ACTSC/STAT 446/846 – Midterm 1 – Fall 2011 – Solutions
1. [12 points] Consider an equitylinked contract as seen in class, in which an investor receives
$1000 max
p
1
,
1 + 0
.
6
p
S
(
T
)
S
(0)

1
PP
(
*
)
at
T
= 2 years, where
S
(
t
) is the value of the index at time
t
. The value of the index is modeled
by a binomial model with two timesteps of length
h
= 1 year,
u
= 1
.
05,
d
= (1
/u
) and
S
(0) = 20.
The continuously compounded interest rate is 3% per year.
(a) [6pts] Compute the value of the equitylinked contract at time 0.
Solution:
The possible index prices at time 2 are
S
(2
,
0) = 18
.
141
, S
(2
,
1) = 20 and
S
(2
,
2) =
22
.
05. Hence the contract’s possible values at time 2 are
V
(2
,
0) =
V
(2
,
1) = 1000 and
V
(2
,
2) =
1061
.
5. We then compute the riskneutral probability
q
=
e
0
.
03

1
/
1
.
05
1
.
05

1
/
1
.
05
= 0
.
79978
.
So that the value of the contract at time 0 is
V
(0) =
e

0
.
03
×
2
(
q
2
×
1061
.
5 + (1

q
2
)
×
1000) = 978
.
81
.
Note: another solution is to write
V
(0) = 1000
e

rT
+30
C
0
, where
C
0
is the price of a call option
on the index with strike 20, using the decomposition
1000 max
p
1
,
1 + 0
.
6
p
S
(
T
)
S
(0)

1
PP
= 1000 + 600
/S
(0)
×
max(0
, S
(
T
)

S
(0))
of the payoF. One then needs to price the call using the binomial tree, getting
C
0
= 1
.
23, and
thus
V
(0) = 1000
e

0
.
06
+ 30
×
1
.
23 = 978
.
81.
(b) [5pts] Determine the composition at time 1—in the case
S
(1) =
uS
(0), i.e., in node (1,1)—of
the portfolio (trading strategy) that replicates the payoF (*).
Solution:
Want
θ
S
(1
,
1) and
θ
B
(1
,
1) such that
θ
S
(1
,
1)
×
22
.
05 +
θ
B
(1
,
1)
e
0
.
06
=1061
.
5
θ
S
(1
,
1)
×
20 +
θ
B
(1
,
1)
e
0
.
06
=1000
therefore
θ
S
(1
,
1) = 61
.
5
/
2
.
05 = 30 and
θ
B
(1
,
1) = (1061