Risk-neutral measure

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In mathematical finance, a risk-neutral measure, (also called an equilibrium measure, or equivalent martingale measure), is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure.^[1] Such a measure exists if and only if the market is arbitrage-free.

Motivating the use of risk-neutral measures

Prices of assets depend crucially on their risk as investors typically demand more profit for bearing more uncertainty. Therefore, today's price of a claim on a risky amount realised tomorrow will generally differ from its expected value. Most commonly, investors are risk-averse and today's price is below the expectation, remunerating those who bear the risk (at least in large financial markets; examples of risk-seeking markets are casinos and lotteries).

To price assets, consequently, the calculated expected values need to be adjusted for an investor's risk preferences (see also Sharpe ratio). Unfortunately, the discount rates would vary between investors and an individual's risk preference is difficult to quantify.

It turns out that in a complete market with no arbitrage opportunities there is an alternative way to do this calculation: Instead of first taking the expectation and then adjusting for an investor's risk preference, one can adjust, once and for all, the probabilities of future outcomes such that they incorporate all investors' risk premia, and then take the expectation under this new probability distribution, the risk-neutral measure. The main benefit stems from the fact that once the risk-neutral probabilities are found, every asset can be priced by simply taking the present value of its expected payoff. Note that if we used the actual real-world probabilities, every security would require a different adjustment (as they differ in riskiness).

The absence of arbitrage is crucial for the existence of a risk-neutral measure. In fact, by the fundamental theorem of asset pricing, the condition of no-arbitrage is equivalent to the existence of a risk-neutral measure. Completeness of the market is also important because in an incomplete market there are a multitude of possible prices for an asset corresponding to different risk-neutral measures. It is usual to argue that market efficiency implies that there is only one price (the "law of one price"); the correct risk-neutral measure to price which must be selected using economic, rather than purely mathematical, arguments.

A common mistake is to confuse the constructed probability distribution with the real-world probability. They will be different because in the real-world, investors demand risk premia, whereas it can be shown that under the risk-neutral probabilities all assets have the same expected rate of return, the risk-free rate (or short rate) and thus do not incorporate any such premia. The method of risk-neutral pricing should be considered as many other useful computational tools—convenient and powerful, even if seemingly artificial.

Formal Definition

Given a finite set $\{1, 2, \ldots, S\}$ of states, suppose there are $N$ securities, given by an $N \times S$ matrix $D$ . The entry $D_{ij}$ represents the payoff of security i in state j. The prices of these securities are given by $q \in \mathbb{R}^n$ . A risk neutral measure is a probability measure Q on the set of states such that there exists an r (the "discount rate") with

q_i = r \mathbb{E}_{Q}(D_i)

for each i, where $D_i$ is the random variable that represents the payoff of security i.

The fundamental theorem of asset pricing states that such a measure can be proven to exist if and only if there is no arbitrage. The proof uses the separating hyperplane version of the Hahn-Banach theorem.

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 = P(0,T) \operatorname{E}_P\left(\frac{dQ}{dP}H_T\right)

where $\frac{dQ}{dP}$ is the Radon–Nikodym derivative of $Q$ with respect to $P$ .^[2]

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.

Example 1 — Binomial model of stock prices

Given a probability space $(\Omega, \mathfrak{F}, \mathbb{P})$ , 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 $S^d \leq (1+R)S \leq S^u$ (else there is arbitrage in the market), 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}.

^[3]

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. Utilizing rules within Itô calculus, one may informally differentiate with respect to $t$ and rearrange the above expression to derive the SDE

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.

Let $\tilde{S}_t$ be the discounted stock price given by $\tilde{S}_t = e^{-rt} S_t$ , then by Ito's lemma we get the SDE:

d\tilde{S}_t = \sigma \tilde{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 $Q$ . Notice the drift of the SDE is r, the risk-free interest rate, implying risk neutrality. Since $\tilde{S}$ and $H$ are $Q$ -martingales we can invoke the martingale representation theorem to find a replicating strategy - a portfolio of stocks and bonds that pays off $H_t$ at all times $t\leq T$ .

Notes

↑ Glyn A. Holton (2005). "Fundamental Theorem of Asset Pricing". riskglossary.com. Retrieved October 20, 2011.
↑ Hans Föllmer; Alexander Schied (2004). Stochastic Finance: An Introduction in Discrete Time (2 ed.). Walter de Gruyter. p. 6. ISBN 978-3-11-018346-7.
↑ Elliott, Robert James; Kopp, P. E. (2005). Mathematics of financial markets (2 ed.). Springer. pp. 48–50. ISBN 978-0-387-21292-0.

External links

Gisiger, Nicolas: Risk-Neutral Probabilities Explained
Tham, Joseph: Risk-neutral Valuation: A Gentle Introduction, Part II

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