Options, Futures, and Other Derivatives

John C. Hull

Chapter 13 Binomial trees - all with Video Answers

Educators

Chapter Questions

Problem 1

A stock price is currently $$\$ 40$$. It is known that at the end of 1 month it will be either $$\$ 42$$ or $$\$ 38$$. The risk-free interest rate is $8 \%$ per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $$\$39$$?

Narayan Hari

Numerade Educator

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Problem 2

Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.

Nick Johnson

Numerade Educator

00:31

Problem 3

What is meant by the "delta" of a stock option?

Amrita Bhasin

Numerade Educator

Problem 4

A stock price is currently $$\$ 50$$. It is known that at the end of 6 months it will be either $$\$ 45$$ or $$\$ 55$$. The risk-free interest rate is $10 \%$ per annum with continuous compounding. What is the value of a 6-month European put option with a strike price of $$\$ 50$$ ?

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04:14

Problem 5

A stock price is currently $$\$ 100$$. Over each of the next two 6-month periods it is expected to go up by $10 \%$ or down by $10 \%$. The risk-free interest rate is $8 \%$ per annum with continuous compounding. What is the value of a 1-year European call option with a strike price of $$\$ 100$$ ?

Narayan Hari

Numerade Educator

Problem 6

For the situation considered in Problem 13.5, what is the value of a 1-year European put option with a strike price of $$\$ 100$$ ? Verify that the European call and European put prices satisfy put-call parity.

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00:31

Problem 7

What are the formulas for $u$ and $d$ in terms of volatility?

Prashansha Kaushik

Numerade Educator

Problem 8

Consider the situation in which stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of the life of the option.

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Problem 9

A stock price is currently $$\$ 50$$. It is known that at the end of 2 months it will be cither $$\$ 53$$ or $$\$ 48$$. The risk-free interest rate is $10 \%$ per annum with continuous compounding. What is the value of a 2-month European call option with a strike price of $$\$49$$? Use no arbitrage arguments.

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Problem 10

A stock price is currently $$\$ 80$$. It is known that at the end of 4 months it will be cither $$\$ 75$$ or $$\$ 85$$. The risk-free interest rate is $5 \%$ per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $$\$80$$? Use no arbitrage arguments.

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02:55

Problem 11

A stock price is currently $$\$ 40$$. It is known that at the end of 3 months it will be either $$\$ 45$$ or $$\$ 35$$. The risk-free rate of interest with quarterly compounding is $8 \%$ per annum. Calculate the value of a 3-month European put option on the stock with an exercise price of $$\$ 40$$. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

Narayan Hari

Numerade Educator

Problem 12

A stock price is currently $$\$ 50$$. Over each of the next two 3-month periods it is expected to go up by $6 \%$ or down by $5 \%$. The risk-free interest rate is $5 \%$ per annum with continuous compounding. What is the value of a 6-month European call option with a strike price of $$\$ 51$$ ?

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Problem 13

For the situation considered in Problem 13.12, what is the value of a 6-month European put option with a strike price of $$\$ 51$$ ? Verify that the European call and European put prices satisfy put-call parity. If the put option were American, would it ever be optimal to exercise it early at any of the nodes on the tree?

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Problem 14

A stock price is currently $$\$ 25$$. It is known that at the end of 2 months it will be either $$\$ 23$$ or $$\$ 27$$. The risk-free interest rate is $10 \%$ per annum with continuous compounding. Suppose $S_T$ is the stock price at the end of 2 months. What is the value of a derivative that pays off $S_T^2$ at this time?

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Problem 15

Calculate $u, d$, and $p$ when a binomial tree is constructed to value an option on a foreign currency. The tree step size is 1 month, the domestic interest rate is $5 \%$ per annum, the foreign interest rate is $8 \%$ per annum, and the volatility is $12 \%$ per annum.

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Problem 16

The volatility of a non-dividend-paying stock whose price is $$\$ 78$$, is $30 \%$. The risk-free rate is $3 \%$ per annum (continuously compounded) for all maturities. Calculate values for $u, d$, and $p$ when a 2 -month time step is used. What is the value a 4-month European call option with a strike price of $$\$ 80$$ given by a two-step binomial tree. Suppose a trader sells 1,000 options ( 10 contracts). What position in the stock is necessary to hedge the trader's position at the time of the trade?

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Problem 17

A stock index is currently 1,500. Its volatility is $18 \%$. The risk-free rate is $4 \%$ per annum (continuously compounded) for all maturities and the dividend yield on the index is $2.5 \%$. Calculate values for $u, d$, and $p$ when a 6 -month time step is used. What is the value a 12 -month American put option with a strike price of 1,480 given by a two-step binomial tree.

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Problem 18

The futures price of a commodity is $$\$ 90$$. Use a three-step tree to value (a) a 9-month American call option with strike price $$\$ 93$$ and (b) a 9-month American put option with strike price $$\$ 93$$. The volatility is $28 \%$ and the risk-free rate (all maturities) is $3 \%$ with continuous compounding.

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Problem 19

The current price of a non-dividend-paying biotech stock is $$\$ 140$$ with a volatility of $25 \%$. The risk-free rate is $4 \%$. For a 3 -month time step:
(a) What is the percentage up movement?
(b) What is the percentage down movement?
(c) What is the probability of an up movement in a risk-neutral world?
(d) What is the probability of a down movement in a risk-neutral world?
Use a two-step tree to value a 6-month European call option and a 6-month European put option. In both cases the strike price is $$\$ 150$$.

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Problem 20

In Problem 13.19, suppose a trader sells 10,000 European call options and the two-step tree describes the behavior of the stock. How many shares of the stock are needed to hedge the 6-month European call for the first and second 3-month period? For the second time period, consider both the case where the stock price moves up during the first period and the case where it moves down during the first period.

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02:55

Problem 21

A stock price is currently $$\$ 50$$. It is known that at the end of 6 months it will be either $$\$ 60$$ or $$\$ 42$$. The risk-free rate of interest with continuous compounding is $12 \%$ per annum. Calculate the value of a 6-month European call option on the stock with an exercise price of $$\$ 48$$. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

Narayan Hari

Numerade Educator

04:14

Problem 22

A stock price is currently $$\$ 40$$. Over each of the next two 3-month periods it is expected to go up by $10 \%$ or down by $10 \%$. The risk-free interest rate is $12 \%$ per annum with continuous compounding. (a) What is the value of a 6-month European put option with a strike price of $$\$ 42$$ ? (b) What is the value of a 6-month American put option with a strike price of $$\$ 42$$ ?

Narayan Hari

Numerade Educator

04:14

Problem 22

A stock price is currently $$$40$$. Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. (a) What is the value of a 6 month European put option with a strike price of $$42$$? (b) What is the value of a 6-month American put option with a strike price of $$42$$?

Narayan Hari

Numerade Educator

04:14

Problem 23

Narayan Hari

Numerade Educator

Problem 24

A stock price is currently $$\$ 30$$. During each 2-month period for the next 4 months it will increase by $8 \%$ or reduce by $10 \%$. The risk-free interest rate is $5 \%$. Use a two-step tree to calculate the value of a derivative that pays off $\left[\max \left(30-S_T, 0\right)\right]^2$, where $S_T$ is the stock price in 4 months. If the derivative is American-style, should it be exercised early?

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Problem 25

Consider a European call option on a non-dividend-paying stock where the stock price is $$\$ 40$$, the strike price is $$\$ 40$$, the risk-free rate is $4 \%$ per annum, the volatility is $30 \%$ per annum, and the time to maturity is 6 months.
(a) Calculate $u, d$, and $p$ for a two-step tree.
(b) Value the option using a two-step tree.
(c) Verify that DerivaGem gives the same answer.
(d) Use DerivaGem to value the option with $5,50,100$, and 500 time steps.

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Problem 26

Repeat Problem 13.25 for an American put option on a futures contract. The strike price and the futures price are $$\$ 50$$, the risk-free rate is $10 \%$, the time to maturity is 6 months, and the volatility is $40 \%$ per annum.

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03:27

Problem 27

Footnote 1 of this chapter shows that the correct discount rate to use for the real-world expected payoff in the case of the call option considered in Figure 13.1 is $55.96 \%$. Show that if the option is a put rather than a call the discount rate is $-70.4 \%$. Explain why the two real-world discount rates are so different.

James Kiss

Numerade Educator

Problem 28

A stock index is currently 990 , the risk-free rate is $5 \%$, and the dividend yield on the index is $2 \%$. Use a three-step tree to value an 18-month American put option with a strike price of 1,000 when the volatility is $20 \%$ per annum. How much does the option holder gain by being able to exercise early? When is the gain made?

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04:10

Problem 29

Calculate the value of 9-month American call option to buy 1 million units of a foreign currency using a three-step binomial tree. The current exchange rate is 0.79 and the strike price is 0.80 (both expressed as dollars per unit of the foreign currency). The volatility of the exchange rate is $12 \%$ per annum. The domestic and foreign risk-free rates are $2 \%$ and $5 \%$, respectively. What position in the foreign currency is initially necessary to hedge the risk?

Narayan Hari

Numerade Educator