176 Mathematics for Finance
Proof
This is an immediate consequence of (8.1):
D(0) =
f(Su)−f(Sd)
u−d+
(1 +u)f(Sd)−(1 +d)f(Su)
(u−d)(1+r)=^1
1+r(
(r−d)f(Su)
(u−d)+(u−r)f(Sd)
u−d)
=
1
1+r(
p∗f(Su)+(1−p∗)f(Sd))
=E∗
(
(1 +r)−^1 f(S(1)))
,
as claimed.
Exercise 8.3
Find the initial value of the portfolio replicating a call option if propor-
tional transaction costs are incurred whenever the underlying stock is
sold. (No transaction costs apply when the stock is bought.) Compare
this value with the case free of such costs. Assume thatS(0) =X= 100
dollars,u=0.1,d=− 0 .1andr=0.05, admitting transaction costs at
c= 2% (the seller receiving 98% of the stock value).Exercise 8.4
LetS(0) = 75 dollars and letu=0.2andd=− 0. 1 .Suppose that you
can borrow money at 12%, but the rate for deposits is lower at 8%. Find
the values of the replicating portfolios for a put and a call. Is the answer
consistent with the put and call prices following from Theorem 8.2?8.1.2 Two Steps.........................................
We begin with two time steps. The stock priceS(2) has three possible values
Suu=S(0)(1 +u)^2 ,Sud=S(0)(1 +u)(1 +d),Sdd=S(0)(1 +d)^2 ,andS(1) has two values
Su=S(0)(1 +u),Sd=S(0)(1 +d),at the nodes of the tree in Figure 8.1 marked by the corresponding sequences
of letters u and d.