Mathematical Modeling in Finance with Stochastic Processes

172 CHAPTER 5. BROWNIAN MOTION

be a sequence of independent, identically distributed Bernoulli random vari-
ables. Note that Var [Yi] = 1, which we will need to use in a moment. Let
Y 0 = 0 for convenience and set

Tn=

∑n

i=0

Yi

be the sequence of sums, which represent the successive net fortunes of our
notorious gambler. As usual, we will sketch the graph ofTnversus time using
linear interpolation between the points (n− 1 ,Tn− 1 ) and (n,Tn) to obtain
a continuous, piecewise linear function. Since this interpolation defines a
function for all time, we could writeWˆ(t), and then for instance,Wˆ(n) =Tn.
NowWˆ(t) is a function defined on [0,∞). This function is piecewise linear
with segments of length

√

2. The notationWˆ(t) reminds us of the piecewise
   linear nature of the function.
   Now we will compress time, and rescale the space in a special way. Let
   Nbe a large integer, and consider the rescaled function

WˆN(t) =

(

1

√

N

)

Wˆ(Nt).

This has the effect of taking a step of size± 1 /

√

N in 1/Ntime unit. That
is,

WˆN(1/N) =

(

1

√

N

)

Wˆ(N· 1 /N) =√T^1

N

=

Y 1

√

N

.

Now consider

WˆN(1) =

Wˆ(N)

√

N

=

TN

√

N

.

According to the Central Limit Theorem, this quantity is approximately
normally distributed, with mean zero, and variance 1. More generally,

WˆN(t) =

Wˆ(Nt)

√

N

=

√

t

Wˆ(Nt)

√

Nt

IfNtis an integer, this will be normally distributed with mean 0 and vari-
ancet. Furthermore,WˆN(0) = 0 andWˆN(t) is a continuous function, and
so is continuous at 0. Altogether, this should be a strong suggestion that
WˆN(t) is an approximation to Standard Brownian Motion. We will define
the very jagged piecewise linear functionWˆN(t) asapproximate Brownian
Motion.