exponential-polynomial families and the term structure of interest rates

International Statistical Institute (ISI) and Bernoulli Society for MathematicalStatistics and Probability

Exponential-Polynomial Families and the Term Structure of Interest RatesAuthor(s): Damir FilipovićSource: Bernoulli, Vol. 6, No. 6 (Dec., 2000), pp. 1081-1107Published by: International Statistical Institute (ISI) and Bernoulli Society for Mathematical Statisticsand ProbabilityStable URL: http://www.jstor.org/stable/3318472 .

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Bernoulli 6(6), 2000, 1081-1107

Exponential-polynomial families and the

term structure of interest rates

DAMIR FILIPOVIC

Department of Mathematics, ETH, Rdmistrasse 101, CH-8092 Ziirich, Switzerland. E-mail: [email protected]

Exponential-polynomial families like the Nelson-Siegel or Svenson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with intertemporal modelling. We characterize the consistent It6 processes which have the property to provide an arbitrage-free interest rate model when representing the parameters of some bounded exponential- polynomial type function. This includes diffusion processes in particular. We show that there is a strong limitation on their choice. Bounded exponential-polynomial families are best not used for modelling the term structure of interest rates.

Keywords: consistent It6 process; diffusion process; exponential-polynomial family; forward rate curve; interest rate model; inverse problem

1. Introduction

The current term structure of interest rates contains all the necessary information for pricing bonds, swaps and forward rate agreements of all maturities. It is used by the central banks as an indicator for their monetary policy.

There are several algorithms for constructing the current forward rate curve from the (finitely many) prices of bonds and swaps observed in the market. Widely used are splines and parametrized families of smooth curves {F(-, z)}z,Z, where ? C RN, N > 1, denotes some finite-dimensional parameter set. By an optimal choice of the parameter z in Z, an optimal fit of the forward curve x ? F(x, z) to the observerd data is attained. Here x > 0 denotes time to maturity. In that sense z represents the current state of the economy taking values in the state space Z1.

Examples are the Nelson and Siegel (1987) family with curve shape

FNs(X, z) = zi + (z2 + Z3X) e-z4x

and the Svenson (1994) family, an extension of Nelson-Siegel,

Fs(x, z) = z1 + (z2 + z3x) e-zsx Z4x e-z6x.

Table 1 gives an overview of the fitting procedures used by some selected central banks. It is taken from the documentation of the Bank for International Settlements (BIS 1999).

Despite the flexibility and low number of parameters of FNs and Fs, their choice is somewhat arbitrary. We shall discuss them from an intertemporal point of view: plenty of

1350-7265 ? 2000 ISI/BS



1082 D. Filipovik

Table 1. Forward rate curve fitting procedures

Central bank Curve fitting procedure

Belgium Nelson-Siegel, Svensson Canada Svensson Finland Nelson-Siegel France Nelson-Siegel, Svensson Germany Svensson Italy Nelson-Siegel Japan Smoothing splines Norway Svensson Spain Nelson-Siegel (before 1995), Svensson Sweden Svensson UK Svensson USA Smoothing splines

cross-sectional data - daily estimations of z - are available. Therefore it would be natural to want to find the stochastic evolution of the parameter z over time. But then there exist economic constraints based on no-arbitrage considerations.

Following Bj6rk and Christensen (1999), instead of FNS and Fs we consider general exponential-polynomial families containing curves of the form

K ni

F(x, z)= Zi,xt e-Z,ni+IX

i= 1 0

that is, linear combinations of exponential functions exp(-zi,ni+x) over some polynomials of degree ni E No. Obviously FNS and Fs are of this type. We then replace z by an It6 process Z =

(Zt)tto taking values in Z. The following questions arise:

* Does F(., Z) provide an arbitrage-free interest rate model? * And what are the conditions on Z for it?

Working in the Heath, Jarrow and Morton (1992) - henceforth HJM - framework with deterministic volatility structure, Bj6rk and Christensen (1999) showed that the exponential- polynomial families are in a certain sense too large to carry an interest rate model. This result has been generalized for the Nelson-Siegel family in Filipovid (1999b), including stochastic volatility structure. Expanding the methods used there, we give in this paper the general result for bounded exponential-polynomial families.

The paper is organized as follows. In Section 2 we introduce the class of It6 processes consistent with a given parametrized family of forward rate curves. Consistent It6 processes provide an arbitrage-free interest rate model when driving the parametrized family. They are characterized in terms of their drift and diffusion coefficients by the HJM drift condition.

By solving an inverse problem we obtain the main result for consistent It6 processes,



Exponential-polynomial families and interest rates 1083

stated in Section 3. It is shown that they are remarkably limited. The proof is divided into several steps, given in Sections 4, 5 and 6.

In Section 7 we extend the notion of consistency to e-consistency when P is not a martingale measure.

The main result is much clearer when restricting to diffusion processes, as shown in Section 8. It turns out that e-consistent diffusion processes driving bounded exponential- polynomial families like Nelson-Siegel or Svensson are very limited: most of the factors are either constant or deterministic. It is shown in Section 9 that there is no non-trivial diffusion process which is e-consistent with the Nelson-Siegel family. Furthermore, we identify the diffusion process which is e-consistent with the Svensson family. It contains just one non-deterministic component. The corresponding short rate model is shown to be the generalized Vasicek model.

We conclude that bounded exponential-polynomial families, in particular FNs and Fs, are best not used for modelling the term structure of interest rates.

2. Consistent Ito processes

For the stochastic background and notation, we refer to Revuz and Yor (1994) and Jacod and Shiryaev (1987). Let (AJ,

, (5t)o0<t<o, IP) be a filtered complete probability space,

satisfying the usual conditions, and let W = (WI, ...,d)Ot< denote a standard d- dimensional (ft)-Brownian motion, d > 1.

Let Z = (Z',..., ZN) denote an RFN-valued It6 process, N > 1, of the form

t d t

Z = Z + bi o rijd

Wj i =1,. . . ,N, 0 < t<0o,

where Zo is 70-measurable, and b and a are progressively measurable processes with values in RWN and RNxd respectively, such that

J(Ibs +

•as12)ds

<o, P-almost surely, for all finite t.

Let F(x, z) be a function in C1,2N(R+ X R N); that is to say, F and the partial derivatives

OF/Ox, OF/Ozi, O2F/OziOzj, which exist for 1 < i,]j < N, are continuous functions on R+ x RfN. Interpreting Zt as the state of the economy at time t, we let x F F(x, Zt) stand for the corresponding term structure of interest rates, meaning that F(x, Zt) denotes the instantaneous forward rate at time t for date t + x.

Notice that

G(x, z)= exp (-J F(r, z)dr)

is in C1,2(l+ X fN) too. Therefore the price processes for zero coupon T-bonds

P(t, T) := G(T - t, Zt), O < t < T < 00, (1)



1084 D. Filipovic

and the process of the savings account

B(t):= exp - G(0, Z)ds , O * t < 00,

form continuous semimartingales. Let Z denote an arbitrary subset of RN. The function F generates in a canonical way a

parametrized set of forward curves {F(., Z)}zI,. We shall refer to Z as the state space of the economy.

Defintion 2.1. Z is called consistent with {F(-, z)}}zE, if the support of Z is contained in Z and

P(t, T) (2) B(t) Jo-~-tT

is a P-martingale, for all T < oo.

Set a = a*, where a* denotes the transpose of a, that is, a'J = at iku

,tk, for

1 < i, j < N and 0 < t < oo. Then a is a progressively measurable process with values in the symmetric non-negative definite N X N matrices.

Using It's formula, the dynamics of (2) can be decomposed into finite variation and local martingale parts. Since consistency is required, the former has to vanish. This is the well-known HJM drift condition and is stated explicitly in the following proposition.

Proposition 2.2. If Z is consistent with {F(., z) }z)E then

o N N x F(x, Z) =-bFZ

F(x, Z)

Nx1i=J1y-0iF xZ

+

a"zJ -----F(x, Z) F(x, Z) - F(r, Z)d , (3)

for all x O0, dt 0 dP-a.s.

Proof Analogous to the proof of Filipovid (1999b, Proposition 3.2). LO

3. Exponential-polynomial families

In this section we introduce a particular class of functions F Our main result characterizes the corresponding consistent Ito processes.

Let K denote a positive integer and let n = (ni, ..., nK) be a vector with components ni E Ro, for 1 < i < K. Write Inl := nl +...+ nK. For a point

z = (Z1,0 ,

-..,

Zl,nl+l, Z2,0, ? ?,

Z2,n2+1,

-..,

ZK,O, ..., ZK,nK+1) E gInj+2K, (4)




define the polynomials pi(z) as

ni

pi(z) = pi(x, z) := zi,UxU

1 i < K. P=0

The function F is now defined as

K

F(x, z) := pi(x, Z) e-zin.i+lx. (5) i=1

Obviously F E C1,'2(( R+ XlIn+2K). Hence the previous section applies with N = Inl + 2K. From an economic point of view, it seems reasonable to restrict to bounded forward rate

curves. Let, therefore, Z denote the set of all z E RN such that supxeR+lF(x, z)j < 00.

Defintion 3.1. The exponential-polynomial family EP(K, n) is defined as the set of forward curves {F(., Z)}zERN .

The bounded exponential-polynomial family BEP(K, n) C EP(K, n) is defined as the set of forward curves {F(-, z)}}zE~.

Clearly FNs(x, z) E BEP(2, (0, 1)) and Fs(x, z) E BEP(3, (0, 1, 1)), if in each case the parameter z is chosen such that the curve is bounded. From now on, the Nelson-Siegel and Svensson families are considered as subsets of BEP(2, (0, 1)) and BEP(3, (0, 1, 1)), respectively.

If two exponents Zi,ni+l and zj,nj+l coincide, the sum (5) defining F reduces to a linear combination of K - 1 exponential functions. Thus for z E Ru we introduce the equivalence relation

i ,,z j: < Zi, n+l = Zj, nj+l (6)

on the set { 1, ..., K} and denote by [i] = [i]z the equivalence class of i. We will use the notation

n[i] = n[i](z):= max{njlj E [i]z},

[iy], =

9•[i],/(Z) := {j E [i]zlnj > p}, 0O Y :

n[i](z),

z[i], =

z[i],U(z) := zJu, 0 *

t n[il(z),

jE9[jl,#,(z)

p[i](z) := pj(z). (7) jE[i]z

In particular p[i](z) = ZoZ[i],,MX , and (5) now reads

F(x, z)= p[i](z) e-zi',i+' x

[i]E{1 ....

K}/~

Observe that for z e we have



1086 D. Filipovid

Zinil= 0 only if p[ij](z)= z[i],o,

(8)

zif,+l < 0 only if p[ij](z) = 0.

We shall write the RN-valued It6 process Z with the same indices as we use for a point z E RN (see (4)),

Z" o' •+ bl, ds + o dWs, ni 1, 1 < i < K. (9)

Its diffusion matrix a consists of the components

d

aio,u;J,v -

oa'ii;A , Ov;, 0 < < ni + 1, O < v < n0 + 1, 1 < i, j < K. A2=1

Notice that, for 1 < i < K,

{zI[il(z) = 0} = U ({zlz,,,nj+l =

zi,i+1 for all j E J}

Jc{ 1..., K} Jai

n z zE

j=U=

0 =U fZiZn,+l

=

Zi,ni+l 1 (10) maU•=0 jEJ IEJc

is not closed in general but nevertheless a Borel set in R1N. We introduce the optional random sets of singular points (t, w)

Ai := {pi(Z) = 0 or p1i](Z) = 0}, 1 < i < K,

K

S:= U {zini+ = zijnj+ l, i,j=1 ifj

K

J:= U {2Zin'+l = ZJnj+l), i,j=l ifj

and the optional random sets of regular points (t, w)

' := (R, x Q)\(3 U ).

Let us recall that for stopping times S and T, a stochastic interval like [S, T] is a subset




of R+ x Q. Hence [S] = [S, S] is the restriction of the graph of the mapping S : Q -4 [0, oo] to the set R+ X Q.

For any stopping time r with [r] E (R+ X Q)\Aj, we define

'(wo) := inf{It > r(wo)l(t, to) E j i},

the debut of the optional set [r, oo[nAi. Observe that in general it is not true that r' > r on {r < oo}. This can be seen from the following example. For

F(x, z) = zl,o e-z,Ix + z2,0 e-z2,lx + Z3,0 e-z3,lx E BEP(3, (0, 0, 0)),

let Zl,o = Z3,0 1, Z,0 = -1, Z3,1 =l+t and Z, '=Z2,1 =-1 for tE[0,1]. Then pl(Zo) = p[l](Zo) = 1 and p[l](Zt) = 0 for all t E (0, 1]. Hence [0] E (R+ X Q)\A1, but ' = 0. However, by continuity of Z we always have

r < r' P-a.s. on {wlj(r(w), w) E _'} . (11)

Recall the fact that there is a one-to-one correspondence between the It6 processes Z starting in Zo (up to indistinguishability) and the equivalence classes of b and cr with respect to the dt 0 dP-nullsets in A+ 0 ~. Hence we may state the following inverse problem to equation (3). Given a family of forward curves, for which choices of coefficients b and ar do we obtain a consistent It6 process Z starting at Zo?

The main result is the following characterization of all consistent It6 processes, which is remarkably restrictive. The proof of the theorem will be given in Sections 5 and 6.

Theorem 3.2. Let K E N, n = (ni, ..., nK) E NK and Z be as above. If Z is consistent with BEP(K, n), then necessarily, for 1 < i < K,

a i,ni+l;i,ni+l = 0, on {pi(Z) , 0}, dt 0 dP-a.s. (12)

bi,n'+l - O, on {pi(Z) 0} 1"

{pn [i(Z) , 0}, dt 0 dP-a.s. (13)

Consequently, Zi,n+l is constant on intervals where pi(Z) , 0 and p[ij(Z) , 0. That is, for P-almost every w,

Z,'ni+l1(t) = Z~ni+l((o), for t E [u, V],

if pi(Zt(o)) = 0 and p[i](Zt(w)) # 0 for t E (u, v). For a stopping time r with [r] C -T', let r'(w):= inf{t > r(w)j(t, w)

_~'} denote the

debut of the optional random set (v U) U) fn [r, oo[. Then we have that r < r' on {r < oo} and

Z ~ •t

= Zi"U e--Z

't + Zi,+lt e-zi t

zi,ni Zi,ni e-zini+l t

on [0, z' - r[, for 0 <• p < ni - 1 and 1 < i < K, up to evanescence.



1088 D. Filipovic

If T' above is replaced by _ and r' is the debut of (U A U U w ) I

[ r, oc[, then r' = oo and in addition

Zi,ni+1 = Zi,ni+l1 r+t r

for 1 < i < K, P-a.s. on {r < oo}.

Remark 3.3. It will be made clear in the proof of the theorem that it is actually sufficient to assume Z to be consistent with EP(K, n) for (12) to hold.

As an immediate consequence we may state the following corollaries. The notation is the same as in the theorem.

Corollary 3.4. If Z is consistent with BEP(K, n) and if the optional random sets {pi(Z) = O} and {pij](Z) = O} have dt 0 dP-measure zero, then the exponent Zi,'ni+l is indistinguishable from Zoni+', 1 < i < K.

Proof If {pi(Z) = O} and {p[ji(Z) = O} have dt 0 dP-measure zero, then {pi(Z) $ 0} n {p[ij](Z) $ 0} = R+ X Q up to a dt 0 dP-nullset. The claim follows using (12) and

(13). O

Corollary 3.5. If Z is consistent with BEP(K,n) and if the following three points are P-a.s.

satisfied.

(i) pi(Zo) $ 0, for all 1 i < K, (ii) there exists no pair of indices i ] j with 7ini+l =

ZJn+l (iii) there exists no pair of indices i j with 2Zo',n"+ =

Z0on;+

then Z and hence the interest rate model F(x, Z) is quasi-deterministic, that is, all randomness remains 7o-measurable. In particular, the exponents Zi,ni+l are indistinguishable from Zoin+1, for 1 < i < K.

Proof If (i), (ii) and (iii) hold P-a.s. then [0] C -T. The claim follows from the second part of the theorem, setting r = 0. O

4. Auxiliary results

For the proof of the main result we need three auxiliary lemmas, presented in this section. First, there is a result on the identification of the coefficients of It6 processes.




Lemma 4.1. Let

t d t

Xt= Xo + J ds + y dWI,

j=1

,' J = YO+ O

S j= o

.? 0

be two It6 processes. Then dt 0 dP-a.s.

d d d

1x=y Z}IX=XY}j)2 1 Y X, -

Yj {IX= Y}} Z Y,j2

j=1 j= j=1

I{X= Y}X = l{x= Y} .

Proof We write (., .) for the scalar product in Rd. Then

(yX, x _ (X XY)1 x ( x _ - x YX) (x - x _ By the occupation times formula (see Revuz and Yor 1994, Corollary (1.6), Chapter VI),

l{x,=,}(y -y, y -y)dsO, for all t<oo, P-a.s.

Hence, by the H61der inequality,

l{x,=,}|(y,x y -) - (y, yr)I ds

< 1{x~_} (y/<,

y ) (y•

- y~', y~ - y~) ds

< J 1({x,=,}(,, y,)

ds 1({x,=-,}(

Y . -

y,, ,x - y ) ds

= 0, for all t < w0, P-a.s.

Thus, by symmetry,

I{x=ry} (y, yx) = 1{x=y} (yx, YY) =

l{x=} (yr, yr), dt 9 dP-a.s.

By continuity of the processes X and Y. there are sequences of stopping times (Sn) and (Tn), Sn < Tn, with [Sm, Tm] n [Sn, TJ] = 0 for all m

- n and

{X = Y} = U [Sn, Tn], up to evanescence. nEN

To see this, let n e N and let S(n, 1) := inf{ t > 0 |Xt - YJ| = 0}. Define T(n, p) := inf{ t > S(n, p)I IXt - Yr| > 0} and inductively



1090 D. Filipovi6

S(n, p + 1):= inf{t > S(n, p)I IXt - Yt| = 0 and sUPS(n,p)<stIXs - YsI > 2-n}?

Then by continuity we have limp-,,,S(n, p) = oc for all n E N and it follows that {X = Y} = Un,pEN[S(n, p), T(n, p)]. Now proceed as in Jacod and Shiryaev (1987, Lemma 1.1.31) to find the sequences (Sn) and (Tn) with the desired properties.

From above we have l{x=y}(yx - Y Y)2 = 0, dt 0 dP-a.s. For any 0 < t < 00, therefore,

SrnAt(s -

YX,)dWs = 0, P-a.s. Hence

TnAt

0 = (X - Y)rAtV - (X - Y)sAt = (fix _

3')ds, IP-a.s. JSnAt

We conclude that

sIx -

TnAt

1{x,=Y}i(3sx - s3)ds = (fix _ -g)ds - 0, for 0 < t < oo, FP-a.s.

O neN E SnAt

Using the same arguments as in the proof of Filipovik (1999b, Proposition 3.2), we derive the desired result. O

Next, we list two results in matrix algebra.

Lemma 4.2. Let y = (yi,j) be an N X d matrix and defne the symmetric non-negative definite N X N matrix a := yy*, that is, ai,j = aj,i = EA=17iyjy. Let I and J denote two arbitrary subsets of { 1,..., N}. Define

aj, = aj,j ai,j. jEJ iEI

Then a1,1 i

0 and lai,jI V< a-•,-ajj.

Proof For 1 < , < d, define y•,A

:= EiElyi,.

Then by definition

aj,JA = =i. (zi,) 9j,) I,=JA. jEJ iEI =1 =1 ie jEJ=1

Hence

d

a,I = (yIA)2 ;> 0,

2=1

and by the Cauchy-Schwarz inequality

d d

laj,j 21 r Z 2 Z= J = 2aJ j. _AN(r12 a a1




Lemma 4.3. Let a = (ai,) be an n x n matrix, n E N, which is diagonally dominant from the right, that is,

n

j=1 j7i

|(aii| set j 0

j=i1 j=n+1

for all 1 < i < n. Then a is regular.

Proof. The proof, involving Gaussian elimination, is a slight modification of an argument given in Schwarz (1986, Theorem 1.5).

By assumption al,l I > Z2j=2al,jl 0, in particular al,1 , 0. If n = 1 we are done. If n > 1, the elimination step

(1) a,1, a, : ai, -- alj, 2 < i, j < n, a 1,1

leads to the (n - 1) X (n - 1) matrix at') = (a1)24i,-jn.

We show that a(') is diagonally dominant from the right. If ai,1 = 0, there is nothing to prove for the ith row. Let ai,1 Z 0, for some 2 < i < n. We have

|aij|l aaid - ai jal,j|, 2 < j < n. a91, 1

Therefore

nnn n n n

_,,

al SI - al,j

aij+ I la j=i+l1 j=2 j=2 a1,1 j=2 a1,1 j=2

j]-i j7-i j]-i j7-i n n

.

j=Proceed inductively1,

j=2

a1,.

Proceed inductively to a(2) t. ,



1092 D. Filipovik

5. The case BEP(1, n)

We will treat the case K = 1 separately, since it represents a key step in the proof of the general BEP(K, n) case. For simplicity we shall skip the index i = 1 and write n = ni E No, p = P1, bi = b1,J, ai = al,i;1,j, etc. In particular, we use the notation of Section 2 with N = n + 2.

Lemma 5.1. Let n E No and Z be as above. If Z is consistent with BEP(1, n), then necessarily

Z = Z~ e-Z~+l't + Z'+lte-zn+l't

Z = Zgn e-z+'

Zn+1- Zn+? + bn+ ds + z i:) dW Q lo,

for 0 < i < n - 1 and 0 < t < oo, P-a.s., where Qo := f{p(Zo) = 0}.

Consequently, if Z is consistent with BEP(1, n), then {p(Z) = 0} = R+ x G0. Hence

{Zn+' I

ZO+l'} C {p(Z) = 0}. Therefore we may state:

Corollary 5.2. If Z is consistent with BEP(1, n), then Z is as in Lemma 5.1 and

F(x, Z)= p(x, Z) e- zo+x

Hence the corresponding interest rate model is quasi-deterministic, that is, all randomness remains 3Fo-measurable.

Proof of Lemma 5.1. Let n E No and let Z be an It6 process, consistent with BEP(1, n). Fix

point (t, w) in R+ x 2. For simplicity we write zi for Zi(w), ai, for a"'(wo) and bi for

tb'(wo). The proof relies on expanding equation (3) at the point z = (zo, ..., z,+1). The terms

involved are

F(xz) = p(x, z) - zn+1 p(x, z) e-zn+ x, (14)

0 x' e-z,+', 0 < i < n, F(x, z) = (15)

0zi -xp(x, z) e-zn+x, i = n + 1,

I 0, < i, 2F(x, z) 2F(x, z) -xi+1 e-z+x, O i n, j = n + 1, (16)

ozizj (

z zi J x2p(, Z) e-zn+1x, i =j n 1.

Finally, the following relation for m E No is useful:




m! x-rm(X) e-z"+x m+1' Zn+1 0,

m e-"+' drl = zn+ (17) x m+l

m+1' zn+1 = 0,

where

m m! xm-k rm(x) Z (m - k)! zk+I

k=O n+l

is a polynomial in x of order m. Let us suppose first that z,n+l 0. Thus, subtracting (a/Ox)F(x, Z) from both sides of

(3), we obtain a null equation of the form

ql(x) e-zn+Ix + q2(x) e-2n+1x = 0, (18)

which has to hold simultaneously for all x > 0. The polynomials ql and q2 depend on the zi, bi and ai,j. Equality (18) implies ql = q2 = 0. This again yields that all coefficients of the qi have to be zero.

To proceed we have to distinguish the two cases p(z) # 0 and p(z) = 0. Let us first suppose the former is true. Then there exists an index i E {0, ..., n} such that zi O0. Set m := max{i < nlzi : 0}. With regard to (15)-(17), it follows that degq2 = 2m + 2. In particular,

Z2m q2(x) = an+l,n+l

2m+2 +

Zn+1

where ... denotes terms of lower order in x. Hence a n+,n+i = 0. But the matrix a has to be non-negative definite, so necessarily

an+l,j = aj,n+I = 0, for all 1 j < n + 1.

In view of Lemma 4.1 (setting Y = 0), since we are characterizing a and b up to dt 0 dP- nullsets, we may assume ai,j = aj,i = 0, for 0 < j < n + 1, for all i > m + 1. Thus the degree of q2 reduces to 2m. Explicitly,

q2(x)= m,m 2m +.. Zn+1

Hence am,m = 0 and so am,j = aj,m = 0, for 0 j < n + 1. Proceeding inductively for i = m - 1, m - 2, ..., 0, we finally obtain that the diffusion matrix a is equal to zero and hence q2 = 0 is fulfilled.

Now we determine the drift b. By Lemma 4.1, we may assume bi = 0 for m + 1 < i : n. With regard to (14) and (15), ql reduces therefore to

ql(x)= -bn+zmxm+i ....

It follows that bn+l = 0 and we are left with



1094 D. Filipovic

m-1

ql(x) = (b. +

zn+Izm)Xm + ? (bi - zi+l + zn+Izi)x i=O

n-1

= (bn + Zn+lZn)Xn +? (bi - zi+l + Zn+lZi)Xi i=O

We now turn to the singular cases. If p(z) = 0, that is zo = ... = Zn = 0, we may assume

aij = aj,i

= bi = 0, O< j < n + 1, for all i < n. But this means that ql = q2 = 0, independently of the choice of bn+l and an+l,n+l.

For the case where z,+1 = 0, we need the boundedness assumption z E . By (8) it follows that z = ... = z, = 0. So by Lemma 4.1 again ai,j

= aj,i = bi = 0, 0 < j < n + 1, for all i > 1. Thus in this case equation (3) reduces to

0 = bo - ao,ox,

and therefore bo = ao,o = 0. Summarizing all cases, we conclude that necessarily

bi = -zn+lZi + z1, 0 < i < n - 1,

bn = -Zn+lzn,

ai,j = 0, for (i, j) (n + 1, n + 1);

while bn+l and an+l,n+l are arbitrary real and non-negative real numbers, respectively, whenever p(z) = 0. Otherwise bn+l = an+l,n+l = 0.

The rest of the proof is analogous to the proof of Filipovi6 (1999b, Proposition 4.1). D

6. The general case BEP(K,n)

Using again the notation of Section 3, we give the proof of Theorem 3.2 for the case K > 2. The exposition is somewhat complicated, which is due to the multidimensionality of the

problem. The idea, however, is simple. For a fixed point (t, wo) E R+ x Q we expand equation (3), which turns out to be a linear combination of linearly independent exponential functions, over the ring of polynomials, equalling zero. Consequently, many of the coefficients have to vanish, which leads to our assertion.

The difficulty is that some exponents may coincide. This causes a considerable number of singular cases which require a separate discussion.

Let K > 2, n = (ni, ..., nK) E NK, and let Z be consistent with BEP(K, n). As in the

proof of Lemma 5.1, we fix a point (t, w) in R+ x Q and use the shorthand notation zj,, for Z''"(w), ai,,;j,v for a';JV"'"(w) and bi,, for b'"(co), etc. Since we are characterizing a and b up to a dt 0 dP-nullset, we assume that (t, o) is chosen outside an exceptional dt 0 dP- nullset. In particular, the lemmas from Section 4 shall apply each time this choice is made.

The strategy is the same as for the case K = 1. Thus we expand equation (3) at the point




S=(z,0, ..., zK,nKl) to obtain a linear combination of (ideally) linearly independent exponential functions over the ring of polynomials

K

Z qi(x)e-z,~'i+x +-

qi,j(x)e-(zii+l,.+zJ,inJ+)x - 0. (19)

i= 1 li- j<-K

Consequently, all polynomials qi and qi,j have to be zero. The main difference between this case and the case K = 1 is that representation (19) may not be unique due to the possibly multiple occurrence of the following singular cases:

(i) zi,ni+l =

Zj,nj+1, for i # j,

(ii) 2zi,ni+l = Zj,nj+l + Zk,nk+l,

(iii) 2zi,ni+l = zJ,nj+l,

(iv) Zi,ni+ = Zj,nj+l + Zk,nk+l,

for some indices 1 < i, j, k < K. However, the lemmas in Section 4 and the boundedness assumption z E are good enough to settle these four cases.

Let us suppose first that pi(z) = 0, for all i E { 1, ..., K}. To settle case (i), let ~ denote the equivalence relation defined in (6). After reparametrization if necessary, we may assume that

{1, ..., K}/1 = {[1], ..., [K]}

and Zl,,,+l < ... <zK,nK+1 for some integer K < K. Write I := {1, ..., K}. In view of

Lemma 4.1 we may assume

aj,nj+l;j,nj+l =

ai,ni+l;i,ni+l and bj,nj+l

= bi,n+l for all j E [i], i E I. (20)

The proof of (12) and (13) is divided into four lemmas.

Lemma 6.1. ai,ni+l;i,ni+l = 0, for all i E I.

Proof Expression (19) takes the form

E 4i(x) e-Zi,ni+X + ~ i,j(x) e-(zi,ni++zj,nj+1)x = 0, (21)

iEI i,jEl

i<-j

for some polynomials qi and qi,j. Taking into account cases (ii)-(iv), this representation may still not be unique. However, if for an index i E I there exist no j, k e I such that

2zi,n,+l = z,nj+1l + Zk,nk+l or

2zi,n,+l =

zj,nj+1 (in particular, zi,,+l #0) then we have

qi,i(x)= ai,ni+l;i,ni+1 ', J 2#m+2+

Zi,ni+l

where um := max {vlv < nj and zj,=, 4 0 for some j E [i]} E No. Hence ai,ni+l;i,ni+1 = 0 and the lemma is proved for the regular case.

For the singular cases observe first that zi,ni,+l = 0 implies ai,n,+l;i,n,+l

= 0, which follows from Lemma 4.1. Now we split I into two disjoint subsets II and I2, where



1096 D. Filipovic

I1 := {i E Izi,n,+l 4 0, and there exist j, k E I such that

2zi,ni+l = z ,nj+l + Zk,nk+l r 2zi,ni+l =

Z,n+l }),

12 := I\I.

Observe that zik,n+, > 0 implies K E 12 and zi,n,+l <0 implies 1 E 12. Since at least one of these events has to happen, the set 12 is not empty. We have shown above that ai,ni+l;i,ni+l = 0, for i E 12. If II is not empty, we will show that for each i E II, the parameter zi,ni+l can be written as a linear combination of zy,n,+ls with j E 12. From this it follows by Lemmas 4.1 and 4.2 that ai,ni+l;i,ni+l = 0 for all i E II and the lemma is completely proved. We proceed as follows. Write I = {il, ..., ir} with zil,nil + <

.. < Zir,nr+l. For each ik E II there exists one linear equation of the form

Zil,nil+1

(,..., , 2, *,...,) Zik,nik+1 = ak,

Zir,nir+1,)

where * stands for 0 or -1, but with at most one -1 on each side of 2. The ak on the right- hand side is 0 or zi,ni+l or zi,ni+l + zj,,n+l for some indices i, j, E 12. Hence we obtain the system of linear equations

2 ? ... zinl+1 a1

q . :.

j

? .. * 2/

Zir,nir,+1 r

By Lemma 4.3, the matrix on the left-hand side is invertible, from which our assertion follows. O

Lemma 6.2. aj,nj+l1;k,v = ak,v;j,nj+l

= 0, for 0 < v < nk, for all 1 < j, k < K.

Proof In view of (20), the lemma follows immediately from Lemmas 6.1 and 4.2. O

Analogous to the notation introduced in (7), we set




a[i],p;k,v aj,p;k,v, jE-7[i],,

for 0 < n[i]0, 0 < v n nk, 1 < k s K, 1 K 9 •

d, i E I, and

a[i],pj;[k],v "

. aj,p;l,v, lE?[k],v jE7[il,,

for 0 < n[i], 0 < v < n[k], i, k E I.

Lemma 6.3. If z[i],, = 0, for i E I and yE {0, ..., n[i]}, then

b[i], , a[i]i],

,l =

a[i],y;k,v = ak,v;[i],, = 0,

for all 0 < v < nk, 1 < k < K.

Proof Notice that a[i],1;[i],1 =

• 1i a ; . Hence Lemma 6.3 follows by Lemmas 4.1 and 4.2. O

Lemma 6.4. bi,ni+l = 0, for all i E I such that p[i](z) # 0.

Proof. Suppose first that z #i,n;+l 0, for all i E I. Let i E I such that p[i](z) , 0, and let us assume there exist no j, k E I with zi,ni+l =

zj,nj+l + Zk,nk+l. What does the polynomial qi

in (21) look like? With regard to (20), Lemmas 6.2 and 4.1, and equalities (14)-(17), the contributing terms are

xpj(x, z)e-zn+x = zj,,ux/u- - zi,ni+l zj,,ux e-zi,ni+lx, (22)

/= 1 ( =0

nj+1

-

( ( j

y(mAinj bj, z• F(x, z)= bj, xl - b ii nIt1 z,,x'1 e

-zi,ni+lx (23)

ad =0 A=0=0

and



1098 D. Filipovi6

>

?I0ajyu;kv

(zJ F(x, z) zkv F(r, z)dy

y(/m Aj

n nk! eZii+x

polynomial (24) - ajZ;k,v nk+1

zin+1knk+1)x (24) '=0 k,nk+l

for 0 < v < nk, for all 1 < k < K and j E [i]. We have used the integer

um := max{A|A <; ni and zi=a 0 for some 1 e [i]}.

Definem -:= max{)jA; f< n[i] and z[il] 4 O} E No. Obviously fim Mm. By Lemma 6.3 we have a[i],,;k,v = 0, for all

m < u fl n[i]. Thus summing the above expressions over j E [i], we

obtain

Si(x) =

-bi,ni+lZ[ i],?,Xm+l + .... (25)

Consequently, bi,,,i+ = 0 in the regular case.

For the singular cases the boundedness assumption z E is essential. We split I into two disjoint subsets J1 and J2, where

J, := {i E II there exist j, k E I, such that zi,ni+l = Zj,nj+l

+ Zk,nk+l

and zy,nj+1 > 0 and Zk,nk+l > 0},

J2 = I\J1.

Notice that in any case 1 E J2. We have shown above that for each i E J2 such that zi,ni+l is not the sum of two other

zj,nj+Is it follows that

bi,n,+1 = 0. We will now show that

bi,ni+l= 0 for all i E J2. Let i E J2 and assume there exist j, k e I with zi,ni+l = zy,nj+l

+ zk,nk+l. Then necessarily one of the summands is strictly less than zero. Without loss of generality zj,n+l < 0. Since z E Z, we have pul(z) = 0 (see (8)). Thus

alu[j,;Uj],, = 0

by Lemma 6.3 and therefore a[j],,;k,v = 0, for all 0 :<y < nU], 0 < v < nk, 1 < k < K. The terms contributing to the polynomial in front of e-Zi,ni+x, that is i + qj,k +..., are those in

(22)-(24) and also

-

al,,;zm,v

F(x, z) F(r, z) r = -at,

x#' e-zjj+lx 7 e-Zk,nk+1?

dr, (26)

aO&ZmY FOx,--- 0 ,v 0 ;m,vv0

for 0 <• < ni, 0 < v < nm, l E [i], m E [k]. However, summing - for fixed j, m and v -

the right-hand side of (26) over 1 e E7I,, gives zero. Hence the terms in (26) do not actually contribute to the polynomial. The same conclusion can be drawn for all j, k I with the property that zi,ni+l= zj,n+l1 + Zk,nk+l. It finally follows, as in the regular case, that

bi,n^+1= 0 for all i E J2. If J1 is not empty, we show that for each i E J1, the parameter zi,ni+ can be written as a

linear combination of zp,,n+ls

with jE J2. From this it follows by Lemma 4.1 that

bi,,+l = 0 for all iE J1. We proceed as follows. Write J1

= {il, ..., ir'} with

zil,n+1 < ...

<Zzi,~,n,,+1*. For each ik E J1 there exists one linear equation of the form




Zil,nil+1

(, ..., *, 1, O ..., 0) Zik,nik+l = k,

Zir',nir,+l

where the * stand for 0 or -1, but at most two of them are -1. The a' on the right-hand side is 0 or zi,oi+l or Zi,ni+l + zj,1j+l for some indices i, j E J2 with zi,ni+l > 0 and zj,nj+l > 0. Obviously a' is of the latter form. Hence we obtain the system of linear equations

1 0 ... 0 Zi,n,1

l \

+ / Zi,ni+l + Zj,n,j+1

Zl 01 1

* . . . *

j

\Zi,,nir,+l +1

for some i, j E J2. On the left-hand side is a lower-triangular matrix, which is therefore invertible. Hence the lemma is proved in the case where zi,ni+l 0 for all i E I.

Assume now that there exists i e I with zi,n,i+ = 0. Then i E J2. We have to make sure that also in this case

bj,nj+l = 0, for all j E J2. Clearly bi,ni+l is zero by Lemma 4.1. The

problem is that zj,,j+1 = zi,ni+l + zj,nj+l for all j E J2. But following the lines above, it is

enough to show a[i],1,;[i],, = 0, for all 0 y <fl n[i]. From the boundedness assumption z E Z we know that p[i](z) = z[i],O (see (8)). Hence a[i],;[i],14 = 0, for 1 ~ f[u n[i]. Suppose there is no pair of indices j, k E I\{i} with

zj,nj+l + Zk,nk+l = 0. Summing the contributing

terms in (22)-(24) over j E [i], we obtain the polynomial in front of eo, i.e.

4i(x) + 4i,i(x) = -a[i],o;[i],ox + .. ., (27)

hence a[i],o;[i],o = 0. If there exist a pair of indices j, k E I\{i} with zj,,n+l + Zk,nk+l = 0,

then one of these summands is strictly less than zero. Arguing as before, the polynomial in front of eo remains of the form (27) and again a[i],o;[i],o = 0. Thus the lemma is completely proved. O

So far we have established (12) and (13) under the hypothesis that pi(z) Z 0, for all i E {1,..., K}. Suppose now that there is an index i E {1,..., K} with pi(z) = 0. By Lemma 4.1, we may assume aji,4;i,4 = bi,, = 0, for all 0 < ~u ni. But then Lemma 4.2 tells us that none of the terms including the index i appears in (19). In particular, ai,ni+l;i,ni+l and bi,ni+l can be chosen arbitrarily without affecting equation (19). This means that we may skip i and proceed, after a reparametrization if necessary, with the remaining index set {1, ..., K - 1} to establish Lemmas 6.1-6.4 as above.

This all has to hold for dt 0 dP-a.e. (t, w). Hence (12) and (13) are fully proved. A closer look to the proof of (12), that is, Lemma 6.1, shows that the boundedness assumption z E was not explicitly used there - whence Remark 3.3.

Next we prove that the exponents Zi,ni+l are locally constant on intervals where pi(Z)



1100 D. Filipovi6

and p[i](Z) do not vanish. Let v > 0 be a rational number and let To:= inf{t > vlpi(Zt) = 0 or p[i](Z) = 0} denote the debut of the optional set [v, oo[nAi. By (12) and (13) and the continuity of Z we have that Zi,'"+1 is PF-a.s. constant on [v, T,], hence P-a.s. constant on every such interval [v, T,]. Since every open interval where pi(Zt) : 0 or p[i](Zt): 0 is covered by a countable union of intervals [v, T,] and by continuity of Z, the assertion follows and the first part of the theorem is proved.

To establish the second part of the theorem, let r be a stopping time with [r] E _' and P(r < oo) > 0. Define the stopping time r'(w):= inf{t > r(wo)(t, o) 0 _' }. By continuity of Z, we conclude that r < t' on {r < oo}. Choose a point (t, wo) in [r, r'[. We use shorthand notation as above.

By definition of _'

we can exclude the singular cases zi,n,+l

= zj,nj+l

or 2zi,n,+l

=

zj,nj+l, for i j. In particular, K = K, hence I = {1, ..., K}. First, we show that the

diffusion matrix for the coefficients of the polynomials pi(z) vanishes.

Lemma 6.5. ai,!,;j,v

= aj,v;i,,u = 0, for 0 a /u

: ni and 0 a v nj, for all i, j E I.

Proof By Lemma 4.1 it is enough to prove that the diagonal ai,,;ij,, vanishes for 0 :/ 'U ni

and i E I. If there is an index i E I with pi(z) = 0 then, arguing as above, ai,4;i, = bi,, = 0, for all 0

/ :; ni, and we may skip the index i. Hence we assume now that there is a

K' : K such that pi(z) Z 0 (and thus zi,n,+1 > 0, since z E ) for all 1 i t K'. Let I' := {1, ..., K'}. To handle the singular cases, we split I' into two disjoint subsets I' and I2, where

If := {i E I'Izi,ni+l > 0, and there exist j, k E I' such that 2zi,ni+l =

Zj,,n+l + Zk,nk+l },

2 : I'\I.

Hence zi,,,;+ = 0 for i E I' implies i e I'. We have already shown in the proof of Lemma 6.4 that in this case ai,;i,, = 0, for all 0 : ;u <; ni. The same follows for i E I' with

zi,ni+l > O0, as was demonstrated for the case K = 1. Now let i E I' and let 1, m E I', such that 1 < m and 2zi,ni+l = Zl,n+1l + Zm,n,+l. Thus

the polynomial in front of e-2zi,ni+1x is qi,i + ql,m +..., and among the contributing terms are also those in (26). If 1 or m is in I', those are all zero. Write I' = {il,..., i,,}

with

zi,ni,+1 < ... <zi, ,nl,+1.

Then necessarily 1 E IV in the above representation for zi,ni, +l. Thus the polynomial in front of e-2Zilil,+'x is qil,il. It follows that ail,1u;i,, = 0, for all 0

t <

nil, as was demonstrated for the case K = 1. Proceeding inductively for

i2, . . .,

ir", we eventually derive that ai,u;i,,u = 0, for all 0 : :u <; ni and i E I'. This

establishes the lemma. O

We are left with the task of determining the drift of the coefficients in pi(z). By (13), we have bi,,,+l -= 0 for all i E I'. Straightforward calculations show that (19) reduces to




K'

qi(x) e -zi'ni+x = 0,

i=1

with

ni-I

qi(x) = (bi,n + zi,n+1zi,nj)xn' + Z(bi,4 - zi,1+l + Zi,ni+lZi,))X .

We conclude that for all 1 < i < K (in particular, if pi(z)= 0)

bi,,4 = zi,41+l - zi,ni+lZi,4, 0 < ni - 1,

bi,n, = -Zi,ni+1Zi,ni. (28)

By continuity of Z, Lemma 6.5 and (28) hold pathwise on the semi-open interval [r(w), r'(w)[ for almost every w. Therefore Z,+. is of the claimed form on [0, r' - r[.

Now replace _' by _ and proceed as above. By (11) we have r < r' on { < 00oo}, and,

since _ C _',

all the above results remain valid. In addition, pi(z) = p[i](z) > 0 and thus

ai,ni+l;i,n,+l = bi,n,+l = 0, for all 1 < i < K, by (12) and (13). Hence

Zlr+. = Zi+T on

[0, r' - r[, for all 1 < i < K, up to evanescence. But this again implies r' = oc by the continuity of Z.

7. E-consistent Ito processes

An It6 process Z is by definition consistent with a family {F(., z)}zEz if and only if D is a martingale measure for the discounted bond price processes. We could generalize this definition and call a process Z e-consistent with {F(., z)}zEZ if there exists an equivalent martingale measure Q. Then obviously consistency implies e-consistency, and e-consistency implies the absence of arbitrage opportunities as is well known.

Where the filtration is generated by the Brownian motion W that is, (7t) = ('"w), we can give the following stronger result:

Proposition 7.1. Let K E N and n = (ni, ..., nK) K. tIf ( = (wt), then any It6 process Z which is e-consistent with BEP(K, n) is of the form stated in Theorem 3.2.

Proof Let Z be an e-consistent It6 process under D, and let Q be an equivalent martingale measure. Since (7t) = (7tw), we know that all P-martingales have the representation property relative to W By Girsanov's theorem it follows, therefore, that Z remains an It6 process under 0, which is consistent with BEP(K, n). The drift coefficients b"', change under Q into

bi,/ ; while

b'i,! = bi',u on

{ai,';ij = 0}, dt 0 dP-a.s. The diffusion matrix a

remains the same. Therefore, and since the measures dt 0 dQ and dt 0 dPW are equivalent on

R+ x 2, the It6 process Z is of the form stated in Theorem 3.2. O



1102 D. Filipovic

Notice that in this case the quasi-deterministic, that is, o0-measurable, expression in Corollaries 3.5 and 5.2 is a purely deterministic one.

8. The diffusion case

The main result from Section 3 is much clearer for diffusion processes. In all applications the generic It6 process Z on (2, ,9 (9t)O~t<oo, P) given by (9) is the solution of a stochastic differential equation

Z't = Z~' J+ b',"(s, Zs)ds + J 'a ";"(s, Zs)d W~, (29) 0

A=1 0

for 0 •< :u ni + 1 and 1 < i < K, where b and a are some Borel measurable mappings

from R+ x RIN to RN and RNxd, respectively. The coefficients b and a could be derived by statistical inference methods from the daily

observations of the diffusion Z made by some central bank. These observations are of course made under the objective probability measure. Hence P is not a martingale measure in applications of this kind.

On the other hand, we want a model for pricing interest rate sensitive securities. Thus the diffusion has to be e-consistent. If we assume that

(7"t)= (7w"), the previous section applies. To stress the fact that 7w"-measurable functions are deterministic, we denote the initial values of the diffusion in (29) with small letters zo'.

Since all reasonable theory for stochastic differential equations requires continuity properties of the coefficients, we shall assume in the following that b(t, z) and a(t, z) are continuous in z. The main result for e-consistent diffusion processes is divided into the two

following theorems. The first one only requires consistency with EP(K, n).

Theorem 8.1. Let K E N, n = (ni, ..., nK) EN,(t) = ( ), and let the diffusion Z, b

and a be as above. If Z is e-consistent with BEP(K, n) or with EP(K, n), then necessarily the exponents are constant,

Zi,ni+l = zi,ni+l -0

for all 1 i K.

Proof The significant difference between this proof and that of Theorem 3.2 is that now the diffusion matrix a and the drift b depend continuously on z.

First, observe that the following sets of singular values, K

A := U{z E RNlpi(z) = 0 or

p(i](z) = 0}

i=1

and

X ":=

{z E RNlzi,,i+- = zj,n;+l + Zk,nk+l for some 1 < i, j, k < K},




are contained in a finite union of hyperplanes of RIN (see (10)). Hence (A U A4) c RN has Lebesgue measure zero. Thus the topological closure of ' := RN\(& U 'y) is RN.

Now let Z be the diffusion in (29), which is e-consistent with either BEP(K, n) or EP(K, n). A closer look at the proof of Lemma 6.4 shows that the boundedness assumption z E Z was not used for the regular case z E 3F (see (25)). Combining this with (12), (13) and Remark 3.3, we conclude that, for any 1 < i < K and 1 < 2 < d,

bi',n+l(t, Zt(w))= a-ini+l;A(t, Zt(w)) = 0, for (t, o) E {Z E f}\N,

where N is an ?*+ 0 .9-measurable dt 0 dP-nullset. By the very definition of the product measure,

0= INdt9dP= RP[Nt]dt,

where Nt := {wl(t, o) E N} e 7. Consequently P[Nt] = 0 for almost every t E R+. Hence, by continuity of b(t, -) and a(t, .),

bin+l(t, ) = ini+l;A(t, ) = 0 (30)

on supp(Zt)n f , for almost every t e R+. Here supp(Zt) denotes the support of the (regular) distribution of Zt, which is by definition the smallest closed set A C N with P[Z E A] = 1. Thus, again by continuity of b(t, -) and o(t, .), equality (29) holds for almost every t E R+ on the closure of supp(Zt) n 3, which is supp(Zt). Hence we may replace the functions b',n'+t(t, ) ad a i'ni+;)(t, .) by zero for almost every t without changing the diffusion Z, whence the assertion follows. [

The sum of two real-valued diffusion processes with coefficients continuous in some argument is again a real-valued diffusion with coefficients continuous in that argument. Consequently, we may assume that the exponents zi,n,+I of the above e-consistent diffusion are mutually distinct. Since otherwise we add the corresponding polynomials to obtain in a canonical way an RNN-valued diffusion Z which is e-consistent with BEP(K, ni) or EP(K, n"), for some K < K, N < N and some ii E NK. Clearly Z provides the same interest rate model as Z and its coefficients are continuous in z.

For the second theorem we have to require e-consistency with BEP(K, n). After a reparametrization if necessary, we may thus assume that

0

1 Zo,n,+l

< ~ < zK,nK+1

0 z0

...K (see (8)). The continuation of Theorem 8.1 is now as follows:

Theorem 8.2. If Z is e-consistent with BEP(K, n), then it is non-trivial only if there exists a pair of indices 1 < i < j * K, such that

. ni+ l " nj+- l 2 -o

Proof If there is no pair of indices 1 < i<j < K such that 2z•,n+ = , then

..' = R+ x Q2. But then Z is deterministic by the second part of Theorem 3.2. O



1104 D. Filipovic

The message of Theorem 8.1 is the following: there is no possibility of modelling the term structure of interest rates by exponential-polynomial families with varying exponents driven by diffusion processes. From this point of view there is no use for daily estimations of the exponents of exponential-polynomial type functions such as FNS or Fs. Once the exponents are chosen, they have to be kept constant. Furthermore, there is a strong restriction on this choice by Theorem 8.2. It will be shown in the next section what this means for FNS and Fs in particular.

Remark 8.3. The boundedness assumption in Theorem 8.2 - that is, e-consistency with BEP(K, n) - is essential for the strong (negative) result to be valid. It can easily be checked that F(x, z) = zo + z1x E EP(1, 1) allows for a non-trivial consistent diffusion process (see Filipovi6 1999a).

Remark 8.4. The choice of an infinite time horizon for traded bonds is not a restriction (see (1)). Indeed, we can limit our considerations to bonds P(t, T) which mature within a given finite time interval [0, T*]. Consequently, the HJM drift condition (3) can only be deduced for x E [0, T* - t], for dt 0 dP-a.e. (t, w) E [0, T*] x ~. But the functions appearing in (3) are analytic in x. Hence, whenever t < T*, relation (3) extends to all x > 0. All conclusions on e-consistent It6 processes

(Zt)o•t1T*

can now be drawn as before.

9. Applications

In this section we apply the results on e-consistent diffusion processes to the Nelson-Siegel and Svensson families, whose curve shapes were given in Section 1.

9.1. The Nelson-Siegel family

In view of Theorem 8.1 we have z4 > 0. Hence it is immediate from Theorem 8.2 that there is no non-trivial e-consistent diffusion. This result has already been obtained in Filipovic (1999b) for e-consistent It6 processes.

9.2. The Svensson family

By Theorems 8.1 and 8.2 there remain the two choices

(i) 2z6 = z5 > 0,

(ii) 2z5 = z6 > 0.

We shall identify the e-consistent diffusion process Z = (Z,..., Z6) in both cases. Let Q be an equivalent martingale measure. Under Q the diffusion Z transforms into a consistent one. Now we proceed as in the proof of Theorem 3.2. The expansion (19) is given by

Q1(x) + Q2(x)e-zsx + Q3(x) e-z6 + 4(x) e-2zsx + 5(x) e-(zS+z6)x + 6(x) e-2z6X = 0,




for polynomials Qi ..., Q6. Explicitly,

Ql(x) = -al,ix +...

Q2(x) = -a,3x2 +...

Q3(x) = -al,4x +...

O4W() a3'-3 X2+... Z5

Q6(x) =

a4,4 x2 + ... Z6

where ... denotes terms of lower order in x. Hence ai,1 = 0 in any case. By the usual arguments (the matrix a is non-negative definite) the degree of Q2 and Q3 reduces to at most 1. Thus in both cases (i) and (ii) it follows that a3,3 = a4,4 = 0. We are left with

Ql(x) = bl,

Q2(x) = (b3 + z3z5)x + b2 - Z3 - a2,2 +25,

Q3(x) = (b4 + 4z6)X - 4, 5

(31)

Q4)- aa2,2 Z5

while Q5 = Q6 = 0. Since in case (i) Q4 must be 0, we have a2,2 = 0 and Z is deterministic. We conclude that there is no non-trivial e-consistent diffusion in case (i).

In case (ii) the condition Q3 + Q4 = 0 leads to

a2,2 = z4z5. (32)

Hence there is a possibility of a non-deterministic consistent diffusion Z. We derive, from (31) and (32), that

bi = 0,

b2 = z3 + Z4 - 2522,

b3 = -zsz3,

b4 = -2Z5 z4

Therefore the dynamics of Z', Z3, ..., Z6 are deterministic. In particular,

z1 = I t -

0Z,

Z = z 0 e-4t, (33)

Zt = z e-24t,

while Zt - zi and Z6

- 2z~. Denoting by W the Girsanov transform of W we have under the

equivalent martingale measure Q



1106 D. Filipovic

Z2 = z2 + (((s) - zoZ2) ds + o2,A(s)dW , (34) t JO0S A1=1 0

where ((t) and o2a(t) are deterministic functions in t, namely

.(t) := z0 e-4t + z e-2 4t

and

By L6vy's characterization theorem (see Revuz and Yor 1994, Theorem (3.6), Chapter IV), the real-valued process

d F 2'2(S) d , 0 W* 1 zt

o 2,A

(• d

Ws 0 *t < 0oo

is an (Yt)-Brownian motion under Q. Hence the corresponding short rates rt = Fs(O, Zt) = z + Z2 satisfy

drt = (P(t) - zort)dt + d(t)dWt,

where q5(t):= "((t)

+ zloz and 6(t):= z e-t. This is just the generalized Vasicek model. It can be easily given a closed-form solution for rt (see Musiela and Rutkowski 1987, p. 293).

Summarizing case (ii), we have found a non-trivial e-consistent diffusion process, which is identified by (33) and (34). Actually ( has to be replaced by a predictable process (Q due to the change of measure. Nevertheless, this is just a one-factor model. The corresponding interest rate model is the generalized Vasicek short rate model. This is very unsatisfactory since Svensson-type functions Fs(x, z) have six factors z1, ..., z6 which are observed. And it is evident that just one of them, z2, can be chosen to be non-deterministic.

10. Conclusions

Bounded exponential-polynomial families such as the Nelson-Siegel and Svensson families

may be well suited for daily estimations of the forward rate curve. They are best not used for intertemporal interest rate modelling by diffusion processes. This is because the exponents have to be kept constant and, moreover, this choice is very restrictive whenever you want to exclude arbitrage possibilities. It has been shown for the Nelson-Siegel family in particular that there exists no non-trivial diffusion process providing an arbitrage-free model. However, there is a choice for the Svensson family, albeit a very limited one, since all parameters but one have to be kept either constant or deterministic.




Acknowledgement

I am greatly indebted to T. Bjrrk for pointing out this extension of my earlier work. I thank B.J. Christensen for his interest in these results and for discussion. I am also grateful to H. Pages from the BIS for his comments and for making available the technical documentation. This paper is part of my doctoral thesis, written under the supervision of E Delbaen. Support from Credit Suisse is gratefully acknowledged.

References

Bank for International Settlements (1999) Zero-coupon yield curves: Technical documentation. Basle: BIS.

Bjork, T. and Christensen, B.J. (1999) Interest rate dynamics and consistent forward rate curves. Math. Finance, 9, 323-348.

FilipoviS, D. (1999a) A general characterization of affine term structure models. Working paper, ETH Ziirich.

Filipovi', D. (1999b) A note on the Nelson-Siegel family. Math. Finance, 9, 349-359. Heath, D., Jarrow, R. and Morton, A. (1992) Bond pricing and the term structure of interest rates: A

new methodology for contingent claims valuation. Econometrica, 60, 77-105. Jacod, J. and Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag. Musiela, M. and Rutkowski, M. (1997) Martingale Methods in Financial Modelling. Berlin: Springer-

Verlag. Nelson, C. and Siegel, A. (1987) Parsimonious modeling of yield curves. J Business, 60, 473-489. Revuz, D. and Yor, M. (1994) Continuous Martingales and Brownian Motion, 2nd edn. Berlin:

Springer-Verlag. Schwarz, H.R. (1986) Numerische Mathematik, 2nd edn. Stuttgart: Teubner. Svensson, L.E.O. (1994) Estimating and interpreting forward interest rates: Sweden 1992-1994. IMF

Working Paper No. 114.

Received November 1998 and revised February 2000



exponential-polynomial families and the term structure of interest rates

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