Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.1
Interest Rate
Markets
Chapter 5
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.2
Types of Rates
? Treasury rates
? LIBOR rates
? Repo rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.3
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.4
Example (Table 5.1,page 95)
Matu rity
(y ears )
Zer o Rate
(% c ont c omp)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.5
Bond Pricing
? To calculate the cash price of a bond we
discount each cash flow at the appropriate
zero rate
? In our example,the theoretical price of a two-
year bond providing a 6% coupon
semiannually is
3 3 3
103 98 39
0 05 0 5 0 058 1 0 0 064 1 5
0 068 2 0
e e e
e
? ? ? ? ? ?
? ?
? ?
? ?
.,,,,,
.,,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.6
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? ? ? ? ? ? ? ?? ? ? ?.,,,,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.7
Par Yield
? The par yield for a certain maturity is the
coupon rate that causes the bond price to
equal its face value.
? In our example we solve
g)c om po u n di n s,a,( w i t h 876g e t to
100
2
100
222
0.2068.0
5.1064.00.1058.05.005.0
.c=
e
c
e
c
e
c
e
c
??
?
?
?
?
?
??
??
??
??????
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.8
Par Yield continued
In general if m is the number of coupon
payments per year,d is the present
value of $1 received at maturity and A is
the present value of an annuity of $1 on
each coupon date
A
mdc )1 001 00( ??
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.9Sample Data for Determining the
Zero Curve (Table 5.2,page 97)
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.10
The Bootstrapping the Zero
Curve
? 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 10.127% with continuous
compounding
? Similarly the 6 month and 1 year rates are
10.469% and 10.536% with continuous
compounding
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.11
The Bootstrap Method
continued
? To calculate the 1.5 year rate we solve
to get R = 0.10681 or 10.681%
? Similarly the two-year rate is 10.808%
9610444 5.10.11 0 5 3 6.05.01 0 4 6 9.0 ??? ?????? Reee
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.12
Zero Curve Calculated from the
Data (Figure 5.1,page 98)
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.13
Forward Rates
The forward rate is the future zero rate
implied by today’s term structure of
interest rates
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.14Calculation of Forward Rates
Table 5.4,page 98
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.15
Formula for Forward Rates
? Suppose that the zero rates for
maturities T1 and T2 are R1 and R2 with
both rates continuously compounded.
? The forward rate for the period between
times T1 and T2 is
R T R T
T T
2 2 1 1
2 1
?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.16
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,It is
where R is the T-year rate
R T R
T
? ?
?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.17
Upward vs Downward Sloping
Yield Curve
? For an upward sloping yield curve:
Fwd Rate > Zero Rate > Par Yield
? For a downward sloping yield curve
Par Yield > Zero Rate > Fwd Rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.18
Forward Rate Agreement
? A forward rate agreement (FRA) is an
agreement that a certain rate will apply
to a certain principal during a certain
future time period
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.19
Forward Rate Agreement
continued (Page 100)
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.20
Theories of the Term Structure
Pages 102
? 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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.21
Day Count Conventions
in the U.S,(Pages 102-103)
Treasury Bonds:
Corporate Bonds:
Money Market Instruments:
Actual/Actual (in period)
30/360
Actual/360
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.22Treasury Bond Price Quotes
in the U.S
Cash price = Quoted price +
Accrued Interest
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.23
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( )?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.24
Treasury Bond Futures
Page 104
Cash price received by party with short
position =
Quoted futures price × Conversion
factor + Accrued interest
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.25
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 6% with semiannual
compounding
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.26
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
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.27
? If Z is the quoted price of a Eurodollar
futures contract,the value 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 (Page 110)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.28
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 (actual/360) and
all contracts are closed out
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.29
Forward Rates and Eurodollar
Futures (Page 111)
? Eurodollar futures contracts last out to
10 years
? For Eurodollar futures we cannot
assume that the forward rate equals the
futures rate
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.30Forward Rates and Eurodollar
Futures continued
A " co n v e x i ty a d j u stm e n t " o f te n m a d e is
For w a r d r a te = Futu r e s r a te
w h e r e is the ti m e to m a tu r i ty o f th e
f u tu r e s co n tr a ct,is the m a tu r i ty o f
th e r a te u n d e r l y i n g th e f u tu r e s co n tr a ct
( 9 0 d a y s l a te r th a n ) a n d is the
sta n d a r d d e v i a tion o f th e sh o r t r a te ch a n g e s
p e r y e a r ( ty p i ca l l y is ab o u t
?
1
2
0 012
2
1 2
1
2
1
?
?
?
t t
t
t
t
,)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.31
? Duration of a bond that provides cash flow c i at time t i is
where B is its price and y is its yield (continuously
compounded)
? This leads to
t
c e
B
i
i
n
i
yt i
?
?
?
?
?
?
?
?
?
1
yD
B
B ????
Duration
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.32
Duration Continued
? When the yield y is expressed with
compounding m times per year
? The expression
is referred to as the,modified duration”
my
yBDB
?
????
1
D
y m1 ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.33
Convexity
The convexity of a bond is defined as
C
B
B
y
c t e
B
B
B
D y C y
i i
yt
i
n
i
? ?
? ? ?
?
?
?
1
1
2
2
2
2
1
2
?
?
?
? ?
so t h a t
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
5.34
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