This preview shows pages 1–2. Sign up to view the full content.
1
Lecture XXIX
Option Pricing using BlackScholes
I.
The European Call Option
A.
Material adapted from C.f. Huang and R. H. Litzenberger
Foundations
for Financial Economics
New York: North–Holland, 1988.
B.
First, we construct the payoff function for a security on which the option is
written as:
()
:0
0otherwise
jj
j
xk
−
∋−
>
=
where
j
x
k
is the payoff a share of security
j
purchased an exercise
price of
k
. The price of the European call option is then defined as
,m
a
x
,
0
1
j
f
k
pSk
S
r
≥−
+
C.
Consider the strategy of selling one share of the security to buy one
European call option written on the security.
1.
The initial cost of the strategy is:
( )
,
1
j
f
k
pSk S
r
−+
+
The possible payoffs of this strategy are
0
i
f
if
j
x
kx k
x k
−
−+=
≥
−+>
<
2.
In the first case, you exercise the option buying back the stock at
the original price while in the second case the investor makes
money because the stock price decreased (you make profit equal
to the decrease in the stock price).
3.
Therefore, to avoid a riskless profit (you can’t make something
for nothing)
( )
,0
1
,
1
j
f
j
f
k
r
k
r
−
+>
+
⇒>
−
+
D.
Starting with a twoperiod economy, we start by assuming a power utility
function for the representative agent:
() ()
11
00
0
1
BB
uz uz
z
z
ρ
−
−
+=
+
−−
where
is the time preference parameter. The arbitrage condition (selling
short a share of stock and purchasing a call option) then implies
0
a
x
,
0
B
j
C
E
C
−
=−
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.
 Fall '08
 Weldon

Click to edit the document details