8.1.5 Consider the simple random walk

in which the summands are independent with Prf D 1g D 1 2 . We are going to stop this random walk when it first drops a units below its maximum to date.
Accordingly, let

(a) Use a first step analysis to show that

Identify the distribution of M .

(c) Let B.t/ be standard Brownian motion,M.t/D maxfB.u/I0utg;Y.t/ D M.t/????B.t/, and D minft 0IY.t/ D ag. Use the invariance principle to argue that M. / has an exponential distribution with mean a.
Note: is a popular strategy for timing the sale of a stock. It calls for keeping the stock as long as it is going up, but to sell it the first time that it drops a units from its best price to date. We have shown that E[M. /] D

a, whence E[B. /] D E[M. /]????a D 0, so that the strategy does not gain a profit, on average, in the Brownian motion model for stock prices.

Transcribed Image Text:

Sn=1++n. So=0,