4.1
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Interest Rates
and Duration(久期 )
Chapter 4
4.2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Types of Rates
Treasury rates（国债利率） —regarded
as risk-free rates
LIBOR rates (London Interbank Offer
rate) (伦敦银行同业放款利率 )–generally
higher than Treasury zero rates
Repo rates (回购利率 )—slightly higher
than the Treasury rates
4.3
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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,
In practice,it is usually called zero-coupon
interest rate (零息票利率 ).
4.4
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example (Table 4.1,page 89)
Ma tu rity
(years)
Ze ro Rate
(% co nt com p)
0,5 5,0
1,0 5,8
1,5 6,4
2,0 6,8
4.5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Bond Pricing
To calculate the cash price of a bond we
discount each cash flow at the appropriate
zero rate
In our example (page 89),the theoretical
price of a two-year bond with a principal of
$100 providing a 6% coupon semiannually is
39.98$1 0 3
333
0.2068.0
5.1064.00.1058.05.005.0
e
eee
4.6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Bond Yield
The bond yield is the discount rate that
makes the 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 in
our example equals its theoretical price of
98.39
The bond yield is given by solving
to get y=0.0676 or 6.76%.
3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y.,,,,
4.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Par Yield
The par yield (面值收益率 ) for a certain
maturity is the coupon rate that causes the
bond price to equal its face value (ie,The
principal),The bond is usually assumed to
provide semiannual coupons.
In our example we solve
c
e
c
e
c
e
c
e
c=,
2 2 2
100
2
100
6 87
0 05 0 5 0 058 1 0 0 064 1 5
0 068 2 0
.,,,,,
.,
to g e t
4.8
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Par Yield (continued)
In general if m is the number of coupon
payments per year,P is the present value of
$1 received at maturity and A is the present
value of an annuity(年金 ) of $1 on each
coupon date
c P mA( )100 100 100100 P
m
cA
4.9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Sample Data (Table 4.2,page 91))
Bond Time to Annual Bond
Principal Maturity Coupon Price
(dollars) (years) (dollars) (dollars)
100 0.25 0 97.5
100 0.50 0 94.9
100 1.00 0 90.0
100 1.50 8 96.0
100 2.00 12 101.6
4.10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Bootstrap Method
(息票剥离法 )
--used to determine zero rates
An amount 2.5 can be earned on 97.5 during
3 months.
The 3-month rate is 4 times 2.5/97.5 or
10.256% with quarterly compounding
This is 10.13% with continuous compounding
Similarly the 6 month and 1 year rates are
10.47% and 10.54% with continuous
compounding
4.11
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Bootstrap Method
（ continued）
To calculate the 1.5 year rate we solve
to get R = 0.1068 or 10.68%
Similarly the two-year rate is 10.81%
（ see the equation on page 91）
4 4 104 960 1047 0 5 0 1054 1 0 1 5e e e R.,,,,
4.12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Zero Curve Calculated from
the Data (Figure 4.1,page 92)
9
10
11
12
0 0,5 1 1,5 2 2,5
Zero
Rate (%)
Maturity (yrs)
10.127
10.469 10.536
10.681 10.808
4.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rates
(远期利率 )
The forward rate is the future zero rate
implied by today’s term structure(期限结构 ) of interest rates
It is determined by the current zero rates
The forward rate is for a specified future
time period
4.14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Calculation of Forward Rates
Table 4.4,page 93
Zero Rate for Forward Rate
an n -year Investment for n th Year
Year (n ) (% per annum) (% per annum)
1 10.0
2 10.5 11.0
3 10.8 11.4
4 11.0 11.6
5 11.1 11.5
4.15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Formula for Forward Rates
Suppose that the zero rates for time
periods T1 and T2 are R1 and R2 with
both rates continuously compounded.
The forward rate RF for the period
between times T1 and T2 is
12
1122)( 221211
TT
TRTRRAeeAe
F
TRTTRTR F
4.16
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Instantaneous Forward Rate
The instantaneous forward rate for a maturity
T is the forward rate that applies for a very
short time period starting at T,Letting
gives rise to
where R is the T-year rate
T
RTRR
F?
TTT 12
4.17
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Upward vs Downward Sloping
Yield Curve (Figure 4.2 and 4.3,page 93)
For an upward sloping yield curve:
Fwd Rate > Zero Rate > Par Yield
(See Figure 4.2 on page 94)
For a downward sloping yield curve
Par Yield > Zero Rate > Fwd Rate
(See Figure 4.3 on page 94)
4.18
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rate Agreement
A forward rate agreement (FRA,远期利率协议 ) is an agreement that a certain
rate RK will apply to a certain principal
during a certain future time period
4.19
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Value of FRA
Case I,If RK will be earned/received for the period
between T1 and T2 on a principal of L,then its value
is
Case II,If RK will be paid for the period between T1
and T2 on a principal of L,then its value is
Example 4.1 (page 96)
22))(( 12 TRFK eTTRRLV
22))(( 12 TRKF eTTRRLV
4.20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rate Agreement
continued (Page 97)
An FRA is equivalent to an agreement
where interest at a predetermined rate,
RK,is exchanged for interest at the
market rate R (see the two cases on
page 97)
An FRA can be valued by assuming that
the forward interest rate is certain to be
realized
4.21
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Theories of the Term Structure
(Pages 97-98)
Expectations Theory,forward rates equal
expected future zero rates
Market Segmentation,short,medium and
long rates determined independently of
each other
Liquidity Preference Theory,forward
rates higher than expected future zero
rates (page 98)
4.22
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Day Count Conventions
in the U.S,(Page 98)
Treasury Bonds:
Corporate Bonds:
Money Market Instruments:
Actual/Actual (in period)
30/360
Actual/360
See the examples on page 99
The day count defines the way that
interest accrues ove time.
Three day count conventions used in USA
4.23
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bond Price Quotes
in the U.S
The quoted price is not the same as
the cash price that is paid by the
purchaser.
Cash price = Quoted price +
Accrued Interest
Example on page 100
4.24
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bond Price Quotes
in the U.S(continued)
Example on page 100
Consider a 11% coupon bond maturing on July 10,2001
Coupons paid semiannually
Actual/actual day count convention
Quoted price=95-16,i.e,$(95+16/32)=$95.50
Suppose it is March 5,1999 NOW and the most recent
coupon date is Jan 10,1999
July10,2000
Case Price=$95.50 + (54/181) x $5.5 = $97.14
Jan 10,99 Mar 5 July 10
54
181 1 year
4.25
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bill Quote in the U.S.
If Y is the cash price of a Treasury bill
that has n days to maturity,the quoted
price is
360 100
n Y( )?
The quoted price/discount rate is not
the same as the rate of return
earned on the treasury bill (page100)
4.26
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Interest Rate Futures
Two types of contracts
1,the underlying is a government bond
Treasury bonds futures (CBOT) 长期国债期货
Treasury notes futures (CBOT) 中期国债期货
Long gilt futures (LIFFE) 金边证券期货
2,the underlying is a shot term Eurodollar or
LIBBOR interest rate
Eurodollar futures (CME)
Euroyen futures (CME and SIMEX)
EuroSwiss futures (LIFFE)
See the quotes from The Wall Street Journal (page 101 & 102)
4.27
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bond Futures
Page 103
Any government bond with more than 15
years to maturity on the first day of the
delivery moth and not callable within 15
years from that can be delivered
Each contract is for $100,000 face value of
bonds
Cash price received by party with short position =
Quoted futures price × Conversion factor +
Accrued interest
4.28
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Conversion Factor
The conversion factor for a bond is
approximately equal to the value of the bond
on the assumption that the yield curve is flat
at 8% with semiannual compounding
The bond maturity and times to the coupon
payment dates are rounded down to the
nearest 3 months for the purpose of
calculating the conversion factor
See example 4.2 & 4.3 on page 104
4.29
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
CBOT
T-Bonds & T-Notes
Factors that affect the futures price:
–Delivery can be made any time
during the delivery month
–Any of a range of
eligible bonds can be delivered
–The wild card play,an option of
delivery that the party with the
short position has
4.30
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Determining the Quoted Futures
Price
Theoretically difficult to determine because of the
factors above
Given that both the cheapest-to-deliver bond and
the delivery date are known
The treasury bond futures contract is a futures
contract on a security providing the holder with
known income
Equation (3.6) can be used
Procedures and illustrative example on page 106
4.31
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Eurodollar Futures
A Eurolldollar is a dollar deposited in a U.S,
or foreign bank outside the U.S.A.
The 3-month Eurodollar interest rate (or 3-
month LIBOR) is the rate of interest earned
on Eurodollars deposited for 3 months by one
bank with another bank
The 3-month Eurodollar futures contract
(CME) is the most popular futures contract on
short-term rates
4.32
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
If Z is the quoted price of a Eurodollar
futures contract,the value (contract
price) of one contract is 10,000[100-
0.25(100-Z)]
A change of one basis point or 0.01 in a
Eurodollar futures quote corresponds to
a contract price change of $25
Eurodollar Futures (continued)
4.33
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Eurodollar Futures (continued)
A Eurodollar futures contract is settled
in cash
When it expires (on the third
Wednesday of the delivery month) Z is
set equal to 100 minus the 90 day
Eurodollar interest rate and all contracts
are closed out
4.34
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rates and Eurodollar
Futures (Page 108)
Eurodollar futures contracts last out to
10 years
For Eurodollar futures lasting beyond
two years we cannot assume that the
forward rate equals the futures rate
4.35
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rates and Eurodollar
Futures (continued )
A " c onv ex i ty adj us tment " of ten mad e i s
F or w ar d r a te = F utur es r a te
w her e i s the ti me to matu r i ty of th e
f utur es c o ntr ac t,i s the ma tur i ty of
the r ate u nder l y i ng the f utur e s c ontr ac t
( 90 day s l ater than ) and i s the
s tandar d d ev i ati on o f the s hor t r ate c ha ng es
per y ear ( ty pi c al l y i s about
1
2
0 012
2
1 2
1
2
1
t t
t
t
t
,)
See example 4.6 on page 108
4.36
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Duration of a bond that provides cash flow c i at time t i is
where B is its price & y is its yield (continuously
compounded)
This leads to
Bect
iyt
i
n
i
i
1
yD
B
B
Duration
4.37
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Duration Continued
When the yield y is expressed with
compounding m times per year
The expression
is referred to as the ―modified duration‖
B BD y
y m
1
D
y m1?
4.38
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Duration Matching
This involves hedging against interest
rate risk by matching the durations of
assets and liabilities
It provides protection against small
parallel shifts in the zero curve
4.39
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Convexity
The convexity of a bond is defined as
so that
2
2
1 yCyD
B
B
B
etc
y
B
B
C
n
i
yt
ii
i?
1
2
2
21
4.40
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assignment
4.3,4.4,4.7,4.9,4.10,4.13,4.15,4.17
4.19,4.21,4.25,4.27
Have a try with the Assignment Questions
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Interest Rates
and Duration(久期 )
Chapter 4
4.2
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Types of Rates
Treasury rates（国债利率） —regarded
as risk-free rates
LIBOR rates (London Interbank Offer
rate) (伦敦银行同业放款利率 )–generally
higher than Treasury zero rates
Repo rates (回购利率 )—slightly higher
than the Treasury rates
4.3
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
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,
In practice,it is usually called zero-coupon
interest rate (零息票利率 ).
4.4
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Example (Table 4.1,page 89)
Ma tu rity
(years)
Ze ro Rate
(% co nt com p)
0,5 5,0
1,0 5,8
1,5 6,4
2,0 6,8
4.5
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Bond Pricing
To calculate the cash price of a bond we
discount each cash flow at the appropriate
zero rate
In our example (page 89),the theoretical
price of a two-year bond with a principal of
$100 providing a 6% coupon semiannually is
39.98$1 0 3
333
0.2068.0
5.1064.00.1058.05.005.0
e
eee
4.6
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Bond Yield
The bond yield is the discount rate that
makes the 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 in
our example equals its theoretical price of
98.39
The bond yield is given by solving
to get y=0.0676 or 6.76%.
3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y.,,,,
4.7
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Par Yield
The par yield (面值收益率 ) for a certain
maturity is the coupon rate that causes the
bond price to equal its face value (ie,The
principal),The bond is usually assumed to
provide semiannual coupons.
In our example we solve
c
e
c
e
c
e
c
e
c=,
2 2 2
100
2
100
6 87
0 05 0 5 0 058 1 0 0 064 1 5
0 068 2 0
.,,,,,
.,
to g e t
4.8
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Par Yield (continued)
In general if m is the number of coupon
payments per year,P is the present value of
$1 received at maturity and A is the present
value of an annuity(年金 ) of $1 on each
coupon date
c P mA( )100 100 100100 P
m
cA
4.9
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Sample Data (Table 4.2,page 91))
Bond Time to Annual Bond
Principal Maturity Coupon Price
(dollars) (years) (dollars) (dollars)
100 0.25 0 97.5
100 0.50 0 94.9
100 1.00 0 90.0
100 1.50 8 96.0
100 2.00 12 101.6
4.10
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Bootstrap Method
(息票剥离法 )
--used to determine zero rates
An amount 2.5 can be earned on 97.5 during
3 months.
The 3-month rate is 4 times 2.5/97.5 or
10.256% with quarterly compounding
This is 10.13% with continuous compounding
Similarly the 6 month and 1 year rates are
10.47% and 10.54% with continuous
compounding
4.11
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
The Bootstrap Method
（ continued）
To calculate the 1.5 year rate we solve
to get R = 0.1068 or 10.68%
Similarly the two-year rate is 10.81%
（ see the equation on page 91）
4 4 104 960 1047 0 5 0 1054 1 0 1 5e e e R.,,,,
4.12
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Zero Curve Calculated from
the Data (Figure 4.1,page 92)
9
10
11
12
0 0,5 1 1,5 2 2,5
Zero
Rate (%)
Maturity (yrs)
10.127
10.469 10.536
10.681 10.808
4.13
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rates
(远期利率 )
The forward rate is the future zero rate
implied by today’s term structure(期限结构 ) of interest rates
It is determined by the current zero rates
The forward rate is for a specified future
time period
4.14
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Calculation of Forward Rates
Table 4.4,page 93
Zero Rate for Forward Rate
an n -year Investment for n th Year
Year (n ) (% per annum) (% per annum)
1 10.0
2 10.5 11.0
3 10.8 11.4
4 11.0 11.6
5 11.1 11.5
4.15
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Formula for Forward Rates
Suppose that the zero rates for time
periods T1 and T2 are R1 and R2 with
both rates continuously compounded.
The forward rate RF for the period
between times T1 and T2 is
12
1122)( 221211
TT
TRTRRAeeAe
F
TRTTRTR F
4.16
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Instantaneous Forward Rate
The instantaneous forward rate for a maturity
T is the forward rate that applies for a very
short time period starting at T,Letting
gives rise to
where R is the T-year rate
T
RTRR
F?
TTT 12
4.17
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Upward vs Downward Sloping
Yield Curve (Figure 4.2 and 4.3,page 93)
For an upward sloping yield curve:
Fwd Rate > Zero Rate > Par Yield
(See Figure 4.2 on page 94)
For a downward sloping yield curve
Par Yield > Zero Rate > Fwd Rate
(See Figure 4.3 on page 94)
4.18
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rate Agreement
A forward rate agreement (FRA,远期利率协议 ) is an agreement that a certain
rate RK will apply to a certain principal
during a certain future time period
4.19
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Value of FRA
Case I,If RK will be earned/received for the period
between T1 and T2 on a principal of L,then its value
is
Case II,If RK will be paid for the period between T1
and T2 on a principal of L,then its value is
Example 4.1 (page 96)
22))(( 12 TRFK eTTRRLV
22))(( 12 TRKF eTTRRLV
4.20
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rate Agreement
continued (Page 97)
An FRA is equivalent to an agreement
where interest at a predetermined rate,
RK,is exchanged for interest at the
market rate R (see the two cases on
page 97)
An FRA can be valued by assuming that
the forward interest rate is certain to be
realized
4.21
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Theories of the Term Structure
(Pages 97-98)
Expectations Theory,forward rates equal
expected future zero rates
Market Segmentation,short,medium and
long rates determined independently of
each other
Liquidity Preference Theory,forward
rates higher than expected future zero
rates (page 98)
4.22
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Day Count Conventions
in the U.S,(Page 98)
Treasury Bonds:
Corporate Bonds:
Money Market Instruments:
Actual/Actual (in period)
30/360
Actual/360
See the examples on page 99
The day count defines the way that
interest accrues ove time.
Three day count conventions used in USA
4.23
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bond Price Quotes
in the U.S
The quoted price is not the same as
the cash price that is paid by the
purchaser.
Cash price = Quoted price +
Accrued Interest
Example on page 100
4.24
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bond Price Quotes
in the U.S(continued)
Example on page 100
Consider a 11% coupon bond maturing on July 10,2001
Coupons paid semiannually
Actual/actual day count convention
Quoted price=95-16,i.e,$(95+16/32)=$95.50
Suppose it is March 5,1999 NOW and the most recent
coupon date is Jan 10,1999
July10,2000
Case Price=$95.50 + (54/181) x $5.5 = $97.14
Jan 10,99 Mar 5 July 10
54
181 1 year
4.25
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bill Quote in the U.S.
If Y is the cash price of a Treasury bill
that has n days to maturity,the quoted
price is
360 100
n Y( )?
The quoted price/discount rate is not
the same as the rate of return
earned on the treasury bill (page100)
4.26
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Interest Rate Futures
Two types of contracts
1,the underlying is a government bond
Treasury bonds futures (CBOT) 长期国债期货
Treasury notes futures (CBOT) 中期国债期货
Long gilt futures (LIFFE) 金边证券期货
2,the underlying is a shot term Eurodollar or
LIBBOR interest rate
Eurodollar futures (CME)
Euroyen futures (CME and SIMEX)
EuroSwiss futures (LIFFE)
See the quotes from The Wall Street Journal (page 101 & 102)
4.27
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Treasury Bond Futures
Page 103
Any government bond with more than 15
years to maturity on the first day of the
delivery moth and not callable within 15
years from that can be delivered
Each contract is for $100,000 face value of
bonds
Cash price received by party with short position =
Quoted futures price × Conversion factor +
Accrued interest
4.28
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Conversion Factor
The conversion factor for a bond is
approximately equal to the value of the bond
on the assumption that the yield curve is flat
at 8% with semiannual compounding
The bond maturity and times to the coupon
payment dates are rounded down to the
nearest 3 months for the purpose of
calculating the conversion factor
See example 4.2 & 4.3 on page 104
4.29
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
CBOT
T-Bonds & T-Notes
Factors that affect the futures price:
–Delivery can be made any time
during the delivery month
–Any of a range of
eligible bonds can be delivered
–The wild card play,an option of
delivery that the party with the
short position has
4.30
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Determining the Quoted Futures
Price
Theoretically difficult to determine because of the
factors above
Given that both the cheapest-to-deliver bond and
the delivery date are known
The treasury bond futures contract is a futures
contract on a security providing the holder with
known income
Equation (3.6) can be used
Procedures and illustrative example on page 106
4.31
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Eurodollar Futures
A Eurolldollar is a dollar deposited in a U.S,
or foreign bank outside the U.S.A.
The 3-month Eurodollar interest rate (or 3-
month LIBOR) is the rate of interest earned
on Eurodollars deposited for 3 months by one
bank with another bank
The 3-month Eurodollar futures contract
(CME) is the most popular futures contract on
short-term rates
4.32
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
If Z is the quoted price of a Eurodollar
futures contract,the value (contract
price) of one contract is 10,000[100-
0.25(100-Z)]
A change of one basis point or 0.01 in a
Eurodollar futures quote corresponds to
a contract price change of $25
Eurodollar Futures (continued)
4.33
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Eurodollar Futures (continued)
A Eurodollar futures contract is settled
in cash
When it expires (on the third
Wednesday of the delivery month) Z is
set equal to 100 minus the 90 day
Eurodollar interest rate and all contracts
are closed out
4.34
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rates and Eurodollar
Futures (Page 108)
Eurodollar futures contracts last out to
10 years
For Eurodollar futures lasting beyond
two years we cannot assume that the
forward rate equals the futures rate
4.35
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Forward Rates and Eurodollar
Futures (continued )
A " c onv ex i ty adj us tment " of ten mad e i s
F or w ar d r a te = F utur es r a te
w her e i s the ti me to matu r i ty of th e
f utur es c o ntr ac t,i s the ma tur i ty of
the r ate u nder l y i ng the f utur e s c ontr ac t
( 90 day s l ater than ) and i s the
s tandar d d ev i ati on o f the s hor t r ate c ha ng es
per y ear ( ty pi c al l y i s about
1
2
0 012
2
1 2
1
2
1
t t
t
t
t
,)
See example 4.6 on page 108
4.36
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Duration of a bond that provides cash flow c i at time t i is
where B is its price & y is its yield (continuously
compounded)
This leads to
Bect
iyt
i
n
i
i
1
yD
B
B
Duration
4.37
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Duration Continued
When the yield y is expressed with
compounding m times per year
The expression
is referred to as the ―modified duration‖
B BD y
y m
1
D
y m1?
4.38
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Duration Matching
This involves hedging against interest
rate risk by matching the durations of
assets and liabilities
It provides protection against small
parallel shifts in the zero curve
4.39
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Convexity
The convexity of a bond is defined as
so that
2
2
1 yCyD
B
B
B
etc
y
B
B
C
n
i
yt
ii
i?
1
2
2
21
4.40
Options,Futures,and Other Derivatives,4th edition? 2000 by John C,Hull
Tang Yincai,? 2003,Shanghai Normal University
Assignment
4.3,4.4,4.7,4.9,4.10,4.13,4.15,4.17
4.19,4.21,4.25,4.27
Have a try with the Assignment Questions
