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### Chapter 13: Interest Rate Forwards and Options

Well, it helps to look at derivatives like atoms. Split them one way and you have heat and energy - useful stuff. Split them another way and you have a bomb. You have to understand the subtleties.

Kate Jennings

Moral Hazard, 2002, p. 8

An Introduction to Derivatives and Risk Management, 7th ed.

Important Concepts

- The notion of a derivative on an interest rate
- Pricing, valuation, and use of forward rate agreements (FRAs), interest rate options, swaptions, and forward swaps

An Introduction to Derivatives and Risk Management, 7th ed.

A derivative on an interest rate:

- The payoff of a derivative on a bond is based on the price of the bond relative to a fixed price.
- The payoff of a derivative on an interest rate is based on the interest rate relative to a fixed interest rate.
- 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 derivative on a bond.
- See Figure 13.1, p. 446 for notional principal of FRAs and interest rate options over time.

An Introduction to Derivatives and Risk Management, 7th ed.

Forward Rate Agreements (FRA)

- Definition
- A forward contract in which the underlying is an interest rate
- A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period
- 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.

An Introduction to Derivatives and Risk Management, 7th ed.

Forward Rate Agreements (continued)

- The Structure and Use of a Typical FRA
- 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

An Introduction to Derivatives and Risk Management, 7th ed.

Forward Rate Agreements (continued)

- 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.

An Introduction to Derivatives and Risk Management, 7th ed.

Zero Rates

- A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

Forward Rates

- The forward rate is the future zero rate implied by today’s term structure of interest rates

Formula for Forward Rates

- Suppose that the zero rates for time periods T1and T2are R1 and R2 with both rates continuously compounded.
- The forward rate for the period between times T1 and T2 is

Calculation of Forward Rates

Zero Rate for

Forward Rate

n

n

an

-year Investment

for

th Year

n

Year (

)

(% per annum)

(% per annum)

1

3.0

2

4.0

5.0

3

4.6

5.8

4

5.0

6.2

5

5.3

6.5

FRA Valuation

- An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate
- An FRA can be valued by assuming that the forward interest rate is certain to be realized

Valuation Formulas

- Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is
- Value of FRA where a fixed rate is paid is
- RFis the forward rate for the period and R2 is the zero rate for maturity T2

Example: FRA Valuation

Suppose that the three-month LIBOR rate is 5% and the six-month LIBOR rate is 5.5% with continuous compounding. Consider an FRA where you will receive a rate 7% measured with quarterly compounding, on a principal of $1 million between the end of month 3 and the end of month 6. The forward rate is 6% percent with continuous compounding or 6.0452 with quarterly compounding. The value of the FRA is

$1,000,000 x (0.07 – 0.0652) x 0.25 x e-0.055 x 0.5 = $2,322

Forward Rate Agreements (continued)

- Applications of FRAs
- FRA users are typically borrowers or lenders with a single future date on which they are exposed to interest rate risk.
- See Table 13.3, p. 452 and Figure 13.2, p. 453 for an example.
- 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.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options

- 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.
- The fixed rate is called the exercise rate.
- Most are European-style.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- The Structure and Use of a Typical Interest Rate Option
- With an exercise rate of X, the payoff of an interest rate call is
- The payoff of an interest rate put is
- The payoff occurs m days after expiration.
- Example: notional principal of $20 million, expiration in 30 days, underlying of 90-day LIBOR, exercise rate of 10 percent.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- The Structure and Use of a Typical Interest Rate Option (continued)
- If LIBOR is 6 percent at expiration, payoff of a call is
- The payoff of a put is
- If LIBOR is 14 percent at expiration, payoff of a call is
- The payoff of a put is

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- Pricing and Valuation of Interest Rate Options
- A difficult task; binomial models are preferred, but the Black model is sometimes used with the forward rate as the underlying.
- When the result is obtained from the Black model, you must discount at the forward rate over m days to reflect the deferred payoff.
- Then to convert to the premium, multiply by (notional principal)(days/360).
- See Table 13.4, p. 456 for illustration.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- Interest Rate Option Strategies
- See Table 13.5, p. 458 and Figure 13.3, p. 459for an example of the use of an interest rate call by a borrower to hedge an anticipated loan.
- See Table 13.6, p. 460 and Figure 13.4, p. 461 for an example of the use of an interest rate put by a lender to hedge an anticipated loan.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- 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.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- Interest Rate Caps, Floors, and Collars (continued)
- Interest Rate Cap
- Each component caplet pays off independently of the others.
- See Table 13.7, p. 463 for an example of a borrower using an interest rate cap.
- To price caps, price each component caplet individually and add up the prices of the caplets.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- Interest Rate Caps, Floors, and Collars (continued)
- Interest Rate Floor
- Each component floorlet pays off independently of the others
- See Table 13.8, p. 464 for an example of a lender using an interest rate floor.
- To price floors, price each component floorlet individually and add up the prices of the floorlets.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- Interest Rate Caps, Floors, and Collars (continued)
- 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.
- See Table 13.9, p. 466 for example.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Options (continued)

- Interest Rate Options, FRAs, and Swaps
- Recall that a swap is like a series of off-market FRAs.
- Now 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.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps

- 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

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- The Structure of a Typical Interest Rate Swaption
- 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):
- 360 day rate: .12 (0.8929)
- 720 day rate: .1328 (0.7901)
- 1080 day rate: .1451 (0.6967)

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- The Structure of a Typical Interest Rate Swaption (continued)
- The rate on 3-year swaps is, therefore,
- So MPK 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?

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- 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 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

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- The Structure of a Typical Interest Rate Swaption (continued)
- In general, the value of a payer swaption at expiration is
- The value of a receiver swaption at expiration is

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- The Equivalence of Swaptions and Options on Bonds
- 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 – 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.

An Introduction to Derivatives and Risk Management, 7th ed.

Swaption and Callable Bonds

- One application of swaptions relates to callable bonds
- Recall callable bond issuer has sold (issued) bonds and purchased a call option
- A receiver swaption is comparable to the embedded call option of a bond
- See Figure 13.5, p. 474

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- Forward Swaps
- 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.
- 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).

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- Forward Swaps (continued)
- 360 days: 0.09 (0.9174)
- 720 days: 0.1006 (0.8325)
- 1080 days: 0.1103 (0.7514)
- 1440 days: 0.12 (0.6757)
- 1800 days: 0.1295 (0.6070)
- We need the forward rates two years ahead for periods of one, two, and three years.

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- Forward Swaps (continued)

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- Forward Swaps (continued)
- The Eurodollar zero coupon (forward) bond prices

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- Forward Swaps (continued)
- The rate on the forward swap would be

An Introduction to Derivatives and Risk Management, 7th ed.

Interest Rate Swaptions and Forward Swaps (continued)

- Applications of Swaptions and Forward Swaps
- Anticipation of the need for a swap in the future
- Swaption can be used
- To exit a swap
- As a substitute for an option on a bond
- Creating synthetic callable or puttable debt
- 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.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

An Introduction to Derivatives and Risk Management, 7th ed.

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