24.1
Interest Rate Derivatives:
More Advanced Models
Chapter 24
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.2
The Two-Factor Hull-White
Model (Equation 24.1, page 571)
dx  (t )  u  ax  dt  1 dz1
du  budt   2 dz2
where x  f (r ) and the correlation
between dz1 and dz2 is 
The short rate reverts to a level dependent
on u, and u itself is mean reverting
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.3
Analytic Results
• Bond prices and European options on
zero-coupon bonds can be calculated
analytically when f(r) = r
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Options on Coupon-Bearing
Bonds
24.4
• We cannot use the same procedure for
options on coupon-bearing bonds as we do in
the case of one-factor models
• If we make the approximate assumption that
the coupon-bearing bond price is lognormal,
we can use Black’s model
• The appropriate volatility is calculated from
the volatilities of and correlations between the
underlying zero-coupon bond prices
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.5
Volatility Structures
• In the one-factor Ho-Lee or Hull-White
model the forward rate S.D.s are either
constant or decline exponentially. All
forward rates are instantaneously
perfectly correlated
• In the two-factor model many different
forward rate S.D. patterns and
correlation structures can be obtained
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.6
Example Giving Humped Volatility
Structure (Figure 24.1, page 572)
a=1, b=0.1, 1=0.01, 2=0.0165, =0.6
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0.00
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.7
Transformation of the
General Model
dx  (t )  u  ax  dt   1 dz1
du  budt   2 dz 2
where x  f (r ) and the correlation
between dz1 and dz 2 is 
We define y  x  u (b  a ) so that
dy  (t )  ay  dt   3 dz 3
du  budt   2 dz 2
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.8
Transformation of the
General Model continued
2

2 1 2
2
2
2
 3  1 
2 
ba
(b  a )
The correlation between dz 2 and
dz 3 is
 1   2 (b  a )
3
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.9
Attractive Features of the
Model
• It is Markov so that a recombining 3dimensional tree can be constructed
• The volatility structure is stationary
• Volatility and correlation patterns similar
to those in the real world can be
incorporated into the model
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.10
HJM Model: Notation
P(t,T ): price at time t of a discount bond
with principal of $1 maturing at T
Wt : vector of past and present values of
interest rates and bond prices at time t
that are relevant for determining bond
price volatilities at that time
v(t,T,Wt ): volatility of P(t,T)
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.11
Notation continued
ƒ(t,T1,T2): forward rate as seen at t for the
period between T 1 and T 2
F(t,T): instantaneous forward rate as seen at
t for a contract maturing at T
r(t): short-term risk-free interest rate at t
dz(t): Wiener process driving term structure
movements
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.12
Modeling Bond Prices
dP (t , T )  r (t ) P (t , T )dt  v (t , T , W t ) P (t , T )dz (t )
We can choose any v function providing
v (t , t , W t )  0
for all t
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.13
Modeling Forward Rates
Equation 24.7, page 575)
dF ( t , T )  m( t , T , W t ) dt  s(t , T , W t ) dz (t )
We must have
T
m( t , T , W t )  s( t , T , W t )  s(t , , W t )d
t
Similar results hold when there is more than
one factor
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Tree For a General Model
A non-recombining tree means that
the process for r is non-Markov
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.14
24.15
The LIBOR Market Model
The LIBOR market model is a model
constructed in terms of the forward
rates underlying caplet prices
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.16
Notation
t k : kth reset date
Fk (t ): forward rate between times t k and t k 1
m(t ): index for next reset date at time t
 k (t ): volatility of Fk (t ) at time t
vk (t ): volatility of P(t, t k ) at time t
 k : t k 1  t k
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.17
Volatility Structure
We assume a stationary volatility structure
where the volatility of Fk (t ) depends only on
the number of accrual periods between the
next reset date and t k [i.e., it is a function only
of k  m(t )]
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
In Theory the L’s can be
determined from Cap Prices
24.18
Define L i as the volatility of Fk (t ) when k  m(t )  i
If  k is the volatility for the (t k ,t k 1 ) caplet. If the model
provides a perfect fit to cap prices we
must have
k
 2k t k   L2k i i 1
i 1
This allows the L ' s to be determined
inductivel y
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.19
Example 24.1 (Page 579)
• If Black volatilities for the first three
caplets are 24%, 22%, and 20%, then
L0=24.00%
L1=19.80%
L2=15.23%
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.20
Example 24.2 (Page 579)
1
2
3
n(%)
15.50
18.25
17.91
17.74 17.27
Ln-1(%)
15.50
20.64
17.21
17.22 15.25
6
7
8
9
10
n(%)
16.79
16.30
16.01
15.76
15.54
Ln-1(%)
14.15
12.98
13.81
13.60
13.40
n
n
4
5
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.21
The Process for Fk in a OneFactor LIBOR Market Model
dFk  L k  m( t ) Fk dz
The drift depends on the world chosen
In a world that is forward risk - neutral
with respect to P(t , ti 1 ), the drift is zero
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.22
Rolling Forward RiskNeutrality (Equation 24.16, page 579)
It is often convenient to choose a world
that is always FRN wrt a bond maturing
at the next reset date. In this case, we
can discount from ti+1 to ti at the i rate
observed at time ti. The process for Fk is
dFk

Fk
i
 i Fi L i m( t ) L k m( t )
j  m( t )
1   i Fi

dt  L k m( t ) dz
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.23
The LIBOR Market Model
and HJM
In the limit as the time between resets
tends to zero, the LIBOR market model
with rolling forward risk neutrality
becomes the HJM model in the
traditional risk-neutral world
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Monte Carlo Implementation
of BGM Cap Model
24.24
(Equation 24.18, page 580)
We assume no change to the drift between
reset dates so that
 i  i Fi (t j ) L i  j L k  j L2k  j 

  k  L k  j   j 
Fk (t j 1 )  Fk (t j ) exp  

1   j Lj
2 
 j  k

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.25
Multifactor Versions of BGM
• BGM can be extended so that there are
several components to the volatility
• A factor analysis can be used to
determine how the volatility of Fk is split
into components
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.26
Ratchet Caps, Sticky Caps,
and Flexi Caps
• A plain vanilla cap depends only on one
forward rate. Its price is not dependent
on the number of factors.
• Ratchet caps, sticky caps, and flexi
caps depend on the joint distribution of
two or more forward rates. Their prices
tend to increase with the number of
factors
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.27
Valuing European Options in
the LIBOR Market Model
There is a good analytic approximation
that can be used to value European
swap options in the LIBOR market
model. See pages 582 to 584.
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.28
Calibrating the LIBOR
Market Model
• In theory the LMM can be exactly calibrated to cap
prices as described earlier
• In practice we proceed as for the one-factor models
in Chapter 23 and minimize a function of the form
n
2
(
U

V
)
 i i P
i 1
• 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 
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.29
Types of Mortgage-Backed
Securities (MBSs)
• Pass-Through
• Collateralized Mortgage
Obligation (CMO)
• Interest Only (IO)
• Principal Only (PO)
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
24.30
Option-Adjusted Spread
(OAS)
• To calculate the OAS for an interest
rate derivative we value it assuming
that the initial yield curve is the
Treasury curve + a spread
• We use an iterative procedure to
calculate the spread that makes the
derivative’s model price = market
price. This is the OAS.
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull