What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed mining company stock is $50 (USD) on the New York Stock Exchange and $60 (CAD) on the Toronto Stock Exchange. Assume that the exchange rate is such that 1 U.S. dollar equals 1.21 Canadian dollars. Explain what is likely to happen to prices as traders take advantage of this opportunity. \
> Why did some companies backdate stock option grants in the United States prior to 2002? What changed in 2002?
> ‘‘Granting stock options to executives is like allowing a professional footballer to bet on the outcome of games.’’ Discuss this viewpoint.
> ‘‘Stock option grants are good because they motivate executives to act in the best interests of shareholders.’’ Discuss this viewpoint.
> Suppose that x is the yield on a perpetual government bond that pays interest at the rate of $1 per annum. Assume that x is expressed with continuous compounding, that interest is paid continuously on the bond, and that x follows the process where a,
> What are the main differences between a typical employee stock option and an American call option traded on an exchange or in the over-the-counter market?
> A company’s CFO says: ‘‘The accounting treatment of stock options is crazy. We granted 10,000,000 at-the-money stock options to our employees last year when the stock price was $30. We estimated the value of each option on the grant date to be $5. At our
> Why was it attractive for companies to grant at-the-money stock options prior to 2005? What changed in 2005?
> A stock price follows geometric Brownian motion with an expected return of 16% and avolatility of 35%. The current price is $38. (a) What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in 6
> What is implied volatility ? How can it be calculated?
> Explain the principle of risk-neutral valuation.
> A company’s stock price is $50 and 10 million shares are outstanding. The company is considering giving its employees 3 million at-the-money 5-year call options. Option exercises will be handled by issuing more shares. The stock price volatility is 25%,
> Use the result in equation (15.17) to determine the value of a perpetual American put option on a non-dividend-paying stock with strike price K if it is exercised when the stock price equals H where H < K. Assume that the current stock price S is greater
> Show that the probability that a European call option will be exercised in a risk-neutral world is, with the notation introduced in this chapter, N(d2)What is an expression for the value of a derivative that pays off $100 if the price of a stock at time
> Consider an American call option on a stock. The stock price is $50, the time to maturity is 15 months, the risk-free rate of interest is 8% per annum, the exercise price is $55, and the volatility is 25%. Dividends of $1.50 are expected in 4 months and
> A stock price is currently $25. It is known that at the end of 2 months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose ST is the stock price at the end of 2 months. What is the value of a d
> Explain carefully why Black’s approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black’s approach understate or overstate the true
> A stock price is currently $50 and the risk-free interest rate is 5%. Use the DerivaGem software to translate the following table of European call options on the stock into a table of implied volatilities, assuming no dividends. Are the option prices con
> With the notation used in this chapter: (a) What is N (x)? (b) Show that , where S is the stock price at time t and (c) (d) where c is the price of a call option on a non-dividend-paying stock. (e) Show that (f ) Show that c satisfies the Black&aci
> A call option on a non-dividend-paying stock has a market price of $212. The stock price is $15, the exercise price is $13, the time to maturity is 3 months, and the risk-free interest rate is 5% per annum. What is the implied volatility?
> Assume that a non-dividend-paying stock has an expected return of and a volatility of . An innovative financial institution has just announced that it will trade a security that pays off a dollar amount equal to ln ST at time T, where ST denotes the va
> A portfolio manager announces that the average of the returns realized in each year of the last 10 years is 20% per annum. In what respect is this statement misleading?
> It has been suggested that the short-term interest rate r follows the stochastic process where a, b, c are positive constants and dz is a Wiener process. Describe the nature of this process.
> The process for the stock price in equation (14.8) is Where µ and σ are constant. Explain carefully the difference between this model and each of the following: Why is the model in equation (14.8) a more appropriate model of stoc
> Stock A and stock B both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of one of stock A and one of stock B follow geometric Brownian motion? Explain you
> Consider a variable S that follows the process For the first three years,and ; for the next three years, and . If the initial value of the variable is 5, what is the probability distribution of the value of the variable at the end of year 6?
> The one-day 99% VaR is calculated for the four-index example in Section 22.2 as $253,385. Look at the underlying spreadsheets on the author’s website and calculate: (a) the one-day 95% VaR, (b) the one-day 95% ES, (c) the one-day 97% VaR, and (d) the one
> Variables X1 and X2 follow generalized Wiener processes, with drift rates and and variances and . What process does X1 +X2 follow if : (a) The changes in X1 and X2 in any short interval of time are uncorrelated? (b) There is a correlation between t
> A company’s cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.5 per quarter and a variance rate of 4.0 per quarter. How high does the company’s initial cash position have to be for the company to
> The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate a minimum variance hedge ratio. (Do not make an adjustment for daily settlement.) Spot price change Futures p
> A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During the last year, the risk-free rate was 5% and equities performed very badly providing a return of 30%. The portfolio manager produced a return of 10% and claims th
> A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6 correlation with gasoline futures price changes. The company will lose $1 million for each 1 cent increase in the price per gallon of the new fuel over the next three mon
> A company has derivatives transactions with Banks A, B, and C that are worth + $20 million, ─ $15 million, and ─ $25 million, respectively, to the company. How much margin or collateral does the company have to provide in each of the following two situ
> What position is equivalent to a long forward contract to buy an asset at K on a certain date and a put option to sell it for K on that date.
> A company enters into a short futures contract to sell 5,000 bushels of wheat for 750 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Under what circumstances could $1,50
> One orange juice futures contract is on 15,000 pounds of frozen concentrate. Suppose that in September 2017 a company sells a March 2019 orange juice futures contract for 120 cents per pound. At the end of December 2017, the futures price is 140 cents; a
> A trader buys a European call option and sells a European put option. The options have the same underlying asset, strike price, and maturity. Describe the trader’s position. Under what circumstances does the price of the call equal the price of the put?
> Suppose that, for a particular three-year derivative entered into by a bank, two outcomes, A and B, are equally likely. Under outcome A, the values of the derivative at the midpoint of the first, second, and third years are 3, 5, and 7, respectively. Und
> Explain carefully the difference between hedging, speculation, and arbitrage.
> Describe how foreign currency options can be used for hedging in the situation considered in Section 1.7 so that (a) ImportCo is guaranteed that its exchange rate will be less than 1.4700, and (b) ExportCo is guaranteed that its exchange rate will be at
> Suppose that in the situation of Table 1.1 a corporate treasurer said: ‘‘I will have £1 million to sell in 6 months. If the exchange rate is less than 1.42, I want you to give me 1.42. If it is greater tha
> A bond issued by Standard Oil some time ago worked as follows. The holder received no interest. At the bond’s maturity the company promised to pay $1,000 plus an additional amount based on the price of oil at that time. The additional amount was equal to
> On May 3, 2016, an investor owns 100 Google shares. As indicated in Table 1.3, the share price is about $696 and a December put option with a strike price of $660 costs $38.10. The investor is comparing two alternatives to limit downside risk. The first
> The current price of a stock is $94, and 3-month European call options with a strike price of $95 currently sell for $4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 cal
> The price of gold is currently $1,200 per ounce. The forward price for delivery in 1 year is $1,300 per ounce. An arbitrageur can borrow money at 3% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold pro
> A bank’s derivatives transactions with a counterparty are worth þ$10 million to the bank and are cleared bilaterally. The counterparty has posted $10 million of cash collateral. What credit exposure does the bank have?
> A U.S. company knows it will have to pay 3 million euros in three months. The current exchange rate is 1.1500 dollars per euro. Discuss how forward and options contracts can be used by the company to hedge its exposure.
> In March, a U.S. investor instructs a broker to sell one July put option contract on a stock. The stock price is $42 and the strike price is $40. The option price is $3. Explain what the investor has agreed to. Under what circumstances will the trade pro
> Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 4% per annum. How could you make money if the June and December futures contracts for a particular year trade at $50 and $56, respectively?
> In Problem 18.12, what does the binomial tree give for the value of a six-month European put option on futures with a strike price of 60? If the put were American, would it ever be worth exercising it early? Verify that the call prices calculated in Prob
> On May 3, 2016, as indicated in Table 1.2, the spot offer price of Google stock is $696.25 and the offer price of a call option with a strike price of $700 and a maturity date of September is $39.20. A trader is considering two alternatives: buy 100 shar
> Explain what is meant by open interest. Why does the open interest usually decline during the month preceding the delivery month? On a particular day, there were 2,000 trades in a particular futures contract. This means that there were 2,000 buyers (goin
> Trader A enters into futures contracts to buy 1 million euros for 1.1 million dollars in three months. Trader B enters in a forward contract to do the same thing. The exchange rate (dollars per euro) declines sharply during the first two months and then
> A trader owns 55,000 units of a particular asset and decides to hedge the value of her position with futures contracts on another related asset. Each futures contract is on 5,000 units. The spot price of the asset that is owned is $28 and the standard de
> Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20 per ounce. What is the effect of the basis on the hedger’s financial
> It is April 7, 2017. The quoted price of a U.S. government bond with a 6% per annum coupon (paid semiannually) is 120-00. The bond matures on July 27, 2033. What is the cash price? How does your answer change if it is a corporate bond?
> The one-year LIBOR rates is 3%, and the LIBOR forward rate for the 1- to 2-year period is 3.2%, respectively. The three-year swap rate for a swap with annual payments is 3.2%. What is the LIBOR forward rate for the 2- to 3-year period if OIS zero rates f
> (a) Company X has been offered the swap quotes in Table 7.3. It can invest for four years at 2.8%. What floating rate can it swap this fixed rate into? (b) Company Y has been offered the swap quotes in Table 7.3. It is confident that it will be able t
> Repeat Problem 19.12 for a financial institution with a portfolio of short positions in put and call options on a currency. Data from Problem 19.12: A company uses delta hedging to hedge a portfolio of long positions in put and call options on a currency
> Suppose that mezzanine tranches of the ABS CDOs, similar to those in Figure 8.3, are resecuritized to form what is referred to as a ‘‘CDO squared.’’ As in the case of tranches create
> Suppose that a stock price has an expected return of 16% per annum and a volatility of 30% per annum. When the stock price at the end of a certain day is $50, calculate the following: (a) The expected stock price at the end of the next day (b) The standa
> A bank has written a call option on one stock and a put option on another stock. For the first option the stock price is 50, the strike price is 51, the volatility is 28% per annum, and the time to maturity is 9 months. For the second option the stock pr
> A company has a long position in a 2-year bond and a 3-year bond, as well as a short position in a 5-year bond. Each bond has a principal of $100 and pays a 5% coupon annually. Calculate the company’s exposure to the 1-year, 2-year, 3-year, 4-year, and 5
> Consider a portfolio of options on a single asset. Suppose that the delta of the portfolio is 12, the value of the asset is $10, and the daily volatility of the asset is 2%. Estimate the 1-day 95% VaR for the portfolio from the delta. Suppose next that t
> Consider a position consisting of a $300,000 investment in gold and a $500,000 investment in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2%, respectively, and that the coefficient of correlation between their returns is
> A company has a position in bonds worth $6 million. The modified duration of the portfolio is 5.2 years. Assume that only parallel shifts in the yield curve can take place and that the standard deviation of the daily yield change (when yield is measured
> A four-step Cox–Ross–Rubinstein binomial tree is used to price a one-year American put option on an index when the index level is 500, the strike price is 500, the dividend yield is 2%, the risk-free rate is 5%, and the volatility is 25% per annum. What
> Estimate delta, gamma, and theta from the tree in Example 21.3. Explain how each can be interpreted. Example 21.3 Consider a 4-month American call option on index futures where the current futures price is 300, the exercise price is 300, the risk-fre
> Answer the following questions concerned with the alternative procedures for constructing trees in Section 21.4: Show that the binomial model in Section 21.4 is exactly consistent with the mean and variance of the change in the logarithm of the stock pri
> The average funding cost for a company is 5% per annum when the risk-free rate is 3%. The company is currently undertaking projects worth $9 million. It plans to increase its size by undertaking $1 million of risk-free projects. What would you expect to
> The current value of the British pound is $1.60 and the volatility of the pound/dollar exchange rate is 15% per annum. An American call option has an exercise price of $1.62 and a time to maturity of 1 year. The risk-free rates of interest in the United
> A 1-year American call option on silver futures has an exercise price of $9.00. The current futures price is $8.50, the risk-free rate of interest is 12% per annum, and the volatility of the futures price is 25% per annum. Use the DerivaGem software with
> Using Table 20.2, calculate the implied volatility a trader would use for an 11-month option with K=S0 ¼ 0:98. Table 20.2 Volatility surface. K/So 0.90 0.95 1.00 1.05 1.10 1 month 14.2 13.0 12.0 13.1 14.5 3 month 14.0 13.0 12.0 13.1 14.
> Consider a European call and a European put with the same strike price and time to maturity. Show that they change in value by the same amount when the volatility increases from a level to a new level within a short period of time. (Hint: Use put–cal
> A futures price is currently $40. The risk-free interest rate is 5%. Some news is expected tomorrow that will cause the volatility over the next 3 months to be either 10% or 30%. There is a 60% chance of the first outcome and a 40% chance of the second o
> A company is currently awaiting the outcome of a major lawsuit. This is expected to be known within 1 month. The stock price is currently $20. If the outcome is positive, the stock price is expected to be $24 at the end of 1 month. If the outcome is nega
> A company’s stock is selling for $4. The company has no outstanding debt. Analysts consider the liquidation value of the company to be at least $300,000 and there are 100,000 shares outstanding. What volatility smile would you expect to see?
> Use the DerivaGem Application Builder functions to reproduce Table 19.2. (In Table 19.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma and theta of the position each week. Calculate the change in the value of the portfolio
> The formula for the price c of a European call futures option in terms of the futures price F0 is given in Chapter 18 as and K, r, T, and are the strike price, interest rate, time to maturity, and volatility, respectively. (a) Prove that (b)
> A deposit instrument offered by a bank guarantees that investors will receive a return during a 6-month period that is the greater of (a) zero and (b) 40% of the return provided by a market index. An investor is planning to put $100,000 in the instrument
> Explain the impact of an increase in default correlation on the risks of the senior tranche of an ABS. What is its impact on the risks of the equity tranche?
> Consider a 1-year European call option on a stock when the stock price is $30, the strike price is $30, the risk-free rate is 5%, and the volatility is 25% per annum. Use the DerivaGem software to calculate the price, delta, gamma, vega, theta, and rho o
> A financial institution has the following portfolio of over-the-counter options on sterling: A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8. (a) What position in the traded option and in sterling would make the portfo
> The strike price of a futures option is 550 cents, the risk-free interest rate is 3%, the volatility of the futures price is 20%, and the time to maturity of the option is 9 months. The futures price is 500 cents. (a) What is the price of the option if i
> It is February 4. July call options on corn futures with strike prices of 260, 270, 280, 290, and 300 cost 26.75, 21.25, 17.25, 14.00, and 11.375, respectively. July put options with these strike prices cost 8.50, 13.50, 19.00, 25.625, and 32.625, respec
> Suppose the USD/euro exchange rate is 1.3000. The exchange rate volatility is 15%. A U.S. company will receive 1 million euros in three months. The euro and USD risk free rates are 5% and 4%, respectively. The company decides to use a range forward contr
> The spot price of an index is 1,000 and the risk-free rate is 4%. The prices of 3-month European call and put options when the strike price is 950 are 78 and 26. Estimate (a) the dividend yield and (b) the implied volatility.
> Suppose that the spot price of the Canadian dollar is U.S. $0.95 and that the Canadian dollar/U.S. dollar exchange rate has a volatility of 8% per annum. The risk-free rates of interest in Canada and the United States are 4% and 5% per annum, respectivel
> The Dow Jones Industrial Average on July 20, 2016, was 18,580 and the price of a September 185 (European) call option on the index was $3.35. Use the DerivaGem software to calculate the implied volatility of this option. Assume the risk-free rate was 0.7
> (a) Hedge funds earn a management fee plus an incentive fee that is a percentage of the profits, if any, that they generate (see Business Snapshot 1.3). How is a fund manager motivated to behave with this type of compensation package? (b) ‘‘Granting opti
> A company has granted 1,000,000 options to its employees. The stock price and strike price are both $20. The options last 10 years and vest after 3 years. The stock price volatility is 30%, the risk-free rate is 5%, and the company pays no dividends. Use
> A company uses delta hedging to hedge a portfolio of long positions in put and call options on a currency. Which of the following would give the most favorable result? (a) A virtually constant spot rate (b) Wild movements in the spot rate Explain your an
> A company has granted 2,000,000 options to its employees. The stock price and strike price are both $60. The options last for 8 years and vest after 2 years. The company decides to value the options using an expected life of 6 years and a volatility of 2
> What is the (risk-neutral) expected life for the employee stock option in Example 16.2? What is the value of the option obtained by using this expected life in Black–Scholes– Merton? Example 16.2 Suppose a company grants stock options that last 8 years
> The appendix derives the key result Show that and use this to derive the Black–Scholes–Merton formula for the price of a European put option on a non-dividend-paying stock. E[max(V – K, 0)] = E(V)N(dj) – KN(dz) %3D
> Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is 6 months, the volatility is 30% per annum, and the risk-free interest rate is 10% per annum. Two equal dividends are expected during the life
> Assume that the stock in Problem 15.30 is due to go ex-dividend in 1 2/1 months. The expected dividend is 50 cents. (a) What is the price of the option if it is a European call? (b) What is the price of the option if it is a European put? (c) If the o
> Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time to maturity is 4 months. (a) What is the price of the option if it
> A financial institution plans to offer a security that pays off a dollar amount equal to ST2 at time T, where ST is the price at time T of a stock that pays no dividends. (a) Use risk-neutral valuation to calculate the price of the security at time t in
> Suppose that observations on a stock price (in dollars) at the end of each of 15 consecutive weeks are as follows: 30:2; 32:0; 31:1; 30:1; 30:2; 30:3; 30:6; 33:0; 32:9; 33:0; 33:5; 33:5; 33:7; 33:5; 33:2 Estimate the stock price volatility. What is the s
> A stock price is currently $50. Assume that the expected return from the stock is 18% and its volatility is 30%. What is the probability distribution for the stock price in 2 years? Calculate the mean and standard deviation of the distribution. Determine
