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Options, Futures, and Other Derivatives

John C. Hull

Chapter 11

Properties of stock options - all with Video Answers

Educators

Chapter Questions

Problem 1

List the six factors that affect stock option prices.

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Problem 2

What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock when the stock price is $$\$ 28$$, the strike price is $$\$ 25$$, and the risk-free interest rate is $8 \%$ per annum?

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Problem 3

What is a lower bound for the price of a 1-month European put option on a nondividend-paying stock when the stock price is $$\$ 12$$, the strike price is $$\$ 15$$, and the riskfree interest rate is $6 \%$ per annum?

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Problem 4

Give two reasons why the early exercise of an American call option on a non-dividendpaying stock is not optimal. The first reason should involve the time value of money. The second should apply even if interest rates are zero.

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Problem 5

"The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put." Explain this statement.

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Problem 6

Why is an American call option on a dividend-paying stock always worth at least as much as its intrinsic value. Is the same true of a European call option? Explain your answer.

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Problem 7

The price of a non-dividend-paying stock is $$\$ 19$$ and the price of a 3-month European call option on the stock with a strike price of $$\$ 20$$ is $$\$ 1$$. The risk-free rate is $4 \%$ per annum. What is the price of a 3-month European put option with a strike price of $$\$ 20$$ ?

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Problem 8

Explain why the arguments leading to put-call parity for European options cannot be used to give a similar result for American options.

Rashmi Sinha
Rashmi Sinha
Numerade Educator

Problem 9

What is a lower bound for the price of a 6-month call option on a non-dividend-paying stock when the stock price is $$\$ 80$$, the strike price is $$\$ 75$$, and the risk-free interest rate is $10 \%$ per annum?

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Problem 10

What is a lower bound for the price of a 2 -month European put option on a nondividend-paying stock when the stock price is $$\$ 58$$, the strike price is $$\$ 65$$, and the riskfree interest rate is $5 \%$ per annum?

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02:10

Problem 11

A 4-month European call option on a dividend-paying stock is currently selling for $$\$ 5$$. The stock price is $$\$ 64$$, the strike price is $$\$ 60$$, and a dividend of $$\$ 0.80$$ is expected in 1 month. The risk-free interest rate is $12 \%$ per annum for all maturities. What opportunities are there for an arbitrageur?

Anand Jangid
Anand Jangid
Numerade Educator
02:10

Problem 12

A 1-month European put option on a non-dividend-paying stock is currently selling for $$\$ 2.50$$. The stock price is $$\$ 47$$, the strike price is $$\$ 50$$, and the risk-free interest rate is $6 \%$ per annum. What opportunities are there for an arbitrageur?

Anand Jangid
Anand Jangid
Numerade Educator

Problem 13

Give an intuitive explanation of why the early exercise of an American put becomes more attractive as the risk-free rate increases and volatility decreases.

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Problem 14

The price of a European call that expires in 6 months and has a strike price of $$\$ 30$$ is $$\$ 2$$. The underlying stock price is $$\$ 29$$, and a dividend of $$\$ 0.50$$ is expected in 2 months and again in 5 months. Risk-free interest rates (all maturities) are $10 \%$. What is the price of a European put option that expires in 6 months and has a strike price of $$\$ 30$$ ?

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Problem 15

Explain the arbitrage opportunities in Problem 11.14 if the European put price is $$\$ 3$$.

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Problem 16

The price of an American call on a non-dividend-paying stock is $$\$ 4$$. The stock price is $$\$ 31$$, the strike price is $$\$ 30$$, and the expiration date is in 3 months. The risk-free interest rate is $8 \%$. Derive upper and lower bounds for the price of an American put on the same stock with the same strike price and expiration date.

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Problem 17

Explain carefully the arbitrage opportunities in Problem 11.16 if the American put price is greater than the calculated upper bound.

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Problem 18

Prove the result in equation (11.7). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to $K$, and (b) a portfolio consisting of an American put option plus one share.)

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Problem 19

Prove the result in equation (11.11). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to $D+K$, and (b) a portfolio consisting of an American put option plus one share.)

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Problem 20

Consider a 5-year call option on a non-dividend-paying stock granted to employees. The option can be exercised at any time after the end of the first year. Unlike a regular exchange-traded call option, the employee stock option cannot be sold. What is the likely impact of this restriction on the early-exercise decision?

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Problem 21

Use the software DerivaGem to verify that Figures 11.1 and 11.2 are correct.

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05:30

Problem 22

What is the impact (if any) of negative interest rates on:
(a) The put-call parity result for European options
(b) The result that American call options on non-dividend-paying stocks should never be exercised early
(c) The result that American put options on non-dividend-paying stocks should sometimes be exercised early.
Assume that holding cash earning zero interest is not possible.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:10

Problem 23

Calls were traded on exchanges before puts. During the period of time when calls were traded but puts were not traded, how would you create a European put option on a nondividend-paying stock synthetically.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 24

The prices of European call and put options on a non-dividend-paying stock with an expiration date in 12 months and a strike price of $$\$ 120$$ are $$\$ 20$$ and $$\$ 5$$, respectively. The current stock price is $$\$ 130$$. What is the implied risk-free rate?

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Problem 25

A European call option and put option on a stock both have a strike price of $$\$ 20$$ and an expiration date in 3 months. Both sell for $$\$ 3$$. The risk-free interest rate is $10 \%$ per annum, the current stock price is $$\$ 19$$, and a $$\$ 1$$ dividend is expected in 1 month. Identify the arbitrage opportunity open to a trader.

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Problem 26

Suppose that $c_1, c_2$, and $c_3$ are the prices of European call options with strike prices $K_1$, $K_2$, and $K_3$, respectively, where $K_3>K_2>K_1$ and $K_3-K_2=K_2-K_1$. All options have the same maturity. Show that
$$
c_2 \leqslant 0.5\left(c_1+c_3\right)
$$
(Hint: Consider a portfolio that is long one option with strike price $K_1$, long one option with strike price $K_3$, and short two options with strike price $K_2$.)

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Problem 27

What is the result corresponding to that in Problem 11.26 for European put options?

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Problem 28

You are the manager and sole owner of a highly leveraged company. All the debt will mature in 1 year. If at that time the value of the company is greater than the face value of the debt, you will pay off the debt. If the value of the company is less than the face value of the debt, you will declare bankruptcy and the debt holders will own the company.
(a) Express your position as an option on the value of the company.
(b) Express the position of the debt holders in terms of options on the value of the company.
(c) What can you do to increase the value of your position?

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Problem 29

Consider an option on a stock when the stock price is $$\$ 41$$, the strike price is $$\$ 40$$, the risk-free rate is $6 \%$, the volatility is $35 \%$, and the time to maturity is 1 year. Assume that a dividend of $$\$ 0.50$$ is expected after 6 months.
(a) Use DerivaGem to value the option assuming it is a European call.
(b) Use DerivaGem to value the option assuming it is a European put.
(c) Verify that put-call parity holds.
(d) Explore using DerivaGem what happens to the price of the options as the time to maturity becomes very large and there are no dividends. Explain your results.

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Problem 30

Consider a put option on a non-dividend-paying stock when the stock price is $$\$ 40$$, the strike price is $$\$ 42$$, the risk-free interest rate is $2 \%$, the volatility is $25 \%$ per annum, and the time to maturity is three months. Use DerivaGem to determine the following:
(a) The price of the option if it is European (use Black-Scholes: European)
(b) The price of the option if it is American (use Binomial: American with 100 tree steps) (c) Point B in Figure 11.7.
11.31. Section 11.1 gives an example of a situation where the value of a European call option decreases as the time to maturity is increased. Give an example of a situation where the same thing happens for a European put option.

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Problem 30

Consider a put option on a non-dividend-paying stock when the stock price is $$\$ 40$$, the strike price is $$\$ 42$$, the risk-free interest rate is $2 \%$, the volatility is $25 \%$ per annum, and the time to maturity is three months. Use DerivaGem to determine the following:
(a) The price of the option if it is European (use Black-Scholes: European)
(b) The price of the option if it is American (use Binomial: American with 100 tree steps) (c) Point B in Figure 11.7.

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04:10

Problem 31

Section 11.1 gives an example of a situation where the value of a European call option decreases as the time to maturity is increased. Give an example of a situation where the same thing happens for a European put option.

Narayan Hari
Narayan Hari
Numerade Educator