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Unformatted text preview: TUTORIAL 2– WEEK 3
ECON3107/ECON5106 – Economics of Finance
ANSWERS
1. Consider the following trades. First, trade 1PA for
GA for BA. These trades can be summarized as
1P A −→ 1
0.3 GA. Then, trade the obtained amount of 1
1
GA −→
0.6BA = 2BA.
0.3
0.3 Hence 1BA costs 0.5PA.
2. One way to determine whether there are arbitrage opportunities is the following. Consider the
impact of trading in a clockwise and anticlockwise direction: Clockwise : 1P A −→ 1
GA −→ 2BA −→ 1.2P A
0.3 This trading sequence generates a proﬁt of 0.2PA. Hence, there are arbitrage opportunities if you
trade in the clockwise direction.
Anticlockwise : 1P A −→ 1
1
0. 3
5
BA −→
2 GA −→
2 PA = 6PA
0. 6
(0.6)
(0.6) If you trade in the anticlockwise direction, you will make a loss. Also, note that if you trade in one
direction followed by the other direction you must break even, i.e., 1.2 × 5/6 = 1 (when there are
no transaction costs).
3. In the previous question it would have been suﬃcient to check the trades in one direction
only. If you ended up with anything other than 1PA, then arbitrage opportunities must exist. In
this question, if you end up with an answer less than 1PA in one direction, this tells you nothing
about what will happen if you trade in the other direction. In such cases, you must check in both
directions.
Clockwise : 1P A −→ 2GA −→ 1.5BA −→ 0.75P A
This trading sequence generates a loss of 0.25PA. Hence there are no arbitrage opportunities if you
trade in the clockwise direction.
Anticlockwise : 1P A −→ 1.5BA −→ 1.5GA −→ 0.6P A
Hence if you trade in the anticlockwise direction, you also make a loss. There are no arbitrage
opportunities.
1 4.
(i) Let Q {states*securities} be the payment matrix of the two securities:
Q:
Good Weather
Bad Weather Bond
20
20 Stock
50
25 Let pS {1*securities} be a vector of security prices:
ps: Bond
18 Stock
30 Then, the vector of the atomic prices patom can be found as
patom = pS · Q −1
20 50 −1
= 18 30
= 0. 3 0 . 6 .
20 25
(ii) Let q {states*1} be a vector of payments for the apple tree:
q:
Good Weather
Bad Weather Tree
70
45 Then, the price of the tree can be calculated as follows:
ptree
70
= patom · q = 0.3 0.6
= 48.
45
(iii) Buy an apple tree and sell one bond and one stock. You make money now while at the same
time being perfectly hedged.
(iv) The discount factor is 0.9 (i.e., the sum of the atomic security prices). It tells us the value of
an apple in the next period in terms of present apples.
(v) Let c {states*1} be a vector of statecontingent payments:
c:
Good Weather
Bad Weather Security
80
100 The portfolio n that provides the desired set of state contingent payments will be given by
n=Q −1 ·c= 20 50
20 25 − 1 80
100 =
6
.
−0.8 In other words, the investor should buy 6 bonds and sell (short) 0.8 stocks. The price of this
payment combination is calculated as follows:
6
p = pS · n = 18 30
= 84.
− 0. 8
2 ...
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This note was uploaded on 07/13/2011 for the course ECON 3107 at University of New South Wales.
 '11
 valentyn
 Economics

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