Mathematical Modeling in Finance with Stochastic Processes

182 CHAPTER 5. BROWNIAN MOTION

probability that the random walk goes up to valueabefore going down to
valuebwhen the step size is ∆xis

P[ toabefore−b] =

b∆x

(a+b)∆x

=

b

a+b

Thus, the probability ofa >0, before hitting−b <0 does not depend on
the step size, and also does not depend on the time interval. Therefore in
passing to the limit the probabilities should remain the same. Here is a place
where it is easier to derive the result from the coin-flipping game and pass to
the limit than to derive the result directly from Wiener process principles.

The Distribution of the Maximum

Lettbe a given time, leta >0 be a given value, then

P

[

max

0 ≤u≤t

W(u)≥a

]

=P[Ta≤t]

=

2

√

2 π

∫∞

a/

√

t

exp(−y^2 /2)dy

Sources

This section is adapted from: Probability Models, by S. Ross, andA First
Course in Stochastic ProcessesSecond Edition by S. Karlin, and H. Taylor,
Academic Press, 1975.

Problems to Work for Understanding

1. Differentiate the c.d.f. ofTato obtain the expression for the p.d.f ofTa.


2. Show thatE[Ta] =∞fora >0.


3. Suppose that the fluctuations of a share of stock of a certain company
   are well described by a Wiener process. Suppose that the company
   is bankrupt if ever the share price drops to zero. If the starting share
   price isA(0) = 5, what is the probability that the company is bankrupt
   byt= 25? What is the probability that the share price is above 10 at
   t= 25?.