 Riskneutral measure

In mathematical finance, a riskneutral measure, is a prototypical case of an equivalent martingale measure. It 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 riskneutral measure.^{[1]}
Contents
Motivating the use of riskneutral 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,^{[2]} investors are riskaverse 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 an investor's risk preferences (see also Sharpe ratio). Unfortunately, the discounted 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 investor's risk premia, and then take the expectation under this new probability distribution, the riskneutral measure. The main benefit stems from the fact that once the riskneutral probabilities are found, every asset can be priced by simply taking its expected payoff. Note that if we used the actual realworld probabilities, every security would require a different adjustment (as they differ in riskiness).
The lack of arbitrage is crucial for existence of a riskneutral measure. In fact, by the fundamental theorem of asset pricing, the condition of noarbitrage is equivalent to the existence of a riskneutral 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 riskneutral measures. It is usual to argue that market efficiency implies that there is only one price (the "law of one price"); the correct riskneutral measure to price with must be selected using economic, rather than purely mathematical, arguments.
A common mistake is to confuse the constructed probability distribution with the realworld probability. They will be different because in the realworld, investors demand risk premia, whereas it can be shown that under the riskneutral probabilities all assets have the same expected rate of return, the riskfree rate (or short rate) and thus do not incorporate any such premia. The method of riskneutral pricing should be considered as many other useful computational tools  convenient and powerful, even if seemingly artificial.
The origin of the riskneutral measure (Arrow securities)
It is natural to ask how a riskneutral measure arises in a market free of arbitrage. Somehow the prices of all assets will determine a probability measure. One explanation is given by utilizing the Arrow security. For simplicity, consider a discrete world with only one future time horizon. In other words, there is the present (time 0) and the future (time 1), and at time 1 the state of the world can be one of finitely many states. An Arrow security corresponding to state n, A_{n}, is one which pays $1 at time 1 in state n and $0 in any of the other states of the world.
What is the price of A_{n} now? It must be positive as there is a chance you will gain $1; it should be less than $1 as that is the maximum expected payoff. Thus the price of each A_{n}, which we denote by A_{n}(0), is strictly between 0 and 1.
Actually, the sum of all the security prices must be equal to the present value of $1, because holding a portfolio consisting of each Arrow security will result in certain payoff of $1. For simplicity, we will consider the interest rate to be 0, so that the present value of $1 is $1.
Thus the A_{n}(0) 's satisfy the axioms for a probability distribution. Each is nonnegative and their sum is 1. This is the riskneutral measure! Now it remains to show that it works as advertised, i.e. taking expected values with respect to this probability measure will give the right price at time 0.
Suppose you have a security C whose price at time 0 is C(0). In the future, in a state n, its payoff will be C_{n}. Consider a portfolio P consisting of C_{n} amount of the Arrow security A_{n}. In the future, whatever state i occurs, then A_{i} pays $1 while the other Arrow securities pay $0, so P will pay C_{i}. In other words, the portfolio P replicates the payoff of C regardless of what happens in the future. The lack of arbitrage opportunities implies that the price of P and C must be the same now, as any difference in price means we can, without any risk, (short) sell the more expensive, buy the cheaper, and pocket the difference. In the future we will need to return the shortsold asset but we can fund that exactly by selling our bought asset, leaving us with our initial profit.
But regarding each Arrow security price as a probability, we see that the portfolio price P(0) is the expected value of C under the riskneutral probabilities. If the interest rate were positive, we would need to discount the expected value appropriately to get the price.
Note that Arrow securities do not actually need to be traded in the market. This is where market completeness comes in. In a complete market, every Arrow security can be replicated using a portfolio of real, traded assets. The argument above still works considering each Arrow security as a portfolio.
In a more realistic model, such as the BlackScholes model and its generalizations, our Arrow security would be something like a double digital option, which pays off $1 when the underlying asset lies between a lower and an upper bound, and $0 otherwise. The price of such an option then reflects the market's view of the likelihood of the spot price ending up in that price interval, adjusted by risk premia, entirely analogous to how we obtained the probabilities above for the onestep discrete world.
Usage
Riskneutral 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
where the riskneutral measure is denoted by Q. This can be restated in terms of the physical measure P as
where is the Radon–Nikodym derivative of Q with respect to P.
Another name for the riskneutral measure is the equivalent martingale measure. If in a financial market there is just one riskneutral measure, then there is a unique arbitragefree price for each asset in the market. This is the fundamental theorem of arbitragefree 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 1to1 relation between a consistent pricing process and an equivalent martingale measure.
Example 1 — Binomial model of stock prices
Given a probability space , consider a singleperiod binomial model. A probability measure is called risk neutral if for all . Suppose we have a twostate 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 (else there is arbitrage in the market), then the riskneutral probability of an upward stock movement is given by the number
 ^{[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
Example 2 — Brownian motion model of stock prices
Suppose our economy consists of 2 assets, a stock and a riskfree bond, and that we use the BlackScholes model. In the model the evolution of the stock price can be described by Geometric Brownian Motion:
where W_{t} is a standard Brownian motion with respect to the physical measure. If we define
Girsanov's theorem states that there exists a measure Q under which is a Brownian motion. is known as the market price of risk. Differentiating and rearranging yields:
Put this back in the original equation:
Let be the discounted stock price given by , then by Ito's lemma we get the SDE:
Q is the unique riskneutral measure for the model. The discounted payoff process of a derivative on the stock is a martingale under Q. Notice the drift of the SDE is r, the riskfree interest rate, implying risk neutrality. Since and H are Qmartingales 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 .
Notes
 ^ Glyn A. Holton (2005). "Fundamental Theorem of Asset Pricing". riskglossary.com. http://www.riskglossary.com/link/EMM.htm. Retrieved October 20, 2011.
 ^ At least in large financial markets. Example of riskseeking markets are casinos and lotteries.
 ^ Elliott, Robert James; Kopp, P. E. (2005). Mathematics of financial markets (2 ed.). Springer. pp. 4850. ISBN 9780387212920.
See also
 Mathematical finance
 Forward measure
 Fundamental theorem of arbitragefree pricing
 Law of one price
 Rational pricing
 Brownian model of financial markets
 Martingale (probability theory)
External links
 Gisiger, Nicolas: RiskNeutral Probabilities Explained
 Tham, Joseph: Riskneutral Valuation: A Gentle Introduction, Part II
Categories: Derivatives (finance)
 Mathematical finance
 Probability theory
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