210 CHAPTER 6. STOCHASTIC CALCULUS
Calculation of the Variance
We can calculate the variance of Geometric Brownian Motion by using the
m.g.f. for the normal distribution, together with the common formula
Var [X] =E[
(X−E[X])^2
]
=E
[
X^2
]
−(E[X])^2
and the previously obtained formula forE[X].
Theorem 5 .Var [z 0 exp(μt+σW(t))] =z 02 exp(2μt+σ^2 t)[exp(σ^2 t)−1]
Proof.First compute:
E[
X(t)^2]
=E
[
z^20 exp(μt+σW(t))^2]
=z 02 E[exp(2μt+ 2σW(t))]
=z 02 exp(2μt)E[exp(2σW(t))]
=z 02 exp(2μt)E[exp(2σW(t)u)]|u=1
=z 02 exp(2μt) exp(4σ^2 tu^2 /2)|u=1
=z 02 exp(2μt+ 2σ^2 t)Therefore,Var [z 0 exp(μt+σW(t))] =z 02 exp(2μt+ 2σ^2 t)−z^20 exp(2μt+σ^2 t)
=z 02 exp(2μt+σ^2 t)[exp(σ^2 t)−1].Note that this has the consequence that the variance starts at 0 and then
increases. The variation of Geometric Brownian Motion starts small, and
then increases, so that the motion generally makes larger and larger swings
as time increases.
Parameter Summary
If a Geometric Brownian Motion is defined by the stochastic differential equa-
tion
dX=rXdt+σXdW X(0) =z 0
then the Geometric Brownian Motion is
X(t) =z 0 exp((r−(1/2)σ^2 )t+σW(t)).