Usage
Risk-neutral measures make it easy to express the value of a derivative in a formula. Suppose at a future time a derivative (e.g., a call option on a stock) pays units, where is a random variable on the probability space describing the market. Further suppose that the discount factor from now (time zero) until time is . Then today's fair value of the derivative is
where the risk-neutral measure is denoted by . This can be re-stated in terms of the physical measure P as
where is the Radon–Nikodym derivative of with respect to .
Another name for the risk-neutral measure is the equivalent martingale measure. If in a financial market there is just one risk-neutral measure, 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 are more such measures, then in an interval of prices no arbitrage is possible. If no equivalent martingale measure exists, arbitrage opportunities do.
In markets with transaction costs, with no numéraire, the consistent pricing process takes the place of the equivalent martingale measure. There is in fact a 1-to-1 relation between a consistent pricing process and an equivalent martingale measure.
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