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Eurodollars/Swaptions

Eurodollars/Swaptions. Sorta Part Deux …. Forward Rate Agreements. Underlying is usually LIBOR Payoff is made at expiration (contrast with swaps) and discounted. For FRA on m-day LIBOR, the payoff is

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Eurodollars/Swaptions

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  1. Eurodollars/Swaptions Sorta Part Deux….

  2. Forward Rate Agreements • Underlying is usually LIBOR • Payoff is made at expiration (contrast with swaps) and discounted. For FRA on m-day LIBOR, the payoff is • 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

  3. Forward Rate Agreements • Some possible payoffs. If LIBOR at expiration is 8 percent, • So the long has to pay $98,039. If LIBOR at expiration is 12 percent, the payoff is • Note the terminology of FRAs: A  B means FRA expires in A months and underlying matures in B months.

  4. Eurodollar Futures • US$ Deposits in foreign banks . • Underlying asset is the 90-day $1,000,000 Eurodollar time deposit interest rate; LIBOR • Allow investor to lock in an IR on $1million for a future 3 month period • IMM Index = 100 – aoy • aoy = add on yield on a 90-day forward Eurodollar time deposit (LIBOR). (F – P) = discount of corresponding T-bill. • If the IMM Index is 95.19, then the futures LIBOR is 4.81%.(100 – 95.19 = 4.81) • LIBOR rises => IMM index falls.

  5. Eurodollar Futures • 1 tick = 0.5 basis point in 3-month futures LIBOR = $12.50 ( for the spot month contract, one basis =$25.). • To speculate that LIBOR will rise, sell Eurodollar futures (declining prices of debt instruments mean rising interest rates). • To profit if short term interest rates fall, buy Eurodollar futures. • Four months prior to the delivery date, a Eurodollar futures contract is ~ to a 4X7 FRA. • 12 months prior to delivery a Eurodollar futures contract ~ 12X15 FRA.

  6. Eurodollar Futures On 1/8/07 investor wants to lock in rate on $5million for three months starting June 20,2007. Buys 5 June 07 eurodollar contracts at 94.79. On June 20, the 3-month LIBOR is 4%, so final settlement price is 96.00. The investor gains 5 x 1,000,000 x0.25 x (96.00-94.79) = 15,125. Plus 4% earned on the $5m for 3 months: 5 x 1,000,000 x0.25 x 0.04 = 50,000. The gain on the futures contract was $65,125. The same interest if the rate had been 5.21% - locked in the initial rate.

  7. Interest Rate Options • Interest Rate Caps, Floors, and Collars • 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. • A combination of interest rate puts used by a lender to hedge a floating-rate loan is called an interest rate floor. The component puts are referred to as floorlets. • A combination of a long cap and short floor at different exercise prices is called an interest rate collar.

  8. Interest Rate Options • Interest Rate Caps, Floors, and Collars • Interest Rate Cap & Floor • Each component caplet/floorlet pays off independently of the others. • To price caps/floors, price each component caplet/floorlet individually and add up the prices of the caplets.

  9. Interest Rate Options • 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. • 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.

  10. Caplet • A caplet is designed to provide insurance against LIBOR rising above a certain level • Suppose RKis the cap rate, L is the principal, and R is the actual LIBOR rate for the period between time t and t+d. The caplet provides a payoff at time t+d of Ld max(R-RK, 0)

  11. Caps • A cap is a portfolio of caplets • Each caplet can be regarded as a call option on a future interest rate with the payoff occurring in arrears • When using Black’s model we assume that the interest rate underlying each caplet is lognormal

  12. Black’s Model for Caps • The value of a caplet, for period [tk, tk+1] is • L: principal • RK : cap rate • dk=tk+1-tk • Fk : forward interest rate • for (tk, tk+1) • sk: forward interest rate vol. • rk: interest rate for maturity tk

  13. When Applying Black’s ModelTo Caps We Must ... • EITHER • Use forward volatilities • Volatility different for each caplet • OR • Use flat volatilities • Volatility same for each caplet within a particular cap but varies according to life of cap Cap Valuation Example

  14. Interest Rate Options • Compare a swap to interest rate options. On a settlement date, the payoff of a long call is: • 0 if LIBOR  X • (LIBOR – X ) if LIBOR > X • The payoff of a short put is: • -(X – LIBOR) if LIBOR  X • 0 if LIBOR > X • 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.

  15. European Swap Options • Definition of a swaption: an option to enter into a swap at a fixed rate. • Payer swaption: an option to enter into a swap as a fixed-rate payer • Receiver swaption: an option to enter into a swap as a fixed-rate receiver • Either it gives the holder the right to pay a prespecified fixed rate and receive LIBOR • Or it gives the holder the right to pay LIBOR and receive a prespecified fixed rate

  16. Interest Rate Swaptions and Forward Swaps • Example: Schmoo Corp 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): • 360 day rate: .12 (0.8929) • 720 day rate: .1328 (0.7901) • 1080 day rate: .1451 (0.6967)

  17. Interest Rate Swaptions and Forward Swaps • The rate on 3-year swaps is, therefore, • So Schmoo could enter into a swap at 12.75 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?

  18. Interest Rate Swaptions and Forward Swaps • Exercise would create a stream of 11.5 percent fixed payments and LIBOR floating receipts. Schmoo could then enter into the opposite swap in the market to receive 12.75 fixed and pay LIBOR floating. The LIBORs offset leaving a three-year annuity of 12.75 – 11.5 = 1.25 percent, or $125,000 on $10 million notional principal. The value of this stream of payments is: $125,000(0.8929 + 0.7901 + 0.6967) = $297,463

  19. Interest Rate Swaptions and Forward Swaps • In general, the value of a payer swaption at expiration is • The value of a receiver swaption at expiration is • Where X = exercise rate, R = swap rate at expiration, and summation term captures the PV factors over the life of the swap.

  20. Interest Rate Swaptions and Forward Swaps • Using the previous 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 – 0.6967 - 0.115(0.8929 + 0.7901 + 0.6967)) • 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.

  21. European Swaptions Alternatively (Hull): The value of the swaption is F0 is the forward swap rate; s is the swap rate volatility; tiis the time from today until the ith swap payment; and the value of the contract is: Where m is the payments per year on the swap and n = years swap will last.

  22. Relationship Between Swaptions and Bond Options • An interest rate swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond • A swaption or swap option is therefore an option to exchange a fixed-rate bond for a floating-rate bond

  23. Illustration of a synthetic fixed rate bond using a swaption– company issues a callable bond but no longer wishes to call it. A fixed bond can be made puttable by selling a payer swaption. Bonds that are puttable may be made non-puttable by buying a payer swaption. (diagram for the fun of it!)

  24. Relationship Between Swaptions and Bond Options • At the start of the swap the floating-rate bond is worth par so that the swaption can be viewed as an option to exchange a fixed-rate bond for par • An option on a swap where fixed is paid & floating is received is a put option on the bond with a strike price of par • When floating is paid & fixed is received, it is a call option on the bond with a strike price of par

  25. Homework • Homework: • Chapter 23: 1, 3, 24,

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