It is natural to ask how a risk-neutral measure arises in a market free of arbitrage.

Risk-neutral measures make it easy to express the value of a derivative in a formula.

Another name for the risk-neutral measure is the equivalent martingale measure.

This technique is also known as the certainty-equivalent or martingale approach, and uses a risk-neutral measure.

The lack of arbitrage is crucial for existence of a risk-neutral measure.

For example, the discounted stock price is a martingale under the risk-neutral measure:

The origin of the risk-neutral measure (Arrow securities)

This is the risk-neutral measure!

In incomplete markets, this is one way of choosing a risk-neutral measure (from the infinite number available) so as to still maintain the no-arbitrage conditions.

The percent moneyness is the implied probability that the derivative will expire in the money, in the risk-neutral measure.

The interpretation of these quantities is somewhat subtle, and consists of changing to a risk-neutral measure with specific choice of numéraire.

The main state variable of the model is the short rate, which is assumed to follow the stochastic differential equation (under the risk-neutral measure):

The fundamental theorem of finance states that the price of assembling such a portfolio will be equal to its expected value under the appropriate risk-neutral 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.

A fundamental concept in derivatives pricing is that of the Risk-neutral measure; this is explained in further depth in the above article.

In fact, by the fundamental theorem of asset pricing, the condition of no-arbitrage is equivalent to the existence of a risk-neutral measure.

In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the 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.

To price a derivative whose payoff is a function of one or more underlying assets, we evaluate the expected value of its discounted payoff under a risk-neutral measure.

Note the expectation of the option payoff is not done under the real world probability measure, but an artificial risk-neutral measure, which differs from the real world measure.

In the language of stochastic processes, the forward price is a martingale under the forward measure, whereas the futures price is a martingale under the risk-neutral measure.

If one uses spot S instead of forward F, in instead of the term there is which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire).

Note that above, the risk neutral formula does not refer to the volatility of the underlying - p as solved, relates to the risk-neutral measure as opposed to the actual probability distribution of prices.

The fundamental theorem of arbitrage-free pricing states that the value of a derivative is equal to the discounted expected value of the derivative payoff where the expectation is taken under the risk-neutral measure .

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.