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Introduction To Derivatives And Risk Management 10th Edition Don M. Chance, Robert Brooks - Solutions
Use the Excel spreadsheet Black Scholes Merton Binomial lOe.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 percent, the option expires in 102 days,
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 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?
Use the Excel spreadsheet Black Scholes Merton Binomial lOe.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 volatility is 30 percent, the option expires in
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 binomial model with u = 4/3 and d = 3/4?
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 model with u = 4/3 and d = 3/4? Calculate p
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?
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 currently trading at $12. If the option is mispriced,
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 initially that the U.S. interest rate is 1
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
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
Explain the differences between a recombining and a non-recombining tree. Why is the former more desirable?
How is the volatility of the underlying stock reflected in 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 examples? Why did we adjust the parameters in one
Describe the three primary ways of incorporating dividends into the binomial model?
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
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
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 0.05, the value of u is 1.15, and the value of d is 0.90. The stock pays a dividend at the
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.
On July 6, the dividend yield on the stock is 2.7 percent. Rework part a of problem 7 using the yield-based dividend adjustment procedure. Calculate this answer by hand and then recalculate it using Black-Scholes-Merton Binomial 1 Oe.xlsm.
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 model. (Ignore dividends on the stock.)
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.
Estimate the implied volatility of the August 165 call. Compare your answer with the one you obtained in problem 12. Use trail and error. Stop when your answer is within 0.01 of the true implied volatility. Use the Excel spreadsheet Black Scholes Merton Binomial 10e.xlsm?
Repeat the last problem using the approximation for an at-the-money call. Compare your answer with the one you obtained in problem 13. Is the approximation a good one? Why or why not?
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 whether the call was
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
A financial institution offers a new over-the-counter option that pays 150 percent of the payoff of a standard European option. Demonstrate, using Black-Scholes-Merton-Binomial lOe. xlsm (or by hand), that the value of this option is simply 1.5 times the value of an ordinary option. Let the stock
Using Black-Scholes-Merton-Binomial lOe.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, and the time to expiration is one year.
Using BlackScholesMertonBinomiallOe.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 percent, and the time to expiration is one year.
Explain each of the following concepts as they relate to call options.a. Deltab. Gammac. Rhod. Vegae. Theta
Using BlackScholesMertonBinomiallOe.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, and the time to expiration is one year.
A stock has a current price of $115.83. A European call option on the stock expires in eight weeks and has N(d,) = 0.33. If volatility changes by 0.03, approximate the amount the call price is expected to change?
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 com-pounded return on the stock is 0.76. Assuming a continuously
Consider a European put option that expires in four 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 risk-free rate of 0.0348, use the
Show how a delta hedge using a position in the stock and a long position in a put would be set up?
(Concept Problem) 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 call. Use Black Scholes Merton
Suppose you subscribe to a service that gives you estimates of the theoretically correct volatilities of stocks. The implied volatility of a particular option is substantially higher than the theoretical volatility. What action should you take? Why?
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 for options on the same stock with the same
What factors contribute to the difficulty of making a delta hedge be truly risk-free?
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 the Black-Scholes-Merton price of the call and the European
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-Merton-Binomial
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?
Suppose the stock pays a $1.10 dividend with an ex-dividend date of September 10. Rework part a of problem 7 using an appropriate dividend-adjusted procedure. Calculate this answer by hand and then recalculate it using Black Scholes-Merton Binomial 10e.xlslm?
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?
Buy one August 165 call contract. Hold it unit 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?
Repeat problem 10, but 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 identify the approximate breakeven stock price?
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?
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?
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?
Buy 100 shares of stock and buy one August 165 nut 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, and 18, determine the profit from the
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 euro put with an exercise price of $1.00 is priced at $0.0435. Construct a simple long position in the put?
Use the information in problem 16 to construct a euro covered Call. Assume that the spot rate at the start is $0.9825?
The Black-Scholes-Merton option pricing model assumes that stock price changes re log normally distributed. Show graphically how this distribution changes when an investor is long the stock and short the call?
Suppose that you wish to buy stock and protect yourself against a downside movement in its price. You consider both a covered call and a protective put. What factors will affect your decision?
The Black-Scholes-Merton option pricing model assumes that stock price changes are log normally distributed. Show graphically how this distribution changes when an investor is long the stock and long the put?
Using Black Scholes Merton Binomial lOe.xlsm, 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 one
Suppose the call price is $14.20 and the put price is $9.30 for stock options, where the exercise price is $100, the risk-free interest rate is 5 percent (continuously compounded), and the time to expiration is one year. Explain how you would create a synthetic stock position and identify the cost.
(Concept Problem) In each case examined in this chapter and in the preceding problems, we did not account for the interest on funds invested. One useful way to observe the effect of interest is to look at a conversion or a reverse conversion. Evaluate the August 165 puts and calls and recommend a
(Concept Problem) Another consideration in evaluating option strategies is the effect of transaction costs. Suppose that purchases and sales of an option incur a brokerage commission of 1 percent of the option's value. Purchases and sales of a share of stock incur a brokerage commission of 0.5
Suppose an investor is considering buying one of two call options on a particular stock with the same maturity. The only difference between the two call options is the strike prices. The rate of return on a call option is its profit divided by the investment (the call price here). Identify the
You have inherited some stock from a wealthy relative. The stock had poor performance recently, and analysts believe it has little growth potential. You would like to write calls against the stock; however, the will stipulates that you must agree not to sell it unless you need the funds for a
We briefly mentioned the synthetic call, which consists of stock and an equal number of puts. Assume that the combined value of the puts and stock exceeds the value of the actual call by less than the present value of the exercise price. Show how an arbitrage profit can be made. Do not use the data
A short position in stock can be protected by holding a call option. Determine the profit equations for this position and identify the breakeven stock price at expiration and maximum and minimum profits?
A short stock can be protected by selling a put. Determine the profit equations for this position and identify the breakeven stock price at expiration and maximum and minimum profits?
Explain the advantages and disadvantages to a covered call writer of closing out the position prior to expiration?
Explain the considerations facing a covered call writer regarding the choice of exercise prices?
The three fundamental profit equations for call, puts, and stock are identified symbolically in this chapter asII = NC [Max (0, ST - X) - C]II = NP [Max (0, S - ST) - P]II = ST - S0.Prepare a single graph showing both the long and short for each profit equation above. Assume the positions are held
Derive the profit equations for a put bull spread. Determine the maximum and minimum profits and the breakeven stock price 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 estimate the maximum and minimum profits and
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. Use the spreadsheet to find the time value in
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 any limitations of this procedure?
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?
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?
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 expiration and the minimum profit. Compare 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 expiration and the minimum profit?
Analyze the August 160/170 box spread. Determine whether a profit opportunity exists. If it does, explain how to exploit it?
Complete the following table with the correct formula related to various spread strategies.
Complete the following table with the correct formula related to various spread strategies.
Explain why a straddle is not necessarily a good strategy when the underlying event is well known to everyone?
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 than the second strategy?
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?
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 reflect a Black-Scholes-Merton implied
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. As in earlier problems, determine the profits
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 exercise price and shorter time to expiration.
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?
One way to create a bull spread positions is by purchasing a low strike call option and selling a high strike call option. Identify a strategy with put options that creates a similar bull spread-shaped profit profile?
One way to create a bear spread positions is by purchasing a high strike put option and selling a low strike put option. Identify a strategy with call options that creates a similar bear spread-shaped profit profile.The following option prices were observed for calls and puts on a stock on July 6
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 special considerations associated with this
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 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 possible stock prices on September 20. Generate a
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. Determine the profits and graph the results.
Solve for the price of a forward contract on a generic asset that expires on September 10 whose spot price as of June 10 is $45, assuming that the annually compounded risk-free rate is 6.01 percent?
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. Determine the value of the futures contract at
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 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 bushel. The risk premium is 0.035 per bushel.a.
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 yield on the index is 2.75 percent. Is the
Suppose there is a commodity in which the expected future spot price is $60. To induce investors to buy futures contracts, a risk premium of $4 is required. To store the commodity for the life of the futures contract would cost $5.50. Find the futures 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 whether the futures and options are priced
Use the following data from January 31 of a particular year for a group of March 480 options on futures constricts to answer parts a through g.Futures price: 483.10Expiration: March 13Risk-free rate: 0.0284 percent (simple)Call price: 6.95Put price: 5.25a. Determine the intrinsic value of the
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