Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.1
Interest Rate Derivatives,
Models of the Short Rate
Chapter 23
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.2
Term Structure Models
? Black’s model is concerned with
describing the probability distribution of
a single variable at a single point in
time
? A term structure model describes the
evolution of the whole yield curve
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.3
Use of Risk-Neutral Arguments
? The process for the instantaneous short
rate,r,in the traditional risk-neutral world
defines the process for the whole zero
curve in this world
? If P(t,T ) is the price at time t of a zero-
coupon bond maturing at time T
? ? P t T E e r T t(,) ? ( )? ? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.4
Equilibrium Models
R en dl em an & B artte r:
V asi cek:
C ox,Ing erso l l,& R oss (C IR ):
dr r dt r dz
dr a b r dt dz
dr a b r dt r dz
? ?
? ? ?
? ? ?
? ?
?
?
( )
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.5Mean Reversion
(Figure 23.1,page 539)
Interestrate
HIGH interest rate has negative trend
LOW interest rate has positive trend
Reversion
Level
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.6Alternative Term Structures
in Vasicek & CIR
(Figure 23.2,page 540)
Zero Rate
Maturity
Zero Rate
Maturity
Zero Rate
Maturity
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.7
Equilibrium vs No-Arbitrage
Models
? In an equilibrium model
today’s term structure is
an output
? In a no-arbitrage model
today’s term structure is
an input
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.8Developing No-Arbitrage
Model for r
A model for r can be made to
fit the initial term structure by
including a function of time in
the drift
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.9
Ho and Lee
dr = q(t )dt + ?dz
? Many analytic results for bond prices
and option prices
? Interest rates normally distributed
? One volatility parameter,?
? All forward rates have the same
standard deviation
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.10
Initial Forward
Curve
Short
Rate
r
r
r
r
Time
Diagrammatic Representation of
Ho and Lee
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.11
Hull and White Model
dr = [q(t ) – ar ]dt + ?dz
? Many analytic results for bond prices
and option prices
? Two volatility parameters,a and ?
? Interest rates normally distributed
? Standard deviation of a forward rate is a
declining function of its maturity
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.12Diagrammatic Representation of
Hull and White
Short
Rate
r
r
r
r
Time
Forward Rate
Curve
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.13
Options on Coupon Bearing
Bonds
? A European option on a coupon-bearing bond
can be expressed as a portfolio of options on
zero-coupon bonds,
? We first calculate the critical interest rate at the
option maturity for which the coupon-bearing
bond price equals the strike price at maturity
? The strike price for each zero-coupon bond is
set equal to its value when the interest rate
equals this critical value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.14
Interest Rate Trees vs Stock
Price Trees
? The variable at each node in an
interest rate tree is the dt-period
rate
? Interest rate trees work similarly
to stock price trees except that
the discount rate used varies
from node to node
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.15Two-Step Tree Example
(Figure 23.6,page 551))
Payoff after 2 years is MAX[100(r – 0.11),0]
pu=0.25; pm=0.5; pd=0.25; Time step=1yr
0.35**
1.11*
0.23
0.00
0.14 3
0.12 1
0.10 0
0.08 0
0.06 0
r P
*,(0.25× 3 + 0.50× 1 + 0.25× 0)e–0.12× 1
**,(0.25× 1.11 + 0.50× 0.23 +0.25× 0)e–0.10× 1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.16Alternative Branching Processes
in a Trinomial Tree
(Figure 23.7,page 552)
(a) (b) (c)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.17An Overview of the Tree
Building Procedure
dr = [q(t ) – ar ]dt + ?dz
1.Assume q(t ) = 0 and r (0) = 0
2.Draw a trinomial tree for r to match the
mean and standard deviation of the
process for r
3.Determine q(t ) one step at a time so that
the tree matches the initial term structure
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.18
Example
? = 0.01
a = 0.1
dt = 1 year
The zero curve is as shown in
Table 23.1 on page 556
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.19The Initial Tree
(Figure 23.8,page 554)
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
r 0.000% 1.732% 0.000% -1.732% 3.464% 1.732% 0.000% -1.732% -3.464%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.20The Final Tree
(Figure 23.9,Page 556)
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
r 3.824% 6.937% 5.205% 3.473% 9.716% 7.984% 6.252% 4.520% 2.788%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.21
Extensions
The tree building procedure can be
extended to cover more general models of
the form:
d?(r ) = [q(t ) – a ?(r )]dt + ?dz
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.22Other Models
? These models allow the initial volatility
environment to be matched exactly
? But the future volatility structure may be quite
different from the current volatility structure
? ? dztdtrtatrd
dztdtr
t
t
trd
)()l n ()()(ln
)()l n (
)(
)(
)(ln
???q?
???
?
?
?
?
?
?
? ?
?q?
:K a r a s i n s k i and B l a c k
:T o y and D e r m a n,B l a c k,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.23
Calibration,a and ? constant
? The volatility parameters a and ? are chosen so that
the model fits the prices of actively traded
instruments such as caps and European swap
options as closely as possible
? We can choose a global best fit value of a and imply
? from the prices of actively traded instruments,This
creates a volatility surface for interest rate derivatives
similar to that for equity option or currency options
(see Chapter 15)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.24Calibration,
a and ? functions of time
? We minimize a function of the form
where Ui is the market price of the ith
calibrating instrument,Vi is the model
price of the ith calibrating instrument
and P is a function that penalizes big
changes or curvature in a and ?
?
?
??
n
i
ii PVU
1
2)(
23.1
Interest Rate Derivatives,
Models of the Short Rate
Chapter 23
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.2
Term Structure Models
? Black’s model is concerned with
describing the probability distribution of
a single variable at a single point in
time
? A term structure model describes the
evolution of the whole yield curve
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.3
Use of Risk-Neutral Arguments
? The process for the instantaneous short
rate,r,in the traditional risk-neutral world
defines the process for the whole zero
curve in this world
? If P(t,T ) is the price at time t of a zero-
coupon bond maturing at time T
? ? P t T E e r T t(,) ? ( )? ? ?
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.4
Equilibrium Models
R en dl em an & B artte r:
V asi cek:
C ox,Ing erso l l,& R oss (C IR ):
dr r dt r dz
dr a b r dt dz
dr a b r dt r dz
? ?
? ? ?
? ? ?
? ?
?
?
( )
( )
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.5Mean Reversion
(Figure 23.1,page 539)
Interestrate
HIGH interest rate has negative trend
LOW interest rate has positive trend
Reversion
Level
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.6Alternative Term Structures
in Vasicek & CIR
(Figure 23.2,page 540)
Zero Rate
Maturity
Zero Rate
Maturity
Zero Rate
Maturity
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.7
Equilibrium vs No-Arbitrage
Models
? In an equilibrium model
today’s term structure is
an output
? In a no-arbitrage model
today’s term structure is
an input
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.8Developing No-Arbitrage
Model for r
A model for r can be made to
fit the initial term structure by
including a function of time in
the drift
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.9
Ho and Lee
dr = q(t )dt + ?dz
? Many analytic results for bond prices
and option prices
? Interest rates normally distributed
? One volatility parameter,?
? All forward rates have the same
standard deviation
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.10
Initial Forward
Curve
Short
Rate
r
r
r
r
Time
Diagrammatic Representation of
Ho and Lee
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.11
Hull and White Model
dr = [q(t ) – ar ]dt + ?dz
? Many analytic results for bond prices
and option prices
? Two volatility parameters,a and ?
? Interest rates normally distributed
? Standard deviation of a forward rate is a
declining function of its maturity
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.12Diagrammatic Representation of
Hull and White
Short
Rate
r
r
r
r
Time
Forward Rate
Curve
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.13
Options on Coupon Bearing
Bonds
? A European option on a coupon-bearing bond
can be expressed as a portfolio of options on
zero-coupon bonds,
? We first calculate the critical interest rate at the
option maturity for which the coupon-bearing
bond price equals the strike price at maturity
? The strike price for each zero-coupon bond is
set equal to its value when the interest rate
equals this critical value
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.14
Interest Rate Trees vs Stock
Price Trees
? The variable at each node in an
interest rate tree is the dt-period
rate
? Interest rate trees work similarly
to stock price trees except that
the discount rate used varies
from node to node
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.15Two-Step Tree Example
(Figure 23.6,page 551))
Payoff after 2 years is MAX[100(r – 0.11),0]
pu=0.25; pm=0.5; pd=0.25; Time step=1yr
0.35**
1.11*
0.23
0.00
0.14 3
0.12 1
0.10 0
0.08 0
0.06 0
r P
*,(0.25× 3 + 0.50× 1 + 0.25× 0)e–0.12× 1
**,(0.25× 1.11 + 0.50× 0.23 +0.25× 0)e–0.10× 1
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.16Alternative Branching Processes
in a Trinomial Tree
(Figure 23.7,page 552)
(a) (b) (c)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.17An Overview of the Tree
Building Procedure
dr = [q(t ) – ar ]dt + ?dz
1.Assume q(t ) = 0 and r (0) = 0
2.Draw a trinomial tree for r to match the
mean and standard deviation of the
process for r
3.Determine q(t ) one step at a time so that
the tree matches the initial term structure
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.18
Example
? = 0.01
a = 0.1
dt = 1 year
The zero curve is as shown in
Table 23.1 on page 556
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.19The Initial Tree
(Figure 23.8,page 554)
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
r 0.000% 1.732% 0.000% -1.732% 3.464% 1.732% 0.000% -1.732% -3.464%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.20The Final Tree
(Figure 23.9,Page 556)
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
r 3.824% 6.937% 5.205% 3.473% 9.716% 7.984% 6.252% 4.520% 2.788%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.21
Extensions
The tree building procedure can be
extended to cover more general models of
the form:
d?(r ) = [q(t ) – a ?(r )]dt + ?dz
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.22Other Models
? These models allow the initial volatility
environment to be matched exactly
? But the future volatility structure may be quite
different from the current volatility structure
? ? dztdtrtatrd
dztdtr
t
t
trd
)()l n ()()(ln
)()l n (
)(
)(
)(ln
???q?
???
?
?
?
?
?
?
? ?
?q?
:K a r a s i n s k i and B l a c k
:T o y and D e r m a n,B l a c k,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.23
Calibration,a and ? constant
? The volatility parameters a and ? are chosen so that
the model fits the prices of actively traded
instruments such as caps and European swap
options as closely as possible
? We can choose a global best fit value of a and imply
? from the prices of actively traded instruments,This
creates a volatility surface for interest rate derivatives
similar to that for equity option or currency options
(see Chapter 15)
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
23.24Calibration,
a and ? functions of time
? We minimize a function of the form
where Ui is the market price of the ith
calibrating instrument,Vi is the model
price of the ith calibrating instrument
and P is a function that penalizes big
changes or curvature in a and ?
?
?
??
n
i
ii PVU
1
2)(
