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

Don M. Chance, Robert Brooks

Chapter 5

Option Pricing Models: The Black-Scholes-Merton Model - all with Video Answers

Educators

Chapter Questions

Problem 1

Suppose that you subscribe to a service that gives you estimates of the theoretically correct volatilities of stocks. You note that the implied volatility of a particular option is substantially higher than the theoretical volatility. What action should you take?

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Problem 2

Explain each of the following concepts as they relate to call options.
a. Delta
b. Gamma
c. Rho
d. Vega
e. Theta

Ronald Prasad
Ronald Prasad
Numerade Educator

Problem 3

A stock is priced at $$\$ 50$$ with a volatility of 35 percent. A call option with an exercise price of $$\$ 50$$ has an expiration in one year. The risk-free rate is 5 percent. Construct a table for stock prices of $$\$ 5,10,15, \ldots, 100$$. Compute the BlackScholes-Merton price of the call and the European lower bound and verify that the former is at least as large as the latter. Use the spreadsheet BSMbin7e.xls or the Windows program BSMbwin7e.exe.

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Problem 4

What factors contribute to the difficulty of making a delta hedge be truly risk-free?

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Problem 5

Which assumption in the Black-Scholes-Merton model did we drop somewhere in the chapter? What did we do in removing that assumption to make the model give the correct option price?

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Problem 6

Let the standard deviation of the continuously compounded return on the stock be 21 percent. Ignore dividends. Answer the following:
a. What is the theoretical fair value of the October 165 call? Calculate this answer by hand and then recalculate it using either BSMbin7e.xls or BSMbwin7e.exe.
b. Based on your answer in part $a$, recommend a riskless strategy.
c. If the stock price decreases by $$\$ 1$$, how will the option position offset the loss on the stock?

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Problem 7

Use the Black-Scholes-Merton European put option pricing formula for the October 165 put option. Repeat parts $a, b$, and $c$ of question 6 with respect to the put.

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Problem 8

Suppose the stock pays a $$\$ 1.10$$ dividend with an ex-dividend date of September 10. Rework part $a$ of problem 6 using an appropriate dividend-adjusted procedure. Calculate this answer by hand and then recalculate it using either BSMbin7e.xls or BSMbwin7e.exe.

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Problem 9

On July 6 , the dividend yield on the stock is 2.7 percent. Rework part $a$ of problem 6 using the yield-based dividend adjustment procedure. Calculate this answer by hand and then recalculate it using either the BSMbin7e.xls or the BSMbwin7e.exe. program.

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Problem 10

Suppose on July 7 the stock will go ex-dividend with a dividend of $$\$ 2$$. Assuming that the options are American, determine whether the July 160 call would be exercised.

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Problem 11

On December 9 , a Swiss franc call option expiring on January 13 had an exercise price of $$\$ 0.46$$. The spot exchange rate was $$\$ 0.4728$$. The U.S. risk-free rate was 7.1 percent, and the Swiss risk-free rate was 3.6 percent. The volatility of the exchange rate was 0.145 . Determine if the call was correctly priced at $$\$ 0.0163$$.

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Problem 12

Following is the sequence of daily prices on the stock for the preceding month of June:
$$
\begin{array}{llll}
\hline \text { Date } & \text { Price } & \text { Date } & \text { Price } \\
6 / 1 & 159.88 & & \\
6 / 2 & 157.25 & 6 / 16 & 162.00 \\
6 / 3 & 160.25 & 6 / 17 & 161.38 \\
6 / 4 & 161.38 & 6 / 19 & 160.88 \\
6 / 5 & 160.00 & 61.38 \\
6 / 8 & 161.25 & 6 / 22 & 163.95 \\
6 / 9 & 159.88 & 6 / 23 & 164.88 \\
6 / 10 & 157.75 & 6 / 24 & 166.13 \\
6 / 11 & 157.63 & 6 / 25 & 167.88 \\
6 / 12 & 156.63 & 6 / 26 & 166.50 \\
6 / 15 & 159.63 & 6 / 29 & 165.38 \\
\hline
\end{array}
$$
Estimate the historical volatility of the stock for use in the Black-Scholes-Merton model. (Ignore dividends on the stock.)

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Problem 13

Estimate the implied volatility of the August 165 call. Compare your answer with that obtained in problem 12. Use trial and error. Stop when your answer is within 0.01 of the true implied volatility. Use the Excel spreadsheet BSMbin7e.xls or the Windows program BSMbwin7e.exe.

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03:22

Problem 14

Repeat problem 13 using the approximation for an at-the-moncy call. Compare your answer with the one you obtained in problem 13. Is the approximation a good one?

Linda Hand
Linda Hand
Numerade Educator
01:14

Problem 15

Explain the difference between a normal and a lognormal distribution as it pertains to stock prices.

Carl David Cepeda
Carl David Cepeda
Numerade Educator

Problem 16

Show how a delta hedge using a position in the stock and a long position in a put would be set up.

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Problem 17

A financial institution offers a new over-the-counter option that pays off $150 \%$ of the payoff of a standard European option. Demonstrate, using BSMbin7e.xls or BSMbwin7e.exe (or by hand), that the value of this option is simply 1.5 times the value of an ordinary option. Let the stock price be 82 , the exercise price be 80 , the risk-free rate (continuously compounded) be 4 percent, the volatility be 40 percent, and the option expire in one year.

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Problem 18

Explain what we mean when we say that the binomial model is a discrete time model and the Black-Scholes-Merton model is a continuous time model.

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Problem 19

What is the most critical variable in the Black-Scholes-Merton model? Explain.

James Kiss
James Kiss
Numerade Educator

Problem 20

Using BSMbin7exls or BSMbwin7e.exe, compute the call and put prices for a stock option, where the current stock price is $$\$ 100$$, the exercise price is $$\$ 100$$, the risk-free interest rate is 5 percent (continuously compounded), the volatility is 30 percent, and the time to expiration is 1 year. Now assume the next instant the company announces an immediate 2 -for-1 stock split. As expected, the stock price falls to $$\$ 50$$. The options exchange rules call for dividing the exercise price by 2 and doubling the number of option contracts held. Verify that the option holders are unharmed by these stock split rules of the options exchange.

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Problem 21

Answer the following questions as they relate to implied volatilities.
a. Can implied volatilities be expected to vary for options on the same stock with the same exercise prices but different expirations?
b. Can implied volatilities be expected to vary for options on the same stock with the same expiration but different exercise prices?
c. Why and how are implied volatilities used to quote options prices?

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01:30

Problem 22

Compare the variables in the binomial model with those in the Black-ScholesMerton model. Note any differences and explain.

Natalie Anderson
Natalie Anderson
Numerade Educator

Problem 23

Consider the right-hand side of the Black-Scholes-Merton formula as consisting of the sum of two terms. Explain what each of those terms represents.

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Problem 24

A stock is selling for $$\$ 100$$ with a volatility of 40 percent. Consider a call option on the stock with an exercise price of 100 and an expiration of one year. The risk-free rate is 4.5 percent. Let the call be selling for its Black-Scholes-Merton value. You construct a delta-hedged position involving the saie of 10,000 calls and the purchase of an appropriate number of shares. You can buy and sell shares and calls only in whole numbers. At the end of the next day, the stock is at $$\$ 99$$. You then adjust your position accordingly to maintain the delta hedge. The following day the stock closes at $$\$ 102$$. In all cases use the spreadsheet BSMbin7e.xls or the Windows program BSMbwin7e.exe to price the call.
a. Compare the amount of moncy you end up with to the amount you would have had if you had invested the money in a risk-free bond. Explain why the target was or was not achieved.
b. Now add another option, one on the same stock with an exercise price of 105 and the same expiration. Reconstruct the problem by delta and gamma hedging. Explain why the target was or was not achieved.

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Problem 25

Suppose that a stock is priced at 80 and has a volatility of 0.35 . You buy a call option with an exercise price of 80 that expires in 3 months. The risk-free rate is 5 percent. Answer the following questions:
a. Determine the theoretical value of the call. Use BSMbin7e.xls or BSMbwin7e.exe.
b. Suppose that the actual call is selling for $$\$ 5$$. Suggest a strategy, but do not worry about hedging the risk. Simply buy or sell 100 calls.
c. After purchasing the call, you investigate your possible profits. You expect to unwind the position one month later, at which time you expect the call to have converged to its Black-Scholes-Merton value. Of course you do not know what the stock price will be, but you can calculate the profits for stock prices over a reasonable range. You expect that the stock will not vary beyond $$\$ 60$$ and $$\$ 100$$. Determine your proft in increments of $$\$ 10$$ of the stock price. Comment on your results.
Note: This problem will prepare you for Chapter 6 .

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