Chapter 9, Principles of Pricing Forwards, Futures, and Options on Futures Video Solutions, An Introduction to Derivatives and Risk Management: With Stock-Trak Coupon

Problem 1

Assume that there is a forward market for a commodity. The forward price of the commodity is $$\$ 45$$. The contract expires in one year. The risk-free rate is 10 percent. Now, six months later, the spot price is $$\$ 52$$. What is the forward contract worth at this time? Explain why this is the correct value of the forward contract in six months even though the contract does not have a liquid market like a futures contract.

Narayan Hari

Numerade Educator

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Problem 2

On a particular day the S\&P 500 futures settlement price was 899.30 . You buy one conuact at around the close of the market at the settlement price. The next day, the contract opens at 899.70 and the setclement price at the close of the day is 899.10. Determine the value of the futures contract at the opening, an instant before the close, and after the close. Remember that the S\&P futures contract has a $$\$ 500$$ multiplier.

Manasvee Singh

Numerade Educator

Problem 3

On July 10 a farmer observes that the spot price of corn is $$\$ 2.735$$ per bushel and the September futures price is $$\$ 2.76$$. The farmer would like a prediction of the spot price in September but believes the market is dominated by hedgers holding long positions in corn. Explain how the farmer would use this information in a forecast of the future price of corn.

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Problem 4

Construct an arbitrage example involving an asset that can be sold short, and use it to explain the cost of carry model for pricing futures.

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Problem 5

Why is the value of a futures or forward contract at the time it is purchased equal to zero? Contrast this with the value of the corresponding spot commodity.

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Problem 6

Use the following data from January 31 of a particular year for a group of March 480 options on futures contracts to answer parts $a$ through $g$.
Futures price: 483.10
Expiration: March 18
Risk-free rate; 0.0284 percent (simple)
Call price: 6.95
Put price: 5.25.
a. Determine the intrinsic value of the call.
b. Determine the time value of the call.
c. Determine the lower bound of the call.
d. Determine the intrinsic value of the put.
e. Determine the time value of the put.
f. Determine the lower bound of the put.
g. Determine whether put-call parity holds.

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Problem 7

On September 12, a stock index futures contract was at 423.70 . The December 400 call was at 26.25, and the put was at 3.25 . The index was at 420.55 . The futures and options expire on December 21. The discrete risk-free rate was 2.75 percent. Determine if the futures and options are priced correctly in relation to each other. If they are not, construct a risk-free portfolio and show how it will earn a rate better than the risk-free rate.

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Problem 8

On a particular day, the September $S \& P 500$ stock index futures was priced at 960.50. The S\&P 500 index was at 956.49. The contract expires 73 days later.
a. Assuming continuous compounding, suppose the risk-free rate is 5.96 percent, and the dividend yield on the index is 2.75 percent. Is the futures overpriced or underpriced?
b. Assuming annual compounding, suppose the risk-free rate is 5.96 percent, and the future value of dividends on the index is $$\$ 5.27$$. Is the futures overpriced or underpriced?

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Problem 9

The following information was available:
Spot rate for Japanese yen: $$\$ 0.009313$$
730 -day forward rate for Japanese yen: $$\$ 0.010475$$ (assume a 365-day year)
U.S. risk-free rate: 7.0 percent
Japanese risk-free rate: 1.0 percent .
a. Assuming annual compounding, determine whether interest rate paricy holds and, if not, suggest a strategy.
b. Assuming continuous compounding, determine whether interest rate parity holds and, if not, suggest a strategy.

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Problem 10

Consider the wheat example in problem 24. The interest forgone on money tied up in a bushel until expiration is 0.03 , and the cost of storing the wheat is 0.0875 per bushel. The risk premium is 0.035 per bushel.
a. What is the expected price of wheat on the spot market in December?
b. Show how the futures price is related to the spot price.
c. Show how the expected spot price at expiration, your answer in part $a$, is related to the futures price today.
d. Show how the expected futures price at expiration is related to the futures price today.
e. Explain who earns the risk premium and why.

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Problem 11

Suppose there is a commodity in which the expected future spot price is $$\$ 60$$. To induce investors to buy futures contracts, a risk premium of $$\$ 4$$ is required. To store the commodity for the life of the futures contract would cost $$\$ 5.50$$. Find the futures price.

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Problem 12

(Concept Problem) Suppose that a futures margin account pays interest but at a rate that is less than the risk-free rate. Consider a trader who buys the asset and sells a futures to form a risk-free hedge. Would the trader believe the futures price should be lower or higher than if the margin account paid interest at the risk-free rate?

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Problem 13

Assume a standard deviation of 8 percent, and use the Black model to determine if the call option in problem 6 is correctly priced. If not, suggest a riskless hedge strategy.

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Problem 14

Using the information in problem 13, calculate the price of the put described in problem 6, using the Black model for pricing puts.

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Problem 14

Suggest an arbitrage strategy and show how it can be used to capture a risk-free profit. Assume that there are no transaction costs. Be sure your answer shows the payoffs at expiration and proves that these payoffs are riskless.
Suppose the U.S. interest rate for the next six months is 1.5 percent (annual compounding). The foreign interest rate is 2 percent (annual compounding). The spot price of the foreign currency in dollars is $$\$ 1.665$$. The forward price is $$\$ 1.664$$. Determine the correct forward price and recommend an arbitrage strategy.

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Problem 15

Suppose you observe a one-year futures price of $$\$ 100$$, the futures option strike price of $$\$ 90$$, and a 5 percent interest rate (annual compounding). If the futures option call price is quoted at, $$\$ 9.40$$, identify any arbitrage and explain how it would be captured.

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Problem 16

The put-call parity rule can be expressed as $C-P=\left(f_0(T)-X\right)(1+r)^{-T}$. Consider the following data: $\mathrm{f}_0(\mathrm{~T})=102, \mathrm{X}=100, \mathrm{r}=0.1, \mathrm{~T}=0.25, \mathrm{C}=4$, and $\mathrm{P}=$ 1.75. A few calculations will show that the prices do not conform to the rule.

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Problem 18

(Concept Problem) Suppose that there is a futures contract on a portfolio of stocks that currently are worth $$\$ 100$$. The futures has a life of 90 days, and during that time the stocks will pay dividends of $$\$ 0.75$$ in 30 days, $$\$ 0.85$$ in 60 days, and $$\$ 0.90$$ in 90 days. The simple interest rate is 12 percent.
a. Find the price of the futures contract assuming that no arbitrage opportunities are present.
b. Find the value of $\theta$, the cost of carry in dollars.

Anand Jangid

Numerade Educator

Problem 19

Identify and provide a brief explanation of the factors that affect the spot price of a storable asset.

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Problem 20

If futures prices are less than spot prices, the explanation usually given is the convenience yield. Explain what the convenience yteld is. Then identify certain assets on which convenience yields are more likely to exist and other assets on which they are not likely to be found.

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An Introduction to Derivatives and Risk Management: With Stock-Trak Coupon

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