Question

For problems 1 to 5, consider two firms, XYZ and Teleco. Firm XYZ mines copper, with fixed costs of $0.50/lb and variable costs of $0.40/lb. Teleco sells the telecommunications equipment and uses copper wire as an input. Suppose Teleco earns a fixed revenue of $6.20 for each unit of wire it uses. The wire price is the price of copper/lb plus $5. The 1-year forward price of copper is $1/lb. The 1-year interest rate is 6%. The 1-year option prices for copper are

C(0.9500) = 0.0649, C(1.0250) = 0.0274, P (0.9500) = 0.0178, P (1.0250) = 0.0509,

C(0.9750) = 0.0500, C(1.0000) = 0.0376, C(1.0500) = 0.0194,

P (0.9750) = 0.0265, P (1.0000) = 0.0376, P (1.0500) = 0.0665

1. Suppose XYZ buys a put option with a strike of $0.95, $1.00, or $1.05. Draw a graph of the hedged profit in each case.

2. Suppose XYZ buys collars with the following strikes. Draw a graph of the hedged profit in each case.

a. $0.95 for the put and $1.00 for the call b. $0.975 for the put and $1.025 for the call

c. $1.05 for the put and $1.05 for the call

3. If Teleco does nothing to mange copper price risk, what is its profit 1 year from now, per pound of copper that it buys? If it hedges the price of wire by buying copper forward, what is its estimated profit 1 year from now? Construct graphs illustrating both unhedged and hedged profit.

4. Compute estimated profit in 1 year if Teleco buys a call option with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case.

5. Compute estimated profit in 1 year if Teleco sells collars with the following strikes:

a. $0.95 for the put and $1.00 for the call b. $0.975 for the put and $1.025 for the call

c. $0.95 for the put and $0.95 for the call

6. A BCD stock pays a $1 dividend every 3 months. The current price is $50 and the first dividend coming 3 months from now. The con- tinuously compounded risk-free rate is 6%. What is the price of the prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? What is the price of a forward contract that expires at the same time?

7. A BCD stock pays an 8% continuous dividend. The current price is $50 and the continuously compounded risk-free rate is 6%. What is the price of a prepaid forward contract that expires 1 year from today? What is the price of a forward contract that expires at the same time?

8. Suppose you are a market maker in S&R index forward contracts. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.

a. What is the no-arbitrage forward price for delivery in 9 months?

b. Suppose a customer wishes to enter a short index futures position. If you take the opposite position, demonstrate how you would hedge your resulting long position using the index and borrowing or lending?

c. Suppose a customer wishes to enter a long index futures position. If you take the opposite position, demonstrate how you would hedge your resulting long position using the index and borrowing or lending?

9. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.

a. Suppose you observe a 6 month forward price of 1135. What arbitrage would you undertake?

b. Suppose you observe a 6 month forward price of 1115. What arbitrage would you undertake?

10. Suppose 10 years from now it becomes possible for money managers to engage in time travel. In particular, suppose that a money manager could travel to January 1981, when the 1-year Treasury bill rate was 12.5%.

a. If time travel were costless, what riskless arbitrage strategy could a money manager undertake by traveling back and forth between January 1981 and January 1982.

b. If many money managers undertook this strategy, what would you expect to happen to interest rates in 1981

c. Since interest rates were 12.5% in January 1981, what can you conclude about whether costless time travel will ever be possible?

11. Suppose S&P 500 index futures price is currently 1200. You wish to purchase four futures contracts on margin

a. What is the notional value of your position?

b. Assuming 10% initial margin, what is the value of the initial margin?

c. Assume there is no interest rate on your margin account and the maintenance margin is 80% of initial margin. What is the greatest S&P 500 index futures price at which will you receive a margin call?

12. Suppose the current exchange rate between Euro and YEN is 0.02. The euro-denominated annual continuously compounded risk-free rate is 4% and the yen-dominated annual continuously compounded risk- free rate is 1%. What are the 6-month euro/yen and yen/euro forward prices?

13. The euro exchange rate is $1.25/euro. The continuously compounded dollar interest rate it 5% and the continuously compounded euro in- terest rate is 4%. Suppose that you borrow euros and lend dollars for 1 year. At what exchange rate will you break even on this position?

14. Natural gas futures prices are $6.85 (Oct), $7.50 (Nov), $8.15 (Dec), $8.20 (Jan), $8.25 (Feb), $8.20 (Mar), and $7.45 (Apr). The effective monthly interest rate is 1%.

a. During which months is storage expected occur?

b. What is your estimate of the expected monthly storage cost dur- ing those months?

15. Suppose the gold price is $300/oz., the 1-year forward price is 310.686, and the continuously compounded risk-free rate is 5%.

a. What is the lease rate?

b. Demonstrate a cash-and-carry strategy that provides the zero cash flow at time 0 and the maturity date. (You borrow to buy gold, sell the gold forward, and lend the gold, earning the lease rate.)

c. What is the return on a cash-and-carry strategy in which gold is not loaned? (You borrow to buy gold and sell the gold forward.)

Answer 1

6. Stock price (S) = $50

Dividend (d) = $1 every 3 months

Riskfree rate (R_f) = 6%

Firstly calculate present value of dividends pays in every 3 months.

Present value of dividend = d * e^{-r * t}

because dividend pays every 3 months, then present value of dividend in this e's value = 2.718281828

= d * e ^{-0.06 * (3/12)} + d * e ^{-0.06 * (6/12)} + d * e ^{-0.06 * (9/12)} + d * e ^{-0.06 * (12/12)}   

= $1 * e ^-0.015 + $1 * e ^-0.03 + $1 * e ^-0.045 + $1 * e ^-0.06

= $1 * 0.9851 + $1 * 0.9704 + $1 * 0.956 + $1 * 0.9418

= $3.8533

Stockholder is entitled to receive dividends. We will get the stock only in one year, value of prepaid forward contract price is :

Prepaid forward contract price = Stock Price - Present value of dividend

= $50 - $3.8533

= $46.1467 or $46.15

Prepaid forward contract price paid today by the stockholder and the underlying asset is received at delivery date.

For calculate price of forward contract that expires at same time we calculate future value of prepaid forward contract price.

Price of forward contract = Prepaid forward Contract price * e^{r * t}

= $46.15 * e ^{0.06 * 1}

= $46.15 * e^0.06

= $46.15 * 1.0618

= $49.002 Or $49.00