Finance

Created 12/3/1996
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Problem Set #7: Options

(Answers)

1. Suppose that you hold a call option on a share of stock, and also "owe" a share of stock--that is, you have sold it short. What is the total payoff to your portfolio on the exercise date if the price of the stock is above the strike price?

Minus the strike price--you have to pay the strike price to get a share of stock, and you then have to deliver the share of stock to the person to whom you sold it short. A call option plus a short sale has a payoff identical to a put option plus a debt of the present value of the strike price.

2. Suppose that you hold a call option on a share of stock, and also "owe" a share of stock--that is, you have sold it short. What is the total payoff to your portfolio on the exercise date if the price of the stock is below the strike price?

Minus the stock price--or minus the strike price plus the value of a put option at expiration. A call option plus a short sale has a payoff identical to a put option plus a debt of the present value of the strike price.

3. Suppose that you are writing a call option on a stock selling for $40 per share with a $30 exercise price. The stock's standard deviation is 6% per month; the option matures in 3 months; the risk-free interest rate is 4% per year. Use appendix table 6 to calculate the value of this call option--what would be a fair price to sell it for?

From table 6, the price is 25.9% of the share price--or $10.36

4. Suppose that you are writing a call option on a stock selling for $40 per share with a $30 exercise price. The stock's standard deviation is 6% per month; the option matures in 3 months; the risk-free interest rate is 4% per year. Use appendix table 7 to calculate what initial hedge position in the stock you should take if you sell the option to a customer and wish to neutralize your risk. Should you buy or should you sell the stock? How much of the stock should you buy or sell?

You should hedge your position by buying almost a full share of the stock--0.997 of a share, according to table 7.

5. If, in question 4, the stock price goes up in the next month what, qualitatively, do you do in order to maintain your hedged position? Do you buy more stock, or sell more stock?

If the stock price goes up you buy more stock in order to maintain your hedge. If the stock price goes down you sell stock to maintain your hedge.

6. Over the coming year the common stock of Dandelion, Inc., will either halve to $50 from its current level of $100, or rise to $200. The 1-year risk-free interest rate is 5%. What is the delta of a one-year call option on Dandelion stock with a strike price of $170? What is the value of such an option?

The delta is 30/150= 0.2; the replicating portfolio is to buy 0.2 share and to borrow $9.52 at the riskfree rate--that will replicate the (0, $30) ultimate payoffs. Thus the value of the option is $20 - $9.52 = $10.48

7. [Problem 19, chapter 20]. In 1988 the Bond Corporation sold some land it owed near Rome for $110 million; it had originally purchased the land for $36 million; as a result the transaction boosted Bond's reported 1988 earnings by $74 million. In 1989 a TV program revealed that Bond had given the purchaser a put option to sell the land back to Bond for $110 million within a year, and that Bond had paid the purchaser $20 million for a call option to buy the land back for $110 million within a year. (a) What happens if the land is worth more than $110 million when the options expire? (b) What happens if the land is worth less than $110 million. (c) What is the implicit risk-free interest rate that would make sense of this transaction? (d) The TV program argued that the land was not really sold--hence no profit should have been reported. Do you agree? Why or why not?

(a) If the land is worth more than $110 million, Bond exercises his call--and buys back the land.
(b) If the land is worht less than $110 million, the purchaser exercises his put--and Bond takes back the land.
(c) Bond has paid $20 million to borrow $110 million for one year--an interest rate of a hair over 18% per year.
(d) On Bond's balance sheet the land is now valued at $110 million rather than $36 million; if in the future the land is truly sold, then the gain reported will be not the sale price minus $36 million but the sale price minus $110 million; the gain from $36 million to $110 million has to be reported as a profit, and why not report it now rather than at some time in the future? The other side of the argument is that maybe there was no profit--maybe the land is still worth only $36 million, we can't tell.

8. [Problem 23, chapter 20]. Consider the following three six-month call options:

Exercise Price	Option Price
$90	$5
$100	$11
$110	$15

Suppose that you can both buy and sell call options. How would you make money by trading in these three options?

Suppose you buy one of the $90 calls and sell one of the $110 calls. You have just cleared $10--you sold the one option for $10 more than it cost you to buy the other. Now look at what happens when the options expire. If the stock price is below $90, neither option is exercised. If the stock price is between $90 and $110, you exercise your option (and earn the difference between the stock price and the $90 strike price). If the stock price is over $90, you exercise your option, collect your stock, turn it over to the person who bought the option from you for $110, and collect the difference between the two strike prices--$20--as profit. Call options with higher prices have to sell for less--otherwise there is a money machine to be taken advantage of.

9. Suppose that the price of Carbonics stock can go up by 15% or down by 13% in the next year from the current stock price of $60, and that only these two outcomes are possible. Suppose, further, that the safe interest rate is 10% per year. What is the value of a call option on Carbonics stock? How much of a hedge in the stock would be required to replicate the option's payoffs? How would these values change if the interest rate were 5% per year?

This problem is... hard to do given the deletion of the sentence containing the strike price from the problem. Full credit for anyone who notes that such a problem is... hard to do without a strike price.

Assume that the strike price is the current price of $60. Then the stock can either go up to $69, or down to $52.50, for a payoff of $9.00 in the good state and zero in the bad state. The option delta is 9/16.5 or 0.5455. The replicating portfolio is to buy .5455 of share for $32.73, to borrow $26.03, and thus the current price of the option is $6.69--that portfolio gives you zero in the bad state and $9 in the good state. If the safe interest rate is 5%, you borrow $27.27 (the present value of the price of 0.5455 of a share in the bad state), and the value of the option is $5.45. Note that the value of the option goes down as the interest rate goes down. Why? Remember that a call option is kind of like an interest free loan of the strike price to you, and with a lower interest rate this interest free loan is worth less.

Associate Professor of Economics Brad DeLong, 601 Evans
University of California at Berkeley; Berkeley, CA 94720-3880
(510) 643-4027 phone (510) 642-6615 fax
delong@econ.berkeley.edu
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