Risk-neutral measure
Encyclopedia
In mathematical finance
, a risk-neutral 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 risk-neutral measure.
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.
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 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 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 lack of arbitrage is crucial for 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 with 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.
. 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, An, 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 An 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 An, which we denote by An(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 An(0) 's satisfy the axioms for a probability distribution. Each is non-negative and their sum is 1. This is the risk-neutral 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 i, its payoff will be Ci. Consider a portfolio P consisting of Ci amount of each Arrow security Ai. In the future, whatever state i occurs, then Ai pays $1 while the other Arrow securities pay $0, so P will pay Ci. 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 short-sold 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 risk-neutral 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 Black-Scholes 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 one-step discrete world.
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.
, consider a single-period binomial model. A probability measure 
is called risk neutral if for all 
.
Suppose we have a two-state economy: the initial stock price
can go either up to 
or down to 
. If the interest rate is 
, and 
(else there is arbitrage
in the market), then the risk-neutral probability of an upward stock movement is given by the number
Given a derivative with payoff
when the stock price moves up and 
when it goes down, we can price the derivative via
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
:
where
is a standard Brownian motion
with respect to the physical measure. If we define
Girsanov's theorem states that there exists a measure
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:
is the unique risk-neutral measure for the model.
The discounted payoff process of a derivative on the stock
is a martingale
under
. Notice the drift of the SDE is r, the risk-free interest rate, implying risk neutrality. Since 
and 
are 
-martingales we can invoke the martingale representation theorem
to find a replicating strategy
- a portfolio of stocks and bonds that pays off
at all times 
.
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
due to the fundamental theorem of asset pricing, which implies that in a complete market
Complete market
In economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs...
a derivative's price is the discounted expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the future payoff under the unique risk-neutral measure.
Motivating the use of risk-neutral measures
Prices of assets depend crucially on their riskRisk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...
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.
To price assets, consequently, the calculated expected values need to be adjusted for an investor's risk preferences (see also Sharpe ratio
Sharpe ratio
The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return per unit of deviation in an investment asset or a trading strategy, typically referred to as risk , named after William Forsyth Sharpe...
). 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
Complete market
In economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs...
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 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 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 lack of arbitrage is crucial for 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
Law of one price
The law of one price is an economic law stated as: "In an efficient market, all identical goods must have only one price."-Intuition:The intuition for this law is that all sellers will flock to the highest prevailing price, and all buyers to the lowest current market price. In an efficient market...
"); the correct risk-neutral 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 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.
The origin of the risk-neutral measure (Arrow securities)
It is natural to ask how a risk-neutral 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 securityArrow Security
An Arrow Security is an instrument with a fixed payout of one unit in a specified state and no payout in other states. It is a type of hypothetical asset used in the Arrow market structure model. In contrast to the Arrow-Debreu market structure model, an Arrow market is a market in which the...
. 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, An, 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 An 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 An, which we denote by An(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 An(0) 's satisfy the axioms for a probability distribution. Each is non-negative and their sum is 1. This is the risk-neutral 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 i, its payoff will be Ci. Consider a portfolio P consisting of Ci amount of each Arrow security Ai. In the future, whatever state i occurs, then Ai pays $1 while the other Arrow securities pay $0, so P will pay Ci. 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 short-sold 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 risk-neutral 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 Black-Scholes model and its generalizations, our Arrow security would be something like a double digital option
Double digital option
A double digital option is a particular variety of option . At maturity, the payoff is 1 if the spot price of the underlying asset is between two numbers, the lower and upper strikes of the option; otherwise, it is 0....
, 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 one-step discrete world.
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 optionCall option
A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...
on a stock
Stock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
) pays
units, where is a random variableRandom variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
on the probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
describing the market. Further suppose that the discount factor from now (time zero) until time
is . Then today's fair value of the derivative iswhere the risk-neutral measure is denoted by
. This can be re-stated in terms of the physical measure P aswhere
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
Fundamental theorem of arbitrage-free pricing
The fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...
. 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
Numéraire
Numéraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
, 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, consider a single-period binomial model. A probability measure is called risk neutral if for all .Suppose we have a two-state economy: the initial stock price
can go either up to or down to . If the interest rate is , and (else there is arbitrageArbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
in the market), then the risk-neutral probability of an upward stock movement is given by the number
Given a derivative with payoff
when the stock price moves up and when it goes down, we can price the derivative viaExample 2 — Brownian motion model of stock prices
Suppose our economy consists of 2 assets, a stockStock
The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
and a risk-free bond
Risk-free bond
A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. In practice, government bonds are treated as risk-free bonds, as governments can raise taxes or indeed print money to repay their domestic...
, and that we use the Black-Scholes model. In the model the evolution of the stock price can be described by Geometric Brownian Motion
Geometric Brownian motion
A geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, also called a Wiener process...
:
where
is a standard Brownian motionBrownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
with respect to the physical measure. If we define
Girsanov's theorem states that there exists a measure
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 lemmaIto's lemma
In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...
we get the SDE:
is the unique risk-neutral measure for the model.The discounted payoff process of a derivative on the stock
is a martingaleMartingale (probability theory)
In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
under
. Notice the drift of the SDE is r, the risk-free interest rate, implying risk neutrality. Since and are -martingales we can invoke the martingale representation theoremMartingale representation theorem
In probability theory, the martingale representation theorem states that a random variable which is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion....
to find a replicating strategy
Replicating strategy
In Finance a Replicating Strategy of a particular financial instrument is a set of liquid, usually exchange-traded assets with the same net payoff....
- a portfolio of stocks and bonds that pays off
at all times .See also
- Mathematical financeMathematical financeMathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
- Forward measureForward measureIn finance, a T-forward measure is a pricing measure absolutely continuous with respect to a risk-neutral measure but rather than using the money market as numeraire, it uses a bond with maturity T...
- Fundamental theorem of arbitrage-free pricingFundamental theorem of arbitrage-free pricingThe fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss...
- Law of one priceLaw of one priceThe law of one price is an economic law stated as: "In an efficient market, all identical goods must have only one price."-Intuition:The intuition for this law is that all sellers will flock to the highest prevailing price, and all buyers to the lowest current market price. In an efficient market...
- Rational pricingRational pricingRational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away"...
- Brownian model of financial marketsBrownian Model of Financial MarketsThe Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and...
- Martingale (probability theory)Martingale (probability theory)In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...
External links
- Gisiger, Nicolas: Risk-Neutral Probabilities Explained
- Tham, Joseph: Risk-neutral Valuation: A Gentle Introduction, Part II