Change of Measure and Girsanov Theorem for Brownian motion. . tinuous time, discuss the Black-Scholes model from a probabilistic perspective and. This section discusses risk-neutral pricing in the continuous-time setting, from stochastic calculus, especially the martingale representation theorem and Girsanov’s i.e. the SDE for σ makes use of another, independent Brownian ( My Derivative Securities notes demonstrated this “by example,” but see. Quadratic variation of continuous martingales 7 The Girsanov Theorem. Probabilistic solution of the Black- Scholes PDE. .. Let Wt be a Brownian motion process and let T be a fixed time. Note that the r.v. ΔWi are independent with EΔWi = 0, EΔW2 i = Δti.
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Girsanov transformations describe how Brownian motion and, more generally, local hirsanov behave under changes of the underlying probability measure. Let us start with a much simpler identity applying to normal random variables.
Suppose that X and are jointly normal random variables defined on a probability space. Ntes is a positive random variable with expectation 1, and a new measure can be defined by for all sets. Writing for expectation under the new measure, then for all bounded random variables Z.
The expectation of a bounded measurable function of Y under the new measure is. So, Y has the same distribution under as has under. That is, when changing to the new measure, Y remains jointly normal with the same covariance matrix, but its mean increases by. Equation 1 follows from a straightforward calculation of the characteristic function of Y with respect to both and. Now consider a standard Brownian motion B and fix a time and a constant.
Then, for all timesthe covariance of and is. Applying 1 to the measure shows that. Under the new measure, B has gained a constant drift of over the interval. Such transformations are widely applied in finance. For example, in the Black-Scholes model of option pricing it is common to work under a risk-neutral measure, which transforms the drift of a financial asset to be the risk-free rate of return.
Girsanov transformations extend this idea to much more general changes of measure, and to arbitrary local martingales. However, as shown belowthe strongest results are obtained for Brownian motion which, under a change of measure, just gains a stochastic drift term. As always, we work under a complete filtered probability space. Consider a new measurefor some strictly positive random variable U with expectation 1. Then and are equivalent.
That is, if and only if for sets. By the Radon-Nikodym theoremall probability measures on equivalent to can be defined in this way, and U is referred to as the Radon-Nikodym derivative of with respect todenoted by.
Conditional expectations with respect to the new measure are related to the original one as follows. Lemma 1 Let be an equivalent measure to. Then, for any bounded random variable Z and sigma-algebrathe conditional expectation is given by. Denote the right-hand-side of 2 by Ywhich is -measurable and satisfies.
So, for any. Given a measure equivalent todefine the martingale. Note that there is symmetry here in exchanging the roles of and. Using Lemma 1 with the simple identity. In particular, is a uniformly integrable martingale with respect to so, if a cadlag version of U is used, then will be a cadlag martingale converging to the limit and is finite.
We can now answer the following question — when is a process X a martingale under the equivalent measure? Lemma 2 Let be an equivalent measure toand be as in 3.
Then, a process X is a -martingale if and only if UX is a -martingale. Then, X is adapted if and only if M is adapted. Also,so X is integrable under if and only if M is integrable under. Using 2 for the conditional expectation. So, if and only if. Lemma 2 can be localized to obtain a condition for a cadlag adapted process to be a local martingale with respect to. First, as the process U defined by 3 is a martingale, it has a cadlag modification whenever it is right-continuous in probability.
In particular, if the filtration is right-continuous then U always has a cadlag modification. Lemma 3 Let be an equivalent measure toand suppose that U given by 3 is cadlag. Then, a cadlag adapted process X is a -local martingale if and only if UX is a -local martingale.
Replacing X by if necessary, suppose that. Given a stopping timewe first show that the stopped process is a martingale if and only if is a martingale which, by Lemma 2is equivalent to being a -martingale.
As U is a nonnegative martingale, optional sampling gives and.
Questions tagged [girsanov]
So is integrable if and only if is. In this case, let M be the difference. Therefore, M is a martingale, and is a martingale if and only if is. So, given stopping timesare -martingales if and only if and, therefore, are -martingales.
Girsanov Transformations | Almost Sure
If X is a local martingale, then Lemma 3 can be used to derive a decomposition of X into the sum of a -local martingale and an FV process defined in terms of the quadratic covariation [ UX ]. Theorem 4 Let be an equivalent measure toand suppose that U given by 3 is cadlag. It is just necessary to show that is a local martingale under. Applying integration by parts. Also, UX -[ UX ] is a local martingale, so. For a local martingale M this is the solution to the stochastic differential equation with initial condition so, by preservation of the local martingale propertyU is a local martingale.
In particular, if the quadratic variation at infinity,is finite then the limit exists and will be strictly positive. If, furthermore, U is a uniformly integrable martingale rather than just a local martingale then, for each time t.
So defines an equivalent measure with U satisfying equation 3. Also, contknuous any -local martingale X, and Theorem 4 shows that is a -local martingale. Applying this with gives the following. Theorem 5 Girsanov transformation Let X be a continuous local martingale, and be a predictable process such that.
If is a uniformly integrable martingale then and the measure is equivalent to. Then, X decomposes as. So, Girsanov transformations allow us to change to an equivalent measure where the local martingale X gains a drift term which is an integral with respect notws its quadratic variation [ X ]. In fact, continuous local martingales always decompose as in 4 under any continuous change of measure.
In the following, it is required that we take a cadlag version of the martingale Uwhich is guaranteed to exist if the filtration is right-continous. However, in these notes we are not assuming that filtrations are right-continuous.
Still, it is always possible to pass to the right-continuous filtration. Then, a continuous process starting at zero will be -adapted if and only if it is -adapted, and the two filtrations define the same space of continuous local martingales starting from 0. So Theorem 6 can be applied to arbitrary equivalent changes of measure on all complete filtered probability spaces.
Theorem 6 Let be an equivalent measure toand suppose that U given by 3 has a cadlag version.
Then, there is a predictable process satisfying andin which case. In general, however, this will not be the case since need only be tie local martingale. Next, by the Kunita-Watanabe inequalityif is a nonnegative process satisfying then. That is, V is absolutely continuous with respect to [ X ].
Newest ‘girsanov’ Questions – Quantitative Finance Stack Exchange
We would like to use a stochastic version of the Radon-Nikodym theorem to imply the existence of a predictable process with. In fact, this is possible as stated below in Lemma 7so as required. It still needs to be shown that is finite and that U satisfies the required decomposition. Next, we show that is finite. Asthe process. As cadlag martingales are semimartingales, and have well defined quadratic variation, is finite.
Then, applying the Kunita-Watanabe inequality again, for any positive constant K. This is finite, since it has been shown that is finite and is a cadlag -martingale tending to the finite limitso is bounded.
Finally, as is finite for all times tis X-integrable. Define the local martingales and. The quadratic covariation is given by. The decomposition of U follows by taking. The following stochastic version of the Radon-Nikodym theorem was used in the proof of Theorem 6which we now prove.
Lemma 7 Let A be a continuous FV process and B be a continuous adapted increasing process such that almost surely for all and bounded nonnegative predictable satisfying. Then, there is a predictable process satisfyingand.
Let us first suppose that A and B have integrable variation and, without loss of generality, assume that. Then, we can define the following finite signed measures on the predictable measurable space. As B is increasing, is a nonnegative measure. If for a predictable set Sthen and, from the condition of the lemma,giving. So, is absolutely continuous with respect to and the Radon-Nikodym derivative exists. This is a predictable process satisfying and for all bounded predictable. This shows that M is a martingale.
As continuous FV local martingales are constantM is identically 0, giving as required.