A two-factor model (Duffie & Kan)

Since the coefficients in SDE1 and SDE2 do not depend on time, the bond price depends ontandT only through their difference = T,t.. We compute its stochastic differential and set thedtte

A two-factor model (Duffie & Kan)

Let us define:

X 1 ( t ) =Interest rate at timet

X 2 ( t ) =Yield at timeton a bond maturing at timet +  0

LetX 1 (0) > 0,X 2 (0) > 0be given, and letX 1 ( t ) andX 2 ( t )be given by the coupled stochastic

differential equations

dX 1 ( t ) = ( a 11 X 1 ( t ) + a 12 X 2 ( t ) + b 1 ) dt +  1

q

1 X 1 ( t ) + 2 X 2 ( t ) + dW 1 ( t ) ; (SDE1)

dX 2 ( t ) = ( a 21 X 1 ( t ) + a 22 X 2 ( t ) + b 2 ) dt +  2

q

1 X 1 ( t ) + 2 X 2 ( t ) + (  dW 1 ( t ) +q

1, 2 dW 2 ( t )) ;

(SDE2) whereW 1 andW 2are independent Brownian motions To simplify notation, we define

Y ( t ) 4

= 1 X 1 ( t ) + 2 X 2 ( t ) + ;

W 3 ( t ) 4

= W 1 ( t ) +q

1, 2 W 2 ( t ) :

ThenW 3is a Brownian motion with

dW 1 ( t ) dW 3 ( t ) =  dt;

and

dX 1 dX 1 =  21 Y dt; dX 2 dX 2 =  22 Y dt; dX 1 dX 2 =  1  2 Y dt:

319

32.1 Non-negativity of Y

dY = 1 dX 1 + 2 dX 2

= ( 1 a 11 X 1 + 1 a 12 X 2 + 1 b 1 ) dt + ( 2 a 21 X 1 + 2 a 22 X 2 + 2 b 2 ) dt

+p

Y ( 1  1 dW 1 + 2  2 dW 1 + 2

q

1, 2  2 dW 2 )

= [( 1 a 11 + 2 a 21 ) X 1 + ( 1 a 12 + 2 a 22 ) X 2 ] dt + ( 1 b 1 + 2 b 2 ) dt

+ ( 21  21 + 2 1 2  1  2 + 22  22 ) 1 2q

Y ( t ) dW 4 ( t ) where

W 4 ( t ) = ( 1  1 + 2  2 ) W 1 ( t ) + 2p

1, 2  2 W 2 ( t )

q

21  21 + 2 1 2  1  2 + 22  22

is a Brownian motion We shall choose the parameters so that:

Assumption 1: For some , 1 a 11 + 2 a 21 = 1 ; 1 a 12 + 2 a 22 = 2 :

Then

dY = [ 1 X 1 + 2 X 2 + ] dt + ( 1 b 1 + 2 b 2, ) dt

+ ( 21  21 + 2 1 2  1  2 + 22  22 ) 1 2p

Y dW 4

= + ( 1 b 1 + 2 b 2, ) dt + ( 21  21 + 2 1 2  1  2 + 22  22 ) 1 2p

Y dW 4 :

From our discussion of the CIR process, we recall thatY will stay strictly positive provided that:

Assumption 2: Y (0) = 1 X 1 (0) + 2 X 2 (0) + > 0 ;

and

Assumption 3: 1 b 1 + 2 b 2, 

1

2 ( 21  21 + 2 1 2  1  2 + 22  22 ) :

Under Assumptions 1,2, and 3,

Y ( t ) > 0 ; 0t <1; almost surely, and (SDE1) and (SDE2) make sense These can be rewritten as

dX 1 ( t ) = ( a 11 X 1 ( t ) + a 12 X 2 ( t ) + b 1 ) dt +  1

q

Y ( t ) dW 1 ( t ) ; (SDE1’)

dX 2 ( t ) = ( a 21 X 1 ( t ) + a 22 X 2 ( t ) + b 2 ) dt +  2

q

Y ( t ) dW 3 ( t ) : (SDE2’)

32.2 Zero-coupon bond prices

The value at timetT of a zero-coupon bond paying $1 at timeT is

B ( t;T ) = IE

"

exp

(

,

Z T

t X 1 ( u ) du

)

F( t )

# :

Since the pair ( X 1 ;X 2 ) of processes is Markov, this is random only through a dependence on

X 1 ( t ) ;X 2 ( t ) Since the coefficients in (SDE1) and (SDE2) do not depend on time, the bond price depends ontandT only through their difference = T,t Thus, there is a functionB ( x 1 ;x 2 ; )

of the dummy variablesx 1 ;x 2and, so that

B ( X 1 ( t ) ;X 2 ( t ) ;T,t ) = IE

"

exp

(

,

Z T

t X 1 ( u ) du

)

F( t )

# :

The usual tower property argument shows that

exp ,

Z t

0 X 1 ( u ) du

B ( X 1 ( t ) ;X 2 ( t ) ;T ,t )

is a martingale We compute its stochastic differential and set thedtterm equal to zero

d

exp

,

Z t

0 X 1 ( u ) du

B ( X 1 ( t ) ;X 2 ( t ) ;T ,t )

= exp

,

Z t

0 X 1 ( u ) du 

,X 1 B dt + B x1 dX 1 + B x2 dX 2,B  dt

+ 1 2 B x1x1 dX 1 dX 1 + B x1x2 dX 1 dX 2 + 1 2 B x2x2 dX 2 dX 2



= exp

,

Z t

0 X 1 ( u ) du 

,X 1 B + ( a 11 X 1 + a 12 X 2 + b 1 ) B x1+ ( a 21 X 1 + a 22 X 2 + b 2 ) B x2

,B 

+ 1 2  21 Y B x1x1 +  1  2 Y B x1x2 + 1 2  22 Y B x2x2

 dt

+  1

p

Y B x1 dW 1 +  2

p

Y B x2 dW 3



The partial differential equation forB ( x 1 ;x 2 ; )is

,x 1 B,B  +( a 11 x 1 + a 12 x 2 + b 1 ) B x1+( a 21 x 1 + a 22 x 2 + b 2 ) B x2+ 1 2  21 ( 1 x 1 + 2 x 2 + ) B x1x1

+  1  2 ( 1 x 1 + 2 x 2 + ) B x1x2 + 1 2  22 ( 1 x 1 + 2 x 2 + ) B x2x2 = 0 : (PDE)

We seek a solution of the form

B ( x 1 ;x 2 ; ) = expf,x 1 C 1 (  ),x 2 C 2 (  ),A (  )g;

valid for all 0and allx 1 ;x 2satisfying

We must have

B ( x 1 ;x 2 ; 0) = 1 ; 8x 1 ;x 2 satisfying (*);

because = 0corresponds tot = T This implies the initial conditions

C 1 (0) = C 2 (0) = A (0) = 0 : (IC)

We want to findC 1 (  ) ;C 2 (  ) ;A (  )for > 0 We have

B  ( x 1 ;x 2 ; ) =

,x 1 C0

1 (  ),x 2 C0

2 (  ),A0

(  )

B ( x 1 ;x 2 ; ) ;

B x1( x 1 ;x 2 ; ) =,C 1 (  ) B ( x 1 ;x 2 ; ) ;

B x2( x 1 ;x 2 ; ) =,C 2 (  ) B ( x 1 ;x 2 ; ) ;

B x1x1( x 1 ;x 2 ; ) = C 21 (  ) B ( x 1 ;x 2 ; ) ;

B x1x2( x 1 ;x 2 ; ) = C 1 (  ) C 2 (  ) B ( x 1 ;x 2 ; ) ;

B x2x2( x 1 ;x 2 ; ) = C 22 (  ) B ( x 1 ;x 2 ; ) :

(PDE) becomes

0 = B ( x 1 ;x 2 ; )

,x 1 + x 1 C0

1 (  ) + x 2 C0

2 (  ) + A0(  ),( a 11 x 1 + a 12 x 2 + b 1 ) C 1 (  )

,( a 21 x 1 + a 22 x 2 + b 2 ) C 2 (  ) + 1 2  21 ( 1 x 1 + 2 x 2 + ) C 21 (  ) +  1  2 ( 1 x 1 + 2 x 2 + ) C 1 (  ) C 2 (  ) + 1 2  22 ( 1 x 1 + 2 x 2 + ) C 22 (  )

= x 1 B ( x 1 ;x 2 ; )

,1 + C0

1 (  ),a 11 C 1 (  ),a 21 C 2 (  ) + 1 2  21 1 C 21 (  ) +  1  2 1 C 1 (  ) C 2 (  ) + 1 2  22 1 C 22 (  )

+ x 2 B ( x 1 ;x 2 ; )



C0

2 (  ),a 12 C 1 (  ),a 22 C 2 (  ) + 1 2  21 2 C 21 (  ) +  1  2 2 C 1 (  ) C 2 (  ) + 1 2  22 2 C 22 (  )

+ B ( x 1 ;x 2 ; )

A0(  ),b 1 C 1 (  ),b 2 C 2 (  ) + 1 2  21 C 21 (  ) +  1  2 C 1 (  ) C 2 (  ) + 1 2  22 C 22 (  )

We get three equations:

C0

1 (  ) = 1 + a 11 C 1 (  ) + a 21 C 2 (  ),1 2  21 1 C 21 (  ), 1  2 1 C 1 (  ) C 2 (  ),1 2  22 1 C 22 (  ) ;

(1)

C 1 (0) = 0;

C0

2 (  ) = a 12 C 1 (  ) + a 22 C 2 (  ),1 2  21 2 C 21 (  ), 1  2 2 C 1 (  ) C 2 (  ),1 2  22 2 C 22 (  ) ; (2)

C 2 (0) = 0;

A0

(  ) = b 1 C 1 (  ) + b 2 C 2 (  ),1 2  21 C 21 (  ), 1  2 C 1 (  ) C 2 (  ),1 2  22 C 22 (  ) ; (3)

A (0) = 0;

We first solve (1) and (2) simultaneously numerically, and then integrate (3) to obtain the function

A (  )

32.3 Calibration

Let 0 > 0be given The value at timetof a bond maturing at timet +  0is

B ( X 1 ( t ) ;X 2 ( t ) ; 0 ) = expf,X 1 ( t ) C 1 (  0 ),X 2 ( t ) C 2 (  0 ),A (  0 )g

and the yield is

,

1

 0 log B ( X 1 ( t ) ;X 2 ( t ) ; 0 ) = 1  0 [ X 1 ( t ) C 1 (  0 ) + X 2 ( t ) C 2 (  0 ) + A (  0 )] :

But we have set up the model so thatX 2 ( t )is the yield at timetof a bond maturing at timet +  0 Thus

X 2 ( t ) = 1  0 [ X 1 ( t ) C 1 (  0 ) + X 2 ( t ) C 2 (  0 ) + A (  0 )] :

This equation must hold for every value ofX 1 ( t )andX 2 ( t ), which implies that

C 1 (  0 ) = 0 ; C 2 (  0 ) =  0 ; A (  ) = 0 :

We must choose the parameters

a 11 ;a 12 ;b 1 ; a 21 ;a 22 ;b 2 ; 1 ; 2 ; ;  1 ;; 2 ;

so that these three equations are satisfied

and

dX dX =  21 Y dt; dX dX =  22 Y dt; dX dX =   Y dt:

319

32.1...

q

21  21 + 2   + 22  22

is a Brownian motion We shall choose the parameters so that:

Assumption 1: For some , a 11 + a 21 = ; a 12 + a 22... data-page="2">

32.1 Non-negativity of Y

dY = dX + dX 2

= ( a 11 X + a 12 X + b ) dt + ( a 21 X + a 22 X + b ) dt

+p

Y