ACTL 3004 / 5109Financial Economics for Insurance and Superannuation

Final ExamS2, 2005

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Question 1 (4 marks)

Consider a world in which there are only two risky assets, A and B. There isalso a risk free asset with return rf . The two risky assets are in equal supply inthe market, and hence the market return rM is defined by

rM =1

2(rA + rB) .

The following information is known: rf = 0.09, 2A = 0.04,

2B = 0.03, AB =

0.02, E [rM ] = 0.18

Find the CAPM expected return for asset A.

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Question 2 (8 marks)

Consider an investor with the following utility function:

u (w) = aw b (max (L w, 0))2

where a and b are positive constants.

(i) [4 marks] List two behavioural properties implied by this utility function.

(ii) [4 marks] How does this investors investment in risky assets (in dollarterms) change as his wealth increases?

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Question 3 (9 marks)

A guarantee contract pays off according to the gains of the stock index S (t),with a guaranteed minimum payout and maximum payout. More precisely, it isa five year contract which pays out 0.8 times the ratio of the terminal and initialvalues of the index. Or it pays out 125% if otherwise it would be less, or 175%if otherwise it would be more. Assume that the Black-Scholes assumptions aresatisfied.

(i) [5 marks] Assume that the index has expected return = 6%, standarddeviation = 20%, dividend yield = 3%, and that the risk free rate is r =6.5%. How much is this contract worth?

(ii) [4 marks] Find the amount of stock and cash that would be required tohedge this guarantee (at time 0).

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Question 4 (7 marks)

(Yr 2012 comment: This was a topic discussed in the 2005 offering of the course)

Consider two bounded random variables, A and B. The random variable A issaid to be Second Order Stochastic Dominant (SSD) over B if z

a

FA (y) dy za

FB (y) dy, for all z

and za

FA (y) dy 0.

(i) [3 marks] Assuming no arbitrage, derive the market price of risk forthis model.

(ii) [6 marks] Construct a self-financing replicating strategy for a derivativewith payoff X and maturity T

(iii) [4 marks] Present an applicable formula for the price of a European calloption with strike price K and maturity at time T .

You may find the following mathematical results useful:

1. Girsanov Theorem

Let W (t) be a P - Brownian Motion. If we set the Radon-Nikodym deriva-tive

dQ

dP= e

T0(t)dW (t) 12

T02(t)dt

then

WQ (t) = W (t) +

t0

(s) ds

will be a Q - Brownian motion.

2. Martingale Representation Theorem:

Consider two Q martingales X (t) , Y (t). There exists a previsible process (t) such that

dX (t) = (t) dY (t) .

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Question 10 (9 marks)

Consider a model where the stock price is modelled by a 2 year trinomial tree.

For example, starting at a current value of $1, in one years time the stock pricecan either

rise to a ratio of 1.2 of the current value stay at the current value drop to a ratio of 0.85 of the current value.

Assume that a risk free bond is available for trade, with r = 0%, i.e. the valueof the bond is $1 at all times.

The following (European) option prices can be observed in the market:

A call option with strike price of 0.9 and maturity at time 1 is currentlytrading at 0.12.

A call option with strike price of 1.3 and maturity at time 2 is currentlytrading at 0.0105.

An fixed strike lookback option with payoff

max

((maxt=0,1,2

S (t)

) 1.1, 0

)and maturity at time 2 is currently trading at 0.0295.

Given the above information, and assuming no arbitrage, find the price of a calloption with a strike price of 1.1 and maturity at time 2.

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Question 11 (10 marks)

Consider the equilibrium asset pricing model

p = E [mx]

where p is the (time 0) market price of an asset, x is the time 1 payoff of theasset, and m is the equilibrium stochastic discount factor.Assume the economy is in equilibrium, and the equilibrium stochastic discountfactor m can be represented as

m = a+ bRx + cRy

where a, b and c are constants. Rx and Ry represent the (random) returns onthe stock market index and the commodities price index respectively. Assumethat Rx and Ry are independent.

(i) [2 marks] Let rf be the risk free rate. Show that

a =1

1 + rf bE [Rx] cE [Ry]

(ii) [8 marks] Using the result in (i), solve for b and c, and hence developformula for E [Ri], the expected return on a stock with (random) return Ri.

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Question 12 (6 marks)

Lionel, your friend from high school, is currently looking at different methodsto price options. One day he says to you the following:

I have looked at the derivation and assumptions of the Capital AssetPricing Model. It is clear that it is not designed to price options.

Discuss the above statement.

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