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 using Black Scholes MertonBinomial10e.xlsm. 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?
> Suppose you are concerned about your firm’s jet fuel exposure. Further, your analysis suggests the best futures contract to hedge jet fuel is unleaded gasoline. The fuel volatility (standard deviation of changes in your firm’s entire jet fuel exposure) i
> On July 1, a portfolio manager holds $1 million face value of Treasury bonds, the 11 l/4s maturing in about 29 years. The price is 107 14/32. The bond will need to be sold on August 30. The manager is concerned about rising interest rates and believes th
> A call option on the euro expiring in six months has an exercise price of $1.00 and is priced at $0.0385. Construct a simple long position in the call.
> A major bread maker is planning to purchase wheat in the near future. Identify and explain the appropriate hedging strategy.
> Assume that there is a forward market for a commodity. The forward price of the commodity is $45. The contract expires in one year. The risk-free rate is 10 percent. Now six months later, the spot price is $52. What is the forward contract worth at this
> Rework the last problem assuming that the index was at 388.14 at expiration. Determine the profit from the arbitrage trade and express it in terms of the profit from the spot and futures sides of the transaction. How does your answer relate to that in pr
> Explain why the Black option on futures pricing model is simply a pricing model for options on instruments with a zero cost of carry.
> Explain why American call options on futures could be exercised early when call options on the spot are not. Assume that there are no dividends.
> On July 10, a farmer observes that the spot price of corn is $2.735 per bushel and the September futures price is $2.76. The farmer would like a prediction of the spot price in September but believes the market is dominated by hedgers holding long positi
> Calculate the net effect that a change in the annually compounded risk-free rate from 6.83 percent to 6.60 percent would make on the price of a commodity futures contract whose spot price as of March 30 was $49.90, assuming that there is a $5.60 storage
> Suppose that a futures margin account pays interest but at a rate that is less than the risk-free rate. Consider a trader who buys the asset and sells a future to form a risk-free hedge. Would the trader believe that the futures price should be lower or
> Suppose there is a futures contract on a portfolio of stocks that currently are worth $100. The futures have a life of 90 days, and during that time, the stocks will pay dividends of $0.75 in 30 days, $0.85 in 60 days, and $0.90 in 90 days. The simple in
> Using the Black–Scholes–Merton option pricing model and the generic carry formula for forward contracts (using continuous compounding), demonstrate that Ce S0,T,X Ce f 0 T ,T,X .
> Buy 100 shares of stock and buy one August 165 put contract. Hold the position until expiration. Determine the profits and graph the results. Determine the breakeven stock price at expiration, the maximum profit, and the maximum loss. For problems 16, 17
> Suppose you observe a one-year futures price of $100, the futures option strike price of $90, and a 5 percent interest rate (annual compounding). If the futures option call price is quoted at $9.40, identify any arbitrage and explain how it would be capt
> Suppose the U.S. interest rate for the next six months is 1.5 percent (annual compounding). The foreign interest rate is 2 percent (annual compounding). The spot price of the foreign currency in dollars is $1,665. The forward price is $1,664. Determine t
> Given an investor holding a long position two days prior to expiration, when will he prefer futures contracts to forward contracts? (Do not assume interest rate certainty.)
> On July 5, a stock index futures contract was at 394.85. The index was at 392.54, the risk-free rate was 2.83 percent, the dividend yield was 2.08 percent, and the contract expired on September 20. Determine whether an arbitrage opportunity was available
> The put–call parity rule can be expressed as C P f 0 T X 1 r T. Consider the following data: f 0 T 102, X 100, r 0 1, T 0 25, C 4, and P 1 75. A few calculations will show that the prices do not conform to the rule. Suggest an arbitrage strategy and show
> Use the following data from January 31 of a particular year for a group of March 480 options on futures contracts to answer parts a through g. Futures price: 483.10 Expiration: March 13 Risk-free rate: 0.0284 percent (simple) Call price: 6.95 Put price:
> On September 12, a stock index futures contract was at 423.70. The December 400 call was at 26.25, and the put was at 3.25. The index was at 420.55. The futures and options expire on December 21. The discrete risk-free rate was 2.75 percent. Determine wh
> The following information was available: spot rate for Japanese yen: $0.009313; 730-day forward rate for Japanese yen: $0.010475 (assume a 365-day year); U.S. risk-free rate: 7.0 percent; Japanese risk-free rate: 1.0 percent a. Assuming annual compoundi
> On a particular day, the September S&P 500 stock index futures was priced at 960.50. The S&P 500 index was at 956.49. The contract expires 73 days later. a. Assuming continuous compounding, suppose the risk-free rate is 5.96 percent and the dividend yie
> On September 26, the spot price of wheat was $3.5225 per bushel and the price of a December wheat futures was $3.64 per bushel. The interest forgone on money tied up in a bushel until expiration is 0.03, and the cost of storing the wheat is 0.0875 per bu
> Repeat the previous problem, but close the position on September 1. Use the spreadsheet to find the profits for the possible stock prices on September 1. Generate a graph and use it to approximate the breakeven stock price.
> Construct an arbitrage example involving an asset that can be sold short and use it to explain the cost of carry model for pricing futures.
> On a particular day, the S&P 500 futures settlement price was 899.30. You buy one contract at the settlement price at around the close of the market. The next day the contract opens at 899.70, and the settlement price at the close of the day is 899.10. D
> Suppose you are expecting the stock price to move substantially over the next three months. You are considering a butterfly spread. Construct an appropriate butterfly spread using the October 160, 165, and 170 calls. Hold the position until expiration. D
> Construct a collar using the October 160 put. First, use the Black–Scholes–Merton model to identify a call that will make the collar have zero up-front cost. Then close the position on September 20. Use the spreadsheet to find the profits for the possibl
> Assume that on March 16, the cheapest bond to deliver on the June Treasury bond futures contract is the 14s, callable in about 19 years and maturing in about 24 years. Coupons are paid on November 15 and May 15. The price of the bond is 161 23/32, and th
> Repeat problem 6, but close the position 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 price.
> Construct a bear money spread using the October 165 and 170 calls. Hold the position until the options expire. Determine the profits and graph the results. Identify the breakeven stock price at expiration and the maximum and minimum profits. Discuss any
> The chapter showed how analyzing a box spread is like a capital budgeting problem using the NPV approach. Consider the internal rate of return method of examining capital budgeting problems and analyze the box spread in that context.
> Many option traders use a combination of a money spread and a calendar spread called a diagonal spread. This transaction involves the purchase of a call with a lower exercise price and longer time to expiration and the sale of a call with a higher exerci
> Another variation of the straddle is called a strangle. A strangle is the purchase of a call with a higher exercise price and a put with a lower exercise price. Evaluate the strangle strategy by examining the purchase of the August 165 put and 170 call.
> Buy 100 shares of stock and write one October 170 call contract. Hold the position until expiration. Determine the profits and graph the results. Identify the breakeven stock price at expiration, the maximum profit, and the maximum loss.
> Pear, Inc. is presently trading at $100 per share; at-the-money one-month calls are trading at $5.43, and puts are trading at $5.01; and at the-money two-month calls are trading at $7.72, and puts are trading at $6.89. At present, these option prices ref
> Explain conceptually the choice of strike prices when it comes to designing a zero-cost collar. Specifically, address the costs and benefits of two strategies. One strategy has a higher put strike price than the second strategy.
> Explain conceptually the choice of strike prices when it comes to designing a call-based bull spread. Specifically, address the costs and benefits of two bull spread strategies. One strategy has the call strike prices further from the current stock price
> Explain why a straddle is not necessarily a good strategy when the underlying event is well known to everyone.
> Analyze the August 160/170 box spread. Determine whether a profit opportunity exists. If it does, explain how to exploit it.
> It is August 20, and you are trying to determine which of two bonds is the cheaper bond to deliver on the December Treasury bond futures contract. The futures price is 89 12/32. Assume that delivery will be made on December 14 and use 7.9 percent as the
> A strip is a variation of a straddle involving two puts and one call. Construct a short strip using the August 170 options. Hold the position until the options expire. Determine the profits and graph the results. Identify the breakeven stock prices at ex
> A slight variation of a straddle is a strap, which uses two calls and one put. Construct a long strap using the October 165 options. Hold the position until expiration. Determine the profits and graph the results. Identify the breakeven stock prices at e
> Identify and discuss four nontraded delivery options related to U.S. Treasury bond futures contracts.
> Buy one August 165 call contract. Hold it until the options expire. Determine the profits and graph the results. Then identify the breakeven stock price at expiration. What is the maximum possible loss on this transaction?
> Buying an at-the-money put has a greater return potential than buying an out-of-the-money put because it is more likely to be in-the-money. Appraise this statement.
> Suppose the stock pays a $1.10 dividend with an ex-dividend date of September 10. Rework part of problem 7 using an appropriate dividend adjusted procedure. Calculate this answer by hand and then recalculate it using BlackScholesMertonBinomial10e.xlsm.
> 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.
> 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.
> Why do higher interest rates lead to higher call option prices but lower put option prices?
> 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.
