- A three-month call option is the right to buy stock at $20. Currently the stock is selling for $22 and the call is selling for $5. You are considering buying 100 shares of the stock ($2,200) or one call option ($500).
- a) If the price of the stock rises to $29 within three months, what would be the profits or losses on each position? What would be the percentage gains or losses?
- b) If the price of the stock declines to $18 within three months, what would be the profits or losses on each position? What would be the percentage gains or losses?
- c) If the price of the stock remained stable at $22, what would be the percentage gains or losses at the expiration of the call option?
- d) If you compare purchasing the stock to purchasing the call, why do the percentage gains and losses differ?

- A particular call is the option to buy stock at $25. It expires in six months and currently sells for $4 when the price of the stock is $26.
- a) What is the intrinsic value of the call? What is the time premium paid for the call?
- b) What will the value of this call be after six months if the price of the stock is $20? $25? $30? $40?
- c) If the price of the stock rises to $40 at the expiration date of the call, what is the percentage increase in the value of the call? Does this example illustrate favorable leverage?
- d) If an individual buys the stock and sells this call, what is the cash outflow (i.e., net cost) and what will the profit on the position be after six months if the price of the stock is $10? $15? $20? $25? $26? $30? $40?
- e) If an individual sell this call naked, what will the profit or loss be on the position after six months if the price of the stock is $20? $26? $40?

- A particular put is the option to sell stock at $40. It expires after three months and currently sells for $2 when the price of the stock is $42.
- a) If an investor buys this put, what will the profit be after three months if the price of the stock is $45? $40? $35?
- b) What will the profit from selling this put be after three months if the price of the stock is $45? $40? $35?
- c) Compare the answers to (a) and (b). What is the implication of the comparison?

- A call option is the right to buy stock at $50 a share. Currently the option has six months to expiration, the volatility of the stock (standard deviation) is 0.30, and the rate of interest is 10 percent (0.1 in Exhibit 18.2).
- a) What is the value of the option according to the Black-Scholes model if the price of the stock is $45, $50, or $55?
- b) What is the value of the option when the price of the stock is $50 and the option expires in six months, three months, or one month?
- c) What is the value of the option when the price of the stock is $50 and the interest rate is 5 percent, 10 percent, or 15 percent?
- d) What is the value of the option when the price of the stock is $50 and the volatility of the stock is 0.40, 0.30, or 0.10?
- e) What generalizations can be derived from the solutions to these problems?

- Black-Scholes demonstrates that the value of a put option increases the longer the time to expiration. Currently the price of a stock is $100 and there are two put options to sell the stock at $100. The three-month option sells for $7.00 and the six-month op-tion sells for $4.50.
- a) What would you do and why?
- b) How much do you earn or lose after three months at the following prices of the underlying stock: $85, $90, $95, $100, $105, and $110? Assume the worst-case scenario.
- c) Is there any reason to anticipate earning a higher return than your answers in (b)?

- The futures price of corn is $2.00. The contracts are for 10,000 bushels, so a contract is worth $20,000. The margin requirement is $2,000 a contract, and the maintenance margin requirement is $1,200. A speculator expects the price of the corn to fall and enters into a contract to sell corn.
- a) How much must the speculator initially remit?
- b) If the futures price rises to $2.13, what must the speculator do?
- c) If the futures price continues to rise to $2.14, how much does the speculator have in the account?

**2.**The futures price of gold is $1,250. Futures contracts are for 100 ounces of gold, and the margin requirement is $5,000 a contract. The maintenance margin requirement is $1,500. You expect the price of gold to rise and enter into a contract to buy gold.

- a) How much must you initially remit?
- b) If the futures price of gold rises to $1,255, what is the profit and percentage return on your position?
- c) If the futures price of gold declines to $1,248, what is the loss and percentage return on the position?
- d) If the futures price falls to $1,238, what must you do?
- e) If the futures price continues to decline to $1,210, how much do you have in your account?
- f) How do you close your position?

**3.**You expect to receive a payment of £1,000,000 in British pounds after six months. The You expect to receive a payment of £1,000,000 in British pounds after six months. The Y pound is currently worth $1.60 (i.e., £1 5 $1.60), but the six-month futures price is $1.56 (i.e., £1 5 $1.56). You expect the price of the pound to decline (i.e., the value of the dollar to rise). If this expectation is fulfilled, you will suffer a loss when the pounds are converted into dollars when you receive them six months in the future

- a) Given the current price, what is the expected payment in dollars?
- b) Given the futures price, how much would you receive in dollars?
- c) If, after six months, the pound is worth $1.35, what is your loss from the decline in the value of the pound?

d)To avoid this potential loss, you decide to hedge and sell a contract for the future delivery of pounds at the going futures price of $1.56. What is the cost to you of this protection from the possible decline in the value of the pound?

- e) If, after hedging, the price of the pound falls to $1.35, what is the maximum amount that you lose? Why is your answer different from your answer to part (c)?
- f) If, after hedging, the price of the pound rises to $1.80, how much do you gain from your position?
- g) How would your answer to part (f) be different if you had not hedged and the price of the pound had risen to $1.80?
- An American portfolio manager owns a bond worth £2,000,000 that will mature in one year. The pound is currently worth $1.65, and the one-year future price is $1.61. If the value of the pound were to fall, the portfolio manager would sustain a loss. If the value of the pound were to rise, the portfolio manager would experience a profit.
- a) What is the expected payment based on the current exchange rate?
- b) What is the expected payment based on the futures exchange rate?
- c) If, after a year, the pound is worth $1.53, what is the loss from the decline in the value of the pound?
- d) If, after a year, the pound is worth $1.72, what is the gain from the increase in the value of the pound?
- e) To avoid the potential loss in part (c) the portfolio manager hedges by selling futures contracts for the delivery of pounds at $1.61. What is the cost of the protection from a decline in the value of the pound?
- f) If, after hedging, the price of the pound falls to $1.53, what is the maximum amount the portfolio manager can lose? Why is this answer different from the answer to part (c) above?
- g) If, after hedging, the price of the pound rises to $1.72, what is the maximum amount the portfolio manager can gain? Why is this answer different from the answer to part (d) above?

- You expect the stock market to decline, but instead of selling stocks short, you decide to sell a stock index futures contract based on an index of New York Stock Exchange common stocks. The index is currently 600 and the contract has a value that is $250 times the amount of the index. The margin requirement is $2,000 and the maintenance margin requirement is $1,000.
- a) When you sell the contract, how much must you put up?
- b) What is the value of the contract based on the index?
- c) If after one week of trading the index stands at 601, what has happened to your position? How much have you lost or profited?
- d) If the index rose to 607, what would you be required to do?
- e) If the index declined to 594 (1 percent from the starting value), what is your percentage profit or loss on your position?
- f) If you had purchased the contract instead of selling it, how much would you have invested?
- g) If you had purchased the contract and the index subsequently rose from 600 to 607, what would be your required investment?
- h) Contrast your answers to parts (d) and (g).
- i) At the expiration of the contract, do you deliver the securities you contracted to sell?