Question: When first issued, a stock provides funds

> Show that the Black–Scholes–Merton formulas for call and put options satisfy put–callparity.

> Consider an American call option on a stock. The stock price is $70, the time to maturity is 8 months, the risk-free rate of interest is 10% per annum, the exercise price is $65, and the volatility is 32%. A dividend of $1 is expected after 3 months and

> The DerivaGem Application Builder functions enable you to investigate how the prices of options calculated from a binomial tree converge to the correct value as the number of time steps increases. (See Figure 21.4 and Sample Application A in DerivaGem.)

> A stock price is currently $50. It is known that at the end of 6 months it will be either $45 or $55. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a 6-month European put option with a strike price of $50?

> Explain what a seven-year swap rate is.

> A bank finds that its assets are not matched with its liabilities. It is taking floating-rate deposits and making fixed-rate loans. How can swaps be used to offset the risk?

> Why is the expected loss to a bank from a default on a swap with a counterparty less than the expected loss from the default on a loan to the counterparty when the loan and swap have the same principal? Assume that there are no other derivatives transact

> It is May 5, 2017. The quoted price of a government bond with a 12% coupon that matures on July 27, 2034, is 110-17. What is the cash price?

> The price of a 90-day Treasury bill is quoted as 10.00. What continuously compounded return (on an actual/365 basis) does an investor earn on the Treasury bill for the 90-day period?

> It is January 30. You are managing a bond portfolio worth $6 million. The duration of the portfolio in 6 months will be 8.2 years. The September Treasury bond futures price is currently 108-15, and the cheapest-to-deliver bond will have a duration of 7.6

> The 350-day LIBOR rate is 3% with continuous compounding and the forward rate calculated from a Eurodollar futures contract that matures in 350 days is 3.2% with continuous compounding. Estimate the 440-day zero rate.

> What is the equation corresponding to equation (19.4) for (a) a portfolio of derivatives on a currency and (b) a portfolio of derivatives on a futures price?

> Suppose that a Eurodollar futures quote is 88 for a contract maturing in 60 days. What is the LIBOR forward rate for the 60- to 150-day period? Ignore the difference between futures and forwards for the purposes of this question.

> It is January 9, 2018. The price of a Treasury bond with a 6% coupon that matures on October 12, 2030, is quoted as 102-07. What is the cash price?

> How can the portfolio manager change the duration of the portfolio to 3.0 years in Problem 6.17? Data from 6.17: On August 1 a portfolio manager has a bond portfolio worth $10 million. The duration of the portfolio in October will be 7.1 years. The Dece

> On August 1, a portfolio manager has a bond portfolio worth $10 million. The duration of the portfolio in October will be 7.1 years. The December Treasury bond futures price is currently 91-12 and the cheapest-to-deliver bond will have a duration of 8.8

> Suppose that a bond portfolio with a duration of 12 years is hedged using a futures contract in which the underlying asset has a duration of 4 years. What is likely to be the impact on the hedge of the fact that the 12-year rate is less volatile than the

> Suppose that the Treasury bond futures price is 101-12. Which of the following four bonds is cheapest to deliver? Bond Price Conversion factor 1 125-05 1.2131 2 142-15 1.3792 3 115-31 1.1149 4 144-02 1.4026

> A U.S. Treasury bond pays a 7% coupon on January 7 and July 7. How much interest accrues per $100 of principal to the bondholder between July 7, 2017, and August 8, 2017? How would your answer be different if it were a corporate bond?

> What rate of interest with continuous compounding is equivalent to 8% per annum with monthly compounding?

> The term structure of interest rates is upward-sloping. Put the following in order of magnitude: (a) The 5-year zero rate (b) The yield on a 5-year coupon-bearing bond (c) The forward rate corresponding to the period between 4.75 and 5 years in the futur

> Assuming that risk-free zero rates are as in Problem 4.5, what is the value of an FRA where the holder will pay LIBOR and receive 4.5% (quarterly compounded) for a three month period starting in one year on a principal of $1,000,000? The forward LIBOR ra

> Investigate what happens as the width of the mezzanine tranche of the ABS in Figure 8.3 is decreased with the reduction of mezzanine tranche principal being divided equally between the equity and senior tranches. In particular, what is the effect on Tabl

> Suppose that risk-free zero interest rates with continuous compounding are as follows: Calculate forward interest rates for the second, third, fourth, fifth, and sixth quarters. Maturity (тоnhs) Rate (% per anmum) 3 3.0 3.2 9 3.4 12 3.5 15 3.6 18 3.7

> The 6-month and 1-year zero rates are both 5% per annum. For a bond that has a life of 18 months and pays a coupon of 4% per annum (with semiannual payments and one having just been made), the yield is 5.2% per annum. What is the bond’s price? What is th

> ‘‘An interest rate swap where 6-month LIBOR is exchanged for a fixed rate of 5% on a principal of $100 million for 5 years involves a known cash flow and a portfolio of nine FRAs.’’ Explain this statement.

> Explain how LIBOR is determined.

> Use the risk-free rates in Problem 4.14 to value an FRA where you will pay 5% (annually compounded) and receive LIBOR for the third year on $1 million. The forward LIBOR rate (annually compounded) for the third year is 5.5%. Risk-free rates in Problem 4.

> Suppose that risk-free zero interest rates with continuous compounding are as follows: Calculate forward interest rates for the second, third, fourth, and fifth years. Maturity (vears) Rate (% per annum) 1 2.0 2 3.0 3 3.7 4 4.2 5 4.5

> Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are, respectively, 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum, with continuous compounding. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months

> A deposit account pays 4% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?

> ‘‘Buying a put option on a stock when the stock is owned is a form of insurance.’’ Explain this statement.

> A bank quotes an interest rate of 7% per annum with quarterly compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?

> Prove the result in equation (11.7). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to K, and (b) a portfolio consisting of an American put option plus one share.)

> Does a perfect hedge always succeed in locking in the current spot price of an asset for a future transaction? Explain your answer.

> In the corn futures contract traded on an exchange, the following delivery months are available: March, May, July, September, and December. Which of the available contracts should be used for hedging when the expiration of the hedge is in (a) June, (b) J

> The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected return on an investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?

> Suppose that, on October 24, 2018, a company sells one April 2019 live cattle futures contract. It closes out its position on January 21, 2019. The futures price (per pound) is 121.20 cents when it enters into the contract, 118.30 cents when it closes ou

> The standard deviation of monthly changes in the spot price of live cattle is (in cents per pound) 1.2. The standard deviation of monthly changes in the futures price of live cattle for the closest contract is 1.4. The correlation between the futures pri

> ‘‘Options and futures are zero-sum games.’’ What do you think is meant by this?

> The CME Group offers a futures contract on long-term Treasury bonds. Characterize the traders likely to use this contract.

> ‘‘When the futures price of an asset is less than the spot price, long hedges are likely to be particularly attractive.’’ Explain this statement.

> On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per share. The investor is interested in hedging against movements in the market over the next month and decides to use an index futures contract. The index futures pr

> A company knows that it is due to receive a certain amount of a foreign currency in 4 months. What type of option contract is appropriate for hedging?

> An American put option on a non-dividend-paying stock has 4 months to maturity. The exercise price is $21, the stock price is $20, the risk-free rate of interest is 10% perannum, and the volatility is 30% per annum. Use the explicit version of the finite

> At the end of one day a clearing house member is long 100 contracts, and the settlement price is $50,000 per contract. The original margin is $2,000 per contract. On the following day the member becomes responsible for clearing an additional 20 long cont

> Explain what a stop–limit order to sell at 20.30 with a limit of 20.10 means.

> ‘‘If the minimum variance hedge ratio is calculated as 1.0, the hedge must be perfect.’’ Is this statement true? Explain your answer.

> Explain what is meant by basis risk when futures contracts are used for hedging.

> Explain why a short hedger’s position improves when the basis strengthens unexpectedly and worsens when the basis weakens unexpectedly.

> What are the most important aspects of the design of a new futures contract?

> The party with a short position in a futures contract sometimes has options as to the precise asset that will be delivered, where delivery will take place, when delivery will take place, and so on. Do these options increase or decrease the futures price?

> A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures price is currently standing at 1080, and each contract is for delivery of $250 times the index. What is

> Show by substituting for the various terms in equation (19.4) that the equation is true for: (a) A single European call option on a non-dividend-paying stock (b) A single European put option on a non-dividend-paying stock (c) Any portfolio of European pu

> Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. W

> An investor enters into a short forward contract to sell 100,000 British pounds for U.S. dollars at an exchange rate of 1.5000 USD per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.4900 and (b) 1.

> Under what circumstances does a minimum variance hedge portfolio lead to no hedging at all?

> What is the difference between entering into a long forward contract when the forward price is $50 and taking a long position in a call option with a strike price of $50?

> What is the difference between a local and a futures commission merchant?

> Distinguish between the terms open interest and trading volume.

> Explain why the linear model can provide only approximate estimates of VaR for a portfolio containing options.

> Suppose that a company has a portfolio consisting of positions in stocks and bonds. Assume that there are no derivatives. Explain the assumptions underlying (a) the linear model and (b) the historical simulation model for calculating VaR.

> A financial institution owns a portfolio of options on the U.S. dollar–sterling exchange rate. The delta of the portfolio is 56.0. The current exchange rate is 1.5000. Derive an approximate linear relationship between the change in the portfolio value an

> Describe three ways of handling instruments that are dependent on interest rates when the model-building approach is used to calculate VaR. How would you handle these instruments when historical simulation is used to calculate VaR?

> Stock A, whose price is $30, has an expected return of 11% and a volatility of 25%. Stock B, whose price is $40, has an expected return of 15% and a volatility of 30%. The processes driving the returns are correlated with correlation parameter. In Excel,

> Use the spreadsheets on the author’s website to calculate the one-day 99% VaR and ES, employing the basic methodology in Section 22.2, if the four-index portfolio considered in Section 22.2 is equally divided between the four indices.

> Suppose that in Problem 22.12 the vega of the portfolio is 2 per 1% change in the annual volatility. Derive a model relating the change in the portfolio value in 1 day to delta, gamma, and vega. Explain without doing detailed calculations how you would u

> A bank has a portfolio of options on an asset. The delta of the options is –30 and the gamma is 5. Explain how these numbers can be interpreted. The asset price is 20 and its volatility is 1% per day. Adapt Sample Application E in the DerivaGem Applicati

> The text calculates a VaR estimate for the example in Table 22.9 assuming two factors. How does the estimate change if you assume (a) one factor and (b) three factors. Table 22.9 Change in portfolio value for a l-basis-point rate move (S milli ons).

> Some time ago a company entered into a forward contract to buy £1 million for $1.5 million. The contract now has 6 months to maturity. The daily volatility of a 6-month zero-coupon sterling bond (when its price is translated to dollars) is 0.06% and the

> Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between their returns is 0.3. Estimate the 5-day 99%

> Explain why the Monte Carlo simulation approach cannot easily be used for American-style derivatives

> Use stratified sampling with 100 trials to improve the estimate of in Business Snapshot 21.1 and Table 21.1. Business Snapshot 21.1 Calculating Pi with Monte Carlo Simulation Suppose the sides of the square in Figure 21.13 are one unit in length. Im

> Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 8 holds.

> ‘‘For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine.’’ Explain this statement.

> Explain carefully the arbitrage opportunities in Problem 11.16 if the American put price is greater than the calculated upper bound. Data from Problem 11.16: The price of an American call on a non-dividend-paying stock is $4. The stock price is $31, the

> Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a m

> Consider an option that pays off the amount by which the final stock price exceeds the average stock price achieved during the life of the option. Can this be valued using the binomial tree approach? Explain your answer.

> Calculate the price of a 9-month American call option on corn futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum, and the volatility is 30% per annum. Use a binomial tree with a

> Explain how the control variate technique is implemented when a tree is used to value American options.

> Provide formulas that can be used for obtaining three random samples from standard normal distributions when the correlation between sample i and sample j is
.

> How would you use the antithetic variable method to improve the estimate of the European option in Business Snapshot 21.2 and Table 21.2? Business Snapshot 21.2 Checking Black-Scholes-Merton in Excel The Black-Scholes-Merton formula for a European ca

> When do the boundary conditions for
and S→∞ affect the estimates of derivative prices in the explicit finite difference method?

> Use the binomial tree in Problem 21.19 to value a security that pays off in 1 year where x is the price of copper. Binomial tree in Problem 21.19: 1.335 0.735 1.093 0.493 0.895 0.895 0.295 0.295 0.733 0.133 0.733 0.133 0.600 0.062 0.600 0.042 0.600 0

> Calculate the price of a 3-month American put option on a non-dividend-paying stock when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% perannum, and the volatility is 45% per annum. Use a binomial tree with a time in

> The spot price of copper is $0.60 per pound. Suppose that the futures prices (dollars per pound) are as follows: 3 months……………….. 0.59 6 months………………… 0.57 9 months ………………..0.54 12 months ………………0.50 The volatility of the price of copper is 40% per annum

> Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate and antithetic variable technique be used to improve numerical efficiency

> How do equations (21.27) to (21.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency?

> How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used?

> A 2-month American put option on a stock index has an exercise price of 480. The current level of the index is 484, the risk-free interest rate is 10% per annum, the dividend yield on the index is 3% per annum, and the volatility of the index is 25% per

> A 1-year American put option on a non-dividend-paying stock has an exercise price of $18. The current stock price is $20, the risk-free interest rate is 15% per annum, and the volatility of the stock price is 40% per annum. Use the DerivaGem software wit

> A 3-month American call option on a stock has a strike price of $20. The stock price is $20, the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $2 is expected in 1.5 months. Use a three-step binomial tree to calculate

> Use a three-time-step binomial tree to value a 9-month American call option on wheat futures. The current futures price is 400 cents, the strike price is 420 cents, the risk-free rate is 6%, and the volatility is 35% per annum. Estimate the delta of the

> A 9-month American put option on a non-dividend-paying stock has a strike price of $49. The stock price is $50, the risk-free rate is 5% per annum, and the volatility is 30% per annum. Use a three-step binomial tree to calculate the option price.

> Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho?

> Explain what is meant by ‘‘crashophobia.’’

> The market price of a European call is $3.00 and its price given by Black–Scholes– Merton model with a volatility of 30% is $3.50. The price given by this Black–Scholes–Merton model for a European put option with the same strike price and time to maturit

> ‘‘Resecuritization was a badly flawed idea. AAA tranches created from the mezzanine tranches of ABSs are bound to have a higher probability of default than the AAA-rated tranches of ABSs.’’ Discuss this point of view.

> Using Table 20.2, calculate the implied volatility a trader would use for an 8-month option with K=S0 ¼ 1:04. 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 14.2

> ‘‘The Black–Scholes–Merton model is used by traders as an interpolation tool.’’ Discuss this view.