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Probability Measure

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Risk-neutral measure and physical measure

In this section, we talk informally about the probability measures used in pricing and hedging financial derivatives.

Risk-neutral measure is a probability measure, it is very important in the world of mathmatical finance. Most commonly, it is used in the valuation of financial derivatives. Under the risk-neutral measure, the future expected value of the financial derivatives is discounted at the risk-free rate.

Physical measure is also a probability measure, it is also called actual measure. Physical measure is used in computations in the actual world, as its name suggested. The most common applications are seen in statistical estimations from historical data and the hedging of portfolios.

The difference between Risk-neutral and Physical measure is, among others, the following

  • Risk-neutral measure is used for option pricing and portfolio replication purposes, wheras Physical measure is used for hedging
  • The expected return in the risk-neural world is the riskless interest rate

However, both risk-neutral measure and the physical measure must agree on the null space, i.e. they have the same probability on the impossibility of events.

To go from the physical measure to the risk-neutral one, we have to subtract an amount of quantity of risk, commonly called the market price of risk.

Mathematical exposition

In mathematical details, consider a probability triple $ (\Omega, A, Q) $, where $ \Omega $ is the sample space, $ A $ is the sets of events, $ Q $ is the probability measure defined on $ A $. There are other probabability measure on the sample space $ \Omega $ as well, such as $ (\Omega, A, P) $. If we denote $ Q $ as the risk-neutral measure and $ P $ as the physical measure. the expectation of a random variable $ X $ that is $ L^1 (\Omega, A) $ under the risk-neutral measure $ Q $ is

$ E^Q(X)=\int_{-\infty} ^ {+\infty} X dQ. \qquad (1)\! $

The expectation of $ X $ under the physical measure $ P $ is

$ E^P(X)=\int_{-\infty} ^ {+\infty} X dP. \qquad (2)\! $

The transformation from $ Q $ to $ P $ is via the density funtion called Radon-Nikodym derivative $ f $. Using equations (1) and (2), we have

$ E^Q(X)=\int_{-\infty} ^ {+\infty} X dQ =\int_{-\infty} ^ {+\infty} X \frac{dQ}{dP}\,dP = \int_{-\infty} ^ {+\infty} Xf dP=E^P(fX). \qquad (3)\! $

In the above formulation, the function $ f=\frac{dQ}{dP}\, $ summarizes the relation between the risk-neutral measure $ Q $ and the physical measure $ P $.