we define then Girsanov's theorem states that there exists a measure under which , is a Brownian motion under Q. Substituting in we have

Q is the unique risk neutral measure for the model. The payoff process of a derivative on the stock 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 at all times .

=Notes=

In fact the rate of return is zero in the measure.

This is true in all risk-averse markets, which consists of all large financial markets. An example of a risk-loving market is a casino. A player could 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 entertain of taking a risk.