- Option Pricing 187
a probabilityP∗such thatV(t)=W(t)+
(
m−r+^12 σ^2)
t/σ (rather than
W(t) itself) becomes a Wiener process underP∗, then the exponential factor
e(m−r+^12 σ
(^2) )t
will be eliminated from the final expression. (The existence of such
a probabilityP∗follows from an advanced result in Stochastic Calculus, the so-
calledGirsanov theorem.) Indeed, sinceV(t) has density√^12 πte−x
2
2 t underP∗,
that is, it is normally distributed with mean 0 and variancet, it follows that
E∗
(
e−rtS(t))
=S(0)E∗
(
eσW(t)+(m−r)t)
=S(0)E∗
(
eσV(t)−(^12) σ (^2) t)
=S(0)
∫∞
−∞eσx−(^12) σ (^2) t√ 1
2 πt
e−
x 22 t
dx
=S(0)
∫∞
−∞√^1
2 πte−(x− 2 σtt)^2
dx=S(0)
∫∞
−∞√^1
2 πte−y 22 t
dy=S(0).The fact thatE∗(e−rtS(t)) =S(0) does not depend on timetis a necessary
condition for the discounted price process e−rtS(t) to be a martingale underP∗.
To show that e−rtS(t) is indeed a martingale underP∗we need in fact to verify
the stronger condition
E∗(
e−rtS(t)|S(u))
=e−ruS(u) (8.7)for anyt≥u≥0, involving the conditional expectation of e−rtS(t)givenS(u).
So far we have dealt with conditional expectation where the condition was
given in terms of a discrete random variable, see Section 3.2.2. Here, however,
the condition is expressed in terms ofS(u), a random variable with continuous
distribution. In this case the precise mathematical meaning of (8.7) is that for
everya> 0
E∗
(
e−rtS(t)1S(u)<a)
=E∗
(
e−ruS(u)1S(u)<a)
, (8.8)
where 1S(u)<ais theindicator random variable,equalto1ifS(u)<aand to 0
ifS(u)≥a.
Exercise 8.13
Verify equality (8.8).Exercise 8.14
Find the density ofW(t) under the risk-neutral probabilityP∗.