Interest Rate Options: Hedging Applications and Pricing

1. Chapter 15  Interest Rate Options: Hedging Applications and Pricing

2. Topics Hedging fixed income and debt positions with interest rate options Hedging a series of cash flows with OTC caps, floors, and other interest-rate derivatives Valuation of interest-rate options with the binomial interest rate tree Pricing interest rate options with the Black Futures Model

3. Hedging By hedging with either exchange-traded futures call options on Treasury or Eurodollar contracts or with an OTC spot call option on a debt security or an interest rate put (floorlet), a fixed-income manager planning to invest a future inflow of cash, can obtain protection against adverse price increases while still realizing lower costs if security prices decrease.

4. Hedging For cases in which bond or money market managers are planning to sell some of their securities in the future, a manager, for the costs of buying an interest rate option, can obtain downside protection if bond prices decrease while earning greater revenues if security prices increase. Similar downside protection from hedging positions using interest rate put options can be obtained by bond issuers, borrowers, and underwriters.

5. Hedging Risk Like futures-hedged position, option-hedged positions are subject to quality, quantity, and timing risk. For OTC options, some of the hedging risk can be minimized, if not eliminated, by customizing the contract. For exchange-traded option-hedged positions the objective is to minimize hedging risk by determining the appropriate number of option contracts.

6. Hedging Risk For direct hedging cases (cases in which the future value of the asset or liability to be hedged, VT, is the same as the one underlying the option contract) the number of options can be determined by using the naïve hedge where: n = VT/X. For cross hedging cases (cases in which the asset or liability to be hedge is not the same as the one underlying the option contract), the number of options can be determined by using the price-sensitivity model defined in Chapter 13.

7. Long Hedging Cases

8. Future CD Purchase Consider the case of a money market manager who is expecting a cash inflow of approximately $985,000 in September that he plans to invest in a 90-day jumbo CD with a face value of $1M and yield tied to the LIBOR. Assume: CDs are currently trading at a spot index of 94.5 (S0 = $98.625 per $100 face value), for YTM = ($100/$98.625)(365/90) - 1 = .057757).

9. Future CD Purchase Suppose the manager would like to earn a minimum rate on his September investment that is near the current rate, with the possibility of a higher yield if short-term rates increase. To achieve these objectives, the manager could take a long position in a September Eurodollar futures call. Suppose the manager buys one September 94 Eurodollar futures call option priced at 4: RD = 6% X = $985,000 C0 = 4($250) =$1,000 nc = VT/X = $985,000/$985,000 = 1 call contract

10. Future CD Purchase Assume the manager will close the futures call at expiration at its intrinsic value (fT – X) if the option is in the money and then buy the $1M, 90-day CD at the spot price. The exhibit shows the effective investment expenditures at expiration (costs of the CD minus the profit from the Eurodollar futures call) and the hedged YTM earned from the hedged investment:

11. Future CD Purchase

12. Future CD Purchase If spot rates (LIBOR) are lower than RD = 6% at expiration, the Eurodollar futures call will be in the money and the manager will be able to profit from the call position, offsetting the higher costs of buying the 90-day, $1M CD. As a result, the manager will be able to lock in a maximum purchase price of $986,000 and a minimum yield on his 90-day CD investment of 5.885% when RD is 6% or less.

13. Future CD Purchase If spot discount rates (LIBOR) are 6% or higher, the call will be worthless. In these cases, though, the manager’s option losses are limited to just the $1,000 premium he paid, while the prices of buying CDs decrease, the higher the rates. As a result, for rates higher than 6%, the manager is able to obtain lower CD prices and therefore higher yields on his CD investment as rates increase.

14. Future CD Purchase Thus, by hedging with the Eurodollar futures call, the manager is able to obtain at least a 5.885% YTM, with the potential to earn higher returns if rates on CDs increase.

15. Future 91-Day T-Bill Investment In Chapter 13, we examined the case of a corporate treasurer who was expecting a $5 million cash inflow in June, which she was planning to invest in T-bills for 91 days. In that example, the treasurer locked in the yield on the T-bill investment by going long in June T-bill futures contracts. Suppose the treasurer expected higher short-term rates in June but was still concerned about the possibility of lower rates. To be able to gain from the higher rates and yet still hedge against lower rates, the treasurer could alternatively buy June call options on T-bill futures.

16. Future 91-Day T-Bill Investment Suppose there is a June T-bill futures call with Exercise price = $987,500 (index = 95, RD = 5) Price = $1,000 (quote = 4; C = (4)($250) = $1,000) June expiration (on both the underlying futures and futures option) occurring at the same time the $5M cash inflow is to be received.

17. Future 91-Day T-Bill Investment To hedge the 91-day investment with this call, the treasurer would need to buy 5 calls at a cost of $5,000:

18. Future 91-Day T-Bill Investment If T-bill rates were lower at the June expiration, then the treasurer would profit from the calls and could use the profit to defray part of the cost of the higher priced T-bills. As shown in the exhibit, if the spot discount rate on T-bills is 5% or less, the treasurer would be able to buy 5.058 spot T-bills (assume perfect divisibility) with the $5M cash inflow and profit from the futures calls, locking in a YTM for the next 91 days of approximately 4.75% on the $5M investment.

19. Future 91-Day T-Bill Investment

20. Future 91-Day T-Bill Investment If T-bill rates are higher, then the treasurer would benefit from lower spot prices while her losses on the call would be limited to just the $5,000 costs of the calls. For spot discount rates above 5%, the treasurer would be able to buy more T-bills, the higher the rates, resulting in higher yields as rates increase.

21. Future 91-Day T-Bill Investment Thus, for the cost of the call options, the treasurer is able to establish a floor by locking in a minimum YTM on the $5M June investment of approximately 4.75%, with the chance to earn a higher rate if short-term rates increase.

22. Hedging a CD Rate with an OTC Interest Rate Put Suppose the ABC manufacturing company was expecting a net cash inflow of $10M in 60 days from its operations and was planning to invest the excess funds in a 90-day CD from Sun Bank paying the LIBOR. To hedge against any interest rate decreases occurring 60 days from the now, suppose the company purchases an interest rate put (corresponding to the bank's CD it plans to buy) from Sun Bank for $10,000.

23. Hedging a CD Rate with an OTC Interest Rate Put Suppose the put has the following terms: Exercise rate = 7% Reference rate = LIBOR Time period applied to the payoff = 90/360 Day Count Convention = 30/360 Notional principal = $10M Payoff made at the maturity date on the CD (90 days from the option’s expiration) Interest rate put’s expiration = T = 60 days (time of CD purchase) Interest rate put premium of $10,000 to be paid at the option’s expiration with a 7% interest: Cost = $10,000(1+(.07)(60/360)) = $10,117

24. Hedging a CD Rate with an OTC Interest Rate Put As shown in the exhibit, the purchase of the interest rate put makes it possible for the ABC company to earn higher rates if the LIBOR is greater than 7% and to lock in a minimum rate of 6.993% if the LIBOR is 7% or less.

25. Hedging a CD Rate with an OTC Interest Rate Put

26. Hedging a CD Rate with an OTC Interest Rate Put For example, if 60 days later the LIBOR is at 6.5%, then the company would receive a payoff (90 day later at the maturity of its CD) on the interest rate put of $12,500: The $12,500 payoff would offset the lower (than 7%) interest paid on the company’s CD of $162,500: At the maturity of the CD, the company would therefore receive CD interest and an interest rate put payoff equal to $175,000:

27. Hedging a CD Rate with an OTC Interest Rate Put With the interest-rate put’s payoffs increasing the lower the LIBOR, the company would be able to hedge any lower CD interest and lock in a hedged dollar return of $175,000. Based on an investment of $10M plus the $10,117 costs of the put, the hedged return equates to an effective annualized yield of 6.993%: On the other hand, if the LIBOR exceeds 7%, the company benefits from the higher CD rates, while its losses are limited to the $10,117 costs of the puts.

28. Short Hedging Cases

29. Hedging Future T-Bond Sale with an OTC T-Bond Put Consider the case of a trust-fund manager who plans to sell ten $100,000 face value T-bonds from her fixed income portfolio in September to meet an anticipated liquidity need. The T-bonds the manager plans to sell pay a 6% interest and are currently priced at 94 (per $100 face value), and at their anticipated selling date in September, they will have exactly 15 years to maturity and no accrued interest. Suppose the manager expects long-term rates in September to be lower and therefore expects to benefit from higher T-bond prices when she sells her bonds, but she is also concerned that rates could increase and does not want to risk selling the bonds at prices lower than 94.

30. Hedging Future T-Bond Sale with an OTC T-Bond Put As a strategy to lock in minimum revenue from the September bond sale if rates increase, while obtaining higher revenues if rates decrease, suppose the manager decides to buy spot T-bond puts from an OTC Treasury security dealer who is making a market in spot T-bond options. Suppose the manger pays the dealer $10,000 for a put option on ten 15-year, 6% T-bonds with an exercise price of 94 per $100 face value and expiration coinciding with the manager’s September sales date. The exhibit shows the manager's revenue from either selling the T-bonds on the put if T-bond prices are less than 94 or on the spot market if prices are equal to or greater than 94.

31. Hedging Future T-Bond Sale with an OTC T-Bond Put

32. Hedging Future T-Bond Sale with an OTC T-Bond Put If the price on a 15-year T-bond is less than 94 at expiration (or rates are approximately 6.60% or more) the manager would be able to realize a minimum net revenue of $930,000 by selling her T-bonds to the dealer on the put contract at X = $940,000 and paying the $10,000 cost for the put; If T-bond prices are greater than 94 (below approximately 6.60%), her put option would be worthless, but her revenue from selling the T-bond would be greater, the higher T-bond prices, while the loss on her put position would be limited to the $10,000 cost of the option.

33. Hedging Future T-Bond Sale with an OTC T-Bond Put Thus, by buying the put option, the trust-fund manager has attained insurance against decreases in bond prices. Such a strategy represents a bond insurance strategy.

34. Hedging Future T-Bond Sale with an OTC T-Bond Put Note: If the portfolio manager were planning to buy long-term bonds in the future and was worried about higher bond prices (lower rates), she could hedge the future investment by buying T-bond spot or futures calls.

35. Managing the Maturity Gap with a Eurodollar Futures Put In Chapter 13, we examined the case of a small bank with a maturity gap problem resulting from making $1M loans in June with maturities of 180 days, financed by selling $1M worth of 90-day CDs at the current LIBOR of 5% and then 90 days later selling new 90-day CDs to finance its June CD debt of $1,012,103: To minimize its exposure to market risk, the bank hedged its $1,012,103 CD sale in September by going short in 1.02491 ($1,012,103/$987,500) September Eurodollar futures contract trading at quoted index of 95 ($987,500).

36. Managing the Maturity Gap with a Eurodollar Futures Put Instead of hedging its future CD sale with Eurodollar futures, the bank could alternatively buy put options on Eurodollar futures. By hedging with puts, the bank would be able to lock in or cap the maximum rate it pays on it September CD.

37. Managing the Maturity Gap with a Eurodollar Futures Put For example, suppose the bank decides to hedge its September CD sale by buying a September Eurodollar futures put with Expiration coinciding with the maturity of its September CD Exercise price of 95 (X = $987,500) Quoted premium of 2 (P = $500)

38. Managing the Maturity Gap with a Eurodollar Futures Put With the September debt from the June CD of $1,012,103, the bank would need to buy 1.02491 September Eurodollar futures puts (assume perfect divisibility) at a total cost of $512.46 to cap the rate it pays on its September CD:

39. Managing the Maturity Gap with a Eurodollar Futures Put If the LIBOR at the September expiration is greater than 5%, the bank will have to pay a higher rate on its September CD, but it will profit from its Eurodollar futures put position, with the put profits being greater, the higher the rate. The put profit would serve to reduce part of the $1,012,103 funds the bank would need to pay on the maturing June CD, in turn, offsetting the higher rate it would have to pay on its September CD.

40. Managing the Maturity Gap with a Eurodollar Futures Put As shown in the exhibit, if the LIBOR is at discount yield of 5% or higher, then the bank would be able to lock in a debt obligation 90 days later of $1,025,435 (allow for slight rounding differences), for an effective 180-day rate of 5.225%. If the rate is less than or equal to 5%, then the bank would be able to finance its $1,012,615.46 debt (June CD of $1,012,103 and put cost of $512.46) at lower rates, while its losses on its futures puts would be limited to the premium of $512.46. As a result, for lower rates the bank would realize a lower debt obligation 90 days later and therefore a lower rate paid over the 180-day period.

41. Managing the Maturity Gap with a Eurodollar Futures Put

42. Managing the Maturity Gap with a Eurodollar Futures Put Thus, for the cost of the puts, hedging the maturity gap with puts allows the bank to lock in a maximum rate on its debt obligation, with the possibility of paying lower rates if interest rates decrease.

43. Hedging a Future Loan Rate with an OTC Interest Rate Call Suppose a construction company plans to finance one of its project with a $10M, 90-day loan from Sun Bank, with the loan rate to be set equal to the LIBOR + 100 BP when the project commences 60 day from now. Furthermore, suppose that the company expects rates to decrease in the future, but is concerned that they could increase.

44. Hedging a Future Loan Rate with an OTC Interest Rate Call To obtain protection against higher rates, suppose the company buys an interest rate call option from Sun Bank for $20,000 with the following terms: Exercise rate = 7% Reference rate = LIBOR Time period applied to the payoff = 90/360 Notional principal = $10M Payoff made at the maturity date on the loan (90 days after the option’s expiration) Interest rate call’s expiration = T = 60 days (time of the loan) Interest rate call premium of $20,000 to be paid at the option’s expiration with a 7% interest: Cost = $20,000(1+(.07)(60/360)) = $20,233

45. Hedging a Future Loan Rate with an OTC Interest Rate Call The exhibit shows the company's cash flows from the call, interest paid on the loan, and effective interest costs that would result given different LIBORs at the starting date on the loan and the expiration date on the option. As shown in Column 6 of the table, the company is able to lock in a maximum interest cost of 8.016% if the LIBOR is 7% or greater at expiration, while still benefiting with lower rates if the LIBOR is less than 7%.

46. Hedging a Future Loan Rate with an OTC Interest Rate Call

47. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge In Chapter 13, we defined the price sensitivity model for hedging debt positions in which the underlying futures contract was not the same as the debt position to be hedge. The model determines the number of futures contracts that will make the value of a portfolio consisting of a fixed-income security and an interest rate futures contract invariant to small changes in interest rates.

48. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge The model also can be extended to hedging with put or call options. The number of options (calls for hedging long positions and puts for short positions) using the price-sensitivity model is:

49. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge Suppose a bond portfolio manager is planning to liquidate part of his portfolio in September. The portfolio he plans to sell consist of a mix of A to AAA quality bonds with a weighted average maturity of 15.25 years, face value of $10M, weighted average yield of 8%, portfolio duration of 10, and current value of $10M. Suppose the manager would like to benefit from lower long-term rates that he expects to occur in the future but would also like to protect the portfolio sale against the possibility of a rate increase.

50. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge To achieve this dual objective, the manager could buy a spot or futures put on a T-bond. Suppose there is a September 95 (X = $95,000) T-bond futures put option trading at $1,156 with the cheapest-to-deliver T-bond on the put’s underlying futures being a bond with a current maturity of 15.25 years, duration of 9.818, and currently priced to yield 6.0%.

51. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge Using the price-sensitivity model, the manager would need to buy 81 puts at a cost of $93,636 to hedge his bond portfolio:

52. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge Suppose that in September, long-term rates were higher, causing the value of the bond portfolio to decrease from $10M to $9.1M and the prices on September T-bond futures contracts to decrease from 95 to 86. In this case, the bond portfolio’s $900,000 loss in value would be partially offset by a $635,364 profit on the T-bond futures puts: ? = 81($95,000 - $86,000) - $93,636 = $635,364. The manager’s hedged portfolio value would therefore be $9,735,364; a loss of 2.6% in value (this loss includes the cost of the puts) compared to a 9% loss in value if the portfolio were not hedged.

53. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge On the other hand, if rates in September were lower, causing the value of the bond portfolio to increase from $10M to $10.5M and the prices on the September T-bond futures contracts to increase from 95 to 100, then the puts would be out of the money and the loss of the options would be limited to the $93,636. In this case, the hedged portfolio value would be $10.406365M – a 4.06% gain in value compared to the 5% gain for an unhedged position.

54. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge The exhibit shows the put-hedged bond portfolio values for a number of pairs of T-bond futures and bond portfolio values (and their associate yields). As shown, for increasing interest rates cases in which the pairs of the T-bond and bond portfolio values are less than 95 (the exercise price) and $10,000,000, respectively, the hedge portfolio losses are between approximately 1% and 2.6%, while for increasing interest rate cases in which the pairs of T-bond prices and bond values are greater than 95 and $10,000,000, the portfolio increases as the bond value increases.

55. Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge

56. Hedging a Series of Cash Flows

57. Hedging a Series of Cash Flows Using Exchange-Traded Options When there is series of cash flows, such as a floating-rate loan or an investment in a floating-rate note, a series or strip of interest rate options can be used. For example, a company with a one-year floating-rate loan starting in September at a specified rate and then reset in December, March, and June to equal the spot LIBOR plus BP, could obtain a maximum rate or cap on the loan by buying a series of Eurodollar futures puts expiring in December, March, and June.

58. Hedging a Series of Cash Flows Using Exchange-Traded Options At each reset date, if the LIBOR exceeds the discount yield on the put, the higher LIBOR applied to the loan will be offset by a profit on the nearest expiring put, with the profit increasing the greater the LIBOR. If the LIBOR is equal to or less than the discount yield on the put, the lower LIBOR applied to the loan will only be offset by the limited cost of the put. Thus, a strip of Eurodollar futures puts used to hedge a floating-rate loan places a ceiling on the effective rate paid on the loan.

59. Hedging a Series of Cash Flows Using Exchange-Traded Options In the case of a floating-rate investment, such as a floating-rate note tied to the LIBOR or a bank’s floating rate loan portfolio, a minimum rate or floor can be obtained by buying a series of Eurodollar futures calls, with each call having an expiration near the reset date on the investment. If rates decrease, the lower investment return will be offset by profits on the calls. If rates increase, the only offset will be the limited cost of the calls.

60. Hedging a Series of Cash Flows Using OTC Caps And Floors Using exchange-traded options to establish interest rate floors and ceiling on floating rate assets and liabilities is subject to hedging risk. As a result, many financial and non-financial companies looking for such interest rate insurance prefer to buy OTC caps or floors that can be customized to meet their specific needs.

61. Floating Rate Loan Hedged with an OTC Cap Example: Suppose the Diamond Development Company borrows $50M from Commerce Bank to finance a two-year construction project. Suppose the loan is for two years, starting on March 1 at a known rate of 8%, then resets every three months -- 6/1, 9/1, 12/1, and 3/1 -- at the prevailing LIBOR plus 150 BP.

62. Floating Rate Loan Hedged with an OTC Cap In entering this loan agreement, suppose the company is uncertain of future interest rates and therefore would like to lock in a maximum rate, while still benefiting from lower rates if the LIBOR decreases.

63. Floating Rate Loan Hedged with an OTC Cap To achieve this, suppose the company buys a cap corresponding to its loan from Commerce Bank for $150,000, with the following terms: The cap consist of seven caplets with the first expiring on 6/1/2003 and the others coinciding with the loan’s reset dates. Exercise rate on each caplet = 8%. NP on each caplet = $50M. Reference Rate = LIBOR. Time period to apply to payoff on each caplet = 90/360. (Typically the day count convention is defined by the actual number of days between reset date.) Payment date on each caplet is at the loan’s interest payment date, 90 days after the reset date. The cost of the cap = $150,000; it is paid at beginning of the loan, 3/1/2003.

64. Floating Rate Loan Hedged with an OTC Cap On each reset date, the payoff on the corresponding caplet would be With the 8% exercise rate (sometimes called the cap rate), the Diamond Company would be able to lock in a maximum rate each quarter equal to the cap rate plus the basis points on the loan, 9.5%, while still benefiting with lower interest costs if rates decrease. This can be seen in the exhibit, where the quarterly interests on the loan, the cap payoffs, and the hedged and unhedged rates are shown for different assumed LIBORs at each reset date on the loan.

65. Floating Rate Loan Hedged with an OTC Cap

66. Floating Rate Loan Hedged with an OTC Cap For the five reset dates from 12/1/2003 to the end of the loan, the LIBOR is at 8% or higher. In each of these cases, the higher interest on the loan is offset by the payoff on the cap, yielding a hedged rate on the loan of 9.5% (the 9.5% rate excludes the $150,000 cost of the cap; the rate is 9.53% with the cost included). For the first two reset dates on the loan, 6/1/2003 and 9/1/2003, the LIBOR is less than the cap rate. At these rates, there is no payoff on the cap, but the rates on the loan are lower with the lower LIBORs.

67. Floating Rate Asset Hedged with an OTC Floor As noted, floors are purchased to create a minimum rate on a floating-rate asset. As an example, suppose the Commerce Bank in the above example wanted to establish a minimum rate or floor on the rates it was to receive on the two-year floating-rate loan it made to the Diamond Company.

68. Floating Rate Asset Hedged with an OTC Floor To this end, suppose the bank purchased from another financial institution a floor for $100,000 with the following terms corresponding to its floating-rate asset: The floor consist of seven floorlets with the first expiring on 6/1/2003 and the others coinciding with the reset dates on the bank’s floating-rate loan to the Diamond Company. Exercise rate on each floorlet = 8%. NP on each floorlet = $50M. Reference Rate = LIBOR. Time period to apply to payoff on each floorlet = 90/360. Payment date on each floorlet is at the loan’s interest payment date, 90 days after the reset date. The cost of the floor = $100,000; it is paid at beginning of the loan, 3/1/2003.

69. Floating Rate Asset Hedged with an OTC Floor On each reset date, the payoff on the corresponding floorlet would be With the 8% exercise rate, Commerce Bank would be able to lock in a minumum rate each quarter equal to the floor rate plus the basis points on the floating-rate asset, 9.5%, while still benefiting with higher returns if rates increase.

70. Floating Rate Asset Hedged with an OTC Floor In the exhibit, Commerce Bank’s quarterly interests received on its loan to Diamond, its floor payoffs, and its hedged and unhedged yields on its loan are shown for different assumed LIBORs at each reset date.

71. Floating Rate Asset Hedged with an OTC Floor

72. Floating Rate Asset Hedged with an OTC Floor For the first two reset dates on the loan, 6/1/2003 and 9/1/2003, the LIBOR is less than the floor rate of 8%. At theses rates, there is a payoff on the floor that compensates for the lower interest Commerce receives on the loan; this results in a hedged rate of return on the bank’s loan asset of 9.5% (the rate is 9.52% with the $100,000 cost of the floor included). For the five reset dates from 12/1/2003 to the end of the loan, the LIBOR equals or exceeds the floor rate. At these rates, there is no payoff on the floor, but the rates the bank earns on its loan are greater, given the greater LIBORs.

73. Collars A collar is combination of a long position in a cap and a short position in a floor with different exercise rates. The sale of the floor is used to defray the cost of the cap. For example, the Diamond Company in our above case could reduce the cost of the cap it purchased to hedge its floating rate loan by selling a floor. By forming a collar to hedge its floating-rate debt, the Diamond Company, for a lower net hedging cost, would still have protection against a rate movement against the cap rate, but it would have to give up potential interest savings from rate decreases below the floor rate.

74. Collars Example: suppose the Diamond Company decided to defray the $150,000 cost of its 8% cap by selling a 7% floor for $70,000, with the floor having similar terms to the cap: Effective dates on floorlet = reset date on loan Reference rate = LIBOR NP on floorlets = $50M Time period for rates = .25

75. Collars By using the collar instead of the cap, the company reduces its hedging cost from $150,000 to $80,000, and as shown in the exhibit can still lock in a maximum rate on its loan of 9.5%. However, when the LIBOR is less than 7%, the company has to pay on the 7% floor, offsetting the lower interest costs it would pay on its loan. For example: When the LIBOR is at 6% on 6/1/2003, Diamond has to pay $125,000 ninety days later on its short floor position. When the LIBOR is at 6.5% on 9/1/2003, the company has to pay $62,500. These payments, in turn, offset the benefits of the respective lower interest of 7.5% and 8% (LIBOR + 150) it pays on its floating rate loan.

76. Collars

77. Collars Thus, for LIBORs less than 7%, Diamond has a floor in which it pays an effective rate of 8.5% (losing the benefits of lower interest payments on its loan) and for rates above 8% it has a cap in which it pays an effective 9.5% on its loan.

78. Corridor An alternative financial structure to a collar is a corridor. A corridor is a long position in a cap and a short position in a similar cap with a higher exercise rate. The sale of the higher exercise-rate cap is used to partially offset the cost of purchasing the cap with the lower strike rate.

79. Corridor For example, the Diamond company, instead of selling a 7% floor for $70,000 to partially finance the $150,000 cost of its 8% cap, could sell a 9% cap for say $70,000. If cap purchasers believe there was a greater chance of rates increasing than decreasing, they would prefer the collar to the corridor as a tool for financing the cap.

80. Reverse Collar A reverse collar is combination of a long position in a floor and a short position in a cap with different exercise rates. The sale of the cap is used to defray the cost of the floor. For example, the Commerce Bank in our above floor example could reduce the $100,000 cost of the 8% floor it purchased to hedge the floating-rate loan it made to the Diamond company by selling a cap. By forming a reverse collar to hedge its floating-rate asset, the bank would still have protection against rates decreasing against the floor rate, but it would have to give up potential higher interest returns if rates increase above the cap rate.

81. Reverse Collar Example: Suppose Commerce sold a 9% cap for $70,000, with the cap having similar terms to the floor. By using the reverse collar instead of the floor, the company would reduce its hedging cost from $100,000 to $30,000, As shown in the exhibit, Commerce would lock in an effective minimum rate on it’s a asset of 9.5% and an effective maximum rate of 10.5%.

82. Reverse Collar

83. Reverse Corridor Instead of financing a floor with a cap, an investor could form a reverse corridor by selling another floor with a lower exercise rate.

84. Barrier Options Barrier options are options in which the payoff depends on whether an underlying security price or reference rate reaches a certain level. They can be classified as either knock-out or knock-in options: Knock-out option is one that ceases to exist once the specified barrier rate or price is reached. Knock-in option is one that comes into existence when the reference rate or price hits the barrier level.

85. Barrier Options Knock-out and knock-in options can be formed with either a call or put and the barrier level can be either above or below the current reference rate or price when the contract is established Down-and-out or up-and-out knock out options Up-and-in or down-and-in knock in options

86. Barrier Options Barrier caps and floors with termination or creation feature are offered in the OTC market at a premium above comparable caps and floors without such features.

87. Barrier Options Down-and-out caps and floors are options that ceases to exist once rates hit a certain level. Example: A two-year, 8% cap that ceases when the LIBOR hits 6.5%, or a two-year, 8% floor that ceases once the LIBOR hits 9%.

88. Barrier Options Up-and-in cap is one that become effective once rates hit a certain level. Example: A two-year, 8% cap that that becomes effective when the LIBOR hits 9% or a two-year, 8% floor that become effective when rates hit 6.5%.

89. Path-Dependent Options In the generic cap or floor, the underlying payoff on the caplet or floorlet depends only on the reference rate on the effective date. The payoff does not depend on previous rates; that is, it is independent of the path the LIBOR has taken. Some caps and floors, though, are structured so that their payoff is dependent on the path of the reference rate.

90. Path-Dependent Options: Average Cap An average cap is one in which the payoff depends on the average reference rate for each caplet. If the average is above the exercise rate, then all the caplets will provide a payoff. If the average is equal or below, the whole cap expires out of the money.

91. Path-Dependent Options: Average Cap Consider a one-year average cap with an exercise rate of 7% with four caplets. If the LIBOR settings turned out to be 7.5%, 7.75%, 7%, and 7.5%, for an average of 7.4375%, then the average cap would be in the money: (.074375 - .07)(.25)(NP). If the rates, though, turned out to be 7%, 7.5%, 6.5, and 6%, for an average of 6.75%, then the cap would be out of the money.

92. Path-Dependent Options: Q-Cap In a cumulative cap (Q-cap), the cap seller pays the holder when the periodic interest on the accompanying floating-rate loan hits or exceeds a specified level. Example: Suppose the Diamond Company in our earlier cap example decided to hedge its two-year floating rate loan (paying LIBOR + 150BP) by buying a Q-Cap from Commerce Bank with the following terms:

93. Path-Dependent Options: Average Cap Q-Cap Terms: The cap consist of seven caplets with the first expiring on 6/1/2003 and the others coinciding with the loan’s reset dates. Exercise rates on each caplet = 8%. NP on each caplet = $50M. Reference Rate = LIBOR. Time period to apply to payoff on each caplet = 90/360. For the period 3/1/2003 to 12/1/2003, the caplet will payoff when the cumulative interest starting from loan date 3/1/2003 on the company’s loan hits $3M. For the period 3/1/2004 to 12/1/2004, the caplet will payoff when the cumulative interest starting from date 3/1/2004 on the company’s loan hits $3M. Payment date on each caplet is at the loan’s interest payment date, 90 days after the reset date. The cost of the cap = $125,000; it is paid at beginning of the loan, 3/1/2003.

94. Path-Dependent Options: Q-Cap The exhibit shows the quarterly interest, cumulative interests, Q-cap payments, and effective interests for assumed LIBORs. In the Q-caps first protection period, 3/1/2003 to 12/1/2003, Commerce Bank will pay the Diamond Company on its 8% caplet when the cumulative interest hits $3M. The cumulative interest hits the $3M limit on reset date 9/1/2003, but on that date the 9/1/2003 caplet is not in the money. On the following reset date, though, the caplet is in the money at the LIBOR of 8.5%. Commerce would, in turn, have to pay Diamond $62,500 (90 days later) on the caplet, locking in a hedged rate of 9.5% on Diamond’s loan.

95. Path-Dependent Options: Q-Cap In the second protection period, 3/1/2004 to 12/1/2004, the assumed LIBOR rates are higher. The cumulative interest hits the $3M limit on reset date 9/1/2004. Both the caplet on that date and the next reset date (12/1/2004) are in the money. As a result, with the caplet payoffs, Diamond is able to obtained a hedged rate of 9.5% for the last two payment periods on its loan.

96. Path-Dependent Options: Q-Cap

97. Path-Dependent Options: Q-Cap When compared to a standard cap, the Q-cap provides protection for the one-year protection periods, while the standard cap provides protection for each period (quarter). As shown in the next exhibit, a standard 8% cap provides more protection given the assumed increasing interest rate scenario than the Q-cap, capping the loan at 9.5% from date 12/1/2003 to the end of the loan and providing a payoff on 5 of the 7 caplets for a total payoff of $687,500. In contrast, the Q-cap pays on only 3 of the 7 caplets for a total payoff of only $500,000. Because of its lower protection limits, a Q-cap cost less than a standard cap.

98. Path-Dependent Options: Q-Cap

99. Exotic Options Q-caps, average caps, knock-in options, and knock-out options are sometimes referred to as exotic options. Exotic option products are non-generic products that are created by financial engineers to meet specific hedging needs and return-risk profiles. The next two slides define some of the popular exotics options used in interest rate management.

100. Exotic Options

101. Exotic Options

102. Currency Options When investors purchase and hold foreign securities or when corporations and governments sell debt securities in external markets or incur foreign debt positions, they are subject to exchange-rate risk. In Chapter 13, we examined how foreign currency futures contracts could be used by financial and non-financial corporations to hedge their international positions. These positions can also be hedge with currency options.

103. Currency Options: Markets In 1982, the Philadelphia Stock Exchange (PHLX) became the first organized exchange to offer trading in foreign currency options. Foreign currency options also are traded on a number of derivative exchanges outside the U.S. In addition to offering foreign currency futures, the International Monetary Market and other futures exchanges also offers options on foreign currency futures.

104. Currency Options: Markets There is also a sophisticated dealer's market. This interbank currency options market is part of the interbank foreign exchange market. In this dealer's market, banks provide tailor-made foreign currency option contracts for their customers, primarily multinational corporations. Compared to exchange-traded options, options in the interbank market are larger in contract size, often European, and are available on more currencies.

105. Currency Options: Hedging With exchange-traded currency options and dealer's options, hedgers, for the cost of the options, can obtain not only protection against adverse exchange rate movements, but (unlike forward and futures positions) benefits if the exchange rates move in favorable directions.

106. Currency Options: Hedging Example: Consider the case of a U.S. fund with investments in Eurobonds that were to pay a principal in British pounds of £10M next September. For the costs of BP put options, the U.S. fund could protect its dollar revenues from possible exchange rate decreases when it converts, while still benefiting if the exchange rate increases.

107. Currency Options: Hedging Suppose there is a September BP put with Exercise price of X = $1.425/£ Price = P = $0.02/£. Contract size of 31,250 British pounds The U.S. fund would need to buy 320 put contracts at a cost of $200,000 to establish a floor for the dollar value of its £10,000,000 receipt in September.

108. Currency Options: Hedging The exhibit shows the dollar cash flows the U.S. Fund would receive in September from converting its receipts of £10,000,000 to dollars at the spot exchange rate (ET) and closing its 320 put contracts at a price equal to the put's intrinsic value (assume the September payment date and option expiration date are the same).

109. Currency Options: Hedging

110. Currency Options: Hedging If the exchange rate is less than X = $1.425/£, the company would receive less than $14,250,000 when it converts its £10,000,000 to dollars; these lower revenues, however, would be offset by the profits from the put position. On the other hand, if the exchange rate at expiration exceeds $1.425/£, the U.S. fund would realize a dollar gain when it converts the £10,000,000 at the higher spot exchange rate, while its losses on the put would be limited to the amount of the premium.

111. Currency Options: Hedging Thus, by hedging with currency put options, the company is able to obtain exchange rate risk protection in the event the exchange rate decreases while still retaining the potential for increased dollar revenues if the exchange rate rises.

112. Currency Options: Hedging Suppose that instead of receiving foreign currency, a U.S. company had a foreign liability requiring a foreign currency payment at some future date. To protect itself against possible increases in the exchange rate while still benefiting if the exchange rate decreases, the company could hedge the position by taking a long position in a currency call option.

113. Currency Options: Hedging Example: Suppose a U.S. company owed £10,000,000, with the payment to be made in September. To benefit from the lower exchange rates and still limit the dollar costs of purchasing £10,000,000 in the event the $/£ exchange rate rises, the company could buy September British pound call options. The exhibit shows the costs of purchasing £10,000,000 at different exchange rates and the profits and losses from purchasing 320 September British pound calls with X = $1.425/£ at $0.02/£ (contract size = £31,250) and closing them at expiration at a price equal to the call's intrinsic value.

114. Currency Options: Hedging

115. Currency Options: Hedging As shown in the table, for cases in which the exchange rate is greater than $1.425/£, the company has dollar expenditures exceeding $14,250,000; the expenditures, though, are offset by the profits from the calls. On the other hand, when the exchange rate is less than $1.425/£, the dollar costs of purchasing £10,000,000 decreases as the exchange rate decreases, while the losses on the call options are limited to the option premium.

116. Pricing Interest Rate Options with a Binomial Interest Tree

117. Pricing Interest Rate Options with a Binomial Interest Tree In Chapter 9, we examined how the binomial interest rate model can be used to price bonds with embedded call and put options, sinking fund arrangements, and convertible clauses. The binomial interest rate tree also can be used to price interest rate options.

118. Valuing T-Bill Options with a Binomial Tree The exhibit shows: A two-period binomial tree for an annualized risk-free spot rate (S) The corresponding prices on a T-bill (B) with a maturity of .25 years and face value of $100 A futures contract (f) on the T-bill, with the futures expiring at the end of period 2.

119. Valuing T-Bill Options with a Binomial Tree The features of the binomial tree: The length of each period is six months (six-month steps) The upward parameter on the spot rate (u) is 1.1 The downward parameter (d) is 1/1.1 = 0.9091 The probability of spot rate increasing in each period is .5 The yield curve is assumed flat.

121. Valuing T-Bill Options with a Binomial Tree Spot T-Bill Prices At the current spot rate of 5%, the price of the T-bill is B0 = 98.79 (= 100/(1.05).25). In period 1, the price is 98.67 when the spot rate is 5.5% (= 100/(1.055).25) and 98.895 when the rate is 4.54545% (= 100/(1.0454545).25). In period 2, the T-bill prices are 98.54, 98.79, and 99 for spot rates of 6.05%, 5%, and 4.13223%, respectively.

122. Valuing T-Bill Options with a Binomial Tree Futures Prices The futures prices are obtained by assuming a risk neutral market. That is: If the market is risk neutral, then the futures price is an unbiased estimator of the expected spot price: ft = E(ST). The futures prices at each node in the exhibit are therefore equal to their expected price next period.

123. Valuing T-Bill Options with a Binomial Tree Values of call and put options on spot and futures T-bills: For European options, the methodology for determining the price is to start at expiration where we know the possible option values are equal to their intrinsic values, IVs. Given the option’s IVs at expiration, we then move to the preceding period and price the option to equal the present value of its expected cash flows for next period. Given these values, we then roll the tree to the next preceding period and again price the option to equal the present value of its expected cash flows. We continue this recursive process to the current period.

124. Valuing T-Bill Options with a Binomial Tree Values of call and put options on spot and futures T-bills: If the option is American, then its early exercise advantage needs to be taken into account by determining at each node whether or not it is more valuable to hold the option or exercise.

125. Valuing T-Bill Options with a Binomial Tree Values of call and put options on spot and futures T-bills: The methodology for valuing American options: Start one period prior to the option’s expiration and constrain the price of the American option to be the maximum of its binomial value (present value of next period’s expected cash flows) or the intrinsic value (i.e., the value from exercising). Roll those values to the next preceding period, and then price the option value as the maximum of the binomial value or the IV. Continue this process to the current period.

126. Valuing T-Bill Options with a Binomial Tree Values of American and European Spot Call Options: The exhibit show the binomial valuation of both a European and an American call on the spot T-bill, each with an exercise prices of 98.75 per $100 face value and expiration of one year. The price of the European call is 0.0787. The price of the American call is 0.08.

128. Valuing T-Bill Options with a Binomial Tree Value of European Futures Call Option: If the call option were on a European T-bill futures contract, instead of a spot T-bill, with the futures and option having the same expiration, then the value of the futures option will be the same as the spot option. That is, at the expiration spot rates of 6.05%, 5%, and 4.13223%, the futures prices on the expiring contract would be equal to the spot prices (98.54, 98.79, and 99), and the corresponding IVs of the European futures call would be 0, .04, and .25 – the same as the spot call’s IV. Thus, when we roll these call values back to the present period, we end up with the price on the European futures call of .0787 – the same as the European spot.

129. Valuing T-Bill Options with a Binomial Tree Value of American Futures Call Option: If the futures call option were American, then the option prices at each node needs to be constrained to be the maximum of the binomial value or the futures option’s IV. In this case, the corresponding prices of the American futures option are the same as the spot option. Thus, the price on the American T-bill futures call is .08 -- the same price as the American spot option.

130. Valuing T-Bill Options with a Binomial Tree Value of American Futures Put Option: The next exhibit shows the binomial valuation of both a European and American T-bill futures put with an exercise price of 98.75 and expiration of one year (two periods). The price of the European futures put is .05. The price of the American futures put is .05.

132. Valuing T-Bill Options with a Binomial Tree Value of American Futures Put Option: Note: The put price of .05 is consistent with the put-call futures parity relation:

133. Valuing a Caplet and Floorlet with a Binomial Tree The price of a caplet or floorlet can also be valued using a binomial tree of the option’s reference rate. Consider an interest rate call and put on the spot rate defined by our binomial tree, with Exercise rate of 5% Time period applied to the payoff of ? = .25 Notional principal of NP = 100.

134. Valuing a Caplet and Floorlet with a Binomial Tree The next exhibit shows the binomial valuation of the interest rate call and the interest rate put. The value of the caplet is 0.06236. The value of the floorlet is 0.05177.

136. Valuing a Cap and Floorwith a Binomial Tree Since a cap is a series of caplets, its price is simply equal to the sum of the values of the individual caplets making up the cap. To price a cap, we can use a binomial tree to price each caplet and then aggregate the caplet values to obtain the value of the cap. Similarly, the value of a floor can be found by summing the values of the floorlets comprising the floor.

137. Valuing T-Bond Options with a Binomial Tree The T-bill underlying the spot or futures T-bill option is a fixed-deliverable bill; that is, the features of the bill (maturity of 91 days and principal of $1M) do not change during the life of the option. In contrast, the T-bond or T-note underlying a T-bond or T-note option or futures option is a specified T-bond or note or the bond from an eligible group that is most likely to be delivered. Because of the specified bond clause on a T-bond or note option or futures option, the first step in valuing the option is to determine the values of the specified T-bond (or bond most likely to be delivered) at the various nodes on the binomial tree, using the same methodology we used in Chapter 9 to value a coupon bond.

138. Valuing T-Bond Options with a Binomial Tree Example: Consider an OTC spot option on a T-bond with 6% annual coupon Face value of $100 3 years left to maturity

139. Valuing T-Bond Options with a Binomial Tree In valuing the bond, assume: 2-period binomial tree of risk-free spot rates Length of each period is one year Upward parameter = u = 1.2 Downward parameter = d = .8333 Current spot rate S0 = 6%

140. Valuing T-Bond Options with a Binomial Tree Binomial Valuation of T-Bond To value the T-bond, we start at the bond’s maturity (end of period 3) where the bond’s value is equal to the principal plus the coupon, 106. We next determine the three possible values in period 2 given the three possible spot rates. Given these values, we next roll the tree to the first period and determine the two possible values. Finally, using the bond values in period 1, we roll the tree to the current period where we determine the value of the T-bond. As shown in the exhibit the value of the T-bond is 99.78.

141. Valuing T-Bond Options with a Binomial Tree Binomial Valuation of T-Bond Futures The exhibit also shows the prices on a two-year futures contract on the three-year, 6% T-bond. The prices are generated by assuming a risk-neutral market. The futures price is f0 = 99.83.

143. Valuing T-Bond Options with a Binomial Tree Values of American and European Spot Call The next exhibit shows the binomial valuation of both a European and an American call on the spot T-bond, each with an exercise prices of 98 per $100 face value, and expiration of two year. The price of the European call is 1.734. The price of the American call is 2.228.

145. Valuing T-Bond Options with a Binomial Tree Value of European Futures Call If the European call were an option on a futures contract on the three-year, 6% T-bond (or if that bond were the most likely to-be-deliver bond on the futures contract), with the futures contract expiring at the same time as the option (end of period 2), then the value of the futures option will be the same as the spot. That is, at expiration the futures prices on the expiring contract would be equal to the spot prices, and the corresponding IVs of the European futures call would be the same as the spot call’s IV. Thus, when we roll these call values back to the present period, we end up with the price on the European futures call being the same as the European spot: 1.734.

146. Valuing T-Bond Options with a Binomial Tree Value of American Futures Call If the futures call were American, then at the spot rate of 5% in period 1, its IV would be 2.88 (= Max[100.88-98,0]), exceeding the binomial value of 2.743. Rolling the 2.88 value to the current period yields a price on the American futures option of 1.798 (= .5(.9328) + .5(2.88)) – this price differs from the American spot option price of 2.228.

147. Valuing T-Bond Options with a Binomial Tree European Spot and Futures Put Values Given the bond’s possible prices at expiration of 97.57, 100, and 101.76, the corresponding IVs of the put are .43, 0, and 0. In period 1, the put’s two possible values are .2006 (= (.5(.43) + .5(0))/1.072) and 0. Rolling these value to the current period yields a price on the European put of .0946 (= (.5(.2006) + .5(0))/1.06).

148. Valuing T-Bond Options with a Binomial Tree American Spot and Futures Put Values If the spot put were American, then its possible prices in period 1 would be .253 and 0, and it current price would be .119 (= (.5(.253) + .5(0))/1.06). If the futures put were American, there would be no exercise advantage in period 1 and thus the price would be equal to its European value of .0946.

149. Arbitrage-Free Features of the Binomial Tree Suppose the binomial tree of spot rates shown in the last exhibit were calibrated to the current spot yield curve. If so, the T-bond and T-bond options would be arbitrage free.

150. Arbitrage-Free Features of the Binomial Tree The binomial price of the three-year 6% bond is equal to the value of a portfolio of three stripped bonds: 1-year zero coupon paying $6, a 2-year zero paying $6, and a 3-year zero paying $106. The next exhibit shows the prices on the 1-year, 2-year, and 3-year stripped securities as 5.66, 5.33, and 88.79, respectively. Summing these prices, we obtain a portfolio value of 99.78 -- the same value of the three-year 6% T-bond. Thus, the bond valuation approach yields a bond value equal to the value of a portfolio of stripped zero-coupon bonds.

152. Arbitrage-Free Features of the Binomial Tree The model also values the option as an arbitrage-free price. In this example, the 1.734 value of the spot European T-bond call is also equal to the value of a replicating portfolio consisting of a 1-year zero and a 2-year zero constructed so that next year the portfolio is worth .9328 if the spot rate is 7.2% and 2.743 if the spot rater is 5%. The replicating portfolio is formed by solving for the number of 1-year zero coupon bonds, n1, and number of 2-year zeros, n2, where:

153. Arbitrage-Free Features of the Binomial Tree Solving for n1 and n2, we obtain n1 = -14.1711 and n2 = 15.35812. This replicating portfolio yields possible cash flows next period that match the option’s cash flows of .9328 and 2.743. The value of this portfolio is also equal to the equilibrium price of the option, 1.734: The two call values in period 1 can also be shown to be equal to the values of their replicating portfolio. Thus, the binomial model yields an arbitrage-free price in which the value of the option is equal to the value of the replicating portfolio.

154. Pricing Interest Rate Options with the Black Futures Model

155. Black Futures Model An extension of the B-S OPM that is sometimes used to price interest rate options is the Black’s futures option model. The model is defined as follows:

156. Black Futures Model where: ?f2 = variance of the logarithmic return of futures prices = V(ln(fn/f0) T = time to expiration expressed as a proportion of a year. Rf = continuously compounded annual risk-free rate (if simple annual rate is R, the continuously compounded rate is ln(1+R)). N(d) = cumulative normal probability; this probability can be looked up in a standard normal probability table or by using the following formula: N(d) = 1 – n(d), for d < 0 N(d) = n(d), for d > 0, where: n(d) = 1 - .5[1 + .196854 (?d?) + .115194 (?d?)2 + .0003444 (?d?)3 + .019527(?d?)4 ]-4 ?d? = absolute value of d.

157. Black Futures Model Consider the European futures T-bill call options in which the futures option has Exercise price of 98.75 Expiration of one year Current futures price = f0 = 98.7876. Assume: Simple risk-free rate is 5%, implying a continuously compound rate of 4.879% (= ln(1.05)) Annualized standard deviation of the futures logarithmic return, ?(ln(fn/f0), is .00158

158. Black Futures Model Using the Black futures model the price of the T-bill futures call would be .0792:

159. Black Futures Model Consider one-year put and call options on a CBOT T-bond futures contract, with each option having an exercise price of $100,000. Suppose: Current futures price = $96,115 Futures volatility is ?(ln(fn/f0) = .10, Continuously compound risk-free rate = .065.

160. Black Futures Model Using the Black futures option model, the price of the call option would be $2,137 and the price of the put would be $5,777:

161. Black Futures Model Note: The call and future prices are also consistent with put-call futures parity

162. Pricing Caplets and Floorlets with the Black Futures Model The Black futures option model also can be extended to pricing caplets and floorlets by: Substituting T* for T in the equation for C* (for a caplet) or P* (for a floorlet), where T* is the time to expiration on the option plus the time period applied to the interest rate payoff time period, ?: T* = T + ?. Using an annual continuous compounded risk-free rate for period T* instead of T. Multiplying the Black adjusted-futures option model by the notional principal times the time period: (NP) ?.

163. Pricing Caplets and Floorlets with the Black Futures Model

164. Pricing Caplets and Floorlets with the Black Futures Model Consider a caplet with Exercise rate of X = 7% NP = $100,000 ? = .25 Expiration = T = .25 year Reference rate = LIBOR.

165. Pricing Caplets and Floorlets with the Black Futures Model Assume Current LIBOR = R = 6% Estimated annualized standard deviation of the LIBOR’s logarithmic return = .2 Continuously compounded risk-free rate = 5.8629%

166. Pricing Caplets and Floorlets with the Black Futures Model Using the Black model, the price of the caplet is 4.34:

167. Pricing a Cap Suppose the caplet represented part of a contract that caps a two-year floating-rate loan of $100,000 at 7% for a three-month period. The cap consist of seven caplets, with expirations of T = .25 years, .5, .75, 1, 1.25, 1.5, and 1.75. The value of the cap is equal to the sum of the values of the caplets comprising the cap. If we assume a flat yield curve such that the continuous rate of 5.8629% applies, and we use the same volatility of .2 for each caplet, then the value of the cap would be $254.38.

168. Pricing a Cap