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3.6 First Passage Time Distribution 109

principle in Section 5.3. Here we develop the first approach; the second is pre­

sented in the next section. In Sections 5.2 and 5.3 of Volume I, we observed

after deriving the distribution of the first passage time for the symmetric

random walk that our answer could easily be modified to obtain the first pas­

sage distribution for an asymmetric random walk. In this section, we work

only with Brownian motion, the continuous-time counterpart of the symmet­

ric random walk. The case of Brownian motion with drift, the continuous-time

counterpart of an asymmetric random walk, is treated in Exercise 3.7. We re­

visit this problem in Chapter 7, where it is solved using Girsanov's Theorem.

The resulting formulas often provide explicit pricing and hedging formulas for

exotic options. Examples of the application of these formulas to such options

are given in Chapter 7.

Just as we began in Section 5.2 of Volume I with a martingale that had

the random walk in the exponential function, we must begin here with a

martingale containing Brownian motion in the exponential function. We fix a

constant u. The so-called exponential martingale corresponding to u, which is

Z(t) = exp { uW(t)-�u^2 t}, (3.6.1)

plays a key role in much of the remainder of this text.

Theorem 3.6.1 (Exponential martingale). Let W(t), t ;:::: 0, be a Brow­

nian motion with a filtmtion :F( t), t ;:::: 0, and let u be a constant. The process

Z(t), t 2: 0, of {3.6. 1} is a martingale.

PROOF: For 0 :::; s :::; t, we have

lE[Z(t)I:F(s)]

= lE [ exp { uW(t)- �u^2 t} I :F(s)]

= lE [ exp { u(W(t)-W(s))} · exp { uW(s)-�u^2 t} I :F(s)]

= exp { uW(s)- �u^2 t} ·lE [ exp { u(W(t)-W(s)) }I :F(s)], (3.6.2)

where we have used "taking out what is known" (Theorem 2.3.2(ii)) for the
last step. We next use "independence" (Theorem 2.3.2(iv)) to write

lE [exp {u(W(t)-W(s))}I:F(s)] = lE [exp {u(W(t)-W(s))}].

Because W(t ) -W(s) is normally distributed with mean zero and variance
t-s , this expected value is exp { �u^2 (t-s)} (see (3.2.13)). Substituting this
into (3.6.2), we obtain the martingale property

lE[Z(t)i:F(s)] = exp { uW(s)- �u^2 s} = Z(s). D