ñòð. 15 |

optionâ€™ price. If volatility rises from 50 to 51 percent, a 1 point rise, and vega is .60, then

s

the optionâ€™ price will rise by 60 cents.

s

17. The reason is that a call option on a non-dividend-paying stock is always worth more alive

than dead, meaning that you will always get more from selling it than exercising it. If you

exercise it, you only get the intrinsic value. If you sell it, you get instrinsic value at a

minimum plus any remaining time value. For a put, however, early exercise can be

optimal. Suppose, for example, the stock price drops to zero. Thatâ€™ as good as it gets, so

s

we would like to go ahead and exercise. It will actually pay to exercise early for some

stock price greater than zero, but no general formula is known for the critical stock price.

B-54 CORRADO AND JORDAN

18. We have to use trial and error to find the answer (there is no other way). Using the option

calculator with the following inputs:

S = current stock price = $100,

K = option strike price = $120,

r = risk-free rate =.06,

Ïƒ = stock volatility = ??,

T = time to expiration = 100 days,

y = dividend yield = .03,

we try different values for the volatility until a call price of $2.57 results. Verify that Ïƒ =

40% does the trick.

19. You can either buy put options or sell call options. In either case, gains or losses on your

stock portfolio will be offset by gains or losses on your option contracts. To calculate the

number of contracts needed to hedge a $150 million portfolio with a beta of 1.4 using an

option contract value of $120,000 (100 times the index) and a delta of .50, we use the

formula from the chapter:

Portfolio beta Ã— Portfolio value

Number of option contracts =

Option delta Ã— Option contract value

Filling in the numbers, we need to sell 1.4 Ã— $150M/(.5 Ã— $120,000) = 3,500 contracts.

FUNDAMENTALS OF INVESTMENTS B-55

20. Using the option calculator with the following inputs:

S = current stock price = $80,

K = option strike price = $75,

r = risk-free rate =.05,

Ïƒ = stock volatility = .40,

T = time to expiration = 365 days, and

y = dividend yield = 3 percent,

results in the following prices and â€œgreeks:â€

Call Put

Value $15.21 $8.92

Delta .640 -.330

Gamma .011 .011

Rho .360 -.353

Eta 3.366 -2.963

Vega .285 .285

Theta .016 .015

B-56 CORRADO AND JORDAN

Chapter 16

Futures Contracts

Answers to Questions and Problems

Core Questions

1. a. Three are visible in Figure 16.1; wheat futures are traded on the Chicago Board of

Trade (CBT), Kansas City Board of Trade (KC), and Minneapolis Grain Exchange

(MPLS). There are two others, the Winnipeg Commodity Exchange (WPG) and

the MidAmerica Commodity Exchange (MCE), not shown in Figure 16.1. Of

these, the largest trading activity occurs in Chicago.

b. 1,000 troy oz..

c. At 5,000 bu. per contract, you must deliver 100,000 bushels.

d. The April contract has the largest open interest and the October contract has the

smallest open interest.

2. a. The settle price is 231.75 cents per bushel. One contract is valued as the contract

size times the per unit price, so 5,000 Ã— $2.3175 = $11,587.5.

b. The settle price is 123-23, or 123.71875% of par value. The value of a position in

10 contracts is 10 Ã— $100,000 Ã— 1.2371875 = $1,237,187.5.

The index futures price was up 6.65 for the day, or $500 Ã— 6.65 = $3,325. For a

c.

position in 25 contracts, this represents a change in value of 25 Ã— $3,325 =

$83,125, which represents a gain to a long position and a loss to a short position.

d. The contract closed down 11 for the day, so a short position would have made a

profit of 10 Ã— 60,000 Ã— $0.0011 = $660.

The contract settled down .25, so a long position loses: 20 Ã— 5,000 Ã— $0.0025 = $250.

3.

The contract settled down .75, so a short position gains: 15 Ã— 5,000 Ã— $0.0075 = $562.50.

4.

The contract settled up 6 points, so a short position loses: 30 Ã— $100,000 Ã— (6/3200) =

5.

$5,625.

6. Long hedge; i.e., buy corn futures. If corn prices do rise, then the futures position will

show a profit, offsetting the losses from higher corn prices when they are purchased.

7. Short the index futures. If the S&P 500 index subsequently declines in a market sell-off,

the futures position will show a profit, offsetting the losses on the portfolio of stocks.

FUNDAMENTALS OF INVESTMENTS B-57

8. Sell the futures. If interest rates rise, causing the value of the bonds to be less at the time

of sale, the corresponding futures hedge will show a profit. Buy yen futures. If the value of

the dollar depreciates relative to the yen in the intervening four months, then the dollar/yen

exchange rate will rise, and the payment required by the importer in dollars will rise. A

long yen futures position would profit from the dollar's depreciation and offset the

importer's higher invoice cost.

9. Sell crude oil futures. Price declines in the oil market would be offset by a gain on the

short position.

Intermediate Questions

10. The total open interest on the D-mark is 61,889 contracts. This is the number of contracts.

Each contract has a long and a short, so the open interest represents either the number of

long positions or the number of short positions. Each contract calls for the delivery of DM

125,000, and the settle price on the March contract is $.5648 per mark, or $.5648 Ã—

125,000 = $70,600. With 61,889 contracts, the total dollar value is about $4.4 billion.

11. If the contract settles down, a long position loses money. The loss per contract is: 42,000

Ã— $.01 = $420, so when the account is marked-to-market and settled at the end of the

trading day, your balance is $580, which is less than the maintenance margin. The

minimum price change for a margin call is $250 = 42,000 Ã— X, or X = $.00595 = 0.595

cents per pound.

12. It is true. Each contract has a buyer and a seller, a long and a short. One side can only

profit at the expense of the other. Including commissions, futures contracts, like most

derivative assets, are actually negative sum gains. This doesnâ€™ make them inappropriate

t

tools, by the way; it just means that, on average and before commissions, they are a break-

even proposition.

Establish your account at an initial margin of 20 Ã— $1,200 = $24,000. Your maintenance

13.

ñòð. 15 |