Risk-neutral assessment

Risk-neutral valuation is a mathematical method for determining the fair price of derivatives . The idea of risk-neutral rating, developed by John Cox and Stephen Ross in 1976 , is that, under certain conditions, the value of a derivative in the real world where users are not risk-neutral must be the same as the value of the same derivative in a hypothetical, risk-neutral world. This relationship is useful because derivatives can be valued more easily using risk-neutral assumptions.

conditions

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

A complete capital market is assumed. As a result, derivatives can be replicated using other financial instruments.
In addition, there must be no arbitrage opportunities (no arbitrage ).

Risk-neutral assessment

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

This property is used to calculate the expected value of derivatives under risk-neutral assumptions and then to discount it to today's value using the risk-free interest rate. In this way you get the fair price for the derivative, which also has to apply in non-risk-neutral worlds.

motivation

In economic terms, the validity of the risk-neutral valuation can be justified by the fact that, assuming a complete capital market, it is possible to construct a dynamic hedge for the derivative to be valued , which completely eliminates the risk. If the fair price of a derivative were to depend on risk premiums in this case, then arbitrage opportunities could be constructed, as a hedger could reap the premiums without being exposed to risk. In other words, risk-neutral valuation of derivatives is possible due to the perfect correlation between the development of the underlying asset and the derivative asset over time.

In contrast to this, the fair price of non-derivatives, such as that of the underlying asset itself, depends on the risk affinity of the market participants and therefore cannot be assessed in a risk-neutral way. Even if the underlying asset is not traded directly, as is the case, for example, in instantaneous interest rate models for the interest rate structure , a hedge cannot be carried out, so that the price of appropriately valued interest rate derivatives depends on the market price of the risk and cannot be valued in a risk-neutral manner.

example

To illustrate this, consider the following, greatly simplified model of a financial market: There is only one share and there are only two trading times and (so-called one-period model with one security). The current share price is also known (all information in euros). For the future price at the time , it is assumed that the share either doubles or halves its value . The current value is therefore viewed as a random variable , although the probability of a rising share price is unknown or does not play a role. For further simplification, assume an interest rate of 0%, i. H. In particular, it should be possible to take out free loans. ${\ displaystyle t = 0}$ ${\ displaystyle t = 1}$ ${\ displaystyle S_ {0} = 4}$ ${\ displaystyle t = 1}$ ${\ displaystyle s_ {u} = 8}$ ${\ displaystyle s_ {d} = 2}$ ${\ displaystyle t = 1}$ ${\ displaystyle S_ {1}}$ ${\ displaystyle p = P (S_ {1} = s_ {u})}$

The risk-neutral probability measure in this model is determined by the fact that the expected value of the future share price with regard to this measure is equal to the current price : ${\ displaystyle P ^ {*}}$ ${\ displaystyle \ operatorname {E} ^ {*} (S_ {1})}$ ${\ displaystyle S_ {0}}$

{\ displaystyle \ operatorname {E} ^ {*} (S_ {1}) = s_ {u} \ cdot p ^ {*} + s_ {d} \ cdot (1-p ^ {*}) = S_ {0 }}

,

where the probability of a rising price denotes below the risk-neutral level. (If the interest rate is different from zero, the rate would also have to be discounted.) The above numerical values result in the equation , that is ${\ displaystyle p ^ {*} = P ^ {*} (S_ {1} = s_ {u}) \ in (0,1)}$ ${\ displaystyle S_ {1}}$ ${\ displaystyle 8p ^ {*} + 2 (1-p ^ {*}) = 4}$

{\ displaystyle p ^ {*} = {\ tfrac {1} {3}}}

as a clearly determined risk-neutral probability of a rising share price.

Another security is now being introduced into this market: a call option with an exercise price on the share as the underlying. The payout of such an option at the time is calculated as , i. H. the buyer of the call receives euros when the stock rises, but euros when the stock falls. According to the risk-neutral valuation, the fair price of the option is given by the expected value of its (discounted) payout in terms of the risk-neutral probability measure, i.e. by ${\ displaystyle K = 5}$ ${\ displaystyle t = 1}$ ${\ displaystyle C = \ max (S_ {1} -K, 0)}$ ${\ displaystyle s_ {u} -K = 8-5 = 3}$ ${\ displaystyle 0}$

{\ displaystyle \ operatorname {E} ^ {*} (C) = (s_ {u} -K) \ cdot p ^ {*} + 0 \ cdot (1-p ^ {*}) = {\ frac {1 } {3}} \ cdot 3 = 1}

.

The fair price of the call option on the share is therefore euros. ${\ displaystyle 1}$

The fact that this is actually a fair price is also shown by looking at the following hedge, for which an investment of 1 euro is also required. You also take out a loan of another euro and buy half a share with the 2 euro. If the rate rises, you get 4 euros and if the rate falls, you get 1 euro, but in any case you have to repay 1 euro credit (interest rate 0%). With this strategy, the same payout of 3 euros or 0 euros results in both cases as when buying a call option.

Applications

An important model that can be derived using the risk-neutral valuation principle is the Black-Scholes model for options. Here the development of the underlying asset over time is shown as a geometric Brownian movement , i.e. H. its logarithm is a Wiener process with drift . The risk-neutral valuation then means that the fair price of an option on the underlying is independent of this drift.

A time-discrete model that uses risk-neutral valuation of derivatives is the binomial model by Cox, Ross and Rubinstein. As a generalization of the above example, it is assumed in each time step that there are only two possible developments for the base value. The probabilities of the two cases are then chosen so that the expected value of the discounted future prices is equal to the current price at any point in time.

literature

John C. Cox, Stephen A. Ross: The valuation of options for alternative stochastic processes . Journal of Financial Economics 3 (1976) 145-166.
Paul Wilmott: Paul Wilmott on quantitative finance , Wiley, 2nd edition (2006)
Martin Baxter, Andrew Rennie: Financial Calculus: An Introduction to Derivative Pricing . Cambridge University Press (1996)