Hull and White model

Chapter 30
Hull and White model
Consider
drt= t,trt dt +  t dW t;
where
t
,
 t
and
 t
are nonrandom functions of
t
.
We can solve the stochastic differential equation. Set
K t=
Z
t
0
udu:
Then
d

e
K t
rt

= e
K t

 trt dt + drt


= e
K t
t dt +  t dW t :
Integrating, we get
e
K t
rt=r0 +
Z
t
0
e
K u
u du +
Z
t
0
e
K u
u dW u;
so
rt= e
,Kt

r0 +
Z
t
0
e
K u
u du +

Z
t
0
e
K u
 u dW u

:
From Theorem 1.69 in Chapter 29, we see that
rt
is a Gaussian process with mean function
m
r
t=e
,Kt

r0 +
Z
t
0
e
K u
u du

(0.1)
and covariance function

r
s; t=e
,Ks,Kt

Z
s^t
0
e
2Ku

2
udu:
(0.2)
The process
rt
is also Markov.
293
294
We want to study
R
T
0
rt dt
.Todothis,wedefine
X t=
Z
t
0
e
Ku
udW u; Y T =
Z
T
0

e
,Kt
Xtdt:
Then
rt= e
,Kt

r0 +
Z
t
0
e
K u
u du

+ e
,K t
X t;
Z
T
0
rt dt =
Z
T
0
e
,K t

r0 +
Z

t
0
e
K u
u du

dt + Y T :
According to Theorem 1.70 in Chapter 29,
R
T
0
rt dt
is normal. Its mean is
IE
Z
T
0
rt dt =
Z
T
0
e
,K t

r0 +
Z
t
0
e
K u

u du

dt;
(0.3)
and its variance is
var

Z
T
0
rt dt
!
= IEY
2
T 
=
Z
T
0
e
2Kv

2
v

Z
T
v
e
,Ky

dy
!
2
dv :
The price at time 0 of a zero-coupon bond paying $1 at time
T
is
B 0;T= IEexp

,
Z
T
0
rt dt

= exp

,1IE
Z
T
0
rt dt +
1
2
,1
2
var

Z
T

0
rt dt
!
= exp

,r0
Z
T
0
e
,K t
dt ,
Z
T
0
Z
t
0
e
,K t+K u
u du dt
+
1
2
Z
T
0
e
2K v


2
v 

Z
T
v
e
,K y
dy
!
2
dv

= expf,r 0C 0;T,A0;Tg;
where
C 0;T=
Z
T
0
e
,Kt
dt;
A0;T=
Z
T
0
Z
t
0
e

,Kt+K u
u du dt ,
1
2
Z
T
0
e
2K v

2
v 

Z
T
v
e
,K y
dy
!
2
dv :
CHAPTER 30. Hull and White model
295
u
t
u = t
T
Figure 30.1: Range of values of
u; t

for the integral.
30.1 Fiddling with the formulas
Note that (see Fig 30.1)
Z
T
0
Z
t
0
e
,K t+K u
u du dt
=
Z
T
0
Z
T
u
e
,K t+K u
u dt du
y = t; v = u=
Z
T
0
e
Kv
v


Z
T
v
e
,Ky
dy
!
dv :
Therefore,
A0;T=
Z
T
0
2
4
e
Kv
v

Z
T
v
e
,Ky
dy
!
,
1
2
e

2K v

2
v 

Z
T
v
e
,K y
dy
!
2
3
5
dv ;
C 0;T=
Z
T
0
e
,Ky
dy ;
B 0;T = exp f,r0C 0;T, A0;Tg:
Consider the price at time
t 2 0;T
of the zero-coupon bond:
B t; T =IE
"
exp


,
Z
T
t
ru du





F t

:
Because
r
is a Markov process, this should be random only through a dependence on
rt
. In fact,
B t; T  = exp f,rtC t; T  , At; T g ;
296
where
At; T =
Z
T
t
2
4
e
Kv

v

Z
T
v
e
,Ky
dy
!
,
1
2
e
2K v

2
v 

Z
T
v
e
,K y
dy
!
2
3
5
dv ;
C t; T = e

Kt
Z
T
t
e
,Ky
dy :
The reason for these changes is the following. We are now taking the initial time to be
t
rather than
zero, so it is plausible that
R
T
0
::: dv
should be replaced by
R
T
t
::: dv:
Recall that
K v =
Z
v
0
udu;
and this should be replaced by
K v  , K t=
Z
v

t
udu:
Similarly,
K y 
should be replaced by
K y  , K t
. Making these replacements in
A0;T
,we
see that the
K t
terms cancel. In
C 0;T
,however,the
K t
term does not cancel.
30.2 Dynamics of the bond price
Let
C
t
t; T 
and
A
t
t; T 
denote the partial derivatives with respect to
t
. From the formula
B t; T  = exp f,rtC t; T  , At; T g ;
we have

dB t; T =Bt; T 
h
,C t; T  drt ,
1
2
C
2
t; T  drt drt , rtC
t
t; T  dt , A
t
t; T  dt
i
= B t; T 

, C t; T t , trt dt
, C t; T t dW t ,
1
2
C
2
t; T 
2
t dt
, rtC
t
t; T  dt , A
t
t; T  dt


:
Because we have used the risk-neutral pricing formula
B t; T =IE
"
exp

,
Z
T
t
ru du





F t

to obtain the bond price, its differential must be of the form
dB t; T =rtBt; T  dt +::: dW t:
CHAPTER 30. Hull and White model
297
Therefore, we must have
,C t; T t , trt ,
1
2
C
2
t; T 
2

t , rtC
t
t; T  , A
t
t; T = rt:
We leave the verification of this equation to the homework. After this verification, we have the
formula
dB t; T =rtBt; T  dt , tC t; T B t; T  dW t:
In particular, the volatility of the bond price is
 tC t; T 
.
30.3 Calibration of the Hull & White model
Recall:
drt= t,trt dt +  t dB t;
K t=
Z
t
0
udu;
At; T =
Z
T
t
2
4
e
Kv
v

Z

T
v
e
,Ky
dy
!
,
1
2
e
2K v

2
v 

Z
T
v
e
,K y
dy
!
2
3
5
dv ;
C t; T =e
Kt
Z
T

t
e
,Ky
dy ;
B t; T  = exp f,rtC t; T  , At; T g :
Suppose we obtain
B 0;T
for all
T 2 0;T


from market data (with some interpolation). Can we
determine the functions
t
,
 t
,and
 t
for all
t 2 0;T


? Not quite. Here is what we can do.
We take the following input data for the calibration:
1.
B 0;T; 0  T  T

;
2.
r0

;
3.
0
;
4.
t; 0  t  T

(usually assumed to be constant);
5.
0C 0;T; 0  T  T

, i.e., the volatility at time zero of bonds of all maturities.
Step 1. From4and5wesolvefor
C 0;T=
Z
T
0
e
,Ky
dy :