In this paper we comment on a recent paper by Bj¨ork and Gombani. In contrast to this paper our starting point is not the Musiela equation but the forward rate dynamics. In our approach we do not need to talk about infinitesimal generators.

Nieuwenhuis, J.W.

English

Research Papers in Economics

Realizations of interest rate models
J.W. Nieuwenhuis
Department of Econometrics
University of Groningen
the Netherlands
January 23, 2001
Abstract
In this paper we comment on a recent paper by Bj¨ ork and Gombani [1]. In contrast to
this paper our starting point is not the Musiela equation but the forward rate dynamics.
In our approach we do not need to talk about innitesimal generators.
As in [1] we consider a bond market living on a stochastic basis (Ω;F;(Ft)t0;Q) carrying
a standard d-dimensional Wiener process W. We assume that the ltration is the usual one
generated by W.
Furthermore our setting is that of Bj¨ ork's paper [2] and Q is assumed to be a martingale
measure of
df (t;T)= ( t;T)dt + (t;T)dW(t);
where f describes the forward rate dynamics.
We have, see [2]
8 T>0 8 t2[0;T]: ( t;T)= ( t;T)
Z T
0
(t;s)ds:
Instead of starting with Musiela's parametrization of
r(t;x): =f ( t;t + x) ;t0 ;x0 ;
we will \derive" it, where the \ " marks mean that we assume that everything is such that
we may freely apply Fubini and the stochastic version of Fubini.
For this latter theorem we refer to Theorem 44, page 158 in Protter's book [3].
This also means that we do not bother about measurability questions. So in that sense we
only consider structural issues. It is far from trivial to ll in the missing details. For readers
1willing to do that we refer to D¨ oberlein's thesis [4]. We also will add now and then additional
assumptions along the way when considered useful.
As said before we will rst of all derive the Musiela parametrization. It really takes some
eort and certainly we need more lines than for instance the one-line proof in Bj¨ ork [2]. We
take these eorts because we were not able to understand this very short proof, and we have
even doubts whether this one-line proof is correct.
Now the promised derivation.
Step 1:
r(t): =f( t;t)=f(0;t)+
Z t
0
 ( s;t)ds +
Z t
0
(s;t)dW(s):
We assume from now on:
80  s  t :
(s;t)= ( s:s)+
Z t
s
D 2 ( s;u)du
(s;t)= ( s;s)+
Z t
s
D 2( s;u)du;
where \D2" means dierentiation after the second argument. And furthermore these assump-
tions hold for every ! 2 Ω. So, for instance, we should read:
8! 2 Ω: ( s;t)(!)= ( s;s)(!)+
Z t
s
D 2 ( s;u)(!)du:
In the sequel, however, we omit the dependence on !.
Now, using these assumptions and applying \Fubini" and \stochastic Fubini" we get:
r(t)=f (0;t)+
Z t
0
 ( s;s)ds +
Z t
0
(s;s)dW(s)
+
Z t
0
Z u
0
D2(s;u)dsdu +
Z t
0
Z u
0
D2(s;u)dW(s)du:
Now recall
f(t;T)=f (0;T)+
Z t
0
 ( s;T)ds +
Z t
0
(s;T)dW(s):
Hence:
2D2f(u;u) − D2f(0;u)=
Z u
0
D 2  ( s;u)ds +
Z u
0
D2(s;u)dW(s):
Therefore we nd:
r(t)=r (0) +
Z t
0
((s;s)+D 2f( s;s))ds +
Z t
0
(s;s)dW(s): ()
Step 2: We take an arbitrary x  0 and consider now
r(t;x): =f ( t;t + x):
We dene
e f(t;T): =f ( t;T + x)
e (s;T): = ( s;T + x)
e (s;T): = ( s;T + x)
Now notice that r(t;x)= e f( t;t): =e r ( t ).
We now use ()t o n dt h ee q u a t i o nf o re r ( t ), and we nd:
r(t;x)=r (0;x)+
Z t
0
(  ( s;s + x)+D 2r( s;x))ds
+
Z t
0
(s;s + x)dW(s); ()
We recall that (t;T)= ( t;T)
R T
0 (t;u)du,0tT , and we dene 0(t;x): = ( t;t + x).
We further dene D(s;x): = ( s;s + x) and we nd:
r(t;x)=r (0;x)+
Z t
0
( D( s;x)+D 2r( s;x))ds +
Z t
0
0(s;x)dW(s)

!

Equation

!

is the parametrization of Musiela.
Given this set-up it is now easy to give a solution of equation

!

.
Recall r(t;x)=f( t;t + x). We know
f(t;T)=f (0;T)+
Z s
0
 ( s;T)ds +
Z s
0
(s;T)dW(s)
3and after some easy calculations we nd:
r(t;x)=r (0;t+x)+
Z t
0
D( s;t − s + x)ds +
Z t
0
0(s;t − s + x)dW(s)

!

So given x −! r (0;x)w ek n o w

 !

,asolution to Musiela's parametrization.
It is now not dicult to show that given an initial curve x −! r(0;x), x  0, Musiela's
parametrization has a unique solution. To that end we consider:
r(t;x)=
Z t
0
D 2 r ( s;x)ds ; 8t  0; 8x  0;
and ask ourselves whether this equation has solutions and if so what these solutions look like.
It is clear that r(0;x)=08x0.
We also know that 8(t;x) 2 IR 2
+ we have:
5r(t;x)
 
1
−1
!
=0 :
Now write  :=
  1
−1

and consider (): =r ((t;x)+). For  2 IR s u ch t h a t ( t;x)+ 2 IR 2
+
we have that () is well-dened and then we have 0()=0 ,s o :
8( t;x) 2 IR
2
+ 9 c 2 IR s u ch t h a t
r ((t;x)+)=c whenever (t;x)+ 2 IR
2
+:
But we know that r(0;x)=0 ,8 x0, hence we see that 8(t;x) 2 IR 2
+ : r ( t;x) = 0, and
this implies that

!

gives the unique solution to Musiela's parametrization given the initial
curve x −! r (0;x), x  0.
From now on we assume that 0(t;x) does not depend on ! 2 Ωa n yl o n g e r .
For xed x  0 we now consider the mapping:
IR + 3 t −!
Z t
0
 0(s;t − s + x)dW(s):
Denition: We say that this mapping allows for an (A;B;C(x)) realization whenever A, B
and C(x) are nite-dimensional real-valued matrices such that A and B are constant 8x  0
and:
4Z t
0
0(s;t − s + x)dW(s)=C ( x ) X ( t )
where X() is the unique strong solution of
dX(t)=AX(t)+BdW(t);X (0) = 0; 8t  0( S )
For instance from Karatzas and Shreve [5] we know that the solution to (S) for given A and
B is given by X(t)=
Rt
0e A ( t − s )B ( s ) dW(s).
We hence nd the following result.
Theorem: The mapping t !
R t
0 0(s;t−s+x)dW(s) allows for an (A;B;C(x)) realization
i
0(s;t − s + x)=C ( x ) e
A ( t − s ) B;8ts0 ;8x0 :
Now recall 0(t;x): = ( t;t + x).
Hence
0(s;t − s + x)=C ( x ) e
A ( t − s )B ,  ( s;t + x)=C( x ) e
A ( t − s )B:
Assume now in addition that (s;y)=e  ( y ), 8s  0, then
e (t)=C (0)e
AtB = C(t)B; hence
e C(t): =C (0)e
At
together with e (t)=C (0)eAtB gives an (A;B;C(x)) realization, and we nd back results de-
rived by Bj¨ ork and Gombani in [1] without the use of systems theoretical results. Therefore
our paper can be seen as a shortcut to some of the results in [1].
Except for the use of \Fubini" and \stochastic Fubini" our paper can be considered as a
rather elementary treatment of Musiela's parametrization and its connection with simple re-
alizations of r(t;t + x).
5References
[1.] Bj¨ ork, T., Gombani., Minimal realizations of interest rate models. Finance and Stochas-
tics 3 (99), pp. 413-432.
[2.] Bj¨ ork, T., Interest Rate Theory, in: Runggaldier, W., (ed.), Financial Mathematics,
Springer Lecture Notes in Mathematics, Vol. 1656, Springer Verlag, 1997.
[3.] Protter, P., Stochastic Integration and Dierential Equations, A new Approach, springer-
Verlag, 1990.
[4.] D¨ oberlein, F., On term structure models generated by semimartingales, Thesis, Tech-
nische Universit¨ at Berlin, Berlin, 1999.
6

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