Risk-neutral measure

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

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[edit] Background

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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 probabilities 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.[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 HT units, where HT 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 1 — Binomial model of stock prices

Suppose that we have a two state economy: the initial stock price S can go either up to Failed to parse (Cannot write to or create math output directory): S^u

or down to Sd.  If the interest rate is R-1>0, and we have the following relation Sd < RS < Su, 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 Xu when the stock price moves upward and Xd when the stock price goes downward, we can price the derivative via

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

[edit] Example 2 — Brownian motion model of stock prices

Suppose that our economy consists of 2 assets, a stock and a risk-free bond, and that our model of the world is the Black-Scholes model. In the model the evolution of the stock price can be described by Geometric Brownian Motion:

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

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

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

Girsanov's theorem states that there exists a measure Q under which \tilde{W}_t is a Brownian motion. \frac{\mu -r}{\sigma} is known as the market price of risk. Differentiating and rearranging gives us:

dW_t = d\tilde{W}_t - \frac{\mu -r}{\sigma}dt.

Putting this back in the original equation gives:

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 Ht at all times t\leq T.

[edit] Notes

  1. ^ In fact the rate of return is equal to the short rate in the measure.
  2. ^ 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.

[edit] See also

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