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

Get *Quantitative Finance for Physicists* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.