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Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 14: Interest Rate Derivatives Derivatives are part of the vital machinery of the bank. We have put it at the heart of the business. Sman Majd, Global Head of OTC Derivatives Deutsche Bank Risk, May 1999

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Copyright © 2001 by Harcourt, Inc. All rights reserved.2 Important Concepts in Chapter 14 n The concept of a derivative on an interest rate as opposed to a derivative on a price n Concepts and applications of forward rate agreements (FRAs), interest rate swaps, interest rate options and swaptions n Pricing derivatives using a binomial term structure

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Copyright © 2001 by Harcourt, Inc. All rights reserved.3 Forward Rate Agreements n Definition: a forward contract where the payoff is determined by an interest rate, usually LIBOR n Payoff is made at expiration based on LIBOR at that time times a notional principal times days/360 where days can be the actual day count or an agreed-upon number like 90, 180, etc. The FRA has an agreed-upon rate that one party will pay while the other pays a rate to be determined later, usually LIBOR. The payoff is u (Notional Principle)(LIBOR - Agreed-upon rate) (Days/360) n In practice payoff is made at expiration, unlike swaps and interest rate options.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.4 Forward Rate Agreements (continued) n See Table 14.1, p. 582 for example. The payoff is u $20,000,000(LIBOR -.10)90/360 n FRA is used to lock in a fixed rate. See Figure 14.1, p. 584 for comparison of outcomes with and without FRA. n Sample outcome. LIBOR = 12%: F FRA payoff: $20,000,000(.12 -.10)(90/360) = $100,000 F Interest on loan: $20,000,000(.13)(90/360) = $650,000 F Effective interest: $650,000 - $100,000 = $550,000 F Effective rate: ($20,550,000/$20,000,000) 365/90 - 1 =.1163

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Copyright © 2001 by Harcourt, Inc. All rights reserved.5 Forward Rate Agreements (continued) n Note that in practice FRA payment is made at end of period based on rate at end of period. Thus, the firm would have received $20,000,000 + $100,000 = $20,100,000 in 30 days and paid back $20,000,000 + $650,000 = $20,650,000 90 days later. This is a rate of ($20,650,000/$20,100,000) 365/90 - 1 =.1157.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.6 Interest Rate Swaps n Definition: an agreement between two parties in which each party makes a series of interest payments to the other at predetermined dates at different rates. u At least one rate must be floating, i.e., determined at a later date. u Plain vanilla swap: one party pays a fixed rate and the other pays a floating rate. u Usually the parties are an end user - someone needing a a derivative for a specific purpose - and a dealer - a firm offering to take the opposite side of a derivatives transaction.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.7 Interest Rate Swaps (continued) n The Structure of an Interest Rate Swap u Example: XYZ currently borrowing at 90-day LIBOR quarterly with rate reset at start of each quarter. It would prefer a fixed rate. Face value of loan is $50 million. Payment is made at end of quarter. It can do a swap with ABS based on LIBOR on 15th of month for one year. ABS pays XYZ fixed rate of 7.5%. Swap payment is u So the payment to XYZ in this problem is

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Copyright © 2001 by Harcourt, Inc. All rights reserved.8 Interest Rate Swaps (continued) n The Structure of an Interest Rate Swap (continued) F See Figure 14.2, p. 586 for pattern of payments F See Table 14.2, p. 586 for example of after-the-fact payments. u Note that only the net payment is exchanged and that notional principal is never exchanged.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.9 Interest Rate Swaps (continued) n Why Companies Use Interest Rate Swaps u To convert floating payments to fixed and vice versa, oftentimes acting on views of future interest rates. This is cheaper and easier than repaying one time of loan and issuing another. u To adjust the structures of cash inflows u To, in their eyes, save money by getting a better rate. For example, you can borrow floating or fixed. You borrow floating and convert to fixed with a swap. But the floating plus swap transaction has credit risk and the fixed only does not.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.10 Interest Rate Swaps (continued) n Transactions Equivalent to Interest Rate Swaps u A swap is identical to F issuing a fixed-rate bond and using the proceeds to buy a floating rate bond, or vice versa F entering into one spot transaction and a series of FRAs with interest paid at the end of each period based on the rate at the beginning of each period

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Copyright © 2001 by Harcourt, Inc. All rights reserved.11 Interest Rate Swaps (continued) n Pricing an Interest Rate Swap at the Outset u Determine the swap rate, which is the fixed rate that equates the present value of the fixed and floating payments. This can be done by treating the payments as a fixed rate bond and a floating rate bond. u See Table 14.3, p. 590 for example. u In general we can use the formula u This is equivalent to determining the coupon on a par bond

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Copyright © 2001 by Harcourt, Inc. All rights reserved.12 Interest Rate Swaps (continued) n Pricing an Interest Rate Swap at the Outset (continued) u In practice the rate is quoted as a spread over the rate on the U. S. Treasury note with equivalent maturity. u Also one rate is quoted when the dealer pays fixed and a higher rate is quoted when the dealer receives fixed.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.13 Interest Rate Swaps (continued) n Valuing an Interest Rate Swap During Its Life u At the outset, the swap has zero value. As rates change and time elapses, value is gained or lost. u Example: See Figure 14.3, p. 592 where we value the swap three months into its life. Note how the floating rate component has a value of par at the next payment date, provided we add the hypothetical notional principals. The new rates and discount factors are F 3 mos, 9.125%: 1/(1+.09125(90/360)) = 0.9777 F 6 mos, 10.000%: 1/(1+.10(270/360)) = 0.9302 F 9 mos, 10.375%: 1/(1+.10375(450/360)) = 0.8852 F 12 mos, 10.625%: 1/(1+.10625(630/360)) = 0.8432

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Copyright © 2001 by Harcourt, Inc. All rights reserved.14 Interest Rate Swaps (continued) n Pricing an Interest Rate Swap During Its Life (continued) u The present value of the fixed payments is F $974,788(0.9777) + $974,788(0.9302) + $974,788(0.8852) + $20,974,788(0.8432) = $20,408,621 u The present value of the floating payments is F $20,900,000(0.9777) = $20,433,930 u So the overall swap value is F $20,433,930 - $20,408,621 = $25,309

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Copyright © 2001 by Harcourt, Inc. All rights reserved.15 Interest Rate Swaps (continued) n Terminating an Interest Rate Swap u reversal or offset u sale or assignment u buy-back or close-out u exercise of a swaption

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Copyright © 2001 by Harcourt, Inc. All rights reserved.16 Interest Rate Options n Interest Rate Calls and Puts u The right to make a fixed interest payment and receive a floating interest payment (call) or the right to receive a fixed interest payment and make a floating interest payment (put) u Fixed payment is determined by the strike rate or exercise rate.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.17 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u Payoff of an interest rate call u (Notional Principal)Max(0, LIBOR - X)(Days/360) F Option expires but payment is made later. This is standard procedure in swaps and interest rate options. u See example in Table 14.4, p. 597 of a firm planning to borrow $20 million in 30 days at a fixed rate and using the call to lock in a maximum fixed rate. The strike rate is 10% and the premium is $50,000. The firm borrow for 90 days at LIBOR plus 100 bps. The option payoff is F $20,000,000Max(0,LIBOR -.10)(90/360)

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Copyright © 2001 by Harcourt, Inc. All rights reserved.18 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u First we compound the premium forward for 30 days at today’s LIBOR (10%) plus 100 bps. F $50,000[1 +.11(30/360)] = $50,458 u The effective proceeds when the loan is taken out is F $20,000,000 - $50,458 = $19,949,542 u Sample outcome: LIBOR = 6% F Loan interest: $20,000,000[.07(90/360)] = $350,000 F Call is out-of-the-money F Total effective interest: $350,000 F Effective rate on loan: ($20,350,000/$19,949,542) 365/90 - 1 =.0839

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Copyright © 2001 by Harcourt, Inc. All rights reserved.19 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u Sample outcome: LIBOR = 12% F Loan interest: $20,000,000[.13(90/360)] = $650,000 F Call is worth $20,000,000(.12 -.10)(90/360) = $100,000 F Total effective interest: $650,000 - $100,000 = $550,000 F Effective rate on loan: ($20,550,000/$19,949,542) 365/90 - 1 =.1278 u See Figure 14.4, p. 598 for comparison of loan rates with and without the call.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.20 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u Payoff of an interest rate put F (Notional Principal)Max(0, X - LIBOR)(Days/360) F Again, option expires but payment is made later. u See example in Table 14.5, p. 599 of a bank planning to lend $10 million in 90 days at a fixed rate and using the put to lock in a minimum fixed rate. The strike rate is 9% and the premium is $26,500. The firm will lend for 180 days at LIBOR plus 150 bps. The option payoff is F $10,000,000Max(0,.09 - LIBOR)(180/360)

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Copyright © 2001 by Harcourt, Inc. All rights reserved.21 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u First we compound the premium forward for 90 days at today’s LIBOR (9%) plus 150 bps. F $26,500[1 +.105(90/360)] = $27,196 u The effective proceeds when the loan is taken out is F $10,000,000 + $27,196 = $10,027,196

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Copyright © 2001 by Harcourt, Inc. All rights reserved.22 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u Sample outcome: LIBOR = 7% F Loan interest: $10,000,000[.085(180/360)] = $425,000 F Put is worth $10,000,000(.09 -.07)(180/360) = $100,000 F Total effective interest: $425,000 + $100,000 = $525,000 F Effective rate on loan: ($10,525,000/$10,027,196) 365/180 - 1 =.1032

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Copyright © 2001 by Harcourt, Inc. All rights reserved.23 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u Sample outcome: LIBOR = 12% F Loan interest: $10,000,000[.135(180/360)] = $675,000 F Put is out-of-the-money F Total effective interest: $675,000 F Effective rate on loan: ($10,675,000/$10,027,196) 365/180 - 1 =.1354 u See Figure 14.5, p. 600 for comparison of loan rates with and without the call.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.24 Interest Rate Options (continued) n Interest Rate Calls and Puts (continued) u Pricing Interest Rate Options F Can use the Black model for pricing options on futures or forward contracts. F See Table 14.6, p. 601 for example. F Black model gives answer in basis points. Then discount and multiply by (Notional principal)(Days/360). F Though widely used in practice, the Black model makes numerous assumptions that are not realistic, especially for long term options. We shall look at a better way for pricing interest rate options later.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.25 Interest Rate Options (continued) n Interest Rate Caps, Floors and Collars u Definition: An interest rate cap is a series of interest rate calls with equivalent strike rates. Each component cap is called a caplet. Each pays off independently of the others. u See Table 14.7, p. 603 for example of firm borrowing at a floating rate and using a cap to put a limit on the interest it pays. The payoff is F $25,000,000Max(0,LIBOR -.10)(Days/360) u Note how each caplet pays off later than its expiration and how cost of cap is factored into the rate on the loan.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.26 Interest Rate Options (continued) n Interest Rate Caps, Floors and Collars (continued) u Definition: An interest rate floor is a series of interest rate puts with equivalent strike rates. Each component put is called a floorlet. Each pays off independently of the others. u See Table 14.8, p. 605 for example of bank lending at a floating rate and using a floor to put a minimum on the interest it receives. Payoff is F $15,000,000Max(0,.08 - LIBOR)(Days/360) u Note how each floorlet pays off later than its expiration and how cost of floor is factored into the rate on the loan.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.27 Interest Rate Options (continued) n Interest Rate Caps, Floors and Collars (continued) u Definition: An interest rate collar is the purchase of a cap and sale of a floor or vice versa. The purchase of the cap puts a maximum on a borrowing rate paid. The sale of the floor gives up gains from falling rates beyond a point. The floor strike may be set so that its price, which is received, offsets the cap price, which is paid. This is called a zero-cost collar. u See Table 14.9, p. 606 for example of firm borrowing and using the collar to put a maximum and minimum rate on its loan.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.28 Interest Rate Options (continued) n Interest Rate Options and Swaps u A given swap payment (receiving fixed and paying LIBOR) is F Swap Rate - LIBOR u Suppose we enter into this swap and also buy a cap and sell a floor, both with strike X. The cap payoff is F 0ifLIBOR X F LIBOR - XifLIBOR > X u The floor payoff is F -(X - LIBOR)ifLIBOR X F 0ifLIBOR > X

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Copyright © 2001 by Harcourt, Inc. All rights reserved.29 Interest Rate Options (continued) n Interest Rate Options and Swaps (continued) u The total payoff is, thus, F Swap Rate - XifLIBOR X F Swap Rate - XifLIBOR > X u If we set the strike rate to equal the swap rate, then the overall transaction is risk-free and all payoffs are zero. Since the swap has no initial outlay, the long cap and short floor premiums must total to zero. Thus, the cap premium must equal the floor premium when the strike rate equals the swap rate. u So a swap to pay fixed and receive floating is equivalent to a long cap and short floor when the strike rate is the swap rate.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.30 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps u Definition: A swaption is an option to enter into a swap. F payer swaption: an option to enter into a swap as a fixed rate payer (and floating rate receiver) F receiver swaption: an option to enter into a swap as a fixed rate receiver (and floating rate payer) F specifies a strike rate, which is the fixed rate at which the option holder can enter into a swap

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Copyright © 2001 by Harcourt, Inc. All rights reserved.31 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u Example: MPK Resources anticipates entering into a three-year, pay fixed-receive floating swap two years from now. It is concerned about interest rates rising by the time it enters into the swap. It enters into a payer swaption, expiring in two years, where the underlying swap has a three year maturity. The strike rate is 11.5 %. The notional principal is $10 million.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.32 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u At expiration, discount factors for LIBOR are.8929 (1 year),.7901 (2 years),.6967 (3 years). Price of swap is found as: F Coupon(.8929) + Coupon(7901) + (Coupon + $10,000,000)(.6967) = $10,000,000 F Solving for Coupon gives $1,274,530 or in other words, about 12.75 %. F The underlying swap has a rate of 12.75 but MPK has an option to enter into a swap paying 11.5. So it exercises the swaption.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.33 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u By exercising the swaption, it enters into a swap, paying 11.5, and receive LIBOR. It can then enter into a swap at the market rate of 12.75, thus, receiving 12.75 and paying LIBOR. F The LIBORs offset so it has created a three-year annuity of 12.75% - 11.5% = 1.25 %. F The value of this annuity is $125,000(0.8929) + $125,000(0.7901) + $125,000(0.6967) = $297,463$125,000(0.8929) + $125,000(0.7901) + $125,000(0.6967) = $297,463

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Copyright © 2001 by Harcourt, Inc. All rights reserved.34 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u Alternatively, F it could simply exercise the swap and continue to hold it until it expires or F the swaption could have been structured to pay off at expiration by a cash settlement. In this case, it would have received $297,463. u Payer swaptions are exercised when the market swap rate is above the strike rate. This is similar to a cap. u Receiver swaptions are exercised when the market swap rate is below the strike rate. This is similar to a floor.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.35 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u We can show that a swaption is equivalent to an option on a bond. Recall from earlier that the formula for the swap rate is F (1 - Final discount factor)/(Sum of Discount factors). u The payoff of a payer swaption is F Max(0,Swap rate - Strike rate)(Sum of discount factors). u Substituting the former into the latter, we obtain F Max(0,1 - Final discount factor - strike rate(Sum of discount factors))

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Copyright © 2001 by Harcourt, Inc. All rights reserved.36 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u A bond with coupon equal to the strike rate, maturity equal to the maturity of a swap and face value of 1.0 is worth the following at the swaption expiration: F Strike rate(Sum of discount factors) + 1.0(Final discount factor) u A put on the bond with exercise price of 1.0 would, therefore, have a payoff of F Max(0,1.0 - Final discount factor - Strike Rate(Sum of discount factors)) u So the swaption has the same payoff as the put on the bond.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.37 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u Characteristics of the swaptions market F Very liquid and widely used by corporations In anticipation of needing a swap at a later dateIn anticipation of needing a swap at a later date In anticipation of needing to offset an existing swap at a later dateIn anticipation of needing to offset an existing swap at a later date F As a substitute for an option on a bond F To create synthetic callable or puttable debt to either add a call or put feature or remove a call or put feature

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Copyright © 2001 by Harcourt, Inc. All rights reserved.38 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u A forward swap is a forward contract to enter into a swap. The long position is committed to paying fixed and the short position is committed to receiving fixed. u Example: Using the MPK situation, the discount factors are F 0.9174 (1 year), 0.8325 (2 years), 0.5713 (3 years), 0.6756 (4 years) and 0.6070 (5 years) F We want two-year ahead forward rates and/or discount factors. Years 2 to 3: 0.7513/0.8325 = 0.9025Years 2 to 3: 0.7513/0.8325 = 0.9025 Years 2 to 4: 0.6756/0.8325 = 0.8115Years 2 to 4: 0.6756/0.8325 = 0.8115 Years 2 to 5: 0.6070/0.8325 = 0.7291Years 2 to 5: 0.6070/0.8325 = 0.7291

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Copyright © 2001 by Harcourt, Inc. All rights reserved.39 Interest Rate Options (continued) n Interest Rate Swaptions and Forward Swaps (continued) u The forward swap rate is the solution to F Coupon(0.9025) + Coupon(0.8115) + (Coupon + $10,000,000)(0.7291) = $10,000,000 F Solving gives Coupon = $1,108,837 or 11.09%. F So MPK is committed to entering into the swap in two years paying 11.09% fixed. F At expiration, the new swap rate is 12.75%. Then MPK locks in a three year annuity of 12.75% - 11.09% = 1.66% or $166,000. The value of this is $166,000(0.8929) + $166,000(0.7901) + $166,000(0.6967) = $395,030$166,000(0.8929) + $166,000(0.7901) + $166,000(0.6967) = $395,030

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Copyright © 2001 by Harcourt, Inc. All rights reserved.40 Modeling the Term Structure and Pricing Interest Rate Derivatives u For proper pricing of interest rate derivatives, we must model the term structure and how it evolves through time. In addition we must make sure our model permits no arbitrage opportunities involving any of the derivatives priced off of this term structure. u We shall look at the Ho-Lee model. See Figure 14.6, p. 614 for a binomial tree based on Ho-Lee. Details of how the tree is built are in the Appendix. In each cell F 1 st number is the one-period rate, F 2 nd number is the number of up and down moves. Each u is an up move, each d is a down move.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.41 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) F 3 rd number is the number of ways you can get to that node. For example, uud, udu and duu are all equivalent ways to go up twice and down once. They will leave you at the same node of the tree. F 4 th number is the probability of getting to that node. u In this example, let the probability of an up move be p =.52 and the probability of a down move be 1-p =.48. The probability of reaching a node is F Qp (# of times the rate has gone up) (1-p) (# of times the rate has gone down).

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Copyright © 2001 by Harcourt, Inc. All rights reserved.42 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Forward Rate Agreements u Note: Difference between in-arrears and delayed settlement F Price of one-year in-arrears FRA is the rate that would be the solution to F.52(.12 - FRA 1 ) +.48(.0874 - FRA 1 ) = 0 F Solution is 10.44 % per $1 of notional principal. Two-year in-arrears FRA is the solution to F.2704(.1347 - FRA 2 ) +.4992(.1017 - FRA 2 ) +.2304(.0696 - FRA 2 ) = 0. F Solution is 10.32 %.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.43 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Forward Rate Agreements (continued) u General Solution where p i is the probability of the i th outcome and r i is the interest rate in the i th outcome. u For delayed settlement FRA, an adjustment is needed. FRA pays off one period later. Thus payoff must be discounted by one-period rate.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.44 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Forward Rate Agreements (continued) u For one period FRA, the rate is the solution to F.52(.12 - FRA 1 )(1.12) -1 +.48(.0874 - FRA 1 )(1.0874) -1 = 0. F The solution is 10.41 %. u For two-period FRA, the rate is the solution to F.2704(.1347 - FRA 2 )(1.1347) -1 +.4992(.1017 - FRA 2 )(1.1017) -1 +.2304(.0696 - FRA 2 )(1.0696) -1 = 0. F The solution is 10.27 %.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.45 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Forward Rate Agreements (continued) u The general formula is

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Copyright © 2001 by Harcourt, Inc. All rights reserved.46 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Interest Rate Swaps u This is simple and requires no more information than the 1-, 2-, 3-, etc. period rates. Let these be r(0,2) =.1043 and r(0,3) =.1041. Of course r(0,1) =.105. Using the same procedure we used earlier, F c(1.105) -1 + c(1.1043) -2 + (c + 1.00)(1.1041) -3 = 1.00 F Solving for c gives c =.1041. F For the two-period swap, c =.1043.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.47 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Interest Rate Options u We price a two-period interest rate call with $1 notional principal. It expires in two periods and pays off one period after that. u See Figure 14.7, p. 618. The option is priced the same basic way we priced stock options in the binomial model. Note that payoff at expiration is discounted one period at the rate at that node. u If the option was a cap, you would have to price each component caplet and add up their prices.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.48 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Swaptions u Recall that exercise of a swaption sets up an annuity equal to the difference between the market rate on the underlying swap at expiration and the strike rate. u Consider a two-period payer swaption where the underlying swap is a two-period swap. Thus, at expiration we have to find the price of a two-period swap. That means we require the two-period rate as well as the one-period spot rate, the latter of which is given in the binomial tree of Figure 14.6.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.49 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Swaptions (continued) u Let us assume these rates are r(2,4) uu =.1335, r(2,4) ud =.1011, and r(2,4) dd =.0691. From the binomial tree of Figure 14.6, we have r(2,3) uu =.1347, r(2,3) ud =.1017 and r(2,3) dd =.0696. The three possible swap prices at time 2 are the solutions to the equations F c uu (1.1347) -1 + (c uu + 1.00)(1.1335) -2 = 1.00. F c ud (1.1017) -1 + (c ud + 1.00)(1.1011) -2 = 1.00. F c dd (1.0696) -1 + (c dd + 1.00)(1.0691) -2 = 1.00.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.50 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Swaptions (continued) u (Remember the c‘s are coupon rates) For any one of the c’s, the solution is u The solutions are c uu =.1336, c ud =.1011, c dd =.0691. u To price the swaption we must determine the value at expiration in each node at expiration where the payoff is a two-year annuity.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.51 Modeling the Term Structure and Pricing Interest Rate Derivatives (continued) n Pricing Swaptions (continued) u See Figure 14.8, p. 620. Note that the swaption is found by successively discounting the terminal payoff back one period at a time, the same way we did all other options in the binomial framework. The solution is.0130 or 1.30 % per $1 of notional principal. u If the swaption were a receiver swaption, it would be priced by changing the payoff accordingly. u American-style swaptions can be easily accommodated. u Forward swaps are priced by pricing the swap as if it were beginning at the forward swap expiration.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.52 Summary

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Copyright © 2001 by Harcourt, Inc. All rights reserved.53 Appendix 14A: the Ho-Lee Model of the Term Structure n Let r(t,x,y) be the rate observed at time t on a bond beginning at time x and maturing at time y. u Spot rates are like r(0,0,1), r(0,0,2), r(0,0,3), etc. u Forward rates are like r(0,1,2), r(0,1,3), etc. u Under no uncertainty, forward rates would evolve into spot rates. F At time 1, r(1,1,2) would be the same as the time zero rate r(0,1,2). Also, r(1,1,3) would be r(0,1,3)

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Copyright © 2001 by Harcourt, Inc. All rights reserved.54 Appendix 14A: the Ho-Lee Model of the Term Structure (continued) n Uncertainty is introduced by letting there be a shift upward or downward such that the forward rate does not equal the original spot rate. u r(1,1,2) = [(1+r(0,0,2)) 2 /(1 + r(0,0,1))] u(1) - 1 or u r(1,1,2) = [(1+r(0,0,2)) 2 /(1 + r(0,0,1))] d(1) - 1 u where u(1), d(1) are up and down shift factors for a one-period bond. Note that (1+r(0,0,2)) 2 /(1+r(0,0,1)) - 1 is the forward rate r(0,1,2). u Similar relationships exist for r(1,1,3) vs. r(0,1,3) except that you will end up taking the square root. See formula in book.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.55 Appendix 14A: the Ho-Lee Model of the Term Structure (continued) n The up and down factors are given by d(i) = [(p + (1-p) d(i) = [(p + (1-p) i ] i u(i) = d(i) i F where i = 1 or 2 in a two period model. i is determined by the volatility according to the formula u u These formulas prevent arbitrage opportunities.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.56 Appendix 14A: the Ho-Lee Model of the Term Structure (continued) Using the example in the chapter, r(0,0,1) =.1050, r(0,0,2) =.1043, r(0,0,3) =.1041. We let Inserting into the formulas for u(i) and d(i) give Using the example in the chapter, r(0,0,1) =.1050, r(0,0,2) =.1043, r(0,0,3) =.1041. We let = 1.03. Inserting into the formulas for u(i) and d(i) give u d(1) =.9849, u(1) = 1.0144 u d(2) =.9701, u(2) = 1.0292 n This combination of parameters implies a volatility of.0148.

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Copyright © 2001 by Harcourt, Inc. All rights reserved.57 Appendix 14A: the Ho-Lee Model of the Term Structure (continued) n The spot rate at time 1 on one-period bonds would be n The spot rate at time 1 on two-period bonds would be

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Copyright © 2001 by Harcourt, Inc. All rights reserved.58 Appendix 14A: the Ho-Lee Model of the Term Structure (continued) n This gives us the possible one- and two-period spot rates at time 1. Applying the up and down factors enables us to fill in the rest of the binomial tree. n To build a tree of n time steps, you must start off with spot rates for bonds of maturities to n+1, because each period you lose one bond as it matures. n The binomial tree we used in the text was the tree of one- period rates. Derivatives could, however, be created on two-period rates as well.

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