Chapter 4, Option Pricing Models: The Binomial Model Video Solutions, An Introduction to Derivatives and Risk Management: With Stock-Trak Coupon

Problem 1

Consider a two-period, two-state world. Let the current stock price be 45 and the risk-free rate be 5 percent. Each period the stock price can go either up by 10 percent or down by 10 percent. A call option expiring at the end of the second period has an exercise price of 40 .
a. Find the stock price sequence.
b. Determine the possible prices of the call at expiration.
c. Find the possible prices of the call at the end of the first period.
d. What is the current price of the call?
e. What is the initial hedge ratio?
f. What are the two possible hedge ratios at the end of the first period?
g. Construct an example showing that the hedge works. Make sure the example illustrates how the hedge portfolio earns the risk-free rate over both periods.
h. What would an investor do if the call were overpriced? If it were underpriced?

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01:08

Problem 2

Find the value of an American put option using the binomial option pricing model. The parameters are $\mathrm{S}=62, \mathrm{X}=70, \mathrm{r}=0.08, \mathrm{u}=1.10$, and $\mathrm{d}=0.95$. There are no dividends. Use $\mathrm{n}=2$ periods.

Breanna Ollech

Numerade Educator

Problem 3

Consider a stock worth $$\$ 25$$ that can go up or down by 15 percent per period. The risk-free rate is 10 percent. Use one binomial period.
a. Determine the two possible stock prices for the next period.
b. Determine the intrinsic values at expiration of a European call option with an exercise price of $$\$ 25$$.
c. Find the value of the option today.
d. Construct a hedge by combining a position in stock with a position in the call. Show that the return on the hedge is the risk-free rate regardless of the outcome, assuming that the call sells for the value you obtained in part c.
e. Determine the rate of return from a riskless hedge if the call is selling for $$\$ 3.50$$ when the hedge is initiated.

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Problem 4

Use the Excel spreadsheet BSMbin7e.xls and determine the value of a call option and a put option on a stock currently priced at 100 , where the risk-free rate is 5 percent (annually compounded), the exercise price is 100 , the volatility is 30 percent, the option expires in one year, and there are no dividends on the stock. Let the number of binomial periods be 25 . Verify that put-call parity holds.

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Problem 5

Use the Excel spreadsheet BSMbin7exls and determine the value of a call option on a stock currently priced at 165.13 , where the risk-free rate is 5.875 percent (annually compounded), the exercise price is 165 , the volatility is 21 percent, the option expires in 102 days, and there are no dividends on the stock. Let the number of binomial periods be $1,5,10,25$, and 50 .

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Problem 6

In this chapter, we obtained the binomial option pricing formula by hedging a short position in the call option with a long position in stock. An alternative way to do this is to combine the stock and a risk-free bond to replicate the call option. Construct a one-period binomial option pricing model in which the stock and a risk-free bond are used to replicace a European call option. Then show that the option pricing formula is the same as the one developed in the text. Hint: Hold $n_s$ shares of stock and issue $B$ dollars of bonds paying r percent so that the value in both outcomes matches precisely the value of the option. Then solve for $\mathrm{n}_4$ and $\mathrm{B}$ and substitute back into the formula for the value of the portfolio today to obtain the formula for $\mathrm{C}$.

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Problem 7

Consider a European call with an exercise price of 50 on a stock priced at 60 . The stock can go up by 15 percent or down by 20 percent each of two binomial periods. The risk-free rate is 10 percent. Determine the price today of the option.
Then construct a risk-free hedge of long stock and short option. At each point in the binomial tree, show the composition and value of the hedge portfolio and demonstrate that the return is the same as the risk-free rate. On any revisions to the hedge portfolio, make the transactions (buying or selling) in stock and not options. You can borrow any additional funds required at the risk-free rate, and any excess funds should be invested at the risk-free rate.

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Problem 8

Construct a table containing the up and down factors for a one-year option with a stock volatility of 55 percent and a risk-free rate of 7 percent for $n=1,5,10,50$, and 100 where $\mathrm{n}$ is the number of binomial periods. Let $\mathrm{u}$ and $\mathrm{d}$ be defined as
$$
\begin{aligned}
& \mathrm{u}=\mathrm{e}^{\sigma \sqrt{\mathrm{T} / \mathrm{n}}} \\
& \mathrm{d}=1 / \mathrm{u}
\end{aligned}
$$

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Problem 9

The binomial model can be used to price unusual features of options. Consider the following scenario. A stock priced at $$\$ 75$$ can go up by 20 percent or down by 10 percent per period for three periods. The risk-free rate is 8 percent. A European call option expiring in three periods has an exercise price of $$\$ 70$$. The parties to the option agree, however, that the maximum payout of this option is $$\$ 40$$. Find the current value of the option. Then determine whether an American version of the option, also limited to a maximum payout of $$\$ 40$$, would have any additional value over the European version. Compare your answers to the value of the option if there were no limitation on the payoff.

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Problem 10

The binomial option pricing model has several advantages, particularly related to illustrating important concepts and practical applications. Identify and discuss three advantages related to illustrating important concepts, and three advantages related to practical applications.

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01:08

Problem 11

Use the binomial model and two time periods to determine the price of the DCRB June 130 American put. Use the appropriate parameters from the information given in the chapter, which was originally given in Chapter 3, and a volatility of 83 .

Breanna Ollech

Numerade Educator

Problem 12

Explain the similarities and differences between pricing an option by its boundary conditions and using an exact option pricing formula.

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00:40

Problem 13

Why are the up and down parameters adjusted when the number of periods is extended? Recall that in introducing the binomial model, we illustrated one-and two-period examples, but we did not adjust the parameters. What is the difference in these two examples, and why did we adjust the parameters in onc case and not in the other?

Hossam Mohamed

Numerade Educator

Problem 14

Describe the components of a hedge portfolio in the binomial option pricing model where the instrument being hedged is, first, a call, and second, a put.

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Problem 15

If the binomial model produces a call option price that is higher than the price at which the option is trading in the market, what strategy is suggested?

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01:30

Problem 16

Discuss how a binomial model accommodates the possibility of early exercise of an option.

Natalie Anderson

Numerade Educator

00:49

Problem 17

What is the principal benefit of a binomial option pricing model?

Amrita Bhasin

Numerade Educator

Problem 18

How is the volatility of the underlying stock reflected in the binomial model?

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02:02

Problem 19

Explain the differences between a recombining and non-recombining tree. Why is the former more desirable?

Hasnat Umar

Numerade Educator

Problem 20

Describe the three primary ways of incorporating dividends into the binomial model.

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02:01

Problem 21

Consider the following binomial option pricing problem involving an American call. This call has two periods to go before expiring. Its stock price is 30 , and its exercise price is 25 . The risk-free rate is .05 , the value of $u$ is 1.15 , and the value of $\mathrm{d}$ is 0.90 . The stock pays a dividend at the end of the first period at the rate of 0.06 . Find the value of the call.

James Kiss

Numerade Educator