Answers to Exercises
2.2 (a) $113.56; (b) $ 68.13.
2.4 Borrow 100000 USD to buy 100000/1.7705 GBP. Then buy (100000/
1.7705)/0.6694 EUR. Exchange the resulting amount to
1.1914[(100000/1.7705)/0.6694] 100525 USD. Return the loan and
enjoy profits of $525 (minus trans action fees).
3.2 (a) 0.157; (b) 1.645; (c) 1.036
3.4 Since aX þ b N(am þ b,(as)
2
), it follows that C
2
¼ a
2
þ b
2
and D ¼
(a þ b C) m.
4.3 (t) ¼ X(0)exp( mt) þ s
Ð
t
0
exp[ m(t s)]dW (s)
5.2 For this process, the AR(2) polynomial (5.1.12) is:1 – 1.2z þ 0.32z
2
¼ 0.
Since its roots, z ¼ (1.2 0.4)/0.64 > 1, are outside the unit circle, the
process is covariance-stationary.
5.3 Linear regression for the dividends in 2000 – 2003 is D ¼ 1.449 þ
0.044n (where n is number of years since 2000). Hence the dividend
growth is G ¼ 4.4%.
7.1 (a) X
*
¼ 0.5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:25 C
p
. Hence there are two fixed points at C <
0.25, one fixed point at C ¼ 0.25, and none for C > 0.25.
(b) X
1
*
0.14645 is attractor with the basin 0 X < X
2
*
where X
2
*
0.85355.
9.1 (a) 1) c ¼ 2.70, p ¼ 0.26; 2) c ¼ 0.58, p ¼ 2.04.
(b) The Black-Scholes option prices do not depend on the stock
growth rate (see discussion on the risk-neutral valuation).
9.2 Since the put-call parity is violated, you may sell a call and a T-bill for
$(8 þ 98) ¼ $106. Simultaneously, you buy a share and a put for $(100
159
þ 3.50) ¼ $103.50 to cover your obligations. Then you have profits of
$(106 103.50) ¼ $2.50 (minus transaction fees).
10.1 (a) E[R] ¼ 0.13, s ¼ 0.159; (b) E[R] ¼ 0.13, s¼ 0.104.
10.2 (a) b
A
¼ 1.43;
(b) For b
A
¼ 1.43, E[R
A
] ¼ 0.083 according to eq(10.2.1). However,
the average return for the given sample of returns is 0.103. Hence
CAPM is violated in this case.
10.3 w
1
¼ (b
21
b
32
b
22
b
31
)/[ b
11
(b
22
b
32
) þ b
21
(b
32
b
12
) þ b
31
(b
12
b
22
)],
w
2
¼ (b
12
b
31
b
11
b
32
)/[b
22
(b
11
b
31
) þ b
12
(b
31
b
21
) þ b
32
(b
21
b
11
)].
10.4 l
1
¼ [ b
22
(R
1
R
f
)b
12
(R
2
R
f
)]/(b
11
b
22
b
12
b
21
), l
2
¼ [b
11
(R
2
R
f
)
10.4 b
21
(R
1
R
f
)]/(b
11
b
22
b
12
b
21
).
11.1 (a) $136760; (b) $78959.
160 Answers
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