Question: Why do higher interest rates lead to

> Use the Black–Scholes–Merton European put option pricing formula for the October 165 put option. Repeat parts a, b, and c of the previous problem with respect to the put.

> Let the standard deviation of the continuously compounded return on the stock is 21 percent. Ignore dividends. Respond to the following: a. What is the theoretical fair value of the October 165 call? Calculate this answer by hand and then recalculate it

> 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, …, 100. Compute th

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

> 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 price but different expirations? b. Can implied volatilities be expected to vary

> Repeat the previous problem, but close the positions on September 20. Use the spreadsheet to find the profits for the possible stock prices on September 20. Generate a graph and use it to identify the approximate breakeven stock prices.

> Suppose 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 t

> Suppose 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 three months. The risk-free rate is 5 percent. Answer the following questions: a. Determine the theoretical value of the c

> On November 1, the one-month LIBOR rate is 4.0 percent and the two-month LIBOR rate is 5.0 percent. Assume that fed funds futures contracts trades at a 25 basis point rate under one-month LIBOR at the start of the delivery month. The December fed funds f

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

> Consider a European put option that expires in weeks with an exercise price of $120 trading on a stock currently priced at $126.30. Assuming an annualized volatility of the continuously compounded return on the stock of 0.78 and a continuously compounded

> A stock has a current price of $132.43. For a particular European put option that expires in three weeks, the probability of the option expiring in-the-money is 63.68 percent and the annualized volatility of the continuously compounded return on the stoc

> A stock has a current price of $115.83. A European call option on the stock expires in eight weeks and has N(d1) 0.33. If volatility changes by 0.03, approximate the amount the call price is expected to change.

> Using BlackScholesMertonBinomial10e.xlsm, compute the call and put prices for a stock option. The current stock price is $100, the exercise price is $100, the risk-free interest rate is 0 percent (continuously compounded), the volatility is 30 percent, a

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

> Using BlackScholesMertonBinomial10e.xlsm, compute the call and put prices for a stock option. The current stock price is $100, the exercise price is $105.1271, the risk-free interest rate is 5 percent (continuously compounded), the volatility is 30 perce

> Construct a long straddle using the October 165 options. Hold until the options expire. Determine the profits and graph the results. Identify the breakeven stock prices at expiration and the minimum profit.

> Buy one October 165 put contract. Hold it until the options expire. Determine the profits and graph the results. Identify the breakeven stock price at expiration. What are the maximum possible gain and loss on this transaction?

> Using BlackScholesMertonBinomial10e.xlsm, compute the call and put prices for a stock option. 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, a

> A financial institution offers a new over the-counter option that pays 150 percent of the payoff of a standard European option. Demonstrate, using BlackScholesMertonBinomial10e. xylem (or by hand), that the value of this option is simply 1.5 times the va

> 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. Y

> Again, like the previous problem, on February 4 of a particular year, the spot rate for U.S. dollars ($) expressed in euros (€) was $0.7873/€. The U.S. interest rate (compounded annually) was 5.36 percent, whereas the euro interest rate (compounded annua

> 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

> Estimate the implied volatility of the August 165 call. Compare your answer with the one you 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 BlackScholesMertonBin

> Following is the sequence of daily prices on the stock for the preceding month of June: Estimate the historical volatility of the stock for use in the Black–Scholes–Merton model. Ignore dividends on the stock. DAT

> 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. Estimate the historical volatility of the stock for use in the Black Scholes– Merton mo

> On July 6, the dividend yield on the stock is 2.7 percent. AZ yield-based dividend adjustment procedure. Calculate this answer by hand and then recalculate it using BlackScholesMertonBinomial10e.xlsm.

> 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.

> Consider a riskless spread with a long position in the August 160 call and a short position in the October 160 call. Determine the appropriate hedge ratio. Then show how a $1 stock price increase would have a neutral effect on the spread value. Discuss a

> 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

> 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

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

> 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 exam

> On February 4 of a particular year, the spot rate for U.S. dollars ($) expressed in euros (€) was $0.7873/€. The U.S. interest rate (compounded semiannually) was 5.36 percent, whereas the euro interest rate (compounded semiannually) was 3.11 percent. Fro

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

> 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. Constr

> Suppose the spot exchange rate for Narnian currency is trading for $2/N and one year later it can go up to $2.5/N, an increase of 25 percent, or down to $1.80/N, a decrease of 10 percent. Assume a call option with an exercise price of $2.05/N. Assume ini

> Consider a stock currently priced at $80. In the next period, the stock can either increase by 30 percent or decrease by 15 percent. Assume a call option with an exercise price of $80 and a risk-free rate of 6 percent. Suppose the call option is currentl

> Why does the binomial model converge to a specific value of the option as the number of time periods increases? To what value does the option converge? When n approaches infinity, to what famous model does the binomial model converge?

> Using the Black–Scholes–Merton model, compute and graph the time value decay of the October 165 call on the following dates: July 15, July 31, August 15, August 31, September 15, September 30, and October 16. Assume that the stock price remains constant.

> Consider three call options identical in every respect except for the maturity of 0.5, 1, and 1.5 years. Specifically, the stock price is $100, the annually compounded risk-free rate is 5 percent, and the strike price is $100. Use a one-period binomial m

> Consider three call options identical in every respect except for the strike price of $90, $100, and $110. Specifically, the stock price is $100, the annually compounded risk-free rate is 5 percent, and time to maturity is one year. Use a one period bino

> Use the Excel spreadsheet BlackScholesMerton Binomial10e.xlsm 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 (compounded annually), the exercise price is 100, the volati

> 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

> 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 (originally given in Chapter 3) and a volatility of 83 percent.

> The S&P 500 index is at 1,371.00, the continuously compounded risk-free rate is 5.12 percent, time to expiration is 55 days, and futures price is 1,376.42. Assuming the futures price is equal to its theoretical fair price and the underlying has a continu

> 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 in each of two binomial periods. The risk-free rate is 10 percent. Determine the price of the option today. Then constr

> Why are the probabilities of stock price movements not used in the model for calculating an option’s price? What variables are used?

> 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.

> Construct a calendar spread using the August and October 170 calls that will profit from high volatility. Close the position on August 1. Use the spreadsheet to find the profits for the possible stock prices on August 1. Generate a graph and use it to es

> 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?

> Critique the following statement made by an options investor: “My call option is very deepen-the-money. I don’t see how it can go any higher. I think I should exercise it.

> 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-of-the-money.

> Suppose you observe a European call option that is priced at less than the value Max [0, S0 − X (1 r) −T]. What type of transaction should you execute to achieve the maximum benefit? Demonstrate that your strategy is correct by constructing a payoff tabl

> (Concept Question) 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

> Put–call parity is a powerful formula that can be used to create equivalent combinations of options, risk-free bonds, and stock. Suppose there are options available on the number of points LeBron James will score in his next game. For example, a call opt

> Why does a market participant not earn a risk premium on a carry arbitrage transaction?

> Consider a call option on the U.S. dollar written in Indian rupees (). The current spot exchange rate is 70/$, the Indian interest rate is 7 percent, and the U.S. interest rate is 5 percent (both annual compounding). If you observe a one-year at-the-mone

> Suppose you observe options on Apple, Inc. stock trading such that the following condition is observed: S0 Pa S0,T,X Ca S0,T,X X 1 r T Explain how an arbitrageur would seek to capitalize on this observation and defend your answer.

> Suppose an American put is trading for $16.50 and an American call is trading for $15, where both options have identical terms. The underlying stock price is $99, and the exercise price is $100. The annual risk-free interest rate is 5 percent, and the ti

> Derive the profit equations for a pit bull spread. Determine the maximum and minimum profits and the breakeven stock price at expiration.

> Suppose a European put has an exercise price of $110 on February 5. The put expires in 45 days. Suppose the appropriate discount rate on Treasury bills maturing in 44 days is 7.615. What is the maximum value of the European put? If the put were instead a

> A non-dividend-paying common stock is trading at $100. Suppose you are considering a European put option with a strike price of $110 and one year to expiration. What is the annually compounded risk-free interest rate where the boundary condition begins t

> Suppose the annually compounded risk-free rate is 5 percent for all maturities. A no dividend-paying common stock is trading at $100. Suppose you are considering a European call option with a strike price of $110. What is the time to maturity of this opt

> What would happen in the options market if the price of an American call were less than the value Max (0, S0 − X)? Would your answer differ if the option were European? Explain.

> Suppose 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 arbit

> Suppose the current stock price is $100, 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 call price is $3.50 for a one-year option. Identify the appropriate

> 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. Th

> What are some potential dangers posed by program trading?

> Repeat problem 13 using American put–call parity, but do not suggest a strategy.

> 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

> The three fundamental profit equations for call, puts, and stock are identified symbolically in this chapter as Prepare a single graph showing both the long and short for each profit equation above. Assume the positions are held to maturity. Therefore,

> 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 European options for

> 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 fo

> Why does the justification for exercising an American call early not hold up when considering an American put? The following option prices were observed for a stock on July 6 of a particular year. Use this information to solve problems 11 through 16. Unl

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

> Suppose you are an individual investor with an options account at a brokerage firm. You purchase 20 call contracts at a price of $2.25 each on an options exchange. Explain who gets your premium.

> Which of the following combinations of exchanges is not part of the same entity? a. Chicago Mercantile Exchange and Chicago Board Options Exchange b. New York Stock Exchange and London International Financial Futures Exchange c. Chicago Board of Trade

> Suppose you observe a one-year cash-or-nothing digital call option trading for $0.48 and an equivalent cash-or-nothing digital put option trading for $0.45. Calculate the implied interest rate (annualized, continuously compounded.

> Explain three operational advantages offered by derivative markets.

> Identify the three ways in which U.S. companies can satisfy the SEC requirement that they disclose how they use derivatives to manage risk.

> Suppose FRM, Inc., issued a zero coupon, equity index-linked note with a five-year maturity. The par value is $1,000, and the coupon payment is stated as 75 percent of the equity index return or as zero. Calculate the cash flow at maturity assuming the e

> Explain the considerations facing a covered call writer regarding the choice of exercise prices.

> What is the most important component of an effective risk management system?

> What are the four objectives of the Dodd–Frank Act?

> Use the Excel spreadsheet BlackScholesMerton Binomial10e.xlsm 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 (compounded annually), the exercise price is 165, the volatility is 21

> How is liquidity a source of risk?

> What is proprietary trading?

> What is a real option? Why is it important in understanding how companies make decisions?

> Describe two problems in using the Black option on futures pricing model for pricing options on Eurodollar futures.

> What are the three ways in which derivatives can be misused?

> Standardization of a derivatives contract on an exchange applies to all of the following except? a. Expiration b. Size of contract c. Type of commodity d. Price of derivative

> What are the major functions of derivative markets in an economy?

> On September 26 of a particular year, the March Treasury bond futures contract settlement price as 94–22. Compare the following two bonds and determine which is the cheaper bond to deliver. Assume that delivery will be made on March 1. Use 5.3 percent as

> Why is delivery important if so few futures contracts end in delivery?

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

> Interpret the following statements about value at risk so that they would be easily understood by a nontechnical corporate executive: a. VAR of $1.5 million, one week, probability 0.01. b. VARof$3.75million, one year, probability 0.05

> Identify three presuppositions for well-functioning financial markets.

> 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 optio

> Why is the bid–ask spread a transaction cost?