TUTORIAL 2 WEEK 3ECON3107/ECON5106 Economics of Finance

ANSWERS

1. Consider the following trades. First, trade 1PA for 10.3GA. Then, trade the obtained amount ofGA for BA. These trades can be summarized as

1PA 10.3

GA 10.3

0.6BA = 2BA.

Hence 1BA costs 0.5PA.

2. One way to determine whether there are arbitrage opportunities is the following. Consider theimpact of trading in a clockwise and anticlockwise direction:

Clockwise : 1PA 10.3

GA 2BA 1.2PA

This trading sequence generates a profit of 0.2PA. Hence, there are arbitrage opportunities if youtrade in the clockwise direction.

Anticlockwise : 1PA 10.6

BA 1(0.6)2

GA 0.3(0.6)2

PA =5

6PA

If you trade in the anticlockwise direction, you will make a loss. Also, note that if you trade in onedirection followed by the other direction you must break even, i.e., 1.2 5/6 = 1 (when there areno transaction costs).

3. In the previous question it would have been sufficient to check the trades in one directiononly. If you ended up with anything other than 1PA, then arbitrage opportunities must exist. Inthis question, if you end up with an answer less than 1PA in one direction, this tells you nothingabout what will happen if you trade in the other direction. In such cases, you must check in bothdirections.

Clockwise : 1PA 2GA 1.5BA 0.75PAThis trading sequence generates a loss of 0.25PA. Hence there are no arbitrage opportunities if youtrade in the clockwise direction.

Anticlockwise : 1PA 1.5BA 1.5GA 0.6PAHence if you trade in the anticlockwise direction, you also make a loss. There are no arbitrageopportunities.

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4.(i) Let Q {states*securities} be the payment matrix of the two securities:Q: Bond Stock

Good Weather 20 50

Bad Weather 20 25

Let pS {1*securities} be a vector of security prices:ps: Bond Stock

18 30

Then, the vector of the atomic prices patom can be found as

patom = pS Q1 =(18 30

)(20 5020 25

)1=

(0.3 0.6

).

(ii) Let q {states*1} be a vector of payments for the apple tree:q: Tree

Good Weather 70

Bad Weather 45

Then, the price of the tree can be calculated as follows:

ptree = patom q =(0.3 0.6

)(7045

)= 48.

(iii) Buy an apple tree and sell one bond and one stock. You make money now while at the sametime being perfectly hedged.

(iv) The discount factor is 0.9 (i.e., the sum of the atomic security prices). It tells us the value ofan apple in the next period in terms of present apples.

(v) Let c {states*1} be a vector of state-contingent payments:c: Security

Good Weather 80

Bad Weather 100

The portfolio n that provides the desired set of state contingent payments will be given by

n = Q1 c =(

20 5020 25

)1(80100

)=

(60.8

).

In other words, the investor should buy 6 bonds and sell (short) 0.8 stocks. The price of thispayment combination is calculated as follows:

p = pS n =(18 30

)( 60.8

)= 84.

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