Chapter 2: FAQs 107
What is Girsanov’s Theorem, and Why
is it Important in Finance?
Short Answer
Girsanov’s theorem is the formal concept underlying
the change of measure from the real world to the risk-
neutral world. We can change from a Brownian motion
with one drift to a Brownian motion with another.
Example
The classical example is to start with
dS=μSdt+σSdWt
withWbeing Brownian motion under one measure (the
real-world measure) and converting it to
dS=rS dt+σSdW ̃t
under a different, the risk-neutral, measure.
Long Answer
First a statement of the theorem. LetWtbe a Brownian
motion with measurePand sample space.If
γtis a previsible process satisfying the constraint
EP
[
exp
(
1
2
∫T
0 γ
2
t
)]
<∞then there exists an equivalent
measureQonsuch that
W ̃t=Wt+
∫t
0
γsds
is a Brownian motion.
It will be helpful if we explain some of the more techni-
cal terms in this theorem.
Sample space: All possible future states or outcomes.
