Girsanov theorem

In probability theory, the Girsanov theorem (named after Igor Vladimirovich Girsanov) describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure.[1]^:607 The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure, which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values, to the risk-neutral measure which is a very useful tool for pricing derivatives on the underlying instrument.

Visualisation of the Girsanov theorem — The left side shows a Wiener process with negative drift under a canonical measure P; on the right side each path of the process is colored according to its likelihood under the martingale measure Q. The density transformation from P to Q is given by the Girsanov theorem.

History

Results of this type were first proved by Cameron–Martin in the 1940s and by Girsanov in 1960.[2] They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).[3]

Significance

Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is an absolutely continuous measure with respect to P then every P-semimartingale is a Q-semimartingale.

Statement

We state the theorem first for the special case when the underlying stochastic process is a Wiener process. This special case is sufficient for risk-neutral pricing in the Black–Scholes model and in many other models (for example, all continuous models).

Let $\{W_{t}\}$ be a Wiener process on the Wiener probability space $\{\Omega ,{\mathcal {F}},P\}$ . Let $\{X_{t}\}$ be a measurable process adapted to the natural filtration of the Wiener process $\{{\mathcal {F}}_{t}^{W}\}$ with $X_{0}=0$ .

Define the Doléans-Dade exponential ${\mathcal {E}}(X)_{t}$ of X with respect to W

{\mathcal {E}}(X)_{t}=\exp \left(X_{t}-{\frac {1}{2}}[X]_{t}\right),

where $[X]_{t}$ is the quadratic variation of $X_{t}$ . If ${\mathcal {E}}(X)_{t}$ is a strictly positive martingale, a probability measure Q can be defined on $\{\Omega ,{\mathcal {F}}\}$ such that we have Radon–Nikodym derivative

{\frac {dQ}{dP}}{\bigg |}_{{\mathcal {F}}_{t}}={\mathcal {E}}(X)_{t}

Then for each t the measure Q restricted to the unaugmented sigma fields ${\mathcal {F}}_{t}^{W}$ is equivalent to P restricted to ${\mathcal {F}}_{t}^{W}$ . Furthermore, if Y is a local martingale under P, then the process

{\tilde {Y}}_{t}=Y_{t}-\left[Y,X\right]_{t}

is a Q local martingale on the filtered probability space $\{\Omega ,{\mathcal {F}},Q,\{{\mathcal {F}}_{t}^{W}\}\}$ .

Corollary

If X is a continuous process and W is Brownian motion under measure P then

{\tilde {W}}_{t}=W_{t}-\left[W,X\right]_{t}

is Brownian motion under Q.

The fact that ${\tilde {W}}_{t}$ is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing the quadratic variation

\left[{\tilde {W}}\right]_{t}=\left[W_{t},W_{t}\right]-2\left[W_{t},[W,X]_{t}\right]+\left[[W,X]_{t},[W,X]_{t}\right]=\left[W\right]_{t}=t

it follows by Lévy's characterization of Brownian motion that this is a Q Brownian motion.

Comments

In many common applications, the process X is defined by

X_{t}=\int _{0}^{t}Y_{s}\,dW_{s}.

If X is of this form, then a sufficient condition for ${\mathcal {E}}(X)$ to be a martingale is Novikov's condition, which requires that

E_{P}\left[\exp \left({\frac {1}{2}}\int _{0}^{T}Y_{s}^{2}\,ds\right)\right]<\infty .

The stochastic exponential ${\mathcal {E}}(X)$ is the process Z, which solves the stochastic differential equation

Z_{t}=1+\int _{0}^{t}Z_{s}\,dX_{s}.\,

The measure Q constructed above is not equivalent to P on ${\mathcal {F}}_{\infty }$ , as this would only be the case if the Radon–Nikodym derivative were a uniformly integrable martingale, which the exponential martingale described above is not (for $\lambda \neq 0$ ).

Application to finance

In finance, Girsanov theorem is used each time one needs to derive an asset's or rate's dynamics under a new probability measure. The most well known case is moving from historic measure P to risk neutral measure Q which is done—in Black–Scholes model—via Radon–Nikodym derivative:

${\frac {dQ}{dP}}={\mathcal {E}}\left(\int _{0}^{\cdot }{\frac {r-\mu }{\sigma }}\,dW_{s}\right)$

where $r$ denotes the instantaneous risk free rate, $\mu$ the asset's drift and $\sigma$ its volatility.

Other classical applications of Girsanov theorem are quanto adjustments and the calculation of forwards' drifts under LIBOR market model.

References

Musiela, M.; Rutkowski, M. (2004). Martingale Methods in Financial Modelling (2nd ed.). New York: Springer. ISBN 3-540-20966-2.
Girsanov, I. V. (1960). "On transforming a certain class of stochastic processes by absolutely continuous substitution of measures". Theory of Probability and Its Applications. 5 (3): 285–301. doi:10.1137/1105027.
Lenglart, É. (1977). "Transformation des martingales locales par changement absolument continu de probabilités". Zeitschrift für Wahrscheinlichkeit. 39 (1): 65–70. doi:10.1007/BF01844873.

Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315. ISBN 978-981-4678-93-3. (See Chapter 10)
Dellacherie, C.; Meyer, P.-A. (1980). Probabilités et potentiel: Théorie de Martingales: Chapitre VII (in French). Paris: Hermann. ISBN 2-7056-1385-4.

External links

Notes on Stochastic Calculus which contains a simple outline proof of Girsanov's theorem.
Papaioannou, Denis (July 14, 2012). "Applied Multidimensional Girsanov Theorem". SSRN 1805984. Cite journal requires |journal= (help) Contains financial applications of Girsanov's theorem.

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[1] Musiela, M.; Rutkowski, M. (2004). Martingale Methods in Financial Modelling (2nd ed.). New York: Springer. ISBN 3-540-20966-2.

[2] Girsanov, I. V. (1960). "On transforming a certain class of stochastic processes by absolutely continuous substitution of measures". Theory of Probability and Its Applications. 5 (3): 285–301. doi:10.1137/1105027.

[3] Lenglart, É. (1977). "Transformation des martingales locales par changement absolument continu de probabilités". Zeitschrift für Wahrscheinlichkeit. 39 (1): 65–70. doi:10.1007/BF01844873.