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Assignment 1

Question 1

i)

The payment matrix according to the specified payouts in the question is as follows:

Q:

B S C

GW 20 60 45

FW 20 30 5

BW 20 5 0

The security prices are:

Ps:

BOND STOCK SECURITY C

18 19 8

To calculate the arbitrage-free price of the atomic securities, multiply the vector of securityprices by the inverse of the payment matrix.

Q-1 =

0.0015 -0.0132 0.0618

-0.059 0.0529 -0.0471

0.0294 -0.0647 0.0353

Ps * inv(Q)=

Patomic = [ 0.15 0.25 0.5]

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ii)

The arbitrage-free price of the apple tree is calculated by multiplying p,the vector ofelemental security prices and q,the vector of payments

P x q=

P:

0.15 0.25 0.5

q:

GW 80

FW 50

BW 25

Therefore through matrix multiplication, the arbitrage-free price of the apple tree is 37present apples.

iii)

The discount factor is calculated by adding the atomic security prices. It represents thepresent value of a one unit payment to be made with certainty at the specified future date.

0.15 + 0.25 + 0.5= 0.9

Due to this calculation, an apple a year from now has a present value of 0.9 applesregardless of the type of transaction.

iv)

Investor!s payout in which he receives 30 apples if the weather is good or fair and 50

apples if the weather is bad.

C:

GA 30

FA 30

BA 50

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The payment matrix of securities {states x securities}

Q:

B S CGW 20 60 45

FW 20 30 5

BW 20 5 0

To compute the portfolio to provide the payouts as specified in matrix C, multiply theinverse of Q with the payouts C

N = Q-1x C

N:

The replicating portfolio to provide the payments in"C!is shown the the left. The negative stock valueindicates the short sell -0.9412 stocks to buy 2.7353worth of bonds and 0.7059 worth of security C toprovide 30 apples in good or fair weather and 50apples in bad weather.

Ps:

BOND STOCK SECURITY C

18 19 8

psxN

By multiplying this portfolio "N with the securities price vector ps gives the price of the

portfolio which is 37 present apples.

v)

The call option!s value derives from the value of the stock at time 1. The option will onlybe exercised if the payout is higher than the price of the option which is 25. Therefore fromthe stock payouts shown on the following page, the option will only be exercised in goodweather and fair weather as there is a positive net inflow of apples. Thus the Net payoffvector is the payout from the stock minus the price of the option except for in bad weatherwhere the option is not exercised.

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BONDS 2.7353

STOCK -0.9412

SECURITY C 0.7059

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Stock!s payoff Net Payoff

60 35

30 5

5 0

Thus to compute the price of the call option, multiply the vector of atomic security prices"Patomic!against the vector of net payoffs. It shows that the European call option is worth6.5 present apples.

P xc =

[0.15 0.25 0.5] =6.5PA

Question 2

i)

Q{states x securities}:

B0 S0 Bg Sg Bb Sb

G 1.05 1.3 -1 -1 0 0

B 1.05 0.9 0 0 -1 -1

GG 0 0 1.05 1.3 0 0

GB 0 0 1.05 0.9 0 0

BG 0 0 0 0 1.05 1.3

BB 0 0 0 0 1.05 0.9

Ps:

B0 S0 Bg Sg Bb Sb

1 1 0 0 0 0

4

35

5

0

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ii)

To compute the atomic security prices the following formula was used.

Patomic= ps x Q

-1

Through the application of the above formula, the vector of atomic security prices are asfollows:

Patomic =

G B Gg Gb Bg Bb

0.3571 0.5952 0.1276 0.2126 0.2126 0.3543

iii)

The european call option will only be exercised if the price of the stock in the secondperiod is above the strike price of 1.05. A profit is attained when the option is exercised atthe option price which is below market price and subsequently selling the stock receivedat market price. Hence in this model, this will occur in every state except in state "bb!.

iv)

The value of the stock at t=2

Gg 1.69

Gb 1.17

Bg 1.17

Bb 0.81

If it costs 1.05 to exercise the call option and it will only be exercised at stock values abovethe option price, the net payoffs will thus be shown in the time-state vector below denoted"C

!

C :

G 0 0

B 0 0

Gg Max(0,1.69-1.05) 0.64

Gb Max(0,1.17-1.05) 0.12

Bg Max(0,1.17-1.05) 0.12

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Bb Max(0,0.81-1.05) 0

Patomic =

G B Gg Gb Bg Bb

0.3571 0.5952 0.1276 0.2126 0.2126 0.3543

To arrive at the arbitrage-free price of the call option, multiply the vector of atomic securityprices with the matrix "C!shown above.

Price = patomicx c = {1x states} x {states x 1} = (0.64 x 0.1276) + (0.12 x 0.2126) + (0.12 x 0.2126) = 0.1327Therefore, arbitrage-free price of the European call option will be $ 0.1327.

v)

Put options are options to sell the underlying stock at a predetermined price and date. Theoption will only be exercised if the value of the underlying stock at the expiration date islower than the exercise price. By simultaneously buying stock at market value andexercising the option to sell the security at a higher price, an investor will be able to attaina profit.

vi)

The value of the stock at the second period:

Gg 1.69

Gb 1.17

Bg 1.17

Bb 0.81

Depending on the state of the world, the put option will only be exercised if the value of thestock is lower than the exercise price of 1.05. Through observation, only the state of "bb!would result in the option being exercised. The net payoff for this particular state would be"1.05-0.81!which is the sum received from exercising the option against the price of thestock being sold at market value.

The payoff matrix "C!below highlights the possible cash flows.

C:

G 0 0

B 0 0

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Gg Max(1.05-1.69,0) 0

Gb Max(1.05-1.17,0) 0

Bg Max(1.05-1.17,0) 0

Bb Max(1.05-0.81,0) 0.24

Patomic =

G B Gg Gb Bg Bb

0.3571 0.5952 0.1276 0.2126 0.2126 0.3543

The cost of providing these cash flows equals "p x c!being the vector of atomic securityprices multiplied by the vector of cash flows.

Price = patomicx c = {1x states} x {states x 1} = 0.24 x 0.3543 = 0.085

According to the calculations above, the cost of providing the European put option and itsassociated payoffs will be $0.085.

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