188 Mathematics for Finance
Above we have identified the risk-neutral probabilityP∗. Now we shall con-
sider a European call option on the stock with strike priceXto be exercised
at timeT. The general formula (8.6) for the price of an option becomes
CE(0) =E∗(
e−rT(S(T)−X)+)
.
Let us compute this expectation. BecauseV(T)=W(t)+
(
m−r+^12 σ^2)t
σfor
t(≥0 is a Wiener process underP∗, the random variableV(T)=W(T)+
m−r+^12 σ^2
)T
σis normally distributed with mean 0 and varianceT,thatis,
it has density√ 21 πTe−x
2
2 T. As a result,CE(0) =E∗(
e−rT(S(T)−X)+)
=E∗
((
S(0)eσV(t)−(^12) σ (^2) T
−Xe−rT
)+)
=
∫∞
−d 2 √T(
S(0)eσx−^12 σ(^2) T
−Xe−rT
) 1
√
2 πTe−x2
2 Tdx=S(0)
∫∞
−d 1√^1
2 πe−y 22
dy−Xe−rT∫∞
−d 2√^1
2 πe−y 22
dy=S(0)N(d 1 )−Xe−rTN(d 2 ),where
d 1 =lnSX(0)+(
r+^12 σ^2)
T
σ√
T
,d 2 =lnSX(0)+(
r−^12 σ^2)
T
σ√
T
, (8.9)
and where
N(x)=∫x−∞√^1
2 πe−y 22
dy=∫∞
−x√^1
2 πe−y 22
dy (8.10)is the normal distribution function.
What we have just derived is the celebrated Black–Scholes formula for Eu-
ropean call options. The choice of time 0 to compute the price of the option
is arbitrary. In general, the option price can be computed at any timet<T,
in which case the time remaining before the option is exercised will beT−t.
Substitutingtfor 0 andT−tforTin the above formulae, we thus obtain the
following result.
Theorem 8.6 (Black–Scholes Formula)
The timetprice of a European call with strike priceXand exercise timeT,
wheret<T,isgivenby
CE(t)=S(t)N(d 1 )−Xe−r(T−t)N(d 2 )