Options, futures and other derivatives

John C. Hull

Chapter 9 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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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 risk-free 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

Explain why an American call option on a dividend-paying stock is 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.

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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 risk-free interest rate is $5 \%$ per annum?

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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?

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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

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. The term structure is flat, with all risk-free interest rates being $10 \%$. 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 carefully the arbitrage opportunities in Problem 9.14 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 9.16 if the American put price is greater than the calculated upper bound.

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Problem 18

Prove the result in equation (9.4).

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Problem 19

Prove the result in equation (9.8).

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Problem 20

Consider a 5 -year employee stock option on a non-dividend-paying stock. 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 9.1 and 9.2 are correct.

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Problem 22

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 23

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
$$
c_2 \leqslant 0.5\left(c_1+c_3\right)
$$

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Problem 24

What is the result corresponding to that in Problem 9.23 for European put options?

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Problem 25

Suppose that 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 26

Consider an option on a stock when the stock price is $$\$ 41$$, the strike price is $$\$ 40$$, the riskfree 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. For this purpose, assume there are no dividends. Explain the results you get.

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