Question: The formula for the price c of

> 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

> 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

> 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

> 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

> A stock price is currently 50. Its expected return and volatility are 12% and 30%, respectively. What is the probability that the stock price will be greater than 80 in 2 years? (Hint: ST > 80 when ln ST > ln 80.)

> A stock whose price is $30 has an expected return of 9% and a volatility of 20%. In Excel, simulate the stock price path over 5 years using monthly time steps and random samples from a normal distribution. Chart the simulated stock price path. By hitting

> If
follows the geometric Brownian motion process in equation (14.6), what is the process followed by (a) y = 2S, (b) y=S2 , (c) y=eS, and (d) y=er(T-t)/S. In each case express the coefficients of dt and dz in terms of
rather than S.

> A company’s cash position, measured in millions of dollars, follows a generalized Wiener process with a drift rate of 0.1 per month and a variance rate of 0.16 per month. The initial cash position is 2.0. (a) What are the probability distributions of the

> Calculate the value of 9-month American call option to buy 1 million units of a foreign currency using a three-step binomial tree. The current exchange rate is 0.79 and the strike price is 0.80 (both expressed as dollars per unit of the foreign currency)

> A stock index is currently 990, the risk-free rate is 5%, and the dividend yield on the index is 2%. Use a three-step tree to value an 18-month American put option with a strike price of 1,000 when the volatility is 20% per annum. How much does the optio

> Footnote 1 of this chapter shows that the correct discount rate to use for the real-world expected payoff in the case of the call option considered in Figure 13.1 is 55.96%. Show that if the option is a put rather than a call the discount rate is 70:4%.

> Repeat Problem 13.25 for an American put option on a futures contract. The strike price and the futures price are $50, the risk-free rate is 10%, the time to maturity is 6 months, and the volatility is 40% per annum. Data from Problem 13.25: Consider a E

> A stock price is currently $30. During each 2-month period for the next 4 months it will increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off
; 2, where ST is the st

> Using a ‘‘trial-and-error’’ approach, estimate how high the strike price has to be in Problem 13.22 for it to be optimal to exercise the option immediately. Data from Problem 13.22: A stock price is currently $40. Over each of the next two three-month pe

> A stock price is currently $40. Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. (a) What is the value of a 6-month European put option wit

> A stock price is currently $50. It is known that at the end of 6 months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a 6-month European call option on the stock with an

> Suppose that a central bank’s policy is to allow an exchange rate to fluctuate between 0.97 and 1.03. What pattern of implied volatilities for options on the exchange rate would you expect to see?

> The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a 3-month time step: (a) What is the percentage up movement? (b) What is the percentage down movement? (c) What is the probability of

> A bank decides to create a five-year principal-protected note on a non-dividend-paying stock by offering investors a zero-coupon bond plus a bull spread created from calls. The risk-free rate is 4% and the stock price volatility is 25%. The low-strike-pr

> Describe the trading position created in which a call option is bought with strike price K2 and a put option is sold with strike price K1 when both have the same time to maturity and K2 > K1. What does the position become when K1 ¼ K2.

> Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30%, and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the following positions: (a) A bull spread using European ca

> Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy.

> Consider a put option on a non-dividend-paying stock when the stock price is $40, the strike price is $42, the risk-free interest rate is 2%, the volatility is 25% per annum, and the time to maturity is three months. Use DerivaGem to determine the follow

> Consider an option on a stock when the stock price is $41, the strike price is $40, the risk-free rate is 6%, the volatility is 35%, and the time to maturity is 1 year. Assume that a dividend of $0.50 is expected after 6 months. (a) Use DerivaGem to valu

> You are the manager and sole owner of a highly leveraged company. All the debt will mature in 1 year. If at that time the value of the company is greater than the face value of the debt, you will pay off the debt. If the value of the company is less than

> What is the result corresponding to that in Problem 11.26 for European put options?

> Suppose that c1, c2, and c3 are the prices of European call options with strike prices K1, K2, and K3, respectively, where K3 > K2 > K1 and K3 K2 ¼ K2 K1. All options have the same maturity. Show that
(Hint: Consider a portfolio that is long one opt

> A trader buys two July futures contracts on frozen orange juice concentrate. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per contract, and the maintenance margin is $4

> A European call option and put option on a stock both have a strike price of $20 and an expiration date in 3 months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in 1 month.

> The prices of European call and put options on a non-dividend-paying stock with an expiration date in 12 months and a strike price of $120 are $20 and $5, respectively. The current stock price is $130. What is the implied risk-free rate?

> Calls were traded on exchanges before puts. During the period of time when calls were traded but puts were not traded, how would you create a European put option on a nondividend- paying stock synthetically.

> On July 20, 2004, Microsoft surprised the market by announcing a $3 dividend. The exdividend date was November 17, 2004, and the payment date was December 2, 2004. Its stock price at the time was about $28. It also changed the terms of its employee stock

> Use DerivaGem to calculate the value of an American put option on a non-dividendpaying stock when the stock price is $30, the strike price is $32, the risk-free rate is 5%, the volatility is 30%, and the time to maturity is 1.5 years. (Choose ‘‘Binomial