11.5 Change of measure

Change of measure is a crucial concept in stochastic processes, allowing us to transform probability measures while preserving key properties. It simplifies complex problems by switching to more convenient measures, making calculations easier to handle.

In finance, change of measure is vital for risk-neutral pricing and the fundamental theorems of asset pricing. It enables us to switch between real-world and risk-neutral measures, facilitating the valuation of financial derivatives and understanding market dynamics.

Importance of change of measure

• Change of measure is a fundamental concept in stochastic processes that allows for the transformation of probability measures while preserving key properties of the underlying process
• Enables the simplification of complex problems by changing the probability measure to a more convenient one, often making calculations more tractable
• Plays a crucial role in various applications, such as financial mathematics, where it is used to switch between real-world and risk-neutral measures for pricing and hedging purposes

Radon-Nikodym theorem

Equivalent probability measures

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• Two probability measures $P$ and $Q$ on a measurable space $(\Omega, \mathcal{F})$ are said to be equivalent if they assign zero probability to the same events
• Formally, $P$ and $Q$ are equivalent if $P(A) = 0 \Leftrightarrow Q(A) = 0$ for all $A \in \mathcal{F}$
• Equivalent measures share the same null sets and have the same support

Absolute continuity

• A probability measure $Q$ is said to be absolutely continuous with respect to another probability measure $P$ if $P(A) = 0 \Rightarrow Q(A) = 0$ for all $A \in \mathcal{F}$
• Intuitively, if an event is impossible under $P$, it must also be impossible under $Q$
• Absolute continuity is a weaker condition than equivalence, as it allows for $Q$ to assign zero probability to events that have positive probability under $P$

Radon-Nikodym derivative

• The Radon-Nikodym theorem states that if $Q$ is absolutely continuous with respect to $P$, then there exists a non-negative $\mathcal{F}$-measurable function $\frac{dQ}{dP}$, called the Radon-Nikodym derivative, such that $Q(A) = \int_A \frac{dQ}{dP} dP$ for all $A \in \mathcal{F}$
• The Radon-Nikodym derivative acts as a density function that relates the two probability measures
• Can be interpreted as the ratio of the likelihood of an event occurring under $Q$ to the likelihood of the same event occurring under $P$

Girsanov theorem

Brownian motion under change of measure

• Girsanov theorem provides a method for changing the probability measure of a Brownian motion process
• Under a new probability measure $Q$, a Brownian motion $W_t$ can be transformed into another Brownian motion $\tilde{W}_t$ with a different drift term
• The change of measure is achieved by multiplying the original probability measure by a specific Radon-Nikodym derivative

Drift transformation

• The drift of the Brownian motion is modified under the new probability measure $Q$
• If the original Brownian motion $W_t$ has drift $\mu$ under measure $P$, then under $Q$, the new Brownian motion $\tilde{W}_t$ has drift $\mu + \frac{d\langle \lambda, W \rangle_t}{dt}$, where $\lambda_t$ is a suitable process
• The change in drift is determined by the Radon-Nikodym derivative process $\lambda_t$

Martingale property preservation

• Girsanov theorem ensures that the martingale property is preserved under the change of measure
• If a process is a martingale under the original measure $P$, it will remain a martingale under the new measure $Q$ after adjusting for the change in drift
• This property is crucial for applications in finance, such as risk-neutral pricing, where martingale methods are employed

Applications in finance

Risk-neutral pricing

• Change of measure is extensively used in the risk-neutral pricing framework
• Under the risk-neutral measure $Q$, the discounted price process of an asset is a martingale
• Enables the valuation of financial derivatives by taking the expectation of the discounted payoff under the risk-neutral measure

Fundamental theorems of asset pricing

• The First Fundamental Theorem of Asset Pricing (FFTAP) states that a market is arbitrage-free if and only if there exists a risk-neutral probability measure $Q$ equivalent to the real-world measure $P$
• The Second Fundamental Theorem of Asset Pricing (SFTAP) states that a market is complete if and only if the risk-neutral measure $Q$ is unique
• Change of measure plays a central role in establishing these fundamental theorems

Martingale measures vs real-world measures

• Martingale measures, such as the risk-neutral measure, are used for pricing and hedging purposes in finance
• Real-world measures, also known as physical measures, describe the actual probabilities of events occurring in the market
• Change of measure techniques allow for the transition between these two types of measures, enabling the use of martingale methods while still incorporating real-world information

Esscher transform

Exponential tilting

• The Esscher transform is a specific type of change of measure that involves exponential tilting of the original probability measure
• Given a probability measure $P$ and a suitable random variable $X$, the Esscher transform defines a new probability measure $Q$ such that $\frac{dQ}{dP} = \frac{e^{\theta X}}{E_P[e^{\theta X}]}$, where $\theta$ is a parameter
• Exponential tilting allows for the modification of the distribution while preserving certain properties

Moment generating functions

• The Esscher transform is closely related to moment generating functions (MGFs)
• The MGF of a random variable $X$ under the original measure $P$ is given by $M_X(\theta) = E_P[e^{\theta X}]$
• The Esscher transform can be expressed in terms of the MGF as $\frac{dQ}{dP} = \frac{e^{\theta X}}{M_X(\theta)}$

Semi-martingale processes

• The Esscher transform is particularly useful when working with semi-martingale processes
• Semi-martingales are a class of stochastic processes that can be decomposed into a martingale part and a finite variation part
• The Esscher transform preserves the semi-martingale property, making it a valuable tool for analyzing and manipulating these processes

Change of numéraire

Money market account as numéraire

• A numéraire is a reference asset or portfolio used to express prices and values in a financial market
• The most common numéraire is the money market account, which accrues interest at the risk-free rate
• Changing the numéraire from the money market account to another asset or portfolio can simplify calculations and provide alternative perspectives on pricing and hedging

Forward measures vs spot measures

• Forward measures are probability measures associated with a specific future date, where the numéraire is typically a zero-coupon bond maturing on that date
• Spot measures, such as the risk-neutral measure, use the money market account as the numéraire
• Changing between forward and spot measures allows for the simplification of certain pricing problems and the derivation of alternative valuation formulas

Simplifying option pricing formulas

• Change of numéraire techniques can be used to simplify option pricing formulas
• For example, the Black-Scholes formula for European call options can be derived using a change of numéraire approach
• By changing the numéraire from the money market account to the underlying asset, the pricing problem becomes more tractable, and the resulting formula is more intuitive

Discrete-time change of measure

Binomial model example

• Change of measure techniques can also be applied in discrete-time models, such as the binomial option pricing model
• In the binomial model, the stock price can move up or down by specific factors over discrete time steps
• The risk-neutral probabilities of up and down moves are determined by the change of measure from the real-world probabilities

Equivalent martingale measures

• In discrete-time models, the concept of equivalent martingale measures (EMMs) is analogous to the risk-neutral measure in continuous-time models
• An EMM is a probability measure under which the discounted price process of an asset is a martingale
• The existence of an EMM ensures the absence of arbitrage opportunities in the discrete-time setting

Cox-Ross-Rubinstein formula derivation

• The Cox-Ross-Rubinstein (CRR) formula for pricing European options in the binomial model can be derived using change of measure techniques
• By changing the measure from the real-world probabilities to the risk-neutral probabilities, the option price can be expressed as the discounted expected payoff under the risk-neutral measure
• The risk-neutral probabilities are determined by solving a system of equations that ensures the martingale property of the discounted stock price process