- Option Pricing 189
with
d 1 =lnSX(0)+(
r+^12 σ^2)
(T−t)
σ√
T−t,d 2 =lnSX(0)+(
r−^12 σ^2)
(T−t)
σ√
T−t. (8.11)
Exercise 8.15
Derive the Black–Scholes formulaPE(t)=Xe−r(T−t)N(−d 2 )−S(t)N(−d 1 ),withd 1 andd 2 given by (8.11), for the price of a European put with
strikeXand exercise timeT.Remark 8.2
Observe that the Black–Scholes formula contains nom. It is a property anal-
ogous to that in Remark 8.1, and of similar practical significance: There is no
need to knowmto work out the price of a European call or put option in
continuous time.
It is interesting to compare the Black–Scholes formula for the price of a
European call with the Cox–Ross–Rubinstein formula. There is close analogy
between the terms. Apart from the obvious correspondence between the con-
tinuous and discrete time discount factors e−rTand (1 +r)−N, the binomial
and normal distribution terms appearing in these formulae are also related
to one another. The precise relationship comes from a version of theCentral
Limit Theorem: It can be shown that the option price given by the Cox–Ross–
Rubinstein formula tends to that in the Black–Scholes formula in the continuous
time limit described in Chapter 3.
Figure 8.2 Option priceCEin continuous and discrete time models as a
function of timeTremaining before the option is exercised