Suppose that 3-month, 6-month, 12-month, 2-year, and 3-year OIS rates are 2.0%, 2.5%, 3.2%, 4.5%, and 5%, respectively. The 3-month, 6-month, and 12-month OISs involve a single exchange at maturity; the 2-year and 3-year OISs involve quarterly exchanges. The compounding frequencies used for expressing the rates correspond to the frequency of exchanges. Calculate the OIS zero rates using continuous compounding. Interpolate between continuously compounded rates linearly to determine rates between 6 month and 12 months, between 12 months and 2 years, and between 2 years and 3 years.
> Suppose that x is the yield to maturity with continuous compounding on a zero-coupon bond that pays off $1 at time T. Assume that x follows the process where a, , and s are positive constants and dz is a Wiener process. What is the process followed by
> Explain why the market maker’s bid–offer spread represents a real cost to options investors.
> Suppose that a stock price S follows geometric Brownian motion with expected return µ and volatility σ: What is the process followed by the variable Sn? Show that Sn also follows geometric Brownian motion.
> A cattle farmer expects to have 120,000 pounds of live cattle to sell in 3 months. The live cattle futures contract traded by the CME Group is for the delivery of 40,000 pounds of cattle. How can the farmer use the contract for hedging? From the farmer’s
> A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle. For what range of stock prices would the straddle lead to a loss?
> Consider the situation in which stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of th
> What are the formulas for u and d in terms of volatility?
> A company that is uncertain about the exact date when it will pay or receive a foreign currency may try to negotiate with its bank a forward contract that specifies a period during which delivery can be made. The company wants to reserve the right to cho
> What is meant by the ‘‘delta’’ of a stock option?
> It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change the beta of the portfolio to 0.5 during the period July 16 to
> Explain why an American option is always worth at least as much as its intrinsic value.
> ‘‘Speculation in futures markets is pure gambling. It is not in the public interest to allow speculators to trade on a futures exchange.’’ Discuss this viewpoint.
> Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date.
> Suppose that a bank buys an option from a client. The option is uncollateralized and there are no other transactions outstanding with the client. The expected values of the option at the midpoint of years 1, 2, and 3 are 6, 5, and 4. The probability of t
> Calculate u, d, and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is 1 month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.
> A stock index currently stands at 300 and has a volatility of 20%. The risk-free interest rate is 8% and the dividend yield on the index is 3%. Use a three-step binomial tree to value a six-month put option on the index with a strike price of 300 if it i
> Explain what KVA measures.
> Explain the difference between the views of financial economists and most practitioners on how MVA and FVA should be calculated.
> (a) Company A has been offered the swap quotes in Table 7.3. It can borrow for three years at 3.45%. What floating rate can it swap this fixed rate into? (b) Company B has been offered the swap quotes in Table 7.3. It can borrow for five years at LIBOR
> Verify that DerivaGem agrees with the price of the bond in Section 4.6. Test how well DV01 predicts the effect of a 1-basis-point increase in all rates. Estimate the duration of the bond from DV01. Use DV01 and Gamma to predict the effect of a 200-basis-
> Explain what MVA and FVA measure.
> ‘‘If most of the call options on a stock are in the money, it is likely that the stock price has risen rapidly in the last few months.’’ Discuss this statement.
> Can a trading rule based on the past history of a stock’s price ever produce returns that are consistently above average? Discuss.
> Explain the meaning of “netting”. Suppose no collateral is posted. Why does a netting agreement usually reduce credit risks to both sides? Under what circumstances does netting have no effect on credit risk?
> A company is trying to decide between issuing debt and equity to fulfill a funding need. What in theory should happen to the return required by equity holders if it chooses (a) debt and (b) equity?
> What is meant by the term ‘‘agency costs’’? How did agency costs play a role in the credit crisis?
> The following table gives the prices of bonds: (a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months. (b) What are the forward rates for the following periods: 6 months to 12 months, 12 months to 18 months, and 18 month
> Show that, if the futures price of a commodity is greater than the spot price during the delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if the futures price is less than the spot price? Explain your answer.
> The 6-month, 12-month, 18-month, and 24-month zero rates are 4%, 4.5%, 4.75%, and 5%, with semiannual compounding. (a) What are the rates with continuous compounding? (b) What is the forward rate for the 6-month period beginning in 18 months? (c) What is
> It is now October 2017. A company anticipates that it will purchase 1 million pounds of copper in each of February 2018, August 2018, February 2019, and August 2019. The company has decided to use the futures contracts traded by the CME Group to hedge it
> Explain how the ‘‘cure period’’ is used in the calculation of CVA.
> What is the difference between the over-the-counter market and the exchange-traded market? What are the bid and offer quotes of a market maker in the over-the-counter or exchange-traded market?
> Use equation (22.1) to show that when the loss distribution is normal, VaR with 99% confidence is almost exactly the same as ES with 97.5% confidence. equation (22.1) ES = u+a- (22.1)| 27 (1 – X)
> The treasurer of a corporation is trying to choose between options and forward contracts to hedge the corporation’s foreign exchange risk. Discuss the advantages and disadvantages of each.
> An American put option to sell a Swiss franc for dollars has a strike price of $0.80 and a time to maturity of 1 year. The Swiss franc’s volatility is 10%, the dollar interest rate is 6%, the Swiss franc interest rate is 3%, and the current exchange rate
> Calculate the price of a six-month European put option on the spot value of the S&P 500. The six-month forward price of the index is 1,400, the strike price is 1,450, the risk-free rate is 5%, and the volatility of the index is 15%.
> A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
> Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
> The futures price of a commodity is $90. Use a three-step tree to value (a) a 9-month American call option with strike price $93 and (b) a 9-month American put option with strike price $93. The volatility is 28% and the risk-free rate (all maturities) is
> Explain the difference between the credit risk and the market risk in a financial contract.
> A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is .5%. Calculate values for u, d, and p when a 6-month time step is used. What i
> The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Calculate values for u, d, and p when a 2-month time step is used. What is the value a 4-month Euro
> Explain how margin accounts protect futures traders against the possibility of default.
> Suppose you call your broker and issue instructions to sell one July hogs contract. Describe what happens.
> Calculate the implied volatility of soybean futures prices from the following information concerning a European put on soybean futures: Current futures price Exercise price Risk-free rate 525 525 6% per annum 5 months Time to maturity Put price 20
> Explain carefully the difference between writing a put option and buying a call option.
> For the situation considered in Problem 13.12, what is the value of a 6-month European put option with a strike price of $51? Verify that the European call and European put prices satisfy put–call parity. If the put option were American, would it ever be
> OIS rates have been estimated as 3.4% for all maturities. The three-month LIBOR rate is 3.5%. For a six-month swap where payments are exchanged every three months the swap rate is 3.6%. All rates are expressed with quarterly compounding. What is the LIBO
> A stock price is currently $50. Over each of the next two 3-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a 6-month European call option with a str
> A stock price is currently $40. It is known that at the end of 3 months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a 3-month European put option on the stock with an exe
> A stock price is currently $80. It is known that at the end of 4 months it will be either $75 or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a 4-month European put option with a strike price of $80?
> A stock price is currently $40. It is known that at the end of 1 month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $39?
> A currency swap has a remaining life of 15 months. It involves exchanging interest at 10% on £20 million for interest at 6% on $30 million once a year. The term structure of riskfree interest rates in the United Kingdom is flat at 7% and the term structu
> Explain how an aggressive bear spread can be created using put options.
> Use put–call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.
> Explain what CVA and DVA measure.
> A trader buys a call option with a strike price of $45 and a put option with a strike price of $40. Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagram showing the variation of the trader’s profit with the asset pr
> What is the difference between a strangle and a straddle?
> What is meant by a protective put? What position in call options is equivalent to a protective put?
> The table below gives Treasury zero rates and cash flows on a Treasury bond. Zero rates are continuously compounded. (a) What is the bond’s theoretical price? (b) What is the bond’s yield assuming it sells for its theo
> What is a lower bound for the price of a 6-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?
> The price of a non-dividend-paying stock is $19 and the price of a 3-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a 3-month European put option with a strike price of $
> What is a lower bound for the price of a 1-month European put option on a non-dividend- paying stock when the stock price is $12, the strike price is $15, and the risk-free interest rate is 6% per annum?
> What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per annum?
> A stock is expected to pay a dividend of $1 per share in 2 months and in 5 months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a
> The price of a European call that expires in 6 months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in 2 months and again in 5 months. Risk-free interest rates (all maturities) are 10%. What i
> The spot price of oil is $50 per barrel and the cost of storing a barrel of oil for one year is $3, payable at the end of the year. The risk-free interest rate is 5% per annum continuously compounded. What is an upper bound for the one-year futures price
> What is a lower bound for the price of a 2-month European put option on a non-dividend- paying stock when the stock price is $58, the strike price is $65, and the risk free interest rate is 5% per annum?
> What is the effect of an unexpected cash dividend on (a) a call option price and (b) a put option price?
> List the six factors that affect stock option prices.
> Explain the statement at the end of Section 12.1 that, when dividends are zero, the principal protected note cannot be profitable for the bank no matter how long it lasts.
> Companies X and Y have been offered the following rates per annum on a $5 million 10-year investment: Company X requires a fixed-rate investment; company Y requires a floating-rate investment. Design a swap that will net a bank, acting as intermediary, 0
> A company declares a 2-for-1 stock split. Explain how the terms change for a call option with a strike price of $60.
> A stock option is on a February, May, August, and November cycle. What options trade on (a) April 1 and (b) May 30?
> An interest rate is quoted as 5% per annum with semiannual compounding. What is the equivalent rate with (a) annual compounding, (b) monthly compounding, and (c) continuous compounding.
> An investor sells a European call option with strike price of K and maturity T and buys a put with the same strike price and maturity. Describe the investor’s position.
> In early 2012, the spot exchange rate between the Swiss Franc and U.S. dollar was 1.0404 ($ per franc). Interest rates in the United States and Switzerland were 0.25% and 0% per annum, respectively, with continuous compounding. The 3-month forward exchan
> Options on General Motors stock are on a March, June, September, and December cycle. What options trade on (a) March 1, (b) June 30, and (c) August 5?
> Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in 4 months. Explain how the terms of the option contract change when there is: (a) a 10% stock dividend; (b) a 10% cash dividend; and (c) a 4-for-
> In Business Snapshot 17.1, what is the cost of a guarantee that the return on the fund will not be negative over the next 10 years? Business Snapshot 17.1 Can We Guarantee that Stocks Will Beat Bonds in the Long Run? It is often said that if you are
> Explain how you would value a swap that is the exchange of a floating rate in one currency for a fixed rate in another currency.
> The price of a stock is $40. The price of a 1-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a 1-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investo
> Explain why FVA can be calculated for a transaction without considering the portfolio to which the transaction belongs, but that the same is not true of MVA.
> Explain the difference between the views of financial economists and most practitioners on how KVA should be calculated.
> ‘‘Nonfinancial companies with high credit risks are the ones that cannot access fixed-rate markets directly. They are the companies that are most likely to be paying fixed and receiving floating in an interest rate swap.’’ Assume that this statement is t
> Describe the profit from the following portfolio: a long forward contract on an asset and a long European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the ti
> After it hedges its foreign exchange risk using forward contracts, is the financial institution’s average spread in Figure 7.11 likely to be greater than or less than 20 basis points? Explain your answer. Figure 7.11 A currency swap
> Explain what is meant by (a) an ABS and (b) an ABS CDO.
> Companies A and B face the following interest rates (adjusted for the differential impact of taxes): Assume that A wants to borrow U.S. dollars at a floating rate of interest and B wants to borrow Canadian dollars at a fixed rate of interest. A financial
> A financial institution has entered into a 10-year currency swap with company Y. Under the terms of the swap, the financial institution receives interest at 3% per annum in Swiss francs and pays interest at 8% per annum in U.S. dollars. Interest payments
> A financial institution has entered into an interest rate swap with company X. Under the terms of the swap, it receives 4% per annum and pays six-month LIBOR on a principal of $10 million for five years. Payments are made every six months. Suppose that c
> Assume that the price of currency A expressed in terms of the price of currency B follows the process where rA is the risk-free interest rate in currency A and rB is the risk-free interest rate in currency B. What is the process fol
> Companies A and B have been offered the following rates per annum on a $20 million five-year loan: Company A requires a floating-rate loan; Company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.1% per annum a
> What is the purpose of the convexity adjustment made to Eurodollar futures rates? Why is the convexity adjustment necessary?
> A Eurodollar futures price changes from 96.76 to 96.82. What is the gain or loss to a trader who is long two contracts?
> How is the conversion factor of a bond calculated by the CME Group? How is it used?
> Explain why the forward interest rate is less than the corresponding futures interest rate calculated from a Eurodollar futures contract.
> If the volatility of a stock is 18% per annum, estimate the standard deviation of the percentage price change in (a) 1 day, (b) 1 week, and (c) 1 month.
> Prove the result in equation (11.11). (Hint: For the first part of the relationship, consider (a) a portfolio consisting of a European call plus an amount of cash equal to D þ K, and (b) a portfolio consisting of an American put option plus one share.)
> What do you think would happen if an exchange started trading a contract in which the quality of the underlying asset was incompletely specified?
