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Options, Futures, and Other Derivatives

John C. Hull

Chapter 21

Basic numerical procedures - all with Video Answers

Educators

Chapter Questions

02:41

Problem 1

Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?

Gaurav Kalra
Gaurav Kalra
Numerade Educator

Problem 2

Calculate the price of a 3-month American put option on a non-dividend-paying stock when the stock price is $$\$ 60$$, the strike price is $$\$ 60$$, the risk-free interest rate is $10 \%$ per annum, and the volatility is $45 \%$ per annum. Use a binomial tree with a time interval of 1 month.

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Problem 3

Explain how the control variate technique is implemented when a tree is used to value American options.

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Problem 4

Calculate the price of a 9-month American call option on corn futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is $8 \%$ per annum, and the volatility is $30 \%$ per annum. Use a binomial tree with a time interval of 3 months.

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Problem 5

Consider an option that pays off the amount by which the final stock price exceeds the average stock price achieved during the life of the option. Can this be valued using the binomial tree approach? Explain your answer.

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Problem 6

"For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine." Explain this statement.

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01:52

Problem 7

Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 8 holds.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
04:49

Problem 8

Use stratified sampling with 100 trials to improve the estimate of $\pi$ in Business Snapshot 21.1 and Table 21.1.

Raymond Matshanda
Raymond Matshanda
Numerade Educator

Problem 9

Explain why the Monte Carlo simulation approach cannot easily be used for Americanstyle derivatives.

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Problem 10

A 9-month American put option on a non-dividend-paying stock has a strike price of $$\$ 49$$. The stock price is $$\$ 50$$, the risk-free rate is $5 \%$ per annum, and the volatility is $30 \%$ per annum. Use a three-step binomial tree to calculate the option price.

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Problem 11

Use a three-time-step binomial tree to value a 9-month American call option on wheat futures. The current futures price is 400 cents, the strike price is 420 cents, the risk-free rate is $6 \%$, and the volatility is $35 \%$ per annum. Estimate the delta of the option from your tree.

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Problem 12

A 3-month American call option on a stock has a strike price of $$\$ 20$$. The stock price is $$\$ 20$$, the risk-free rate is $3 \%$ per annum, and the volatility is $25 \%$ per annum. A dividend of $$\$ 2$$ is expected in 1.5 months. Use a three-step binomial tree to calculate the option price.

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Problem 13

A 1-year American put option on a non-dividend-paying stock has an exercise price of $$\$ 18$$. The current stock price is $$\$ 20$$, the risk-free interest rate is $15 \%$ per annum, and the volatility of the stock price is $40 \%$ per annum. Use the DerivaGem software with four 3 -month time steps to estimate the value of the option. Display the tree and verify that the option prices at the final and penultimate nodes are correct. Use DerivaGem to value the European version of the option. Use the control variate technique to improve your estimate of the price of the American option.

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Problem 14

A 2 -month American put option on a stock index has an exercise price of 480 . The current level of the index is 484 , the risk-free interest rate is $10 \%$ per annum, the dividend yield on the index is $3 \%$ per annum, and the volatility of the index is $25 \%$ per annum. Divide the life of the option into four half-month periods and use the tree approach to estimate the value of the option.

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Problem 15

How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used?

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Problem 16

Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate and antithetic variable technique be used to improve numerical efficiency? Explain why it is necessary to calculate six values of the option in each simulation trial when both the control variate and the antithetic variable technique are used.

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05:25

Problem 17

How do equations (21.27) to (21.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency?

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator

Problem 18

An American put option on a non-dividend-paying stock has 4 months to maturity. The exercise price is $$\$ 21$$, the stock price is $$\$ 20$$, the risk-free rate of interest is $10 \%$ per annum, and the volatility is $30 \%$ per annum. Use the explicit version of the finite difference approach to value the option. Use stock price intervals of $$\$ 4$$ and time intervals of 1 month.

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Problem 19

The spot price of copper is $$\$ 0.60$$ per pound. Suppose that the futures prices (dollars per pound) are as follows:
$$
\begin{array}{rr}
\hline 3 \text { months } & 0.59 \\
6 \text { months } & 0.57 \\
9 \text { months } & 0.54 \\
12 \text { months } & 0.50 \\
\hline
\end{array}
$$
The volatility of the price of copper is $40 \%$ per annum and the risk-free rate is $6 \%$ per annum. Use a binomial tree to value an American call option on copper with an exercise price of $$\$ 0.60$$ and a time to maturity of 1 year. Divide the life of the option into four 3-month periods for the purposes of constructing the tree. (Hint: As explained in Section 18.7, the futures price of a variable is its expected future price in a riskneutral world.)

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01:56

Problem 20

Use the binomial tree in Problem 21.19 to value a security that pays off $x^2$ in 1 year where $x$ is the price of copper.

Lauren Shelton
Lauren Shelton
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01:12

Problem 21

When do the boundary conditions for $S=0$ and $S \rightarrow \infty$ affect the estimates of derivative prices in the explicit finite difference method?

Carson Merrill
Carson Merrill
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03:37

Problem 22

How would you use the antithetic variable method to improve the estimate of the European option in Business Snapshot 21.2 and Table 21.2?

Heather Duong
Heather Duong
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Problem 23

A company has issued a 3-year convertible bond that has a face value of $$\$ 25$$ and can be exchanged for two of the company's shares at any time. The company can call the issue, forcing conversion, when the share price is greater than or equal to $$\$ 18$$. Assuming that the company will force conversion at the earliest opportunity, what are the boundary conditions for the price of the convertible? Describe how you would use finite difference methods to value the convertible assuming constant interest rates. Assume there is no risk of the company defaulting.

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04:26

Problem 24

Provide formulas that can be used for obtaining three random samples from standard normal distributions when the correlation between sample $i$ and sample $j$ is $\rho_{i, j}$.

Chris Trentman
Chris Trentman
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04:10

Problem 25

An American put option to sell a Swiss franc for dollars has a strike price of $$\$ 0.80$$ and a time to maturity of 1 year. The Swiss franc's volatility is $10 \%$, the dollar interest rate is $6 \%$, the Swiss franc interest rate is $3 \%$, and the current exchange rate is 0.81 . Use a threestep binomial tree to value the option. Estimate the delta of the option from your tree.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 26

A 1-year American call option on silver futures has an exercise price of $$\$ 9.00$$. The current futures price is $$\$ 8.50$$, the risk-free rate of interest is $12 \%$ per annum, and the volatility of the futures price is $25 \%$ per annum. Use the DerivaGem software with four 3-month time steps to estimate the value of the option. Display the tree and verify that the option prices at the final and penultimate nodes are correct. Use DerivaGem to value the European version of the option. Use the control variate technique to improve your estimate of the price of the American option.

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Problem 27

A 6-month American call option on a stock is expected to pay dividends of $$\$ 1$$ per share at the end of the second month and the fifth month. The current stock price is $$\$ 30$$, the exercise price is $$\$ 34$$, the risk-free interest rate is $10 \%$ per annum, and the volatility of the part of the stock price that will not be used to pay the dividends is $30 \%$ per annum. Use the DerivaGem software with the life of the option divided into six time steps to estimate the value of the option. Compare your answer with that given by Black's approximation (see Section 15.12).

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04:10

Problem 28

The current value of the British pound is $$\$ 1.60$$ and the volatility of the pound/dollar exchange rate is $15 \%$ per annum. An American call option has an exercise price of $$\$ 1.62$$ and a time to maturity of 1 year. The risk-free rates of interest in the United States and the United Kingdom are $6 \%$ per annum and $9 \%$ per annum, respectively. Use the explicit finite difference method to value the option. Consider exchange rates at intervals of 0.20 between 0.80 and 2.40 and time intervals of 3 months.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 29

Answer the following questions concerned with the alternative procedures for constructing trees in Section 21.4:
(a) Show that the binomial model in Section 21.4 is exactly consistent with the mean and variance of the change in the logarithm of the stock price in time $\Delta t$.
(b) Show that the trinomial model in Section 21.4 is consistent with the mean and variance of the change in the logarithm of the stock price in time $\Delta t$ when terms of order $(\Delta t)^2$ and higher are ignored.
from each node. Assume that the branching is from $S$ to $S u, S m$, or $S d$ with $m^2=u d$. Match the mean and variance of the change in the logarithm of the stock price exactly.
(c) Construct an alternative to the trinomial model in Section 21.4 so that the probabilities are $1 / 6,2 / 3$, and $1 / 6$ on the upper, middle, and lower branches emanating

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Problem 30

The DerivaGem Application Builder functions enable you to investigate how the prices of options calculated from a binomial tree converge to the correct value as the number of time steps increases. (See Figure 21.4 and Sample Application A in DerivaGem.) Consider a put option on a stock index where the index level is 900 , the strike price is 900 , the risk-free rate is $5 \%$, the dividend yield is $2 \%$, and the time to maturity is 2 years.
(a) Produce results similar to Sample Application A on convergence for the situation where the option is European and the volatility of the index is $20 \%$.
(b) Produce results similar to Sample Application A on convergence for the situation where the option is American and the volatility of the index is $20 \%$.
(c) Produce a chart showing the pricing of the American option when the volatility is $20 \%$ as a function of the number of time steps when the control variate technique is used.
(d) Suppose that the price of the American option in the market is 85.0. Produce a chart showing the implied volatility estimate as a function of the number of time steps.

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Problem 30

The DerivaGem Application Builder functions enable you to investigate how the prices of options calculated from a binomial tree converge to the correct value as the number of time steps increases. (See Figure 21.4 and Sample Application A in DerivaGem.) Consider a put option on a stock index where the index level is 900 , the strike price is 900 , the risk-free rate is $5 \%$, the dividend yield is $2 \%$, and the time to maturity is 2 years.
(a) Produce results similar to Sample Application A on convergence for the situation where the option is European and the volatility of the index is $20 \%$.
(b) Produce results similar to Sample Application A on convergence for the situation where the option is American and the volatility of the index is $20 \%$.
(c) Produce a chart showing the pricing of the American option when the volatility is $20 \%$ as a function of the number of time steps when the control variate technique is used.
(d) Suppose that the price of the American option in the market is 85.0. Produce a chart showing the implied volatility estimate as a function of the number of time steps.

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01:19

Problem 31

Estimate delta, gamma, and theta from the tree in Example 21.3. Explain how each can be interpreted.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
00:59

Problem 32

How much is gained from exercising early at the lowest node at the 9-month point in Example 21.4?

James Kiss
James Kiss
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Problem 33

A four-step Cox-Ross-Rubinstein binomial tree is used to price a one-year American put option on an index when the index level is 500 , the strike price is 500 , the dividend yield is $2 \%$, the risk-free rate is $5 \%$, and the volatility is $25 \%$ per annum. What is the option price, delta, gamma, and theta? Explain how you would calculate vega and rho.

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