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.: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.
Results of this type were first proved by Cameron–Martin in the 1940s and by Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977).
Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure absolutely continuous with respect to P then every P-semimartingale is a Q-semimartingale.
Statement of theorem
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 (e.g. all continuous models).
Given an adapted process with define
where is the stochastic exponential (or Doléans exponential) of X with respect to W, i.e.
Then for each t the measure Q restricted to the unaugmented sigma fields is equivalent to P restricted to
Furthermore if Y is a local martingale under P then the process
is a Q local martingale on the filtered probability space .
If X is a continuous process and W is Brownian motion under measure P then
is Brownian motion under Q.
The fact that is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing the quadratic variation
it follows by Lévy's characterization of Brownian motion that this is a Q Brownian motion.
In many common applications, the process X is defined by
For X of this form then a sufficient condition for to be a martingale is Novikov's condition which requires that
The stochastic exponential is the process Z which solves the stochastic differential equation
The measure Q constructed above is not equivalent to P on 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 ).
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:
where denotes the instantaneous risk free rate, the asset's drift and its volatility.
Other classical applications of Girsanov theorem are quanto adjustments and the calculation of forwards' drifts under LIBOR market model.
- 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 de martingales locales par changement absolue continu de probabilités". Zeitschrift für Wahrscheinlichkeit 39 (1): 65–70. doi:10.1007/BF01844873.
- Dellacherie, C.; Meyer, P.-A. (1980). Probabilités et potentiel: Théorie de Martingales: Chapitre VII. Paris: Hermann. ISBN 2-7056-1385-4. (French)