Equivalent Martingale Measures and the Girsanov TheoremJack Chua March 10, 2009

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Introduction

This is part of the second and third lecture in quantitative nance for the University of Chicago's Blue Chips Investments Club. In the last lecture, we went over some basic building blocks in stochastic calculus, culminating in the derivation of the Black-Scholes-Merton partial dierential equation. Today, we will use these tools to introduce the basics of martingale theory, with particular reference to probability theory, equivalent martingale measures and the Girsanov change of measure theorem.2 Probability Spaces

A probability space is a triplet (, F, P ) dened by a state space , sigma algebra F , and probability measure P . A probability space is said to be equipped with a ltration of the sigma algebra that is augmented as time goes by, since more information is obtained about the past. A state space is the set of all possible states. A sigma algebra F is the event space, i.e. the set of measurable or observable events which are subsets of . A probability measure P is a measure that assigns a probability to any measurable event A F .

Denition. A measure P is said to be a probability measure if all of the following are true:1. P () = 1 2. P ( i=1

Ai ) =

i=1

P (Aj ) for Ai pairwise disjoint F of subsets of is a sigma algebra if all of the

3. 0 P (Ai ) 1 i.

Denition. A collectionfollowing are true: 1. F 2. A F = Ac F 3. {Ai }iJ F =

iJ

Ai F .

likely possible states, i.e. = {Suu , Sud , Sdu , Sdd }. We can dene the ltration for each time step:F2 F1 F0 = 2 = {, , {Suu , Sud }, {Sdu , Sdd }} = {, }

Example. Consider a two-period binomial asset pricing model with four equally

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where is the null set and Ft is the ltration at time t representing the information that is available after t periods. For example, after 0 periods, the ltration equals either the null set or the entire sample space - anything can happen or nothing will happen. After 1 period, the ltration is ner but yet complete, since not every subset of is measurable by P; the probability measure is P ({Suu , Sud }) = P ({Sdu , Sdd }) = 1 . After 2 periods, the ltration is 2 the power set generated by . This represents complete information in our twoperiod model, as every possible subset of is now measurable by P. Note that F can be viewed as a family of nested information sets, i.e.F0 F1 Fi1 Fi

which models information ow as time goes on. Often, = (0, ). In this case, F is the Borel sigma algebra, the smallest sigma algebra that contains all open intervals, and P might be the lognormal distribution.3 Continuous-Time Martingales

Note. In the nancial markets, it is never always as simple as two periods.

Denition. Let

St be a random price process during a nite time interval t [0, T ]. St is said to be adapted to the ltration if St is a Ft -measurable function t T. It is also said to be a non-anticipating process, or one that

cannot see into the future. P if t > 0,

Denition. A process St is a martingale w.r.t.1. St is It -adapted 2. E|St | < 3. Et [ST ] = St t < T with probability 1.

Ft and probability measure

Property 3 is of vast importance. It means that the best forecast of an unobserved future value of a martingale is its last observation. This will turn out to be monumentally useful if we can convert a nancial asset into a martingale. There are two general steps: Find a probability distribution P s.t. bond or stock prices discounted by

the risk-free rate become martingales. This can be done with Doob-Meyer decomposition and similar detrending techniques.P Et [exp(ru)St+u ] > St

Transform the probability distribution. For example, if one had

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we can try to nd an equivalent probability P s.t.P Et [exp(ru)St+u ] = St

and thus we have a martingale. These are called equivalent martingale measures and can be done by use of the Girsanov theorem.4 Girsanov Theorem

Before stating the theorem, some additional background needs to be provided. We can summarize two methods for changing the mean of a random variable. 1. Subtraction. For example, given a random variable Z N (, 1), we can deneZ= Z N (0, 1) 1

a transformed variable with zero mean. 2. Using an equivalent measure. Given a random variable Z with probability measure P , Z P = N (, 1), we obtain a new probability via the RadonNikodym derivative (Z) and obtain a new probability P s.t. Z P = N (0, 1). The Girsanov Theorem attempts to do both by dening the conditions under which the Radon-Nikodym derivative exists. It then constructs a new probability distribution and a new transformed variable that eliminates the stochastic drift term (insofar we are dealing with an Ito process).

Theorem (Girsanov). Let Wt be a Wiener process on the probability space {, F, P }. Let Xt be a measurable process adapted to the ltration of the Wiener process {FtW }. Given the adapted process St dene the stochastic exponential of X w.r.t W t t 1 2 t = E(X)t = exp Xu dW u Xu du 20 0

Under certain conditions, t is a martingale. Then a probability measure P can be dened on {, F} s.t. we have the Radon-Nikodym derivativedP = t dP

then Wt = Wt 0t Xu du is a Wiener process w.r.t. Ft and the probability measure PT given by PT (A) = E P [1A T ], with A being an event determined by FT and 1A the indicator function for the event. 4

Before we start with the proof, we need to be more specic concerning the construction of P using the Radon-Nikodym derivative. Here are a few useful results.

Theorem (Novikov). If the conditionE e21

T0

|Xu |2 du