University of Michigan Department of Mathematics

Homework Answers

Homework I: (1) 20c increase. (2) go long in 275 of commodity. (3) (a) hedge ratio=0.836; (b) coefficient of correlation= 0.914. (4) 3.57 percent per annum, continuous interest.
(5) (a) 6.22 percent per annum, continuous interest; (b) 5.44 percent per annum, continuous interest; (c) 6.49 percent per annum, continuous interest. (6) yield = 5.5 percent.
(7) (a) price= 90.58 dollars, duration= 4.274 years; (b) increase of 1.28 percent. (8) (c) A decreases by 23.8 percent and B by 25.7 percent; (d) A.
Homework II: (1) 1.968 million dollars. (2) X invests floating, Y invests fixed and they pay their interest to the bank. The bank pays 8.3 percent to X and LIBOR plus 0.3 percent to Y.
(3) 0.76 million dollars. (4) 680,000 dollars.
(5) X borrows in the U.K. and Y in the U.S. and the interest on these loans is paid by the bank. X pays the bank 11.25 percent and Y pays 11.75 percent interest.
(6) (a) forward price= 50 dollars, value of contract=0; (b) forward price=47.96 dollars, value of contract=1.997 dollars. (7) profit= 7,100 dollars. (8) (b) interest=0.68 percent.
Homework III: (1) 3.66 dollars. (2) 56 cents. (3) 25 cents. (4) 2.50 dollars. (5) between 2.40 and 3 dollars.
(6) between 40 and 55 dollars. (7) between 56 and 64 dollars. (8) between 40 and 60 dollars.
Homework IV: (1) S=38.4 with probability 0.619, S= 25.6 with probability 0.381. (2) (a) volatility=0.25; (b) rate of interest= 7.32 percent; (c) volatility=0.4334.
(3) price lies between 37.9 and 67 dollars. (4) (a) probability=0.0427; (b) 8.5 years from now. (5) 5.06 dollars. (6) 6.39 dollars. (7) 2.20 dollars. (8) 2.59 dollars.
Homework V: (2)(a) 79 cents; (b) 17 cents. (3) 3.9 dollars. (4) 500 put contracts with strike price 1080.
(5) 2.6 cents. (6)(a) 0.5 cents; (b) 1.6 cents; (c) 1.6 cents. (7) between 2 and 2.02. (8) (a) 1.18; (b) 1.29.
Homework VI: (1)(a) 34.7 percent; (b) 31.1 percent. (2)(a) p=1-lambda; (b) 16.67. (3) 0.58 percent. (4) 19.7 percent.
(5) 35.6 percent. (6) (a) 1.565 dollars; (b) 20.8 percent. (7) 1.98 dollars. (8) for E=40, expected payoff=1.419, implied volatility=18.0 percent;
for E=38, expected payoff=2.646, implied volatility= 18.9 percent; for E=42, expected payoff=0.730 , implied volatility= 18.8 percent.
Homework VII: (1)(a) short on 456 AUD; (b) short on futures contract on 470 AUD; (c) short on forward contract on 494 AUD.
(2)(a) short on 2400 of stock; (b) short on 3900 of stock and long on 2500 of option. (3)(a) -0.4; (b) 0.04; (c) 0.1.
(4)(a) short on 1950 sterling and long on 4000 of option; (b) short on 1710 sterling, long on 3200 of the option of part (a) and long on 2400 of the option of part (b).
(5)(a) short on 30,000 euros; (b) long on 2400 euros; (c) the bank lost 36 dollars. (6) 8.51 dollars. (7) 4.33.
(8)(a) 1st moment=-0.1, 2nd moment=36.03, 3rd moment=-32.42; (b) 14.08; (c) 14.50.
Homework VIII: (1) delta=-488.6. (2) delta=0.6274, gamma=0.05, theta=-2.175, vega=11.35. (3) No, loss of 6600 dollars. (4)(a) S.D.=537; (b) VaR=3957.
(5) VaR=63,220, diversification benefit=7438. (6)(a) VaR=3.62; (b) VaR=3.78. (7) VaR=76,248. (8) allocate 49 dollars to 5 year bond and 605 to 7 year bond.