D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 1 Chapter 13: Interest Rate Forwards and Options If a deal was mathematically complex in 1993 and 1994, that was considered innovation. But this year, what took you forward with clients wasn’t the math – it was bringing them the most efficient application of a product. Mark Wells Risk, January, 1996, p. R15

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 2 Important Concepts in Chapter 13 n The notion of a derivative on an interest rate n Pricing, valuation, and use of forward rate agreements (FRAs), interest rate options, swaptions, and forward swaps

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 3 n A derivative on an interest rate: u The payoff of a derivative on a bond is based on the price of the bond relative to a fixed price. u The payoff of a derivative on an interest rate is based on the interest rate relative to a fixed interest rate. u In some cases these can be shown to be the same, particularly in the case of a discount instrument. In most other cases, however, a derivative on an interest rate is a different instrument than a different on a bond. n See Figure 13.1, p. 467 for notional principal of FRAs and interest rate options over time. Figure 13.1, p. 467Figure 13.1, p. 467

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 4 Forward Rate Agreements u Definition F A forward contract in which the underlying is an interest rate u An FRA can work better than a forward or futures on a bond, because its payoff is tied directly to the source of risk, the interest rate.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 5 Forward Rate Agreements (continued) u The Structure and Use of a Typical FRA F Underlying is usually LIBOR F Payoff is made at expiration (contrast with swaps) and discounted. For FRA on m-day LIBOR, the payoff is F Example: Long an FRA on 90-day LIBOR expiring in 30 days. Notional principal of $20 million. Agreed upon rate is 10 percent. Payoff will be

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 6 Forward Rate Agreements (continued) F Some possible payoffs. If LIBOR at expiration is 8 percent, F So the long has to pay $98,039. If LIBOR at expiration is 12 percent, the payoff is F Note the terminology of FRAs: A  B means FRA expires in A months and underlying matures in B months.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 7 Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs F Let F be the rate the parties agree on, h be the expiration day, and the underlying be an m-day rate. L 0 (h) is spot rate on day 0 for h days, L 0 (h+m) is spot rate on day 0 for h + m days. Assume notional principal of $1. F To find the fixed rate, we must replicate an FRA: Short a Eurodollar maturing in h+m days that pays 1 + F(m/360). This is a loan that can be paid off early or transferred to another partyShort a Eurodollar maturing in h+m days that pays 1 + F(m/360). This is a loan that can be paid off early or transferred to another party Long a Eurodollar maturing in h days that pays $1Long a Eurodollar maturing in h days that pays $1

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 8 Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F On day h, Loan we owe has a market value ofLoan we owe has a market value of Pay if off early. Collect $1 on the ED we hold. So total cash flow isPay if off early. Collect $1 on the ED we hold. So total cash flow is

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 9 Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) This can be rearranged to getThis can be rearranged to get F This is the payoff of an FRA so this strategy is equivalent to an FRA. With no initial cash flow, we set this to zero and solve for F: F This is just the forward rate in the LIBOR term structure. See Table 13.1, p. 471 for an example. Table 13.1, p. 471Table 13.1, p. 471

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 10 Forward Rate Agreements (continued) u The Pricing and Valuation of FRAs (continued) F Now we determine the market value of the FRA during its life, day g. If we value the two replicating transactions, we get the value of the FRA. The ED we hold pays $1 in h – g days. For the ED loan we took out, we will pay 1 + F(m/360) in h + m – g days. Thus, the value is F See Table 13.2, p. 472 for example. Table 13.2, p. 472Table 13.2, p. 472

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 11 Forward Rate Agreements (continued) u Applications of FRAs F FRA users are typically borrowers or lenders with a single future date on which they are exposed to interest rate risk. F See Table 13.3, p. 473 and Figure 13.2, p. 474 for an example. Table 13.3, p. 473Figure 13.2, p. 474Table 13.3, p. 473Figure 13.2, p. 474 F Note that a series of FRAs is similar to a swap; however, in a swap all payments are at the same rate. Each FRA in a series would be priced at different rates (unless the term structure is flat). You could, however, set the fixed rate at a different rate (called an off-market FRA). Then a swap would be a series of off-market FRAs.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 12 Interest Rate Options u Definition: an option in which the underlying is an interest rate; it provides the right to make a fixed interest payment and receive a floating interest payment or the right to make a floating interest payment and receive a fixed interest payment. u The fixed rate is called the exercise rate. u Most are European-style.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 13 Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option F With an exercise rate of X, the payoff of an interest rate call is F The payoff of an interest rate put is F The payoff occurs m days after expiration. F Example: notional principal of $20 million, expiration in 30 days, underlying of 90-day LIBOR, exercise rate of 10 percent.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 14 Interest Rate Options (continued) u The Structure and Use of a Typical Interest Rate Option (continued) F If LIBOR is 6 percent at expiration, payoff of a call is F The payoff of a put is F If LIBOR is 14 percent at expiration, payoff of a call is F The payoff of a put is

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 15 Interest Rate Options (continued) u Pricing and Valuation of Interest Rate Options F A difficult task; binomial models are preferred, but the Black model is sometimes used with the forward rate as the underlying. F When the result is obtained from the Black model, you must discount at the forward rate over m days to reflect the deferred payoff. F Then to convert to the premium, multiply by (notional principal)(days/360). F See Table 13.4, p. 478 for illustration. Table 13.4, p. 478Table 13.4, p. 478

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 16 Interest Rate Options (continued) u Interest Rate Option Strategies F See Table 13.5, p. 479 and Figure 13.3, p. 480 for an example of the use of an interest rate call by a borrower to hedge an anticipated loan. Table 13.5, p. 479Figure 13.3, p. 480Table 13.5, p. 479Figure 13.3, p. 480 F See Table 13.6, p. 481 and Figure 13.4, p. 483 for an example of the use of an interest rate put by a lender to hedge an anticipated loan. Table 13.6, p. 481Figure 13.4, p. 483Table 13.6, p. 481Figure 13.4, p. 483

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 17 Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars F A combination of interest rate calls used by a borrower to hedge a floating-rate loan is called an interest rate cap. The component calls are referred to as caplets. F A combination of interest rate floors used by a lender to hedge a floating-rate loan is called an interest rate floor. The component puts are referred to as floorlets. F A combination of a long cap and short floor at different exercise prices is called an interest rate collar.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 18 Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Cap Each component caplet pays off independently of the others.Each component caplet pays off independently of the others. See Table 13.7, p. 485 for an example of a borrower using an interest rate cap.See Table 13.7, p. 485 for an example of a borrower using an interest rate cap.Table 13.7, p. 485Table 13.7, p. 485 To price caps, price each component caplet individually and add up the prices of the caplets.To price caps, price each component caplet individually and add up the prices of the caplets.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 19 Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Floor Each component floorlet pays off independently of the othersEach component floorlet pays off independently of the others See Table 13.8, p. 486 for an example of a lender using an interest rate floor.See Table 13.8, p. 486 for an example of a lender using an interest rate floor.Table 13.8, p. 486Table 13.8, p. 486 To price floors, price each component floorlet individually and add up the prices of the floorlets.To price floors, price each component floorlet individually and add up the prices of the floorlets.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 20 Interest Rate Options (continued) u Interest Rate Caps, Floors, and Collars (continued) F Interest Rate Collars A borrower using a long cap can combine it with a short floor so that the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar.A borrower using a long cap can combine it with a short floor so that the floor premium offsets the cap premium. If the floor premium precisely equals the cap premium, there is no cash cost up front. This is called a zero-cost collar. The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap.The exercise rate on the floor is set so that the premium on the floor offsets the premium on the cap. By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate.By selling the floor, however, the borrower gives up gains from falling interest rates below the floor exercise rate. See Table 13.9, p. 487 for example.See Table 13.9, p. 487 for example.Table 13.9, p. 487Table 13.9, p. 487

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 21 Interest Rate Options (continued) u Interest Rate Options, FRAs, and Swaps F Recall that a swap is like a series of off-market FRAs. F Now compare a swap to interest rate options. On a settlement date, the payoff of a long call is 0if LIBOR  X0if LIBOR  X LIBOR – Xif LIBOR > XLIBOR – Xif LIBOR > X F The payoff of a short put is - (X – LIBOR)if LIBOR  X- (X – LIBOR)if LIBOR  X 0if LIBOR > X0if LIBOR > X F These combine to equal LIBOR – X. If X is set at R, which is the swap fixed rate, the long cap and short floor replicate the swap.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 22 Interest Rate Swaptions and Forward Swaps u Definition of a swaption: an option to enter into a swap at a fixed rate. F Payer swaption: an option to enter into a swap as a fixed- rate payer F Receiver swaption: an option to enter into a swap as a fixed-rate receiver

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 23 Interest Rate Swaptions and Forward Swaps (continued) u The Structure of a Typical Interest Rate Swaption F Example: MPK considers the need to engage in a $10 million three-year swap in two years. Worried about rising rates, it buys a payer swaption at an exercise rate of 11.5 percent. Swap payments will be annual. At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses):At expiration, the following rates occur (Eurodollar zero coupon bond prices in parentheses): –360 day rate:.12 (0.8929) –720 day rate:.1328 (0.7901) –1080 day rate:.1451 (0.6967)

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 24 Interest Rate Swaptions and Forward Swaps (continued) u The Structure of a Typical Interest Rate Swaption (continued) The rate on 3-year swaps is, therefore,The rate on 3-year swaps is, therefore, So MPK could enter into a swap at percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?So MPK could enter into a swap at percent in the market or exercise the swaption and enter into a swap at 11.5 percent. Obviously it would exercise the swaption. What is the swaption worth?

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 25 Interest Rate Swaptions and Forward Swaps (continued) u The Structure of a Typical Interest Rate Swaption (continued) Exercise would create a stream of 11.5 percent fixed payments and LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of – 11.5 = 1.25 percent, or $125,000 on $10 million notional principal. The value of this stream of payments isExercise would create a stream of 11.5 percent fixed payments and LIBOR floating receipts. MPK could then enter into the opposite swap in the market to receive fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of – 11.5 = 1.25 percent, or $125,000 on $10 million notional principal. The value of this stream of payments is $125,000( ) = $297,463

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 26 Interest Rate Swaptions and Forward Swaps (continued) u The Structure of a Typical Interest Rate Swaption (continued) F In general, the value of a payer swaption at expiration is F The value of a receiver swaption at expiration is

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 27 Interest Rate Swaptions and Forward Swaps (continued) u The Equivalence of Swaptions and Options on Bonds F Using the above example, substituting the formula for the swap rate in the market, R, into the formula for the payoff of a swaption gives Max(0,1 – ( ))Max(0,1 – ( )) F This is the formula for the payoff of a put option on a bond with 11.5 percent coupon where the option has an exercise price of par. So payer swaptions are equivalent to puts on bonds. Similarly, receiver swaptions are equivalent to calls on bonds.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 28 Interest Rate Swaptions and Forward Swaps (continued) u Pricing Swaptions F We do not cover this advanced topic here, but note that based on the previous result, we would price swaptions using models for pricing options on bonds.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 29 Interest Rate Swaptions and Forward Swaps (continued) u Forward Swaps F Definition: a forward contract to enter into a swap; a forward swap commits the parties to entering into a swap at a later date at a rate agreed on today. F Example: The MPK situation previously described. Let MPK commit to a three-year pay-fixed, receive-floating swap in two years. To find the fixed rate at the time the forward swap is agreed to, we need the term structure of rates for one through five years (Eurodollar zero coupon bond prices shown in parentheses).

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 30 Interest Rate Swaptions and Forward Swaps (continued) u Forward Swaps (continued) 360 days:.09 (0.9174)360 days:.09 (0.9174) 720 days:.1006 (0.8325)720 days:.1006 (0.8325) 1080 days:.1103 (0.7514)1080 days:.1103 (0.7514) 1440 days:.12 (0.6757)1440 days:.12 (0.6757) 1800 days:.1295 (0.6070)1800 days:.1295 (0.6070) F We need the forward rates two years ahead for periods of one, two, and three years.

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 31 Interest Rate Swaptions and Forward Swaps (continued) u Forward Swaps (continued)

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 32 Interest Rate Swaptions and Forward Swaps (continued) u Forward Swaps (continued) F The Eurodollar zero coupon (forward) bond prices

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 33 Interest Rate Swaptions and Forward Swaps (continued) u Forward Swaps (continued) F The rate on the forward swap would be

D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 13: 34 Interest Rate Swaptions and Forward Swaps (continued) u Applications of Swaptions and Forward Swaps F Anticipation of the need for a swap in the future F Swaption can be used To exit a swapTo exit a swap As a substitute for an option on a bondAs a substitute for an option on a bond Creating synthetic callable or puttable debtCreating synthetic callable or puttable debt F Remember that forward swaps commit the parties to a swap but require no cash payment up front. Options give one party the choice of entering into a swap but require payment of a premium up front.

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