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An Introduction to Derivatives and Risk Management: With Stock-Trak Coupon

Don M. Chance, Robert Brooks

Chapter 3

Principles of Option Pricing - all with Video Answers

Educators


Chapter Questions

Problem 1

Suppose that you observe a European call option that is priced at less than the value $\operatorname{Max}\left[0, S_0-X(1+r)^{-T}\right]$. What type of transaction should you execute to achieve the maximum benefit? Demonstrate that your strategy is correct by constructing a payoff table showing the outcomes of expiration.

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01:06

Problem 2

Critique the following statement, made by an options investor: "My call option is very deep-in-the-money. I don't see how it can go any higher. I think I should exercise it."

Hubert Agamasu
Hubert Agamasu
Numerade Educator

Problem 3

Explain why an option's time value is greatest when the stock price is near the exercise price and why it nearly disappears when the option is deep-in-or out-ofthe-money.

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04:10

Problem 4

What would happen in the options market if the price of an American call were less than the value $\operatorname{Max}\left(0, \mathrm{~S}_0-\mathrm{X}\right)$ ? Would your answer differ if the option were European?

Narayan Hari
Narayan Hari
Numerade Educator

Problem 5

Consider an option that expires in 68 days. The bid and ask discounts on the Treasury bill maturing in 67 days are 8.20 and 8.24 , respectively. Find the approximate risk-free rate.

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Problem 6

Why do higher interest rates lead to higher call option prices but lower put option prices?

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Problem 7

The value $\operatorname{Max}\left[0, X(1+r)^{-T}-S_0\right]$ was shown to be the lowest possible value of a European put. Why is this value irrelevant for an American put?

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Problem 8

Call prices are directly related to the stock's volatility, yet higher volatility means that the stock price can go lower. How would you resolve this apparent paradox?

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Problem 9

Why does the justification for exercising an American call early not hold up when considering an American put?

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Problem 10

In this chapter, we did not learn how to obtain the exact price of a call without knowing the price of the put and using put-call parity. In one special case, however, we can obtain an exact price for a call. Assume that the option has an infinite maturity. Then use the maximum and minimum values we learned in this chapter to obtain the prices of European and American calls.

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Problem 11

Why might two calls or puts alike in all respects but time to expiration have approximately the same price?

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Problem 12

Why might two calls or puts alike in all respects but exercise price have approximately the same price?

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Problem 13

Suppose a European put price exceeds the value predicted by put-call parity. How could an investor profit? Demonstrate that your strategy is correct by constructing a payoff table showing the outcomes at expiration.

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Problem 14

Examine the following pairs of puts, which differ only by exercise price. Determine if any violate the rules regarding relationships between American options that differ only by exercise price.
a. August 155 and 160
b. October 160 and 170

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Problem 15

Examine the following pairs of calls, which differ only by exercise price. Determine whether any violate the rules regarding relationships between American options that differ only by exercise price.
a. August 155 and 160
b. October 160 and 165

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Problem 16

Compute the intrinsic values, time values, and lower bounds of the following calls. Identify any profit opportunities that may exist. Treat these as American options for purposes of determining the intrinsic values and time values and European options for the purpose of determining the lower bounds.
a. July 160
b. October 155
c. August 170

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Problem 17

Compute the intrinsic values, time values, and lower bounds of the following puts. Identify any profit opportunities that may exist. Treat these as American options for purposes of determining the intrinsic values and time values and as European options for the purpose of determining the lower bounds.
a. July 165
b. August 160
c. October 170

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Problem 18

Check the following combinations of puts and calls, and determine whether they conform to the put-call parity rule for European options. If you see any violations, suggest a strategy.
a. July 155
b. August 160
c. October 170

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Problem 19

Repcat Question 18 using American put-calt parity, but do not suggest a strategy.

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Problem 20

Suppose that the current stock price is $$\$ 100$$, the exercise price is $$\$ 100$$, the annualiy compounded interest rate is 5 percent, the stock pays a $$\$ 1$$ dividend in the next instant, and the quoted call price is $$\$ 3.50$$ for a one year option. Identify the appropriate arbitrage opportunity and show the appropriate arbitrage strategy.

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Problem 21

Suppose Congress decides that investors should not profit when stock prices go down so it outlaws short selling. Congress has not figured out options, however, so there are no restrictions on option trading. Explain how to accomplish the equivalent of a short sale by using options.

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Problem 22

Put-call parity is a powerful formula that can be used to create equivalent combinations of options, risk-free bonds, and stock. Suppose that there are options available on the number of points Shaquille O'Neal will score in his next game. For example, a call option with an exercise price of 32 would pay off $\operatorname{Max}\left(0, S_0-32\right)$, where $S_0$ is the number of points Shaq has recorded by the end of the game. Thus, if he scores 35 , call holders receive for each call. If he scores less than 32 , call holders receive nothing. A put with an exercise price of 32 would pay off $\operatorname{Max}\left(0,32-\mathrm{S}_0\right)$. If Shaq scores more than 32 , put holders receive nothingIf he scores 28 , put holders receive $$\$ 4$$ for each put. Obviously there is no way to actually buy a position in the underlying asset, a point. However, put-call parity shows that the underlying asset can be recreated from a combination of puts, calls, and risk-free bonds. Show how this would be done, and give the formula for the price of a point.

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Problem 23

Suppose that the current stock price is $$\$ 90$$, the exercise price is $$\$ 100$$, the annually compounded interest rate is 5 percent, the stock pays a $$\$ 1$$ dividend in the next instant, and the quoted put price is $\$ 6$ for a one year option. Identify the appropriate arbitrage opportunity and show the appropriate arbitrage strategy.

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Problem 24

On December 9 of a particular year, a January Swiss franc call option with an exercise price of 46 had a price of 1.63. The January 46 put was at 0.14 . The spot rate was 47.28. All prices are in cents per Swiss franc. The option expired on January 13. The U.S. risk-free rate was 7.1 percent, while the Swiss risk-free rate was 3.6 percent. Do the following:
a. Determine the intrinsic value of the call.
b. Determine the lower bound of the call.
c. Determine the time value of the call.
d. Determine the intrinsic value of the put.
e. Determine the lower bound of the put.
f. Determine the time value of the put.
g. Determine whether put-call parity holds.

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