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Derivatives Markets 4th edition Rober L. Macdonald - Solutions

Make the same assumptions as in the previous problem.a. What is the price of a standard European put with 2 years to expiration?b. Suppose you have a compound call giving you the right to pay $2 1 year from today to buy the option in (a). For what stock prices in 1 year will you exercise this
Consider the hedging example using gap options, in particular the assumptions and prices in Table 14.4.a. Implement the gap pricing formula. Reproduce the numbers in Table 14.4.b. Consider the option withK1= $0.8 andK2 = $1. If volatility were zero, what would the price of this option be? What do
Problem 12.11 showed how to compute approximate Greek measures for an option. Use this technique to compute delta for the gap option in Figure 14.3, for stock prices ranging from $90 to $110 and for times to expiration of 1 week, 3 months, and 1 year. How easy do you think it would be to hedge a
Consider the gap put in Figure 14.4. Using the technique in Problem 12.11, compute Vega for this option at stock prices of $90, $95, $99, $101, $105, and $110, and for times to expiration of 1 week, 3 months, and 1 year. Explain the values you compute. Discuss.
Let S = $40, σ = 0.30, r = 0.08, T = 1, and δ = 0. Also let Q = $60, σQ = 0.50, δQ = 0.04, and ρ = 0.5. What is the price of a standard 40-strike call with S as the underlying asset? What is the price of an exchange option with S as the underlying asset and 0.667 × Q as the strike price?
Let S = $40, σ = 0.30, r = 0.08, T = 1, and δ = 0. Also let Q = $60, σQ = 0.50, δQ = 0, and ρ = 0.5. In this problem we will compute prices of exchange calls with S as the price of the underlying asset and Q as the price of the strike asset.a. Vary δ from 0 to 0.1. What happens to the price
Let S = $40, σ = 0.30, r = 0.08, T = 1, and δ = 0. Also let Q = $40, σQ = 0.30, δQ = 0, and ρ = 1. Consider an exchange call with S as the price of the underlying asset and Q as the price of the strike asset. a. What is the price of an exchange call with S as the underlying asset and Q as the
XYZ wants to hedge against depreciations of the euro and is also concerned about the price of oil, which is a significant component of XYZ's costs. However, there is a positive correlation between the euro and the price of oil: The euro appreciates when the price of oil rises. Explain how an
Suppose you observe the prices {5, 4, 5, 6, 5}. What are the arithmetic and geometric averages? Now you observe {3, 4, 5, 6, 7}. What are the two averages? What happens to the difference between the two measures of the average as the standard deviation of the observations increases?
A chooser option (also known as an as-you-like-it option) becomes a put or call at the discretion of the owner. For example, consider a chooser on the S&R index for which both the call, with value C(St , K, T − t), and the put, with value P(St , K, T − t), have a strike price of K. The index
Suppose that S = $100, σ = 30%, r = 8%, and δ = 0. Today you buy a contract which, 6 months from today, will give you one 3-month to expiration at-the-money call option. (This is called a forward start option.) Assume that r, σ, and δ are certain not to change in the next 6 months.a. Six months
You wish to insure a portfolio for 1 year. Suppose that S = $100, σ = 30%, r = 8%, and δ = 0. You are considering two strategies. The simple insurance strategy entails buying one put option with a 1-year maturity at a strike price that is 95% of the stock price. The rolling insurance strategy
Suppose that S = $100, K = $100, r = 0.08, σ = 0.30, δ = 0, and T = 1. Construct a standard two-period binomial stock price tree using the method in Chapter 10.a. Consider stock price averages computed by averaging the 6-month and 1-year prices. What are the possible arithmetic and geometric
Using the information in the previous problem, compute the prices ofIn previous problem Suppose that S = $100, K = $100, r = 0.08, σ = 0.30, δ = 0, and T = 1. Construct a standard two-period binomial stock price tree using the method in Chapter 10.a. An Asian arithmetic average strike call.b. An
Repeat Problem 14.3, except construct a three-period binomial tree. Assume that Asian options are based on averaging the prices every 4 months. a. What are the possible geometric and arithmetic averages after 1 year? b. What is the price of an Asian arithmetic average price call? c. What is the
Let S = $40, K = $45, σ = 0.30, r = 0.08, T = 1, and δ = 0. a. What is the price of a standard call? b. What is the price of a knock-in call with a barrier of $44? Why? c. What is the price of a knock-out call with a barrier of $44? Why?
Let S = $40, K = $45, σ = 0.30, r = 0.08, δ = 0, and T = {0.25, 0.5, 1, 2, 3, 4, 5, 100}. a. Compute the prices of knock-out calls with a barrier of $38. b. Compute the ratio of the knock-out call prices to the prices of standard calls. Explain the pattern you see.
Repeat the previous problem for up-and-out puts assuming a barrier of $44. a. Compute the prices of knock-out calls with a barrier of $38. b. Compute the ratio of the knock-out call prices to the prices of standard calls. Explain the pattern you see.
Let S = $40, K = $45, σ = 0.30, r = 0.08, and δ = 0. Compute the value of knockout calls with a barrier of $60 and times to expiration of 1 month, 2 months, and so on, up to 1 year. As you increase time to expiration, what happens to the price of the knock-out call? What happens to the price of
Consider a 5-year equity-linked note that pays one share of XYZ at maturity. The price of XYZ today is $100, and XYZ is expected to pay its annual dividend of $1 at the end of this year, increasing by $0.50 each year. The fifth dividend will be paid the day before the note matures. The appropriate
Compute λ if the dividend on the CD is 0 and the payoff is $1300 − max(0, 1300 − S5.5) + λ × max(0, S5.5 − 2600) and the initial price is to be $1300.
Consider the equity-linked CD example in Section 15.3. a. What happens to the value of the CD as the interest rate, volatility, and dividend yield change? In particular, consider alternative volatilities of 20% and 40%, interest rates of 0.5% and 7%, and dividend yields of 0.5% and 2.5%. b. For
Use the information in Table 15.5.a. What is the price of a bond that pays one barrel of oil 2 years from now? b. What annual cash payment would the bond have to make in order to sell for $20.90?
Using the information in Table 15.5, suppose we have a bond that pays one barrel of oil in 2 years.a. Suppose the bond pays a fractional barrel of oil as an interest payment after 1 year and after 2 years, in addition to the one barrel after 2 years. What payment would the bond have to make in
Using the information in Table 15.5, assume that the volatility of oil is 15%.a. Show that a bond that pays one barrel of oil in 1 year sells today for $19.2454. b. Consider a bond that in 1 year has the payoff S1 + max(0, K1 ˆ’ S1) ˆ’ max(0, S1ˆ’ K2). Find the strike
Swaps often contain caps or floors. In this problem, you are to construct an oil contract that has the following characteristics: The initial cost is zero. Then in each period, the buyer pays the market price of oil if it is between K1 and K2; otherwise, if S K2, the buyer pays K2 (there is a floor
You have been asked to construct an oil contract that has the following characteristics: The initial cost is zero. Then in each period, the buyer pays S − F, with a cap of $21.90 − F and a floor of $19.90 − F. Assume oil volatility is 15%. What is F?
Suppose the effective semiannual interest rate is 3%.a. What is the price of a bond that pays one unit of the S&P index in 3 years?b. What semiannual dollar coupon is required if the bond is to sell at par?c. What semiannual payment of fractional units of the S&P index is required if the
Consider again the Netscape PEPS discussed in this chapter and assume the following: the price of Netscape is $39.25, Netscape is not expected to pay dividends, the interest rate is 7%, and the 5-year volatility of Netscape is 40%. What is the theoretical value of the PEPS?
A DECS contract pays two shares if ST < 27.875, 1.667 shares if the price is above ST > 33.45, and $27.875 and $55.75 otherwise. The quarterly dividend is $0.87. Value this DECS assuming that S = $26.70, σ = 35%, r = 9%, and T = 3.3 and that the underlying stock pays a quarterly dividend of
A stock purchase contract with a zero initial premium calls for you to pay for one share of stock in 3 years. The stock price is $100 and the 3-year interest rate is 3%.a. If you expect the stock to have a zero dividend yield, what price in 3 years would you agree to pay for the stock?b. If the
Value the M&I stock purchase contract assuming that the 3-year interest rate is 3% and the M&I volatility is 15%. How does your answer change if volatility is 35%? Discuss.
Use information from Table 15.5.a. What is the price of a bond that pays one unit of the S&P index in 2 years?b. What quarterly dollar coupon is required if the bond is to sell at par?c. What quarterly payment of fractional units of the S&P index is required if the bond is to sell at par?
Assume that the volatility of the S&P index is 30%. a. What is the price of a bond that after 2 years pays S2 + max(0, S2 − S0)? b. Suppose the bond pays S2 + [λ × max(0, S2 − S0)]. For what λ will the bond sell at par? Discuss.
Assume that the volatility of the S&P index is 30%. a. What is the price of a bond that after 2 years pays S0 + max(0, S2 − S0)? b. Suppose the bond pays S0 + [λ × max(0, S2 − S0)] in year 2. For what λ will the bond sell at par? Discuss.
Assume that the volatility of the S&P index is 30% and consider a bond with the payoff S2 + λ × [max(0, S2 − S0) − max(0, S2 − K)]. a. If λ = 1 and K = $1500, what is the price of the bond? b. Suppose K = $1500. For what λ will the bond sell at par? c. If λ = 1, for what K will the bond
Explain how to synthetically create the equity-linked CD in Section 15.3 by using a forward contract on the S&P index and a put option instead of a call option. Discuss.
Consider the equity-linked CD in Section 15.3. Assuming that profit for the issuing bank is zero, draw a graph showing how the participation rate, γ , varies with the coupon, c. Repeat assuming the issuing bank earns profit of 5%. Discuss.
Compute the required semiannual cash dividend if the expiration payoff to the CD is $1300 − max(0, 1300 − S5.5) and the initial price is to be $1300.
There is a single debt issue with a maturity value of $120. Compute the yield on this debt assuming that it matures in 1 year, 2 years, 5 years, or 10 years. What debt-to-equity ratio do you observe in each case? Discuss.
Assume there are 20 shares outstanding. Compute the value of the warrant and the share price for each of the following situations. a. Warrants for 2 shares expire in 5 years and have a strike price of $15. b. Warrants for 15 shares expire in 10 years and have a strike of $20. Discuss.
A firm has outstanding a bond with a 5-year maturity and maturity value of $50, convertible into 10 shares. There are also 20 shares outstanding. What is the price of the warrant? The share price? Suppose you were to compute the value of the convertible as a risk-free bond plus an option, valued
Suppose a firm has 20 shares of equity, a 10-year zero-coupon debt with a maturity value of $200, and warrants for 8 shares with a strike price of $25. What is the value of the debt, the share price, and the price of the warrant?
Suppose a firm has 20 shares of equity and a 10-year zero-coupon convertible bond with a maturity value of $200, convertible into 8 shares. What is the value of the debt, the share price, and the price of the warrant?
Using the assumptions of Example 16.4, and the stock price derived in Example 16.5 suppose you were to perform a “naive” valuation of the convertible as a riskfree bond plus 50 call options on the stock. Howdoes the price you compute compare with that computed in Example 16.5?
Consider Panels B and D in Figure 16.4. Using the information in each panel, compute the share price at each node for each bond issue.In figure 16.4.
As discussed in the text, compensation options are prematurely exercised or canceled for a variety of reasons. Suppose that compensation options both vest and expire in 3 years and that the probability is 10% that the executive will die in year 1 and 10% in year 2. Thus, the probability that the
XYZ Corp. compensates executives with 10-year European call options, granted at the money. If there is a significant drop in the share price, the company’s board will reset the strike price of the options to equal the new share price. The maturity of the repriced option will equal the remaining
Suppose that top executives of XYZ are told they will receive at-the-money call options on 10,000 shares each year for the next 3 years. When granted, the options have 5 years to maturity. XYZ’s stock price is $100, volatility is 30%, and r = 8%. Estimate the value of this promise.
Suppose that S = $100, σ = 30%, r = 6%, t = 1, and δ = 0. XYZ writes a European put option on one share with strike price K = $90. a. Construct a two-period binomial tree for the stock and price the put. Compute the replicating portfolio at each node. b. If the firm were synthetically creating
There is a single debt issue. Compute the yield on this debt assuming that it matures in 1 year and has a maturity value of $127.42, 2 years with a maturity value of $135.30, 5 years with a maturity value of $161.98, or 10 years with a maturity value of $218.65. (The maturity value increases with
Firm A has a stock price of $40 and has made an offer for firm B where A promises to pay $60/share for B, as long as A’s stock price remains between $35 and $45. If the price of A is below $35, A will pay 1.714 shares, and if the price of A is above $45, A will pay 1.333 shares. The deal is
Firm A has a stock price of $40, and has made an offer for firm B where A promises to pay 1.5 shares for each share of B, as long as A’s stock price remains between $35 and $45. If the price of A is below $35, A will pay $52.50/share, and if the price of A is above $45, A will pay $67.50/share.
The strike price of a compensation option is generally set on the day the option is issued. On November 10, 2000, the CEO of Analog Devices, Jerald Fishman, received 600,000 options. The stock price was $44.50. Four days later, the price rose to $63.25 after an earnings release: Maria Tagliaferro
Four years after the option grant, the stock price for Analog Devices was about $40. Using the same input as in the previous problem, compute the market value of the options granted in 2000, assuming that they were issued at strikes of $44.50 and $63.25. Discuss.
Suppose that a firm offers a 3-year compensation option that vests immediately. An employee who resigns has two years to decide whether to exercise the option. Compute annual compensation option expense using the stock price tree in the example in Appendix 16.A. Verify that the present value of the
There are four debt issues with different priorities, each promising $30 at maturity. a. Compute the yield on each debt issue assuming that all four mature in 1 year, 2 years, 5 years, or 10 years. b. Assuming that each debt issue matures in 5 years, what happens to the yield on each when you vary
Suppose there is a single 5-year zero-coupon debt issue with a maturity value of $120. The expected return on assets is 12%. What is the expected return on equity? The volatility of equity? What happens to the expected return on equity as you vary A, σ, and r?
Repeat the previous problem for debt instead of equity. Suppose there is a single 5-year zero-coupon debt issue with a maturity value of $120. The expected return on assets is 12%. What is the expected return on equity? The volatility of equity? What happens to the expected return on equity as
In this problem we examine the effect of changing the assumptions in Example 16.1. a. Compute the yield on debt for asset values of $50, $100, $150, $200, and $500. How does the yield on debt change with the value of assets? b. Compute the yield on debt for asset volatilities of 10% through 100%,
The firm is considering an investment project costing $1. What is the amount by which the project’s value must exceed its cost in order for shareholders to be willing to pay for it? Repeat for project values of $10 and $25.
Nowsuppose the firm finances the project by issuing debt that has lower priority than existing debt. How much must a $1, $10, or $25 project be worth if the shareholders are willing to fund it?
Now suppose the firm finances the project by issuing debt that has higher priority than existing debt. How much must a $10 or $25 project be worth if the shareholders are willing to fund it?
Suppose you have a project that will produce a single widget. Widgets today cost $1 and the project costs $0.90. The risk-free rate is 5%. Under what circumstances would you invest immediately in the project? What conditions would lead you to delay the project? Discuss.
Consider a project that in one year pays $50 if the economy performs well (the stock market goes up) and that pays $100 if the economy performs badly (the stock market goes down). The probability of the economy performing well is 60%, the effective annual risk-free rate is 6%, the expected return
Verify the binomial calculations in Figure 17.3.In figure 17.3
A project costing $100 will produce perpetual net cash flows that have an annual volatility of 35% with no expected growth. If the project existed, net cash flows today would be $8. The project beta is 0.5, the effective annual risk-free rate is 5%, and the effective annual risk premium on the
A project has certain cash flows today of $1, growing at 5% per year for 10 years, after which the cash flow is constant. The risk-free rate is 5%. The project costs $20 and cash flows begin 1 year after the project is started. When should you invest and what is the value of the option to invest?
Consider the oil project with a single barrel, in which S = $15, r = 5%, δ = 4%, and X = $13.60. Suppose that, in addition, the land can be sold for the residual value of R = $1 after the barrel of oil is extracted. What is the value of the land? Discuss.
Verify in Figure 17.2 that if volatility were 30% instead of 50%, immediate exercise would be optimal.In figure 17.2
Consider the last row of Table 17.1. What is the solution for S∗ and S∗ when Ks = kr = 0? (This answer does not require calculation.) In the following five problems, assume that the spot price of gold is $300/oz, the effective annual lease rate is 3%, and the effective annual risk-free rate is
A mine costing $275 will produce 1 ounce of gold on the day the cost is paid. Gold volatility is zero. What is the value of the mine? Discuss.
A mine costing $1000 will produce 1 ounce of gold per year forever at a marginal extraction cost of $250, with production commencing 1 year after the mine opens. Gold volatility is zero. What is the value of the mine?
Repeat Problems 17.17 and 17.18 assuming that the annual volatility of gold is 20%. In problem 17.17 A mine costing $275 will produce 1 ounce of gold on the day the cost is paid. In problem 17.18 A mine costing $1000 will produce 1 ounce of gold per year forever at a marginal extraction cost of
You have a project costing $1.50 that will produce two widgets, one each the first and second years after project completion. Widgets today cost $0.80 each, with the price growing at 2% per year. The effective annual interest rate is 5%. When will you invest? What is the value today of the project?
Repeat Problem 17.18 assuming that the volatility of gold is 20% and that once opened, the mine can be costlessly shut down forever. What is the value of the mine? What is the price at which the mine will be shut down?In problem 17.18A mine costing $1000 will produce 1 ounce of gold per year
Repeat Problem 17.18 assuming that the volatility of gold is 20% and that once opened, the mine can be costlessly shut down once, and then costlessly reopened once. What is the value of the mine? What are the prices at which the mine will be shut down and reopened?In problem 17.18A mine costing
Consider again the project in Problem 17.2, only suppose that the widget price is unchanging and the cost of investment is declining at 2% per year. When will you invest? What is the value today of the project?In problem 17.2You have a project costing $1.50 that will produce two widgets, one each
Consider the widget investment problem outlined in Section 17.1. Show the following in a spreadsheet.a. Compute annual widget prices for the next 50 years.b. For each year, compute the net present value of investing in that year.c. Discount the net present value for each year back to the present.
Again consider the widget investment problem in Section 17.1. Verify that with S = $50, K = $30, r = 0.04879, σ = 0, and δ = 0.009569, the perpetual call price is $30.597 and exercise optimally occurs when the present value of cash flows is $152.957. What happens to the value of the project and
The stock price of XYZ is $100. One million shares of XYZ (a negligible fraction of the shares outstanding) are buried on a tiny, otherwise worthless plot of land in a vault that would cost $50 million to excavate. If XYZ pays a dividend, you will have to dig up the shares to collect the
Repeat Problem 17.6, only assume that after the stock is excavated, the land has an alternative use and can be sold for $30m. In problem 17.6 The stock price of XYZ is $100. One million shares of XYZ (a negligible fraction of the shares outstanding) are buried on a tiny, otherwise worthless plot
Consider the widget investment problem of Section 17.1 with the following modification. The expected growth rate of the widget price is zero. (This means there is no reason to consider project delay.) Each period, the widget price will be $0.25 with probability 0.5 or $2.25 with probability 0.5.
To answer this question, use the assumptions of Example 17.1 and the risk-neutral valuation method (and risk-neutral probability) described in Example 17.2.a. Compute the value of a claim that pays the square root of the cash flow in period 1.b. Compute the value of a claim that pays the square of
You draw these five numbers randomly from a normal distribution with mean−8 and variance 15: {−7, −11, −3, 2, −15}. What are the equivalent draws from a standard normal distribution?
What is Pr(St < $98) for t = 1? How does this probability change when you change t?
Let t = 1. What is E(St | St < $98)? What is E(St | St < $120)? How do both expectations change when you vary t from 0.05 to 5? Let σ = 0.1. Does either answer change? How?
Let KT = S0erT. Compute Pr(ST KT) for a variety of T s from 0.25 to 25 years. How do the probabilities behave? How do you reconcile your answer with the fact that both call and put prices increase with time?
Consider Pr(St < K), equation (18.23), and E(St|St < K), equation (18.28). Verify that it is possible to pick parameters such that changes in t can have ambiguous effects on Pr(St < K) (experiment with very short and long times to maturity, and set α > 0.5σ2). Is the effect of t on E(St|St
You draw these five numbers from a standard normal distribution: {−1.7, 0.55, −0.3, −0.02, .85}. What are the equivalent draws from a normal distribution with mean 0.8 and variance 25?
Suppose x1∼ N(1, 5) and x2 ∼ N(−2, 2). The covariance between x1 and x2 is 1.3. What is the distribution of x1 + x2? What is the distribution of x1− x2?
Suppose x1∼ N(2, 0.5) and x2 ∼ N(8, 14). The correlation between x1 and x2 is −0.3. What is the distribution of x1+ x2? What is the distribution of x1− x2?
Suppose x1∼ N(1, 5), x2 ∼ N(2, 3), and x3 ∼ N(2.5, 7), with correlations ρ1,2 = 0.3, ρ1,3 = 0.1, and ρ2,3 = 0.4. What is the distribution of x1+ x2 + x3? x1+ (3 × x2) + x3? x1 + x2 + (0.5× x3)?
If x ∼ N(2, 5), what is E(ex)? What is the median of ex?
Suppose you observe the following month-end stock prices for stocks A and B:For each stock:a. Compute the mean monthly continuously compounded return. What is the annual return?b. Compute the mean monthly standard deviation. What is the annual standard deviation?c. Evaluate the statement: €œThe
What is Pr(St > $105) for t = 1? How does this probability change when you change t? How does it change when you change σ?
What is E(St | St > $105) for t = 1? How does this expectation change when you change t, σ, and r?
For stocks 1 and 2, S1 = $40, S2 = $100, and the return correlation is 0.45. Let r = 0.08, σ1 = 0.30, σ2 = 0.50, and δ1= δ2 = 0. Generate 1000 1-month prices for the two stocks. For each stock, compute the mean and standard deviation of the continuously compounded return. Also compute the
Assume S0 = $100, r = 0.05, σ = 0.25, δ = 0, and T = 1. Use Monte Carlo valuation to compute the price of a claim that pays $1 if ST > $100, and 0 otherwise.
Let h = 1/52. Simulate both the continuously compounded actual return and the actual stock price, St+h. What are the mean, standard deviation, skewness, and kurtosis of both the continuously compounded return on the stock and the stock price? Use the same random normal numbers and repeat for h = 1.

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