Cox-Ingersoll-Ross model

The Cox-Ingersoll-Ross model is the simplest one which avoids negative interest rates.. The Kolmogorov forward equation KFE is a partial differential equation in the “forward” variables

Cox-Ingersoll-Ross model

In the Hull & White model,r ( t )is a Gaussian process Since, for eacht,r ( t )is normally distributed, there is a positive probability thatr ( t ) < 0 The Cox-Ingersoll-Ross model is the simplest one which avoids negative interest rates

We begin with a d-dimensional Brownian motion( W 1 ;W 2 ;::: ;W d ) Let > 0and > 0 be constants Forj = 1 ;::: ;d, letX j (0)2IRbe given so that

X 21 (0) + X 22 (0) + ::: + X 2d (0)0 ;

and letX jbe the solution to the stochastic differential equation

dX j ( t ) =,

1

2 X j ( t ) dt + 1 2  dW j ( t ) :

X jis called the Orstein-Uhlenbeck process It always has a drift toward the origin The solution to

this stochastic differential equation is

X j ( t ) = e,1 2t

X j (0) + 1 2 Z t

0 e 1 2u dW j ( u )

:

This solution is a Gaussian process with mean function

m j ( t ) = e,1 2t X j (0)

and covariance function

 ( s;t ) = 14  2 e,1 2(s+t)Z s^t

0 e u du:

Define

r ( t ) 4

= X 21 ( t ) + X 22 ( t ) + ::: + X 2d ( t ) :

Ifd = 1, we haver ( t ) = X 21 ( t )and for eacht,IPfr ( t ) > 0g= 1, but (see Fig 31.1)

IP

There are infinitely many values oft > 0for whichr ( t ) = 0



= 1

303

t r(t) = X (t)

x x

1

2

( X (t), X (t) ) 1

1 2

2

Figure 31.1:r ( t )can be zero.

Ifd2, (see Fig 31.1)

IPfThere is at least one value oft > 0for whichr ( t ) = 0g = 0 :

Letf ( x 1 ;x 2 ;::: ;x d ) = x 21 + x 22 + ::: + x 2d Then

f xi = 2 x i ; f xixj =

(

2 ifi = j;

0 ifi6= j:

It ˆo’s formula implies

dr ( t ) =Xd

i=1 f xi dX i + 1 2Xd

i=1 f xixi dX i dX i

=Xd

i=1 2 X i



,

1

2 X i dt + 1

2  dW i ( t )

+Xd

i=1

1

4  2 dW i dW i

=,r ( t ) dt + Xd

i=1 X i dW i + d 2

4 dt

= d 2

4 ,r ( t )

!

dt + q

r ( t )Xd

i=1

X i ( t )

p

r ( t ) dW i ( t ) :

Define

W ( t ) =Xd

i=1

0 X i ( u )

p

r ( u ) dW i ( u ) :

ThenW is a martingale,

dW =Xd

i=1

X i

p

r dW i ;

dW dW =Xd

i=1

X 2i

r dt = dt;

soW is a Brownian motion We have

dr ( t ) = d 2

4 ,r ( t )

!

dt + q

r ( t ) dW ( t ) :

The Cox-Ingersoll-Ross (CIR) process is given by

dr ( t ) = ( ,r ( t )) dt + q

r ( t ) dW ( t ) ;

We define

d = 4

 2 > 0 :

Ifdhappens to be an integer, then we have the representation

r ( t ) =Xd

i=1 X 2i ( t ) ;

but we do not requiredto be an integer Ifd < 2(i.e., < 1

2  2), then

IPfThere are infinitely many values oft > 0for whichr ( t ) = 0g = 1 :

This is not a good parameter choice

Ifd2(i.e., 1 2  2), then

IPfThere is at least one value oft > 0for whichr ( t ) = 0g = 0 :

With the CIR process, one can derive formulas under the assumption thatd =  42 is a positive integer, and they are still correct even whendis not an integer

For example, here is the distribution ofr ( t )for fixedt > 0 Letr (0)0be given Take

X 1 (0) = 0 ; X 2 (0) = 0 ; ::: ; X d,1 (0) = 0 ; X d (0) =q

r (0) :

Fori = 1 ; 2 ;::: ;d,1,X i ( t )is normal with mean zero and variance

 ( t;t ) =  2

4 (1,e,t ) :

X d ( t )is normal with mean

m d ( t ) = e,

1

2tq

r (0)

and variance ( t;t ) Then

r ( t ) =  ( t;t ) dX ,1

i=1

X i ( t )

p

 ( t;t )

Chi-square withd,1 = 4 ,2

2 degrees of freedom

Normal squared and independent of the other term

(0.1)

Thusr ( t )has a non-central chi-square distribution.

Ast! 1,m d ( t )!0 We have

r ( t ) =  ( t;t )Xd

i=1

X i ( t )

p

 ( t;t )

:

Ast!1, we have ( t;t ) = 2

4, and so the limiting distribution of r ( t )is 2

4 times a chi-square withd =  42 degrees of freedom The chi-square density with 4 2 degrees of freedom is

f ( y ) = 1

2 2= 2

,

2

2

2 e,y=2 :

We make the change of variabler = 2

4 y The limiting density forr ( t )is

p ( r ) = 4

 2 : 1

2 2= 2

,

2

2





4

 2 r

2

e,

2r

=

2

 2



2 1 ,

2

2

2 e,

2r :

We computed the mean and variance ofr ( t )in Section 15.7

Consider a Markov process governed by the stochastic differential equation

dX ( t ) = b ( X ( t )) dt +  ( X ( t )) dW ( t ) :

-h

Figure 31.2: The functionh ( y )

Because we are going to apply the following analysis to the case X ( t ) = r ( t ), we assume that

X ( t )0for allt

We start at X (0) = x  0 at time 0 Then X ( t ) is random with density p (0 ;t;x;y )(in they

variable) Since 0 andxwill not change during the following, we omit them and writep ( t;y )rather thanp (0 ;t;x;y ) We have

IEh ( X ( t )) =Z

1

0 h ( y ) p ( t;y ) dy

for any functionh

The Kolmogorov forward equation (KFE) is a partial differential equation in the “forward” variables

tandy We derive it below

Leth ( y )be a smooth function ofy0which vanishes neary = 0and for all large values ofy(see Fig 31.2) It ˆo’s formula implies

dh ( X ( t )) =h

h0

( X ( t )) b ( X ( t ))+ 1 2 h00

( X ( t ))  2 ( X ( t ))i

dt + h0

( X ( t ))  ( X ( t )) dW ( t ) ;

so

h ( X ( t )) = h ( X (0))+Z t

0

h

h0( X ( s )) b ( X ( s ))+ 1 2 h00( X ( s ))  2 ( X ( s ))i

ds +

0 h0

( X ( s ))  ( X ( s )) dW ( s ) ; IEh ( X ( t )) = h ( X (0))+ IEZ t

0

h

h0( X ( s )) b ( X ( s )) dt + 1 2 h00( X ( s ))  2 ( X ( s ))i

ds;

or equivalently,

Z

1

0 h ( y ) p ( t;y ) dy = h ( X (0))+Z t

0

Z 1

0 h0( y ) b ( y ) p ( s;y ) dy ds +

1 2

0

Z 1

0 h00( y )  2 ( y ) p ( s;y ) dy ds:

Differentiate with respect totto get

Z

1

0 h ( y ) p t ( t;y ) dy =Z

1

0 h0( y ) b ( y ) p ( t;y ) dy + 1 2Z

1

0 h00( y )  2 ( y ) p ( t;y ) dy:

Integration by parts yields

Z

1

0 h0

( y ) b ( y ) p ( t;y ) dy = h ( y ) b ( y ) p ( t;y )

y=1

y=0

=0

, Z 1

0 h ( y ) @

@y ( b ( y ) p ( t;y )) dy;

Z

1

0 h00( y )  2 ( y ) p ( t;y ) dy = h0( y )  2 ( y ) p ( t;y )

y=1

y=0

=0

, Z 1

0 h0( y ) @

@y



 2 ( y ) p ( t;y )

dy

=,h ( y ) @

@y



 2 ( y ) p ( t;y )

y=1

y=0

=0

+Z 1

0 h ( y ) @ 2

@y 2



 2 ( y ) p ( t;y )

dy:

Therefore,

Z

1

0 h ( y ) p t ( t;y ) dy =,

Z 1

0 h ( y ) @

@y ( b ( y ) p ( t;y )) dy + 1 2

Z 1

0 h ( y ) @ 2

@y 2



 2 ( y ) p ( t;y )

dy;

or equivalently,

Z

1

0 h ( y )

"

p t ( t;y ) + @

@y ( b ( y ) p ( t;y )), 1 2 @ 2

@y 2



 2 ( y ) p ( t;y )

dy = 0 :

This last equation holds for every functionhof the form in Figure 31.2 It implies that

p t ( t;y ) + @

@y (( b ( y ) p ( t;y )),1 2 @ 2

@y 2



 2 ( y ) p ( t;y )

If there were a place where (KFE) did not hold, then we could takeh ( y ) > 0 at that and nearby points, but takehto be zero elsewhere, and we would obtain

Z 1

0 h

"

p t + @

@y ( bp ),1 2 @ 2

@y 2 (  2 p )

#

dy6= 0 :

If the processX ( t )has an equilibrium density, it will be

p ( y ) = lim t

!1

p ( t;y ) :

In order for this limit to exist, we must have

0 = lim t

!1

p t ( t;y ) :

Lettingt! 1in (KFE), we obtain the equilibrium Kolmogorov forward equation

@

@y ( b ( y ) p ( y )),1 2 @ 2

@y 2



 2 ( y ) p ( y )

= 0 :

When an equilibrium density exists, it is the unique solution to this equation satisfying

p ( y )0 8y0 ;

Z 1

0 p ( y ) dy = 1 :

We computed this to be

p ( r ) = Cr2 ,2

2 e,

2r ;

where

C =

2

 2



2 1 ,

2

2

We compute

p0( r ) = 2 , 2

 2 :p ( r r ) ,

2

 2 p ( r )

= 2  2 r



,1 2  2,r

p ( r ) ;

p00

( r ) =,

2

 2 r 2



,1 2  2,r

p ( r ) + 2  2 r (, ) p ( r ) + 2  2 r



,1 2  2,r

p0

( r )

= 2  2 r



,

1

r ( ,1 2  2,r ), + 2  2 r ( ,1 2  2,r ) 2

p ( r )

We want to verify the equilibrium Kolmogorov forward equation for the CIR process:

@

@r (( ,r ) p ( r )),

1

2 @ 2

@

@r (( ,r ) p ( r )) =,p ( r ) + ( ,r ) p0( r ) ;

@ 2

@r 2 (  2 rp ( r )) = @

@r (  2 p ( r ) +  2 rp0

( r ))

= 2  2 p0

( r ) +  2 rp00

( r ) :

The LHS of (EKFE) becomes

,p ( r ) + ( ,r ) p0( r ), 2 p0( r ),1 2  2 rp0( r )

= p ( r )



+ 1 r ( , 1 2  2

2

 2 r ( ,1 2  2

= p ( r )

( ,1 2  2

 2 r ( ,1 2  2,r ) + 1 r ( ,

1

2  2

2

 2 r ( ,

1

2  2

= 0 ;

as expected

The interest rate processr ( t )is given by

dr ( t ) = ( ,r ( t )) dt + q

r ( t ) dW ( t ) ;

wherer (0)is given The bond price process is

B ( t;T ) = IE

"

exp

(

,

t r ( u ) du

)

#

:

Because

exp ,

0 r ( u ) du

B ( t;T ) = IE

"

exp

(

,

0 r ( u ) du

)

#

;

the tower property implies that this is a martingale The Markov property implies thatB ( t;T )is random only through a dependence onr ( t ) Thus, there is a functionB ( r;t;T )of the three dummy variablesr;t;Tsuch that the processB ( t;T )is the functionB ( r;t;T )evaluated atr ( t ) ;t;T, i.e.,

B ( t;T ) = B ( r ( t ) ;t;T ) :

,

B ( r ( t ) ;t;T )is a martingale, its differential has nodtterm We com-pute

d

exp ,

0 r ( u ) du

B ( r ( t ) ;t;T )

= exp

,

0 r ( u ) du 

1

2 B rr ( r ( t ) ;t;T ) dr ( t ) dr ( t ) + B t ( r ( t ) ;t;T ) dt

:

The expression in[ ::: ]equals

=,rB dt + B r ( ,r ) dt + B r p

r dW

+ 1 2 B rr  2 r dt + B t dt:

Setting thedtterm to zero, we obtain the partial differential equation

,rB ( r;t;T )+ B t ( r;t;T )+ ( ,r ) B r ( r;t;T )+ 1 2  2 rB rr ( r;t;T ) = 0 ;

0t < T; r0 : (4.1) The terminal condition is

B ( r;T;T ) = 1 ; r0 :

Surprisingly, this equation has a closed form solution Using the Hull & White model as a guide,

we look for a solution of the form

B ( r;t;T ) = e,rC(t;T),A(t;T) ;

whereC ( T;T ) = 0 ; A ( T;T ) = 0 Then we have

B t = (,rC t,A t ) B;

B r =,CB; B rr = C 2 B;

and the partial differential equation becomes

0 =,rB + (,rC t,A t ) B,( ,r ) CB + 1 2  2 rC 2 B

= rB (,1,C t + C + 1

2  2 C 2 ),B ( A t + C )

We first solve the ordinary differential equation

,1,C t ( t;T ) + C ( t;T ) + 1 2  2 C 2 ( t;T ) = 0; C ( T;T ) = 0 ;

and then set

A ( t;T ) = Z T

t C ( u;T ) du;

soA ( T;T ) = 0and

A t ( t;T ) =,C ( t;T ) :

It is tedious but straightforward to check that the solutions are given by

C ( t;T ) = sinh( ( T,t ))

cosh( ( T,t )) + 1 2 sinh( ( T,t )) ;

A ( t;T ) =,

2

 2 log

2

4

1

2(T,t)

cosh( ( T,t )) + 1 2 sinh( ( T,t ))

3

where

= 1 2q

2 + 2  2 ; sinh u = e u,e,u

2 ; cosh u = e u + e,u

Thus in the CIR model, we have

IE

"

exp

(

,

t r ( u ) du

)

#

= B ( r ( t ) ;t;T ) ;

where

B ( r;t;T ) = expf,rC ( t;T ),A ( t;T )g; 0t < T; r0 ;

andC ( t;T )andA ( t;T )are given by the formulas above Because the coefficients in

dr ( t ) = ( ,r ( t )) dt + q

r ( t ) dW ( t )

do not depend ont, the functionB ( r;t;T )depends ontandT only through their difference =

T ,t Similarly,C ( t;T ) andA ( t;T ) are functions of = T ,t We writeB ( r; )instead of

B ( r;t;T ), and we have

B ( r; ) = expf,rC (  ),A (  )g;  0 ; r0 ;

where

C (  ) = cosh( sinh( )+ 1 )

2 sinh( ) ;

A (  ) =,

2

 2 log

2

4

1

2

cosh( )+ 1 2 sinh( )

3

= 1 2q

2 + 2  2 :

We have

B ( r (0) ;T ) = IE exp

(

,

0 r ( u ) du

)

:

Nowr ( u ) > 0for eachu, almost surely, soB ( r (0) ;T )is strictly decreasing inT Moreover,

B ( r (0) ; 0) = 1 ;

T!1

B ( r (0) ;T ) = IE exp

, Z 1

0 r ( u ) du

= 0 :

But also,

B ( r (0) ;T ) = expf,r (0) C ( T ),A ( T )g;

so

r (0) C (0)+ A (0) = 0 ;

lim

T!1

[ r (0) C ( T )+ A ( T )] =1;

and

r (0) C ( T )+ A ( T )

is strictly inreasing inT

The value at timetof an option on a bond in the CIR model is

v ( t;r ( t )) = IE

"

exp

(

,

t r ( u ) du

)

( B ( T 1 ;T 2 ),K ) +

#

;

whereT 1is the expiration time of the option,T 2is the maturity time of the bond, and0tT 1

T 2 As usual,expn

,

0 r ( u ) duo

v ( t;r ( t ))is a martingale, and this leads to the partial differential equation

,rv + v t + ( ,r ) v r + 1 2  2 rv rr = 0 ; 0t < T 1 ; r0 :

(wherev = v ( t;r ).) The terminal condition is

v ( T 1 ;r ) = ( B ( r;T 1 ;T 2 ),K ) + ; r0 :

Other European derivative securities on the bond are priced using the same partial differential equa-tion with the terminal condiequa-tion appropriate for the particular security

Process time scale: In this time scale, the interest rater ( t )is given by the constant coefficient CIR equation

dr ( t ) = ( ,r ( t )) dt + q

r ( t ) dW ( t ) :

Real time scale: In this time scale, the interest rater ^ (^ t )is given by a time-dependent CIR equation

d r ^ (^ t ) = (^ (^ t ) ^ (^ t )^ r (^ t )) d ^ t + ^  (^ t )q

^

r (^ t ) d W ^ (^ t ) :

-6

^

t : Real time

t = ' (^ t )

.

.

-

A pe-riod of high inter-est rate volatility

Figure 31.3: Time change function.

There is a strictly increasing time change functiont = ' (^ t )which relates the two time scales (See Fig 31.3)

LetB ^ (^ r; ^ t; T ^ )denote the price at real time^ tof a bond with maturityT ^when the interest rate at time

^

tisr ^ We want to set things up so

^

B (^ r; ^ t; T ^ ) = B ( r;t;T ) = e,rC(t;T),A(t;T) ;

wheret = ' (^ t ) ; T = ' ( ^ T ), andC ( t;T )andA ( t;T )are as defined previously

We need to determine the relationship betweenr ^andr We have

B ( r (0) ; 0 ;T ) = IE exp

(

,

0 r ( t ) dt

)

;

B (^ r (0) ; 0 ; T ^ ) = IE exp

(

,

0 ^ r (^ t ) d t ^

)

:

WithT = ' (^ T ), make the change of variablet = ' (^ t ),dt = '0(^ t ) d ^ tin the first integral to get

B ( r (0) ; 0 ;T ) = IE exp

(

,

0 r ( ' (^ t )) '0(^ t ) d t ^

)

;

and this will beB (^ r (0) ; 0 ; T ^ )if we set

^

r (^ t ) = r ( ' (^ t )) '0(^ t ) :

31.7 Calibration

^

B (^ r (^ t ) ; ^ t; T ^ ) = B ' ^ r (^0(^ t t ) ) ;' (^ t ) ;' (^ T )

!

= exp

(

'0(^ t ) ,A ( ' (^ t ) ;' (^ T ))

)

= expn

;

where

^

C (^ t; T ^ ) = C ( ' (^ t ) ;' (^ T ))

'0(^ t )

^

A (^ t; T ^ ) = A ( ' (^ t ) ;' (^ T ))

do not depend on^ tandT ^ only throughT ^,^ t, since, in the real time scale, the model coefficients are time dependent

Suppose we know^ r (0)andB ^ (^ r (0) ; 0 ; T ^ )for allT ^2[0 ; T ^] We calibrate by writing the equation

^

B (^ r (0) ; 0 ; T ^ ) = expn

,r ^ (0) ^ C (0 ; T ^ ),A ^ (0 ; T ^ )o

;

or equivalently,

,log ^ B (^ r (0) ; 0 ; T ^ ) = ^ r (0)

'0(0) C ( ' (0) ;' (^ T )) + A ( ' (0) ;' (^ T )) :

Take; and so the equilibrium distribution ofr ( t )seems reasonable These values determine the functionsC;A Take '0(0) = 1 (we justify this in the next section) For each T ^, solve the equation for' ( ^ T ):

,log ^ B (^ r (0) ; 0 ; T ^ ) = ^ r (0) C (0 ;' (^ T )) + A (0 ;' (^ T )) : (*) The right-hand side of this equation is increasing in the' (^ T ) variable, starting at 0 at time0and having limit1at1, i.e.,

^

r (0) C (0 ; 0)+ A (0 ; 0) = 0 ;

lim

T!1

[^ r (0) C (0 ;T )+ A (0 ;T )] =1:

Since0 ,log ^ B (^ r (0) ; 0 ; T ^ ) <1;(*) has a unique solution for eachT ^ ForT ^ = 0, this solution

is' (0) = 0 IfT ^ 1 < T ^ 2, then

,log ^ B ( r (0) ; 0 ; T ^ 1 ) <,log ^ B ( r (0) ; 0 ; T ^ 2 ) ;

so' (^ T 1 ) < ' (^ T 2 ) Thus'is a strictly increasing time-change-function with the right properties

31.8 Tracking down'

0

Result for general term structure models:

,

@

@T log B (0 ;T )

T=0 = r (0) :

Justification:

B (0 ;T ) = IE exp

(

,

0 r ( u ) du

)

:

(

,

0 r ( u ) du

)

,

@

@T log B (0 ;T ) = IE



r ( T ) e,

0 r(u) du

IEe,

0 r(u) du

,

@

@T log B (0 ;T )

T=0 = r (0) :

In the real time scale associated with the calibration of CIR by time change, we write the bond price as

^

B (^ r (0) ; 0 ; T ^ ) ;

thereby indicating explicitly the initial interest rate The above says that

,

@

@ T ^ log ^ B (^ r (0) ; 0 ; T ^ )

^T=0 = ^ r (0) :

The calibration of CIR by time change requires that we find a strictly increasing function'with

' (0) = 0such that

,log ^ B (^ r (0) ; 0 ; T ^ ) = 1 '0(0)^ r (0) C ( ' (^ T )) + A ( ' (^ T )) ; T ^0 ; (cal) whereB ^ (^ r (0) ; 0 ; T ^ ), determined by market data, is strictly increasing inT ^, starts at 1 whenT ^ = 0, and goes to zero asT ^! 1 Therefore,,log ^ B (^ r (0) ; 0 ; T ^ )is as shown in Fig 31.4

Consider the function

^

r (0) C ( T )+ A ( T ) ;

HereC ( T )andA ( T )are given by

C ( T ) = cosh( sinh( )+ 1 )

2 sinh( ) ;

A ( T ) = ,

2

 2 log

2

4

1

2T

cosh( )+ 1

2 sinh( )

3

= 1 2q

2 + 2  2 :

!

dt + q

r ( t ) dW ( t ) :

The Cox-Ingersoll-Ross (CIR) process is given by

dr ( t ) = ( ,r... :

Surprisingly, this equation has a closed form solution Using the Hull & White model as a guide,

we look for a solution of the form

B ( r;t;T ) = e,rC(t;T),A(t;T)