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D . M. C ha nc e A n I nt roduct ion to D er ivat ives and R is k Managem ent, 6th ed . C h. 13 : 1

Chapter 13: Interest Rate Forwards and

Options

If a deal was mathematically complex in 1993 and 1994, thatwas considered innovation. But this year, what took you

forward with clients wasnt the math it was bringing them

the most efficient application of a product.

Mark Wells

Risk, January, 1996, p. R15


Important Concepts in Chapter 13

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


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 different on a bond.

See Figure 13.1, p. 467 for notional principal of FRAs and

interest rate options over time.

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Forward Rate Agreements

Definition

A forward contract in which the underlying is an

interest rate

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.


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

+ 0)LIBOR(m/361

0)rate)(m/36uponAgreed-(LIBORPrincipal)(Notional

+ 60)LIBOR(90/31

0).10)(90/36-(LIBOR00)($20,000,0



Some possible payoffs. If LIBOR at expiration is 8percent,

So the long has to pay $98,039. If LIBOR atexpiration is 12 percent, the payoff is

Note the terminology of FRAs: A B means FRAexpires in A months and underlying matures in Bmonths.

039,98$(90/360)08.1

0).10)(90/36-(.0800)($20,000,0 =

+

087,97$(90/360)12.1

0).10)(90/36-(.1200)($20,000,0 =

+

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The Pricing and Valuation of FRAs

Let F be the rate the parties agree on, h be the

expiration day, and the underlying be an m-day rate.L0(h) is spot rate on day 0 for h days, L0(h+m) isspot rate on day 0 for h + m days. Assume notionalprincipal of $1.

To find the fixed rate, we must replicate an FRA:

Short a Eurodollar maturing in h+m days thatpays 1 + F(m/360). This is a loan that can bepaid off early or transferred to another party

Long a Eurodollar maturing in h days that pays$1



The Pricing and Valuation of FRAs (continued)

On day h,

Loan we owe has a market value of

Pay if off early. Collect $1 on the ED we hold.

So total cash flow is

(m)(m/360)L1

F(m/360)1

h+

+

(m)(m/360)L1

F(m/360)11

h+

+




This can be rearranged to get

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:

This is just the forward rate in the LIBOR term

structure. See Table 13.1, p. 471 for an example.

(m)(m/360)L1

F)(m/360)(m)(L

h

h

+

+

+++=

m

3601

(h/360)L1

m)/360)m)((h(hL1F

0

0

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D. M. Chance An Introduction to Derivatives and Risk Management, 6th ed. Ch. 13: 10



Now we determine the market value of the FRA

during its life, day g. If we value the two replicatingtransactions, 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

See Table 13.2, p. 472 for example.

+++

+

+=

g)/360)mg)((hm(hL1

F(m/360)1

g)/360)g)((h(hL1

1VFRA

gg



Applications of FRAs

FRA users are typically borrowers or lenders with asingle future date on which they are exposed tointerest rate risk.

See Table 13.3, p. 473 and Figure 13.2, p. 474 for anexample.

Note that a series of FRAs is similar to a swap;however, in a swap all payments are at the samerate. Each FRA in a series would be priced atdifferent rates (unless the term structure is flat).You could, however, set the fixed rate at a differentrate (called an off-market FRA). Then a swapwould be a series of off-market FRAs.


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.

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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.

( )X)(m/360)LIBORMax(0,Principal)(Notional

( )60)LIBOR)(m/3-XMax(0,Prinicpal)(Notional



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

( ) 0$)(90/360)10.Max(0,.0600)($20,000,0 =

( ) 000,200$)(90/360)06.Max(0,.1000)($20,000,0 =

( ) 000,200$)(90/360)10.Max(0,.1400)($20,000,0 =

( ) 0$)(90/360)14.Max(0,.1000)($20,000,0 =



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, youmust 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. 478 for illustration.

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Interest Rate Option Strategies

See Table 13.5, p. 479 and Figure 13.3, p. 480 for an

example of the use of an interest rate call by aborrower to hedge an anticipated loan.

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.



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 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.

A combination of a long cap and short floor at

different exercise prices is called an interest rate collar.



Interest Rate Caps, Floors, and Collars (continued)

Interest Rate Cap

Each component caplet pays off independently of

the others.

See Table 13.7, p. 485 for an example of aborrower using an interest rate cap.

To price caps, price each component caplet

individually and add up the prices of the caplets.

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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



(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

=

n

1i

i0 )(tB360

daysX)-RMax(0,Principal)(Notional

=

n

1i

i0 )(tB360

daysR)-XMax(0,Principal)(Notional



(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 - .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.

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(continued) Forward Swaps (continued)

1238.1080

3601

360).1006(720/1

/360).1295(18001yearsThree

1161.720

3601

360).1006(720/1

60).12(1440/31yearsTwo

1080.360

3601360).1006(720/1

/360).1103(10801yearOne

=

+

+=

=

+

+=

=

+

+

=




The Eurodollar zero coupon (forward) bond prices

7292.0)360/1080(1238.1

1)1800,720(B

8116.0)360/720(1161.1

1)1440,720(B

9025.0)360/360(1080.1

1)1080,720(B

0

0

0

=+

=

=+

=

=+

=




The rate on the forward swap would be

1108.0

0.72920.81160.9025

0.7292-1=

++

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(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.


Summary


(Return to text slide)

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