Chapter

4

Interest

RatesOptions,

Futures,

and

Other

Derivatives

8th

Edition,Copyright

?

John

C.

Hull

2012

1Types

of

RatesOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

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20122Treasury

ratesLIBOR

ratesRepo

ratesTreasury

RatesOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

20123Rates

on

instruments

issued

by

agovernment

in

its

own

currencyLIBOR

and

LIBIDLIBOR

is

the

rate

of

interest

at

which

abank

is

prepared

to

deposit

money

withanother

bank.

(The

second

bank

musttypically

have

a

AA

rating)LIBOR

is

compiled

once

a

day

by

theBritish

Bankers

Association

on

all

majorcurrencies

for

maturities

up

to

12monthsLIBOIptDioniss,

Futthurees,raandtOethewrhDeircivhatiaveAs

8Athbank

ispreEpdiatiroen,dCotpoyripghaty?

Joohnn

Cd.

eHupllo2s01i2

ts

f4rom

antherRepo

RatesRepurchase

agreement

is

an

agreementwhere

a

financial

institution

that

ownssecurities

agrees

to

sell

them

today

forX

and

buy

them

bank

in

the

future

for

aslightly

higher

price,

YThe

financial

institution

obtains

a

loanThe

rate

of

interest

is

calculated

fromthe

difference

between

X

and

Y

andisOptions,

Futures,

andOther

Derivatives

8thknoEwdintiaons,

Ctophyerigrhte?pJoohnrCa.

tHuell

20125The

Risk-Free

RateThe

short-term

risk-free

ratetraditionally

used

by

derivativespractitioners

is

LIBORThe

Treasury

rate

is

considered

to

beartificially

low

for

a

number

of

reasons(See

Business

Snapshot

4.1)As

will

be

explained

in

later

chapters:Eurodollar

futures

and

swaps

are

usedto

extend

the

LIBOR

yield

curve

beyondOptions,

Futures,

and

Other

Derivatives

8thoneEdyiteiaonr,

Copyright

?

John

C.Hull20126Measuring

Interest

RatesOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

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20127The

compounding

frequency

usedfor

an

interest

rate

is

the

unit

ofmeasurementThe

difference

between

quarterlyand

annual

compounding

isanalogous

to

the

differencebetween

milesandkilometersImpact

of

CompoundingOptions,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

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20128

When

we

compound

m

times

per

yearat

rate

R

an

amount

A

grows

toA(1+R/m)m

in

oneyearCompounding

frequencyValue

of

$100

in

one

year

at

10%Annual

(m=1)110.00Semiannual

(m=2)110.25Quarterly

(m=4)110.38Monthly

(m=12)110.47Weekly

(m=52)110.51Daily

(m=365)110.52ContinuousCompounding

(Page

79)Options,

Futures,

andOther

Derivatives

8thconEtdiitinoun,oCuopsylriyghtc?oJmohpn

oC.uHnuldle20d12disc9

ount

rateIn

the

limit

as

we

compound

more

andmore

frequently

we

obtain

continuouslycompounded

interest

rates$100

grows

to

$100eRT

when

investedat

a

continuously

compounded

rate

Rfor

time

T$100

received

at

time

T

discounts

to$100e-RT

at

time

zero

when

theConversion

Formulas

(Page

79)DefineRc

:

continuously

compounded

rateRm:

same

rate

with

compounding

mtimes

per

yearOptions,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

Hull

201210ExamplescomOpptoiounns,dFiutnurges,

and

Other

Derivatives

8thEdition,

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?

John

C.

Hull

20121110%

with

semiannual

compounding

isequivalent

to

2ln(1.05)=9.758%

withcontinuous

compounding8%

withcontinuous

compounding

isequivalent

to

4(e0.08/4

-1)=8.08%

withquarterly

compoundingRates

used

in

option

pricing

are

nearlyalways

expressed

with

continuousZero

RatesOptions,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

Hull

201212

A

zero

rate

(or

spot

rate),

for

maturityT

is

the

rate

of

interest

earned

on

aninvestment

that

provides

a

payoff

onlyat

time

TExample

(Table

4.2,

page

81)Options,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

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201213?Maturity

(years)Zero

rate

(cont.

comp.0.55.01.05.81.56.42.06.8Bond

PricingTo

calculate

the

cash

price

of

a

bond

wediscount

each

cash

flow

at

theappropriate

zero

rateIn

our

example,

the

theoretical

price

ofa

two-year

bond

providing

a

6%

couponsemiannually

isOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

201214Bond

YieldThe

bond

yield

is

the

discount

rate

thatmakes

the

present

value

of

the

cashflows

on

the

bond

equal

to

the

marketprice

of

the

bondSuppose

that

the

market

price

of

thebond

in

our

example

equals

itstheoretical

price

of

98.39The

bond

yield

(continuouslycompounded)

is

given

by

solvingOptions,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

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201215Par

YieldThe

par

yield

for

a

certain

maturity

isthe

coupon

rate

that

causes

the

bondprice

to

equal

its

face

value.In

our

example

we

solveOptions,

Futures,

andOther

Derivatives

8thEdition,

Copyright

?

John

C.

Hull

201216Par

Yield

continued

In

general

if

m

is

the

number

ofcoupon

payments

per

year,

d

is

thepresent

value

of

$1

received

at

maturityandA

is

the

present

value

of

an

annuityof

$1

on

each

coupon

date

(in

our

example,

m

=

2,

d

=0.87284,

and

A

=

3.70027)Options,

Futures,

andOther

Derivatives

8thEdition,

Copyright

?

John

C.

Hull

201217Data

to

Determine

Zero

Curve

(Table

4.3,

page

82)Bond

PrincipalTime

toMaturity

(yrs)Coupon

peryear

($)*Bond

price

($)1000.25097.51000.50094.91001.00090.01001.50896.01002.0012101.6*

Half

the

stated

coupon

is

paid

each

yearOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

2012

18The

Bootstrap

MethodOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

201219An

amount

2.5

can

be

earned

on

97.5during

3

months.Because

100=97.5e0.10127×0.25

the3-month

rate

is

10.127%

withcontinuous

compoundingSimilarly

the

6

month

and

1

year

ratesare

10.469%

and

10.536%

withcontinuous

compoundingThe

Bootstrap

Method

continuedTo

calculate

the

1.5

year

rate

we

solve?to

get

R

=

0.10681

or

10.681%Similarly

the

two-year

rate

is

10.808%Options,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

Hull

201220Zero

Curve

Calculated

from

the

Data

(Figure

4.1,

page

84)?Zero

Rate(%)Options,

Futures,

andOther

Derivatives

8thEdition,

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John

C.

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201221Maturity

(yrs)10.4610.12

9710.53610.68110.808Forward

RatesOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

201222The

forward

rate

is

the

future

zero

rateimplied

by

today’s

term

structure

ofinterest

ratesFormulafor

Forward

RatesSuppose

that

the

zero

rates

for

timeperiods

T1

and

T2

are

R1

and

R2

withboth

rates

continuously

compounded.The

forward

rate

for

the

period

betweentimes

T1

and

T2

isThiOsptifonosr,

Fmuutulreas,iansd

Oothnelr

Dyeraivpatpivreos

8xtih

mately

truewheEnditriaont,

eCospyarirghet?nJoohtn

Ce.

xHupllr2e01s2

sed

2w3

ithApplication

of

the

FormulaOptions,

Futures,

andOther

Derivatives

8thEdition,

Copyright

?

John

C.

Hull

2012

24Year

(n)Zero

rate

for

n-yearinvestment(%

per

annum)Forwardrate

for

nthyear(%

per

annum)13.024.05.034.65.845.06.255.56.5Instantaneous

Forward

RateThe

instantaneous

forward

rate

for

amaturity

T

is

the

forward

rate

thatapplies

for

a

very

short

time

periodstarting

at

T.

It

is?where

R

is

the

T-year

rateOptions,

Futures,

andOther

Derivatives

8thEdition,

Copyright

?

John

C.

Hull

201225Upward

vs

Downward

Sloping

Yield

CurveOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

201226For

an

upward

sloping

yield

curve:Fwd

Rate

>

Zero

Rate

>

Par

YieldFor

a

downward

sloping

yield

curvePar

Yield

>

Zero

Rate

>

Fwd

RateForward

Rate

AgreementOptions,

Futures,

andOther

Derivatives

8thEdition,

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?

John

C.

Hull

201227A

forward

rate

agreement

(FRA)

is

anOTC

agreement

that

a

certain

rate

willapply

to

a

certain

principal

during

acertain

future

time

periodForward

Rate

Agreement:

Key

ResultsOptions,

Futures,

andOther

Derivatives

8thEpdriteiosn,enCotpyrvigahlt

?uJeohon

Cf.

Htulhle201d2iffe28rencetheAn

FRA

is

equivalent

to

an

agreementwhere

interest

at

a

predetermined

rate,RK

is

exchanged

for

interest

at

themarket

rateAn

FRA

can

be

valued

by

assuming

thatthe

forward

LIBOR

interest

rate,

RF

,

iscertain

to

be

realizedThis

means

that

the

value

of

an

FRA

isValuation

Formulascash

flow

is

RF(T2

–T1)

at

time

T2If

the

period

to

which

an

FRA

applieslasts

from

T1

to

T2,

we

assume

that

RFand

RK

are

expressed

with

acompounding

frequency

correspondingto

the

length

of

the

period

between

T1and

T2With

an

interest

rate

of

RK,

the

interestcash

flow

is

RK

(T2–T1)

at

time

T2WitOhptiaonns,iFunttureesr,eansdtOthrear

DteerivoatfiveRsF8t,h

the

interestEdition,

Copyright

?

John

C.

Hull

201229Valuation

Formulas

continuedWhen

the

rate

RK

will

be

received

on

aprincipal

of

L

the

value

of

the

FRA

is

thepresent

value

ofreceived

at

time

T2When

the

rate

RK

will

be

received

on

aprincipal

of

L

the

value

of

the

FRA

is

thepresent

value

ofOptions,

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andOther

Derivatives

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John

C.

Hull

201230ExampleAn

FRA

entered

into

some

time

agoensures

that

a

company

will

receive

4%(s.a.)

on

$100

million

for

six

monthsstarting

in

1

yearForward

LIBOR

for

the

period

is

5%(s.a.)The

1.5

year

rate

is

4.5%

withcontinuous

compoundingOptions,

Futures,

andOther

Derivatives

8thTheEdivtaioln,uCeopoyrfightth?

eJohFn

RC.AHul(l

i20n12

$

mi3l1

lions)

isExample

continuedIf

the

six-month

interest

rate

in

oneyear

turns

out

to

be

5.5%

(s.a.)

therewill

be

a

payoff

(in

$

millions)

ofin

1.5

yearsThe

transaction

might

be

settled

at

theone-year

point

for

an

equivalent

payoffofOptions,

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andOther

Derivatives

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John

C.

Hull

201232Duration

(page

89-90)Duration

of

a

bond

that

providescash

flow

ci

at

time

tiis??

where

B

is

its

price

and

y

is

itsyield

(continuously

compounded)Options,

Futures,

andOther

Derivatives

8thEdition,

Copyright

?

John

C.

Hull

201233Key

Duration

RelationshipDuration

is

important

because

it

leads

tothe

following

key

relationship

betweenthe

change

in

the

yield

on

the

bond

andthe

change

in

its

priceOptions,

Futures,

andOther

Derivatives

8thEdition,

Copyright

?

John

C.

Hull

201234Key

Duration

Relationship

continuedWhen

the

yield

y

is

expressed

withcompounding

m

times

per

yearThe

expression??Optioinss,

Fruteufrees,rarndeOdthetroDeraivsattivhese8t“hEdition,

Copyright

?

John

C.

Hull

2012duration”modified35Bond

PortfoliosThe

duration

for

a

bond

portfolio

is

theweighted

average

duration

of

the

bondsin

the

portfolio

with

weightsproportional

to

pricesThekeyduration

relationship

for

a

bondportfolio

describes

the

effect

of

smallparallel

shifts

in

the

yield

curveWhat

exposures

remain

if

duration

of

aportOpftioonlsi,

Foutourfes,aasndsOethtesr

Deerqivuataivless

8tthhe

duration

oEdition,

Copyright

?

John

C.

Hull

2012a

portfolio

of

liabiliti