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Chapter
4
Interest
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?JohnC.
Hull
20121Types
of
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20122Treasury
ratesLIBOR
ratesRepo
ratesTreasury
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20123
Rates
on
instruments
issued
by
a
governmentin
its
own
currencyLIBOR
and
LIBIDOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20124
LIBOR
is
the
rate
of
interest
at
which
a
bankis
prepared
to
deposit
money
with
anotherbank.
(The
second
bank
must
typically
havea
AA
rating)
LIBOR
is
compiled
once
a
day
by
the
BritishBankers
Association
on
all
major
currenciesfor
maturities
up
to
12
months
LIBID
is
the
rate
which
a
AA
bank
is
preparedto
pay
on
deposits
from
anther
bankRepo
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20125
Repurchase
agreement
is
an
agreementwhere
a
financial
institution
that
ownssecurities
agrees
to
sell
them
today
for
X
andbuy
them
bank
in
the
future
for
a
slightlyhigher
price,
YThe
financial
institution
obtains
a
loan.
Therate
of
interest
is
calculated
from
thedifference
between
X
and
Y
and
is
known
asthe
repo
rateThe
Risk-Free
RateOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20126
The
short-term
risk-free
rate
traditionallyused
by
derivatives
practitioners
is
LIBOR
The
Treasury
rate
is
considered
to
beartificially
low
for
anumber
of
reasons
(SeeBusiness
Snapshot
4.1)As
will
be
explained
in
later
chapters:
Eurodollar
futures
and
swaps
are
used
to
extendthe
LIBOR
yield
curve
beyond
one
year
The
overnight
indexed
swap
rate
is
increasinglybeing
used
instead
of
LIBOR
as
the
risk-free
rateMeasuring
Interest
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20127
The
compounding
frequency
used
foran
interest
rate
is
the
unit
ofmeasurement
The
difference
between
quarterly
andannual
compounding
is
analogous
tothe
difference
between
miles
andkilometersImpact
of
CompoundingOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20128When
we
compound
m
times
per
year
at
rate
R
anamount
A
grows
to
A(1+R/m)m
in
one
yearCompoundingfrequencyValue
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)110.52ContinuousCompoundingOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
20129(Page
79)
In
the
limit
as
we
compound
more
and
morefrequently
we
obtain
continuously
compoundedinterest
rates
$100
grows
to
$100eRT
when
invested
at
acontinuously
compounded
rate
R
for
time
T
$100
received
at
time
T
discounts
to
$100e-RT
attime
zero
when
the
continuously
compoundeddiscount
rate
is
RConversion
Formulas
(Page
79)DefineRc
:
continuously
compounded
rateRm:
same
rate
with
compounding
m
times
peryearOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201210ExamplesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201211
10%
with
semiannual
compounding
isequivalent
to
2ln(1.05)=9.758%
withcontinuous
compounding8%
with
continuous
compounding
isequivalent
to
4(e0.08/4
-1)=8.08%
with
quarterlycompounding
Rates
used
in
option
pricing
are
nearlyalways
expressed
with
continuouscompoundingZero
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201212A
zero
rate
(or
spot
rate),
for
maturity
T
is
therate
of
interest
earned
onan
investment
thatprovides
a
payoff
only
at
time
TExample
(Table
4.2,
page
81)Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201213Maturity
(years)Zero
rate
(cont.
comp.0.55.01.05.81.56.42.06.8Bond
Pricing
To
calculate
the
cash
price
of
a
bond
wediscount
each
cash
flow
at
the
appropriatezero
rate
In
our
example,
the
theoretical
price
of
a
two-year
bond
providing
a
6%
couponsemiannually
isOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201214Bond
Yield
The
bond
yield
is
the
discount
rate
that
makesthe
present
value
of
the
cash
flows
on
the
bond
equal
to
the
market
price
of
the
bond
Suppose
that
the
market
price
of
the
bond
inour
example
equals
its
theoretical
price
of98.39
The
bond
yield
(continuously
compounded)
isgiven
by
solvingto
get
y=0.0676
or
6.76%.Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201215Par
Yield
The
par
yield
for
a
certain
maturity
is
thecoupon
rate
that
causes
the
bondprice
toequal
its
face
value.In
our
example
we
solveOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201216Par
Yield
continuedIn
general
if
m
is
thenumberof
couponpayments
per
year,
d
is
the
present
value
of$1
received
at
maturity
and
A
is
the
presentvalue
of
an
annuity
of
$1
on
eachcoupon
date(in
our
example,
m
=
2,
d
=
0.87284,andA
=3.70027)Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201217Data
to
Determine
Zero
Curve(Table
4.3,
page
82)Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201218Bond
PrincipalTime
toMaturity
(yrs)Coupon
peryear
($)*Bond
price
($)1000.25097.51000.50094.91001.00090.01001.50896.01002.0012101.6*
Half
the
stated
coupon
is
paid
each
yearThe
Bootstrap
MethodOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201219
An
amount
2.5
can
be
earned
on
97.5
during3
months.
Because
100=97.5e0.10127×0.25
the
3-monthrate
is
10.127%with
continuouscompounding
Similarly
the
6month
and1
year
rates
are10.469%
and10.536%with
continuouscompoundingThe
Bootstrap
Method
continuedTo
calculate
the
1.5
year
rate
we
solveto
get
R
=
0.10681
or10.681%Similarly
the
two-year
rate
is
10.808%Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201220Zero
Curve
Calculated
from
the
Data
(Figure
4.1,page84)ZeroRate(%)Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201221Maturity
(yrs)10.12710.46910.53610.68110.808Forward
RatesOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201222
The
forward
rate
is
the
future
zero
rateimplied
by
today’s
term
structure
of
interestratesFormulafor
Forward
Rates
Suppose
that
the
zero
rates
for
time
periods
T1
andT2
are
R1
and
R2
with
both
rates
continuouslycompounded.The
forward
rate
for
the
period
between
times
T1
andT2
is
This
formula
is
only
approximately
true
when
ratesare
not
expressed
with
continuous
compoundingOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201223Application
of
the
FormulaOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201224Year
(n)Zero
rate
for
n-yearinvestment(%
per
annum)Forward
rate
for
nthyear(%
per
annum)13.024.05.034.65.845.06.255.56.5Instantaneous
Forward
Rate
The
instantaneous
forward
rate
for
a
maturityT
is
the
forward
rate
that
applies
for
a
veryshort
time
period
starting
at
T.
It
iswhere
R
is
the
T-year
rateOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201225Upward
vs
Downward
Sloping
Yield
CurveOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201226
For
an
upward
sloping
yield
curve:Fwd
Rate
>
Zero
Rate
>
Par
Yield
For
a
downward
sloping
yield
curvePar
Yield
>
Zero
Rate
>
Fwd
RateForward
Rate
AgreementOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201227
A
forward
rate
agreement
(FRA)
is
an
OTCagreement
that
a
certain
rate
will
apply
to
acertain
principal
during
a
certain
future
timeperiodForward
Rate
Agreement:
Key
ResultsOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201228An
FRA
is
equivalent
to
an
agreement
where
interestat
a
predetermined
rate,
RK
is
exchanged
for
interest
atthe
market
rate
An
FRA
can
be
valued
byassumingthat
the
forwardLIBOR
interest
rate,
RF
,
is
certain
to
be
realizedThis
means
that
the
value
of
an
FRA
is
the
presentvalue
of
the
difference
between
the
interest
that
wouldbe
paid
at
interest
at
rate
RF
and
the
interest
that
woulbe
paid
at
rate
RKValuation
FormulasOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201229
If
the
period
to
which
an
FRA
applies
lastsfrom
T1
to
T2,
we
assume
that
RF
and
RK
areexpressed
with
a
compounding
frequencycorresponding
to
the
length
of
the
periodbetween
T1
and
T2
With
an
interest
rate
of
RK,
the
interest
cashflow
is
RK
(T2
–T1)
at
time
T2
With
an
interest
rate
of
RF,
the
interest
cashflow
is
RF(T2
–T1)
at
time
T2Valuation
Formulas
continuedWhen
the
rate
RK
will
be
received
on
a
principal
of
Lthe
value
of
the
FRA
is
the
present
value
ofreceived
at
time
T2When
the
rate
RK
will
be
received
on
a
principal
of
Lthe
value
of
the
FRA
is
the
present
value
ofreceived
at
time
T2Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201230Example
An
FRA
entered
into
some
time
ago
ensuresthat
a
company
will
receive
4%
(s.a.)
on
$100million
for
six
months
starting
in
1
yearForward
LIBOR
for
the
period
is
5%
(s.a.)
The
1.5
year
rate
is
4.5%
with
continuouscompoundingThe
value
of
the
FRA
(in
$
millions)
isOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201231Example
continued
If
the
six-month
interest
rate
in
one
year
turnsout
to
be
5.5%
(s.a.)
there
will
be
a
payoff
(in$
millions)
ofin
1.5
years
The
transaction
might
be
settled
at
the
one-year
point
for
an
equivalent
payoff
ofOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201232Duration
(page
89-90)Duration
of
a
bond
that
provides
cash
flow
ciat
time
tiiswhereB
is
its
price
and
y
is
its
yield(continuously
compounded)Options,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201233Key
Duration
Relationship
Duration
is
important
because
it
leads
to
thefollowing
key
relationship
between
thechange
in
the
yield
onthebondand
the
change
in
its
priceOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201234Key
Duration
Relationshipcontinued
When
the
yield
y
is
expressed
withcompounding
m
times
per
yearThe
expressionis
referred
to
as
the
“modified
duration”O(jiān)ptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201235Bond
PortfoliosOptions,
Futures,
and
Other
Derivatives
8th
Edition,Copyright
?
John
C.
Hull
201236
The
duration
for
a
bond
portfolio
is
the
weightedaverage
duration
of
the
bonds
in
the
portfolio
withweights
proportional
to
prices
Thekeyduration
relationship
for
a
bond
portfoliodescribes
the
effect
of
small
parallel
shifts
in
the
yicurve
What
exposures
remain
if
duration
of
a
portfolio
ofassets
equals
the
duration
of
a
portfolio
of
liabilitieConvexityThe
convexity,
C,
of
a
bond
is
defined
asThis
leads
to
a
more
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