Options, Futures, and Other Derivatives (10th Edition) 作业一

第2章

Problem 2.3.

Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a margin call? What happens if you do not meet the margin call? 

There will be a margin call when $1,000 has been lost from the margin account. This will occur when the price of silver increases by 1,000/5,000  $0.20. The price of silver must therefore rise to $17.40 per ounce for there to be a margin call. If the margin call is not met, your broker closes out your position.

 

Problem 2.10.

Explain how margin accounts protect futures traders against the possibility of default. 
Margin is money deposited by a trader with his or her broker. It acts as a guarantee that the trader can cover any losses on the futures contract. The balance in the margin account is adjusted daily to reflect gains and losses on the futures contract. If losses lead to the balance in the margin account falling below a certain level, the trader is required to deposit a further margin. This system makes it unlikely that the trader will default. A similar system of margin accounts makes it unlikely that the trader’s broker will default on the contract it has with the clearing house member and unlikely that the clearing house member will default with the clearing house.

 

Problem 2.23.

Suppose that on October 24, 2018, a company sells one April 2019 live-cattle futures contracts. It closes out its position on January 21, 2019. The futures price (per pound) is 121.20 cents when it enters into the contract, 118.30 cents when it closes out its position, and 118.80 cents at the end of December 2018. One contract is for the delivery of 40,000 pounds of cattle. What is the total profit? How is it taxed if the company is (a) a hedger and (b) a speculator? Assume that the company has a December 31 year end. 

 

The total profit is

40,000×(1.2120−1.1830) = $1,160

If the company is a hedger this is all taxed in 2019. If it is a speculator

40,000×(1.2120−1.1880) = $960

is taxed in 2018 and

40,000×(1.1880−1.1830) = $200

is taxed in 2019.

 

 

Problem 2.31.

Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 4% per annum. How could you make money if the June and December futures contracts for a particular year trade at $50 and $56?

You could go long one June oil contract and short one December contract. In June you take delivery of the oil borrowing $50 per barrel at 4% to meet cash outflows. The interest accumulated in six months is about 50×0.04×1/2 or $1 per barrel. In December the oil is sold for $56 per barrel which is more than the $51 that has to be repaid on the loan. The strategy therefore leads to a profit. Note that this profit is independent of the actual price of oil in June and December. It will be slightly affected by the daily settlement procedures.

 

第3章

Problem 3.3.

Explain what is meant by a perfect hedge. Does a perfect hedge always lead to a better outcome than an imperfect hedge? Explain your answer.

A perfect hedge is one that completely eliminates the hedger’s risk. A perfect hedge does not always lead to a better outcome than an imperfect hedge. It just leads to a more certain outcome.

Consider a company that hedges its exposure to the price of an asset. Suppose the asset’s price movements prove to be favorable to the company. A perfect hedge totally neutralizes the company’s gain from these favorable price movements. An imperfect hedge, which only partially neutralizes the gains, might well give a better outcome.

Problem 3.7.

A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures is currently standing at 1080, and each contract is for delivery of $250 times the index. What is the hedge that minimizes risk? What should the company do if it wants to reduce the beta of the portfolio to 0.6?

The formula for the number of contracts that should be shorted gives

 

Rounding to the nearest whole number, 89 contracts should be shorted. To reduce the beta to 0.6, half of this position, or a short position in 44 contracts, is required.

 

 

Problem 3.13.

 “If the minimum-variance hedge ratio is calculated as 1.0, the hedge must be perfect." Is this statement true? Explain your answer. 

The statement is not true. The minimum variance hedge ratio is

 

It is 1.0 when  and. Since  the hedge is clearly not perfect.

 

Problem 3.30.

It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change beta of the portfolio to 0.5 during the period July 16 to November 16. The index is currently 2,000, and each contract is on $250 times the index.

What position should the company take?

Suppose that the company changes its mind and decides to increase the beta of the portfolio from 1.2 to 1.5. What position in futures contracts should it take? 

The company should short

 

or 140 contracts.

The company should take a long position in

 

or 60 contracts.

 

第5章

Problem 5.6.

Explain carefully the meaning of the terms convenience yield and cost of carry. What is the relationship between futures price, spot price, convenience yield, and cost of carry? 

Convenience yield measures the extent to which there are benefits obtained from ownership of the physical asset that are not obtained by owners of long futures contracts. The cost of carry is the interest cost plus storage cost less the income earned. The futures price, , and spot price, , are related by

 

where  is the cost of carry,  is the convenience yield, and  is the time to maturity of the futures contract.

 

Problem 5.9.

A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 5% per annum with continuous compounding.

What are the forward price and the initial value of the forward contract?

Six months later, the price of the stock is $45 and the risk-free interest rate is still 5%. What are the forward price and the value of the forward contract? 

The forward price, , is given by equation (5.1) as:

F₀=40e^0.05×1= 42.05

or $42.05. The initial value of the forward contract is zero.

The delivery price  in the contract is $42.05. The value of the contract, , after six months is given by equation (5.5) as:

f = 45−42.05e^−0.05×0.5=3.99

 

 

i.e., it is $3.99. The forward price is:

45e^0.05×0.5=46.14

 

or $46.14.
Problem 5.14.

The two-month interest rates in Switzerland and the United States are, respectively, 1% and 2% per annum with continuous compounding. The spot price of the Swiss franc is $1.0500. The futures price for a contract deliverable in two months is also $1.0500. What arbitrage  opportunities does this create? 

The theoretical futures price is

1.0500e^{(0.02-0.01)×2/12}= 1.0518

The actual futures price is too low. This suggests that a Swiss arbitrageur should sell Swiss francs for US dollars and buy Swiss francs back in the futures market.

Problem 5.15.

The spot price of silver is $25 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 5% per annum for all maturities, calculate the futures price of silver for delivery in nine months. 

The present value of the storage costs for nine months are

0.06+0.06e^-0.05×0.25+0.06e^-0.05×0.5= 0.178

or $0.178. The futures price is from equation (5.11) given by  where

 

i.e., it is $26.14 per ounce.

Problem 5.30.

A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a six-month forward contract on the stock.

 What are the forward price and the initial value of the forward contract?

Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract? 

 

The present value, , of the income from the security is given by:

 

From equation (5.2) the forward price, , is given by:

 

or $50.01. The initial value of the forward contract is (by design) zero. The fact that the forward price is very close to the spot price should come as no surprise. When the compounding frequency is ignored the dividend yield on the stock equals the risk-free rate of interest.

In three months:

 

The delivery price, , is 50.01. From equation (5.6) the value of the short forward contract, , is given by

 

and the forward price is