The riskneutral probability is:
105
1 12
12
1 12
0 59
.
/ .
.
/ .
.


=
396
275
475
330
229
Thus, the put price tree is:
059 0
0 41 27 6
105
10 75
.
.
.
.
.
x
x
+
=
The price of the twoperiod European put on Bullmart is thus 10.75p.
Put payoff:
max (300475,0)
= 0
max (300330,0)
= 0
max (300229,0)
= 71
059 0
0 41 71
105
27 6
.
.
.
.
x
x
+
=
0
Q3:
You have estimated from financial market data that the annual volatility of XYZ is 15%.
Today the price of XYZ is $100. Assume that XYZ does not pay any dividends. You are
interested in valuing a call option on XYZ maturing in two year’s time. Assume that the annual
interest rate is 10%.
a)
Construct a two period binomial tree (t= 0,1,2) for XYZ (that is, let
D
t =1)
100
b)
Now consider a call option on XYZ with exercise price EX = $110 that expires in period 2.
Find the replicating portfolio for this option first at time t=1 and then at t=0.
First note that:
C
uu
= max {135110;0} = 25
C
du
= max {100110;0} = 0
C
dd
= max {74110;0} = 0
116
86
135
100
74
࠵? = ࠵?
%,’(
= 1,16
d = 1/u
=
0,86
Now, at time 1 if the share price is 116 the replicating portfolio is:
h
C
C
S
S
H
HH
LH
HH
LH
=


=
=
24 6
34 6
71
.
.
.
B
h
S
C
r
H
H
LH
LH
f
=

+
=
=
1
71 100
11
64 5
.
*
.
.
So C
u
= .71 x 116  64.5 = $ 18
If share price is 86, there is no hope that the call option will end up in the money and so C
d
= 0.
We can now find a replicating portfolio at time 0 such that its value at time 1 is sufficient to allow us to
purchase the replicating portfolio at time 1. The time 0 replicating portfolio must have:
h
C
C
S
S
H
L
H
L
0
17 9
116
86 2
6
=


=

=
.
.
.
B
h S
C
r
L
L
f
0
0
1
6
86 2
11
=

+
=
=
. *
.
.
$47
So C
0
= .6 x 100  47 = $ 13
c) What are the rebalancing trades?
If the share price goes up to 116, we need to increase our hedge ratio. We purchase (.71.6) =.11 shares
and borrow .11 * 116 = (64.5  47*1.1) = $12.8 to do so.
If the share price drops to 86 we sell all our shares and pay back our loan.
d)
Price the same call option using the “risk neutral pricing” method.
In this case
p
r
d
u
d
=
+


=


=
1
11
862
116
862
8
.
.
.
.
.
We get:
C
x
H
=

=
.
(
.
)
.
.
8
134 6
110
11
17 9
;
C
L
=
0
;
C
x
0
8 17 9
11
13
=
=
.
.
.
;
Q4: You have estimated from financial market data that the annual volatility of QWE is also
15%. Similarly to the company in the previous example, the price of QWA is also $100. One
difference exists, however, in that QWE is expected to pay a $6 dividend at time 1. Remember
that the option holders are not entitled to receive the dividend.
a)
How does the binomial tree for QWE’s stock look like in the presence of this known $6 dollar
dividend?
100
b)
Use risk neutral pricing method to value a call option on QWE’s stock assuming that the
exercise price of the option is still $110. Explain the difference between QWE’s option price
and the price of the XYZ’s option.
As before,
p
r
d
u
d
=
+


=


=
1
11
862
116
862
8
.
.
.
.
.
127.6
94.6
68.8
1166
= 110
866
= 80
92.8
We get:
C
x
H
=

=
.
(
.
)
.
.
8
132 2
110
11
161
;
C
L
=
0
;
C
x
0
8 161
11
117
=
=
.
.
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 Summer '14
 sam
 Derivatives, Valuation