MATH4060 FINANCIAL MATHEMATICS 2021-2022
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MATH4060-E1
A LEVEL 4 MODULE, SEMESTER SEMESTER 2021-2022
FINANCIAL MATHEMATICS
1. In this question use the values of r0 , and r3 in the table below that correspond to the SEVENTH (PENULTIMATE) digit of your 8-digit Student ID number.
7th digit
0
1
2
3
4
5
6
7
8
9
r0
0.01
0.05
0.02
0.03
0.04
0.04
0.02
0.03
0.01
0.05
r
0.03
0.02
0.01
0.04
0.05
0.02
0.03
0.01
0.05
0.04
A five year short forward contract on 5000 bushels of oats is entered into when the commodity price is 14 cents per bushel and the continuously compounded interest rate is r0 per annum.
(a) What is the forward price for the contract at date of entry
(b) Three years later the price is 15 cents per bushel and the continuously compounded interest rate is
r3 per annum. What is the current value of the original short forward contract
(c) Suppose there is a five year forward contact on 5000 bushels of oats with a forward price of $800 when the oat price is 14 cents per bushel and the continuously compounded interest rate is r0 per annum. How would you exploit the arbitrage opportunity
[10 marks]
2. In this question use the values of p in the table below that correspond to the SEVENTH (PENULTIMATE) digit of your 8-digit Student ID number.
7th digit
0
1
2
3
4
5
6
7
8
9
p
2
4
7
6
5
3
9
1
10
8
The term structure of a bond is given below:
T
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
P(0, T)
1.00
0.98
0.96
0.86
0.85
0.82
0.75
0.71
0.69
0.63
0.61
0.52
0.50
Consider a fixed coupon bond with nominal value $10000 that pays coupons with p% of the nominal value per annum in quarterly coupons. The first coupon is to be paid after 1.5 years and the maturity date is 3 years. A floating rate note with the same nominal value and the same coupon dates also exists.
(a) Investor A purchases the fixed coupon bond at time 0. What would be a fair price for this bond
(b) Investor B purchases the floating rate note at time 0. What would be a fair forward swap rate for the
floating rate note
(c) Suppose investors A and B agree to swap their bonds at time 0. What additional transaction must take place at each coupon date to make this a fair swap
[10 marks]
3. In this question use the values of K1 , K2 and K3 in the table below that correspond to the SEVENTH (PENULTIMATE) digit of your 8-digit Student ID number.
7th digit
0
1
2
3
4
5
6
7
8
9
K1
30
30
30
30
30
40
40
40
40
40
K2
50
50
50
60
60
50
50
50
60
60
K3
70
80
90
70
80
70
80
90
70
80
(a) An investor constructs a portfolio with payoff function at time T given by
vT (sT) = s(2)T(K)1(K)1 sT i(i)f(f) K(K)2(1) s(s)T(T) K(K)3(2)
i) Sketch this payoff function for 20 < sT < 100. ii) Explain when an investor might choose to purchase this portfolio. iii) Show how this payoff function can be recreated using a portfolio of European call and put options. (b) The current stock price is s0 = 50. The European put option premium for strike price K = 30 is D = 1.403. The European call option premiums are shown below. Strike price, K 30 40 50 60 70 80 90 Call price, 26.841 20.997 16.355 12.743 9.958 7.817 6.169 i) Calculate the European put option price for strike price K2 . ii) Use part (b)(i) and the values in the table to calculate the total cost (premium) at time 0 for the portfolio of options considered in part (a). iii) For what values of sT will the holder of the portfolio make a profit [20 marks] 4. In this question use the value of K and option type in the table below that correspond to the SEVENTH (PENULTIMATE) digit of your 8-digit Student ID number. 7th digit 0 1 2 3 4 5 6 7 8 9 K 106 96 98 104 102 106 96 98 104 102 Option Call Call Call Call Call Put Put Put Put Put Consider a stock with current price S0 = 100, drift u = 0.001 and volatility G = 0.072. Consider a European option which matures in 4 years. See the table for the strike price, K, and option type. (a) A one-step binary tree model for this scenario is shown below.
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