Risk-neutral measure

Risk-neutral valuation is a financial mathematical method to determine the fair price of derivatives. The idea of ​​risk-neutral valuation, which was developed in 1976 by John Cox and Stephen Ross is that must be the same under certain conditions the value of a derivative in the real world, not behave risk-neutral in the users to the value of the same derivative in a hypothetical risk-neutral world. This relationship is useful because can rate derivatives under simple risk-neutral assumptions.

Conditions

In order for a risk- neutral valuation is possible, the following conditions must be met:

• It is believed Full capital market. This has the consequence that derivatives can replicate through other financial instruments.
• In addition, there must be no arbitrage opportunities give ( no-arbitrage condition).

Risk Neutral Rate

To determine the present value of a derivative in a non- risk-neutral world, it is necessary to discount future cash flows at an interest rate that differs from the risk-free interest rate because it includes a risk premium. This is problematic because the correct risk premium, on which depends the fair price, is in practice often difficult to determine. In a risk-neutral world, however, any future cash flows are discounted by the risk free rate.

This property is exploited to calculate the expected value of derivatives under risk-neutral assumptions and then to discount it at the risk- free rate on the present value. In this way, the fair price is obtained for the derivative, which must apply equally in non-risk neutral worlds.

Motivation

Economically, can the validity of the risk- neutral valuation justified by the fact that it is possible under the assumption of a perfect capital market, for the derivative to be evaluated to construct a dynamic hedge transaction in which the risk is completely eliminated. If the fair price of a derivative would depend in this case of risk premiums, arbitrage opportunities could construct as a hedger could pocket the premium without having to be exposed to risk. In other words, risk-neutral evaluation of derivatives is possible due to the perfect correlation between the temporal development of the underlying and the derivative value.

In contrast, the fair price of non- derivatives, such as the value of the underlying itself, depending on the risk appetite of market participants and therefore not assessable risk neutral. Even if the Underlying is not traded directly, as is the case for example for the short-rate models of the term structure, a hedge is not feasible, so the price of appropriately weighted interest rate derivatives depends on the market price of risk and non-risk- neutral is measurable.

Example

To illustrate the following, highly simplified model of a financial market to be considered: There exists only a single share and there are only two trade time points (so-called one-period with a security). The current share price is (all amounts in EUR ) with known. For the future course of the time, assume that the stock either their value doubled or halved. The value of time is thus viewed as a random variable, but the probability of a rising share price is unknown or irrelevant. To further simplify a 0% interest rate would be adopted, that is, it should be possible in particular to take free credit.

The risk-neutral probability measure is determined in this model is that the expected value of the future stock price with respect to this measure is equal to the current exchange rate:

The probability of a rising rate referred to under the risk-neutral measure. (For a non-zero interest rate, the price would have to be additionally discounted. ) With the above numerical values ​​results in the equation, ie

Uniquely determined as risk-neutral probability of a rise in share price.

There will now be another securities introduced in this market: a call option with strike price on the stock as the base value. The payoff of such an option at the moment is calculated as, that is, the buyer of the call receives € if the stock rises, but euro if the stock falls. According to the risk-neutral valuation of the fair price of the option is given by the expectation value of their ( discounted ) payoff with respect to the risk-neutral probability measure, ie by

The fair price of the call option on the stock is thus EUR.

That this is in fact the fair price, also shows the consideration of the following hedge transaction for which an investment of 1 € is also necessary. It is in addition to a loan of a further euro and buys with the 2 euro a half share. If the price goes up, you get 4 euros and with decreasing rate 1 Euro, but must in any case even 1 euro loan to repay ( interest rate 0 %). This strategy thus results in both cases the same payoff of 3 Euros or 0 Euros like buying a call option.

Applications

An important model that can be derived with the principle of risk- neutral valuation, the Black- Scholes model for options. Here, the temporal evolution of the underlying asset is represented as a geometric Brownian motion, ie its logarithm is a Wiener process with drift. Risk neutral then has the consequence that the fair price of an option, regardless of this drift to the Underlying is.

A discrete-time model using risk-neutral valuation of derivatives, the binomial model of Cox, Ross and Rubinstein. This is - as a generalization of the above example - adopted in each time step, that there are only two possible developments of the underlying. The probabilities of the two cases are then chosen so that at any time the expected value of the discounted future prices is equal to the current price.