Mathematical Modeling in Finance with Stochastic Processes

158 CHAPTER 5. BROWNIAN MOTION

p+q = 1, sop → 1 /2. The analytical formulation of the problem is as
follows. Letδbe the size of the individual steps, letrbe the number of steps
per unit time. We ask what happens to the random walk in the limit where
δ→0,r→∞, andp→ 1 /2 in such a manner that:

(p−q)·δ·r→c

and
4 ·p·q·δ^2 ·r→D.

Probabilistic Solution of the Limit Question

In our accelerated random walk, consider thenth step at timet=n/rand
consider the position on the linex=k·δ. Let

vk,n=P[Tn=k]

be the probability that thenth step is at positionk. We are interested in the
probability of finding the walk at given instanttand in the neighborhood of
a given pointx, so we investigate the limit ofvk,nasn/r→t, andk·δ→x.
Remember that the random walk can only reach an even-numbered posi-
tion after an even number of steps, and an odd-numbered position after an
odd number of steps. Therefore in all casesn+kis even and (n+k)/2 is
an integer. Likewisen−kis even and (n−k)/2 is and integer. We reach
positionkat time step nif the walker takes (n+k)/2 steps to the right
and (n−k)/2 steps to the left. The mix of steps to the right and the left
can be in any order. So the walk reaches positionkat stepnwith binomial
probability

vk,n=

(

n

(n+k)/ 2

)

·p(n+k)/^2 ·q(n−k)/^2

From the Central Limit Theorem

vk,n∼(1/(

√

2 ·π·p·q))·exp(−[(n+k)/ 2 −n·p]^2 /(2·π·n·pq))

= (1/(

√

2 π·pq))·exp(−[k−n(p−q)]^2 /(8πnpq))

∼((2δ)/(

√

2 πDt)) exp(−[x−ct]^2 /(2Dt))