Options, Futures, and Other Derivatives

John C. Hull

Chapter 18 Futures options and Black’s model - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain the difference between a call option on yen and a call option on yen futures.

Check back soon!

00:47

Problem 2

Why are options on bond futures more actively traded than options on bonds?

Amrita Bhasin

Numerade Educator

01:04

Problem 3

"A futures price is like a stock paying a dividend yield." What is the dividend yield?

Bobby Barnes

University of North Texas

A futures price is currently 50 . At the end of six months it will be either 56 or 46 . The risk-free interest rate is $6 \%$ per annum. What is the value of a six-month European call option on the futures with a strike price of 50 ?

Check back soon!

View

Problem 5

How does the put-call parity formula for a futures option differ from put-call parity for an option on a non-dividend-paying stock?

Rashmi Sinha

Numerade Educator

Problem 6

Consider an American futures call option where the futures contract and the option contract expire at the same time. Under what circumstances is the futures option worth more than the corresponding American option on the underlying asset?

Check back soon!

Problem 7

Calculate the value of a five-month European futures put option when the futures price is $$\$ 19$$, the strike price is $$\$ 20$$, the risk-free interest rate is $12 \%$ per annum, and the volatility of the futures price is $20 \%$ per annum.

Check back soon!

Problem 8

Suppose you buy a put option contract on October gold futures with a strike price of $$\$ 1,400$$ per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $$\$ 1,380$$ ?

Check back soon!

Problem 9

Suppose you sell a call option contract on April live cattle futures with a strike price of 130 cents per pound. Each contract is for the delivery of 40,000 pounds. What happens if the contract is exercised when the futures price is 135 cents?

Check back soon!

Problem 10

Consider a two-month futures call option with a strike price of 40 when the risk-free interest rate is $10 \%$ per annum. The current futures price is 47 . What is a lower bound for the value of the futures option if it is (a) European and (b) American?

Check back soon!

Problem 11

Consider a four-month futures put option with a strike price of 50 when the risk-free interest rate is $10 \%$ per annum. The current futures price is 47 . What is a lower bound for the value of the futures option if it is (a) European and (b) American?

Check back soon!

Problem 12

A futures price is currently 60 and its volatility is $30 \%$. The risk-free interest rate is $8 \%$ per annum. Use a two-step binomial tree to calculate the value of a six-month European call option on the futures with a strike price of 60 . If the call were American, would it ever be worth exercising it early?

Check back soon!

Problem 13

In Problem 18.12, what does the binomial tree give for the value of a six-month European put option on futures with a strike price of 60 ? If the put were American, would it ever be worth exercising it early? Verify that the call prices calculated in Problem 18.12 and the put prices calculated here satisfy put-call parity relationships.

Check back soon!

Problem 14

A futures price is currently 25 , its volatility is $30 \%$ per annum, and the risk-free interest rate is $10 \%$ per annum. What is the valuc of a nine-month European call on the futures with a strike price of 26 ?

Check back soon!

Problem 15

A futures price is currently 70 , its volatility is $20 \%$ per annum, and the risk-free interest rate is $6 \%$ per annum. What is the valuc of a five-month European put on the futures with a strike price of 65 ?

Check back soon!

Problem 16

Suppose that a one-year futures price is currently 35. A one-year European call option and a one-ycar Furopcan put option on the futures with a strike price of 34 are both priced at 2 in the market. The risk-free interest rate is $10 \%$ per annum. Identify an arbitrage opportunity.

Check back soon!

Problem 17

"The price of an at-the-moncy European futures call option always equals the price of a similar at-the-moncy European futures put option." Explain why this statement is true.

Check back soon!

Problem 18

Suppose that a futures price is currently 30 . The risk-free interest rate is $5 \%$ per annum. A three-month American futures call option with a strike price of 28 is worth 4 . Calculate bounds for the price of a three-month American futures put option with a strike price of 28.

Check back soon!

Problem 19

Show that, if $C$ is the price of an American call option on a futures contract when the strike price is $K$ and the maturity is $T$, and $P$ is the price of an American put on the same futures contract with the same strike price and exercise date, then
$$
F_0 e^{-r T}-K<C-P<F_0-K e^{-r T}
$$
where $F_0$ is the futures price and $r$ is the risk-free rate. Assume that $r>0$ and that there is no difference between forward and futures contracts. (Hint: Use an analogous approach to that indicated for Problem 17.12.)

Check back soon!

Problem 20

Calculate the price of a three-month European call option on the spot value of silver. The three-month futures price is $$\$ 12$$, the strike price is $$\$ 13$$, the risk-free rate is $4 \%$ and the volatility of the price of silver is $25 \%$.

Check back soon!

01:28

Problem 21

A corporation knows that in three months it will have $$\$ 5$$ million to invest for 90 days at LIBOR minus 50 basis points and wishes to ensure that the rate obtained will be at least $6.5 \%$. What position in exchange-traded options should it take to hedge?

Narayan Hari

Numerade Educator

04:14

Problem 22

A futures price is currently 40 . It is known that at the end of three months the price will be either 35 or 45 . What is the value of a three-month European call option on the futures with a strike price of 42 if the risk-free interest rate is $7 \%$ per annum?

Narayan Hari

Numerade Educator

Problem 23

The futures price of an asset is currently 78 and the risk-free rate is $3 \%$. A six-month put on the futures with a strike price of 80 is currently worth 6.5 . What is the value of a sixmonth call on the futures with a strike price of 80 if both the put and call are European? What is the range of possible values of the six-month call with a strike price of 80 if both put and call are American?

Check back soon!

Problem 24

Use a three-step tree to value an American futures put option when the futures price is 50 , the life of the option is 9 months, the strike price is 50 , the risk-free rate is $3 \%$, and the volatility is $25 \%$.

Check back soon!

Problem 25

It is February 4. July call options on corn futures with strike prices of 260, 270, 280, 290, and 300 cost $26.75,21.25,17.25,14.00$, and 11.375 , respectively. July put options with these strike prices cost $8.50,13.50,19.00,25.625$, and 32.625 , respectively. The options mature on June 19, the current July corn futures price is 278.25 , and the risk-free interest rate is $1.1 \%$. Calculate implied volatilities for the options using DerivaGem. Comment on the results you get.

Check back soon!

Problem 26

Calculate the implicd volatility of soybean futures prices from the following information concerning a European put on soybean futures:
$$
\begin{array}{lc}
\hline \text { Current futures price } & 525 \\
\text { Exercise price } & 525 \\
\text { Risk-free rate } & 6 \% \text { per annum } \\
\text { Time to maturity } & 5 \text { months } \\
\text { Put price } & 20 \\
\hline
\end{array}
$$

Check back soon!

Problem 27

Calculate the price of a six-month European put option on the spot value of the $S\&P 500$ . The six-month forward price of the index is 1,400 , the strike price is 1,450 , the risk-free rate is $5 \%$, and the volatility of the index is $15 \%$.

Check back soon!

Problem 28

The strike price of a futures option is 550 cents, the risk-free interest rate is $3 \%$, the volatility of the futures price is $20 \%$, and the time to maturity of the option is 9 months. The futures price is 500 cents.
(a) What is the price of the option if it is a European call?
(b) What is the price of the option if it is a European put?
(c) Verify that put-call parity holds.
(d) What is the futures price for a futures-style option if it is a call?
(e) What is the futures price for a futures-style option if it is a put?

Check back soon!

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28