Mathematical Modeling in Finance with Stochastic Processes

208 CHAPTER 6. STOCHASTIC CALCULUS

Vocabulary

1. Geometric Brownian Motionis the continuous time stochastic pro-
   cessz 0 exp(μt+σW(t)) whereW(t) is standard Brownian Motion.


2. A random variableXis said to have thelognormaldistribution (with
   parametersμandσ) if log(X) is normally distributed (log(X)∼N(μ,σ^2 )).
   The p.d.f. forXis

fX(x) =

1

√

2 πσx

exp((− 1 /2)[(ln(x)−μ)/σ]^2 ).

Mathematical Ideas

Geometric Brownian Motion

Geometric Brownian Motionis the continuous time stochastic process
X(t) =z 0 exp(μt+σW(t)) whereW(t) is standard Brownian Motion. Most
economists prefer Geometric Brownian Motion as a model for market prices
because it is always positive, in contrast to Brownian Motion, even Brow-
nian Motion with drift. Furthermore, as we have seen from the stochastic
differential equation for Geometric Brownian Motion, the differential relative
change in Geometric Brownian Motion is a combination of a deterministic
proportional growth term similar to inflation or interest rate growth plus a
random relative change. See Itˆo’s Formula and Stochastic Calculus. On a
short time scale this is a sensible economic model.

Theorem 3 .At fixed timet, Geometric Brownian Motionz 0 exp(μt+σW(t))
has a lognormal distribution with parameters (ln(z 0 ) +μt) andσ

√

t.

Proof.

FX(x) =P[X≤x]

=P[z 0 exp(μt+σW(t))≤x]

=P[μt+σW(t)≤ln(x/z 0 )]

=P[W(t)≤(ln(x/z 0 )−μt)/σ]

=P

[

W(t)/

√

t≤(ln(x/z 0 )−μt)/(σ

√

t)

]

=

∫(ln(x/z 0 )−μt)/(σ√t)

−∞

1

√

2 π

exp(−y^2 /2)dy