In fact, by the fundamental theorem of asset pricing, the condition of no-arbitrage is equivalent to the existence of a risk-neutral measure. The absence of arbitrage is crucial for the existence of a risk-neutral measure. Note that if we used the actual real-world probabilities, every security would require a different adjustment (as they differ in riskiness). 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.

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. Unfortunately, the discount rates would vary between investors and an individual's risk preference is difficult to quantify. To price assets, consequently, the calculated expected values need to be adjusted for an investor's risk preferences (see also Sharpe ratio). 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). Therefore, today's price of a claim on a risky amount realised tomorrow will generally differ from its expected value. Prices of assets depend crucially on their risk as investors typically demand more profit for bearing more risk. Motivating the use of risk-neutral measures This is not strictly necessary to make use of these techniques. It is also worth noting that in most introductory applications in finance, the pay-offs under consideration are deterministic given knowledge of prices at some terminal or future point in time. The concept of a unique risk-neutral measure is most useful when one imagines making prices across a number of derivatives that would make a unique risk-neutral measure, since it implies a kind of consistency in one's hypothetical untraded prices, and theoretically points to arbitrage opportunities in markets where bid/ask prices are visible. This means that you try to find the risk-neutral measure by solving the equation where current prices are the expected present value of the future pay-offs under the risk-neutral measure. It is the implied probability measure (solves a kind of inverse problem) that is defined using a linear (risk-neutral) utility in the payoff, assuming some known model for the payoff.Relevant means those instruments that are causally linked to the events in the probability space under consideration (i.e. An implied probability measure, that is one implied from the current observable/posted/traded prices of the relevant instruments.The risk-neutral measure would be the measure corresponding to an expectation of the payoff with a linear utility. Typically this transformation is the utility function of the payoff. The probability measure of a transformed random variable.The easiest way to remember what the risk-neutral measure is, or to explain it to a probability generalist who might not know much about finance, is to realize that it is:

Such a measure exists if and only if the market is arbitrage-free.Ī risk-neutral measure is a probability 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. 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. ( October 2011) ( Learn how and when to remove this template message) Please help to improve this article by introducing more precise citations. This article includes a list of general references, but it lacks sufficient corresponding inline citations.

Risk neutral probability method