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Risk-neutral measure: Information from Answers.com

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

Idea

In an actual economy, prices of assets depend crucially on their risk. Investors typically demand payment for bearing uncertainty. Therefore, today's price of a claim on a risky amount realised tomorrow will generally differ from its expected value. Most commonly,^[1] investors are risk-averse and today's price is below the expectation, remunerating those who bear the risk.

To price assets, consequently, the calculated expected values need to be adjusted for the risk involved (see also Sharpe ratio).

It turns out, under certain weak conditions (absence of arbitrage) there is an alternative way to do this calculation: Instead of first taking the expectation and then adjusting for risk, one can first adjust the probabilities of future outcomes such that they incorporate the effects of risk, and then take the expectation under those different probabilities. Those adjusted, 'virtual' probabilities are called risk-neutral probabilities, they constitute the risk-neutral measure.

It is important to note that clearly the probabilities over asset outcomes in the real world cannot be impacted; the constructed probabilities are counterfactual. They are only computed because the second way of pricing, called risk-neutral pricing, is often much simpler to calculate than the first.

The main benefit stems from the fact that once the risk-neutral probabilities are found, every asset can be priced by simply taking its expected payoff (i.e. calculating as if investors were risk neutral). If we used the real-world, physical probabilities every security would require a different adjustment (as they differ in riskiness).

Note that under the risk-neutral measure all assets have the same expected rate of return, the risk-free rate (or short rate). This does not imply the assumption that investors were risk neutral. On the contrary, the point is to price given exactly the risk aversion we observe in the physical world. Towards that aim, we hypothesize about parallel universes where everybody is risk neutral. The risk-neutral measure is the probability measure of that parallel universe where all claims have exactly the prices they have in our real world.

Mathematically, adjusting the probabilities is a measure transformation to an equivalent martingale measure; it is possible if there are no arbitrage opportunities. If the markets are complete, the risk-neutral measure is unique.

Often, the physical measure is called P, and the risk-neutral one Q. The term physical measure is often abused to denote the Lebesgue measure, occasionally, the measure induced by the corresponding normal density with respect to the Lebesgue measure.

Usage

Risk-neutral measures make it easy to express the value of a derivative in a formula. Suppose at a future time T a derivative (e.g., a call option on a stock) pays 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. 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.

Example 1 — Binomial model of stock prices

Given a probability space $(\Omega, \mathbb{P},\mathfrak{F})$ , consider a single-period binomial model. A probability measure $\mathbb{P^{*}}$ is called risk neutral if for all $i \in \{0,...,d\}$ $\pi^{i}=\mathbb{E}_{\mathbb{P}^{*}}(S^{i}/(1+r))$ . Suppose 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>0, and S^d < = (1 + R)S < = S^u, then the risk-neutral probability of an upward stock movement is given by the number

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

Given a derivative with payoff X^u when the stock price moves up and X^d when it goes down, we can price the derivative via

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

Example 2 — Brownian motion model of stock prices

Suppose our economy consists of 2 assets, a stock and a risk-free bond, and that we use 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 W_t is a standard Brownian motion with respect to the physical measure. If we define

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

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 yields:

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

Put this back in the original equation:

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

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 $t\leq T$ .

Notes

^ At least in large financial markets. Example of risk-seeking markets are casinos and lotteries.

External links

Aslanidi, Konstantin: Basics of derivative pricing
Gisiger, Nicolas: Risk-Neutral Probabilities Explained

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