Solution to Problem Set 8
Investments
Prof. PierreOlivier Weill
1. The option price tree is:
t
= 0
t
= 1
t
= 2
10
9.52
6.80
10
4.76
0
This is seen as follows:
(a) If the stock price is 144 or 108 at maturity, then the option is worth 10. If the
stock price is 81 then the option is worthless.
(b) Suppose, at time 1, the stock price is 120. Then, next year the option will be
worth 10 for sure. This payoff can be replicated by saving 10/1.05=9.52 at the
riskfree rate of 5%. Hence, at time 1 in the upperbranch scenario, the option
value is 9.52.
(c) Suppose at time 1, the stock price is 90. Then, next year the option will be
worth either 10 or 0, depending on whether the stock price goes up or down.
Suppose that we buy Δ stocks and short a zerocoupon bond (ZCB) with face
value
F
. Then, if the stock price goes up, the portfolio will be worth
Δ
*
108

F
. If the stock price goes down, the portfolio will be worth Δ
*
81

F
.
To match the value of the option, we must choose and Δ and
y
such that:
Δ
*
108

F
= 10
(1)
Δ
*
81

F
= 0
(2)
The option’s delta is
Δ =
C
+

C

S
+

S

=
10

0
108

81
= 0
.
37
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Miyakawa
 ZCB

Click to edit the document details