a note on the nelson–siegel family

Mathematical Finance, Vol. 9, No. 4 (October 1999), 349–359

A NOTE ON THE NELSON–SIEGEL FAMILY

DAMIR FILIPOVIC

Department of Mathematics, ETH, Zurich

We study a problem posed in Björk and Christensen (1999): Does there exist any nontrivial interestrate model that is consistent with the Nelson–Siegel family? They show that within the Heath–Jarrow–Morton framework with deterministic volatility structure the answer is no.

In this paper we give a generalized version of this result including stochastic volatility structure.For that purpose we introduce the class ofconsistent state space processes, which have the property toprovide an arbitrage-free interest rate model when representing the parameters of the Nelson–Siegelfamily. We characterize the consistent state space Itô processes in terms of their drift and diffusioncoefficients. By solving an inverse problem we find their explicit form. It turns out that there exists nonontrivial interest rate model driven by a consistent state space Itô process.

KEY WORDS: interest rate model, Nelson–Siegel family, consistent state space Itô process, inverseproblem.

1. INTRODUCTION

Bjork and Christensen (1999) introduce the following concept: LetM be an interest ratemodel andG a parameterized family of forward curves.M andG are called consistent, ifall forward rate curves that may be produced byM are contained withinG, provided thatthe initial forward rate curve lies inG. Under the assumption of a deterministic volatilitystructure and working under a martingale measure, they show that within the Heath–Jarrow–Morton (henceforth HJM) framework there exists no nontrivial forward rate model, that isconsistent with the Nelson–Siegel family{F( . , z)}. The curve shape ofF( . , z) is givenby the well-known expression

F(x, z) = z1+ z2e−z4x + z3xe−z4x,(1.1)

introduced by Nelson and Siegel (1987).For an optimal todays’ choice of the parameterz ∈ R4, expression (1.1) represents the

current term structure of interest rates—that is,x ≥ 0 denotes time to maturity. This methodof fitting the forward curve is widely used among central banks, see the documentation ofthe Bank for International Settlements (BIS 1999).

From an economic point of view it seems reasonable to restrictz to the state spaceZ := {z= (z1, . . . , z4) ∈ R4 | z4 > 0}.

The corresponding term structure of the bond prices is given by

G(x, z) := exp

(−∫ x

0F(η, z)dη

).

ThenG ∈ C∞([0,∞)× Z).-

I thank T. Bjork for his interest in this result and for encouraging me to write this note. Support from CreditSuisse is gratefully acknowledged.

Initial manuscript received January 1998; final revision received May 1999.Address correspondence to the author at Departement Mathematik, ETH, Rämistrasse 101, CH-8092 Zürich,

Switzerland; e-mail: [email protected].

c© 1999 Blackwell Publishers, 350 Main St., Malden, MA 02148, USA, and 108 Cowley Road, Oxford,OX4 1JF, UK.

349

350 DAMIR FILIPOVI C

In order to imply a stochastic evolution of the forward rates, we introduce in Section 2some state space processZ = (Zt )0≤t<∞ with values inZ and ask whetherF( . , Z)provides an arbitrage-free interest rate model. We callZ consistent, if the correspondingdiscounted bond prices are martingales (see Section 3). Solving an inverse problem wecharacterize in Section 4 the class of consistent state space Itô processes. Since a diffusionis a special Itô process, the very important class of consistent state space diffusion processesis characterized as well. Still we are able to derive a more general result. It turns out thatall consistent Itô processes have essentially deterministic dynamics. The correspondinginterest rate models are in turn trivial.

Consistent state space Itô processes are, by definition, specified under a martingale mea-sure. This seems to be a restriction at first and one may ask whether there exists any ItôprocessZ under some objective probability measure inducing a nontrivial arbitrage-freeinterest rate modelF( . , Z). However if the underlying filtration is not too large we showin Section 5 that our (negative) result holds for Itô processes modeled under any probabilitymeasure provided that there exists an equivalent martingale measure. Hence under therequirement of absence of arbitrage there exists no nontrivial interest rate model driven byIto processes and consistent with the Nelson–Siegel family.

Using the same ideas, still larger classes of consistent processes like Itô processes withjumps could be characterized.

2. THE INTEREST RATE MODEL

For the stochastic background and notations we refer to Revuz and Yor (1994) and Jacodand Shiryaev (1987). Let(Ä,F, (Ft )0≤t<∞,P) be a filtered complete probability space,satisfying the usual conditions, and letW = (W1

t , . . . ,Wdt )0≤t<∞ denote a standardd-

dimensional(Ft )-Brownian motion,d ≥ 1.We assume as given an Itô processZ = (Z1, . . . , Z4) with values in the state spaceZ

of the form

Zit = Zi

0+∫ t

0bi

s ds+d∑

j=1

∫ t

0σ i j

s dWjs , i = 1, . . . ,4, 0≤ t <∞,(2.1)

whereZ0 is F0-measurable, andb, σ are progressively measurable processes with valuesin R4, resp.R4×d, such that

∫ t

0

(|bs| + |σs|2)

ds<∞, P-a.s., for all finite t .(2.2)

Z could be for instance the (weak) solution of a stochastic differential equation, but ingeneralZ is not Markov.

Define by

r (t, x) := F(x, Zt )

the instantaneous forward rateat timet for datet + x.

A NOTE ON THE NELSON–SIEGEL FAMILY 351

It is shown in Delbaen and Schachermayer (1994, Sec. 7) that traded assets have to followsemimartingales. Hence it is important for us to observe that the price at timet for a zerocoupon bondwith maturityT

P(t, T) := G(T − t, Zt ), 0≤ t ≤ T <∞,

and theshort rates

r (t,0) = limx→0

r (t, x) = F(0, Zt ) = − ∂

∂xG(0, Zt ), 0≤ t <∞,

form continuous semimartingales, by the smoothness ofF andG. Therefore the same holdsfor the process of thesavings account:

B(t) := exp

(∫ t

0r (s,0)ds

), 0≤ t <∞.

3. CONSISTENT STATE SPACE PROCESSES

We are going to define consistency in our context, which slightly differs from that in Bj¨orkand Christensen (1999). We focus on the state space processZ, which follows an Ito processor may follow some more general process.

DEFINITION 3.1. Z is calledconsistentwith the Nelson–Siegel family, if

(P(t, T)

B(t)

)0≤t≤T

is aP-martingale, for allT <∞.

The next proposition is folklore in the case whereZ follows a diffusion process—thatis, if bt (ω) = b(t, Zt (ω)) andσt (ω) = σ(t, Zt (ω)) for Borel mappingsb andσ from[0,∞)×Z into the corresponding spaces. That case usually leads to a PDE including thegenerator ofZ. The standard procedure is then to find a solutionu (the term structure ofbond prices) to this PDE on(0,∞)×Z with initial conditionu(0, . ) = 1. It is well knownin the financial literature thatZ is necessarily consistent with the corresponding forwardrate curve family.

In contrast we ask the other way round and are more general as to what concernsZ.Our aim is,given F, to deriveconditions on b andσ beingnecessaryfor consistency ofZwith {F( . , z)}z∈Z . But the coefficientsb andσ are progressively measurable processes.HenceZ given by equation (2.1) is not Markov; that is, there is no infinitesimal generator.By the nature ofb andσ such conditions can therefore only be stateddt ⊗ dP-a.s. (notethat equation (3.1) below is not a PDE). On the other hand the argument mentioned inthe diffusion case works just in one direction: consistency ofZ with a forward curvefamily G = {v( . , z)}z∈Z does not imply validity of the PDE condition foru(x, z) =exp(− ∫ x

0 v(η, z)dη) in general.


Actually one could reparameterizef (t, T) := r (t, T − t), for 0 ≤ t ≤ T < ∞, andwork within the HJM framework. Equation (3.1) below corresponds to the well-knownHJM drift condition for( f (t, T))0≤t≤T . But as soon asZ is not an Ito process anymore,this connection fails and one has to proceed as in the following proof (see Remark 3.3),which therefore is given in its full form.

Seta := σσ ∗, whereσ ∗ denotes the transpose ofσ , that is,ai jt =

∑dk=1 σ

ikt σ

jkt , for

1≤ i, j ≤ 4 and 0≤ t <∞. Thena is a progressively measurable process with values inthe symmetric nonnegative definite 4× 4-matrices.

PROPOSITION3.2. Z is consistent with the Nelson–Siegel family only if

∂

∂xF(x, Z) =

4∑i=1

bi ∂

∂ziF(x, Z)(3.1)

+ 1

2

4∑i, j=1

ai j

(∂2

∂zi ∂zjF(x, Z)− ∂

∂ziF(x, Z)

∫ x

0

∂

∂zjF(η, Z)dη

− ∂

∂zjF(x, Z)

∫ x

0

∂

∂ziF(η, Z)dη

),

for all x ≥ 0, dt⊗ dP-a.s.

Proof. For f ∈ C2(Z) we set

At (ω) f (z) :=4∑

i=1

bit (ω)

∂ f

∂zi(z)+ 1

2

4∑i, j=1

ai jt (ω)

∂2 f

∂zi ∂zj(z), 0≤ t <∞, z ∈ Z.

Using Ito’s formula we get forT <∞

P(t, T) = P(0, T)+∫ t

0

(AsG(T − s, Zs)− ∂

∂xG(T − s, Zs)

)ds

+∫ t

0σ ∗s∇zG(T − s, Zs)dWs, 0≤ t ≤ T, P-a.s.,

where∇z denotes the gradient with respect to(z1, z2, z3, z4), and

1

B(t)= 1+

∫ t

0

1

B(s)

∂

∂xG(0, Zs)ds, 0≤ t <∞, P-a.s.

For 0≤ t ≤ T define

H(t, T) := 1

B(t)

[At G(T − t, Zt )− ∂

∂xG(T − t, Zt )+ ∂

∂xG(0, Zt )G(T − t, Zt )

]


and the local martingale

M(t, T) :=∫ t

0

1

B(s)σ ∗s∇zG(T − s, Zs)dWs.

Integration by parts then yields

1

B(t)P(t, T) = P(0, T)+

∫ t

0H(s, T)ds+ M(t, T), 0≤ t ≤ T, P-a.s.

Let us suppose now thatZ is consistent. Then necessarily forT <∞∫ t

0H(s, T)ds= 0, ∀t ∈ [0, T ], P-a.s.(3.2)

Sinceb andσ are progressive andG is smooth,H( . , T) is progressively measurable on[0, T ] ×Ä. We claim that equation (3.2) yields

H( . , T) = 0, on [0, T ] ×Ä, dt ⊗ dP-a.s.(3.3)

Proof of equation (3.3). DefineN := {(t, ω) ∈ [0, T ] ×Ä | H(t, T)(ω) > 0}. ThenN isaB ⊗ F-measurable set. SinceH( . , T) is positive onN we can use Tonelli’s theorem

∫N

H(t, T)(ω)dt ⊗ dP =∫Ä

(∫Nω

H(t, T)(ω)dt

)dP(ω) = 0,

whereNω := {t | (t, ω) ∈ N} ∈ B and we have used equation (3.2) and the inner regularityof the measuredt. We therefore conclude thatN hasdt ⊗ dP-measure zero. By using asimilar argument for−H( . , T) we have proved equation (3.3).

Note that equation (3.3) holds for allT <∞, where thedt ⊗ dP nullset depends onT .But sinceH(t, T) is continuous inT , a standard argument yields

H(t, t + x)(ω) = 0, ∀x ≥ 0, for dt ⊗ dP-a.e.(t, ω).

Multiplying this equation withB(t) and using again the full form, we obtain

AG(x, Z)− ∂

∂xG(x, Z)+ ∂

∂xG(0, Z)G(x, Z) = 0, ∀x ≥ 0, dt ⊗ dP-a.s.(3.4)

Differentiation gives

∫ x

0A F(η, Z)dη − 1

2

4∑i, j=1

ai j

(∫ x

0

∂

∂ziF(η, Z)dη

)(∫ x

0

∂

∂zjF(η, Z)dη

)− F(x, Z)+ F(0, Z) = 0, ∀x ≥ 0, dt ⊗ dP-a.s.,


where we have divided byG(x, Z), sinceG > 0 on [0,∞) × Z. Differentiating withrespect tox finally yields (3.1).

REMARK 3.3. Definition 3.1 can be extended in a natural way to a wider class of statespace processesZ. We mention here just two possible directions:

a) Z a time homogeneous Markov process with infinitesimal generatorL. The corre-sponding version of Proposition 3.2 can be formulated in terms of equation (3.4),whereA has to be replaced byL. The difficulty here consists of checking whetherG goes well together withZ; that is,x 7→ G(x, . ) has to be a nice mapping from[0,∞) into the domain ofL.

b) Z an Ito process with jumps. Again one could reformulate Proposition 3.2 onthe basis of equation (3.4) by adding the corresponding stochastic integral withrespect to the compensator of the jump measure, which we assume to be absolutelycontinuous with respect todt. The jump measure could be implied for example bya homogeneous Poisson process.

4. THE CLASS OF CONSISTENT ITO PROCESSES

Equation (3.1) characterizesb anda, respectivelyσ , just up to adt ⊗ dP nullset. But thestochastic integral in equation (2.1) is (up to indistinguishability) defined on the equiva-lence classes with respect to thedt ⊗ dP nullsets. Hence this is enough to determine theprocessZ, givenZ0, uniquely (up to indistinguishability). On the other handZ cannot berepresented in the form of equation (2.1) with integrands that differ fromb andσ on a setwith dt ⊗ dP-measure strictly greater than zero (the characteristics ofZ are unique up toindistinguishability). Therefore it makes sense to pose the followinginverse problemonthe basis of equation (3.1): For which choices of coefficientsb andσ do we get a consistentstate space Itˆo processZ starting inZ0?

The answer is rather remarkable.

THEOREM4.1. Let Z be a consistent Ito process. Then Z is of the form

Z1t = Z1

0

Z2t = Z2

0e−Z40t + Z3

0te−Z40t

Z3t = Z3

0e−Z40t

Z4t = Z4

0 +(∫ t

0b4

s ds+d∑

j=1

∫ t

0σ 4 j

s dWjs

)1Ä0

for all 0≤ t <∞, whereÄ0 := {Z20 = Z3

0 = 0}.

REMARK 4.2. OnÄ0 the processesZ2 andZ3 are zero. HenceF(x, Zt ) = Z10 onÄ0;

in other words, the processZ4 has no influence onF( . , Zt ) onÄ0. So it holds that thecorresponding interest rate model is of the form

r (t, x) = F(x, Zt )

= Z10 +

(Z2

0e−Z40t + Z3

0te−Z40t)

e−Z40x + Z3

0e−Z40t xe−Z4

0x


= Z10 + Z2

0e−Z40(t+x) + Z3

0(t + x)e−Z40(t+x)

= r (0, t + x), ∀t, x ≥ 0,

and is therefore quasi deterministic—that is, all randomness remainsF0-measurable.

REMARK 4.3. One can show a similar (negative) result even for the wider class ofstate space Itô processes with jumps on a finite or countable mark space; see Remark 3.3.However allowing for more general exponential-polynomial families{F( . , z)} there exist(although very restricted) consistent Itô processes providing a nontrivial interest rate model(see Filipovic 1998).

Proof. Let Z be a consistent Itô process given by equation (2.1). The proof of thetheorem relies on expanding equation (3.1).

First we subtract∂∂x F(x, Z) from both sides of equation (3.1) to obtain a null equation.

We then fix a point(t, ω) in [0,∞)×Ä. For simplicity we write(z1, z2, z3, z4) for Zt (ω),ai j for ai j

t (ω) andbi for bit (ω). Notice thatZt (ω) ∈ Z; that is,z4 > 0. Observe then that

our null equation is in fact of the form

p1(x)+ p2(x)e−z4x + p3(x)e

−2z4x = 0,(4.1)

which has to hold simultaneously for allx ≥ 0. The expressionsp1, p2, and p3 denotesome polynomials inx, which depend on thezi ’s, bi ’s, andai j ’s. Since the functions{1,e−z4x,e−2z4x} are independent over the ring of polynomials, equation (4.1) can only besatisfied if each of thepi ’s is 0. This again yields that all coefficients of thepi ’s have to bezero. To proceed in our analysis we list all terms appearing in equation (3.1):

∂

∂xF(x, z) = (−z2z4+ z3− z3z4x)e−z4x,

∇zF(x, z) = (1,e−z4x, xe−z4x, (−z2x − z3x2)e−z4x

),

∂2

∂zi ∂zjF(x, z) = 0, for 1≤ i, j ≤ 3,

∂

∂z4∇zF(x, z) = (

0,−xe−z4x,−x2e−z4x, (z2x2+ z3x3)e−z4x).

Finally we need the relation

∫ x

0ηme−z4η dη = −qm(x)e

−z4x + m!

zm+14

, m= 0,1,2, . . . ,

where

qm(x) =m∑

k=0

m!

(m− k)!

xm−k

zk+14

is a polynomial inx of orderm.


First we shall analyzep1. The terms that contribute top1 are those containing

∂

∂z1F(x, z) and

∂

∂z1F(x, z)

∫ x

0

∂

∂zjF(η, z)dη, for 1≤ j ≤ 4.

Actually p1 is of the form

p1(x) = a11x + · · · + b1,

where· · · denotes terms of zero order inx containing the factorsa1 j = aj 1, for 1≤ j ≤ 4.It follows thata11 = 0. But the matrix(ai j ) has to be nonnegative definite, so necessarily

a1 j = aj 1 = 0, for all 1≤ j ≤ 4,

and therefore alsob1 = 0. Thusp1 is done.The contributing terms top3 are those containing∂

∂ziF(x, z)

∫ x0

∂∂zj

F(η, z)dη, for 2 ≤i, j ≤ 4. But observe that the degree ofp3 and p2 depends on whetherz2 or z3 are equalto zero or not. Hence we have to distinguish between the four cases:

i. z2 6= 0, z3 6= 0 iii. z2 = 0, z3 6= 0ii. z2 6= 0, z3 = 0 iv. z2 = z3 = 0.

Case i. The degree ofp3 is 4. The fourth-order coefficient containsa44; that is,

p3(x) = a44z 2

3

z4x4+ · · · ,

where· · · denotes terms of lower order inx. Hence

a4 j = aj 4 = 0, for 1≤ j ≤ 4.

The degree ofp3 reduces to 2. The second-order coefficient isa33/z4. Hencea3 j = aj 3 = 0, for 1 ≤ j ≤ 4. It remains thatp3(x) = (a22/z4). Thus thediffusion matrix(ai j ) is zero. This implies that(σi j ) is zero, independent of thechoice ofd (the number of Brownian motions in equation (2.1)). Now we can writep2:

p2(x) = −b4z3x2+ (b3− b4z2+ z3z4)x + b2+ z2z4− z3.

It follows thatb4 = 0 and

b2 = z3− z2z4,

b3 = −z3z4.

For the other three cases we will need the following lemma, which is a direct consequenceof the occupation times formula, see Revuz and Yor (1994, Cor. 1.6, Chap. VI).


LEMMA 4.4. Using the same notation as in equation (2.1), it holds for1≤ i ≤ 4 that

aii 1{Zi=0} = bi 1{Zi=0} = 0, dt ⊗ dP-a.s.

As a consequence, since we are characterizinga andb up to adt⊗dP nullset, we may andwill assume thatzi = 0 impliesai j = aji = bi = 0, for 1≤ j ≤ 4 andi = 2,3.

Case ii. We have∇zF(x, z) = (1,e−z4x, xe−z4x,−z2xe−z4x). Hence the degree ofp3 is

2. Sincea3 j = aj 3 = b3 = 0, for 1≤ j ≤ 4, the second-order coefficient comesfrom

−a44∂

∂z4F(x, z)

∫ x

0

∂

∂z4F(η, z)dη = a44

z 22

z4x2e−2z4x + · · · ,

where· · · denotes terms of lower order inx. Hencea4 j = aj 4 = 0, for 1≤ j ≤ 4.The polynomialp3 reduces top3(x) = (a22/z4). It follows that also in this casethe diffusion matrix(ai j ) is zero. From Case i we derive immediately that now

p2(x) = −b4z2x + b2+ z2z4;

henceb2 = −z2z4 andb4 = 0.Case iii. Sincea2 j = aj 2 = b2 = 0, for 1 ≤ j ≤ 4, the zero-order coefficient ofp2

reduces to−z3. We conclude thatz2 = 0 impliesz3 = 0, so this case does notenterdt ⊗ dP-a.s.

Case iv. In this caseai j = bk = 0, for all (i, j ) 6= (4,4) andk 6= 4. Also ∂∂z4

F(x, z) =∂∂x F(x, z) = 0. Hencep2(x) = p3(x) = 0, independent of the choice ofb4 anda44.

Summarizing the four cases we conclude that equation (3.1) implies

b1 = 0 b3 = −z3z4

b2 = z3− z2z4 ai j = 0, for (i, j ) 6= (4,4).

Whereasb4 anda44 are arbitrary real, respectively nonnegative real, numbers wheneverz2 = z3 = 0. Otherwiseb4 = a44 = 0.

This has to hold fordt ⊗ dP-a.e.(t, ω). So Z is uniquely (up to indistinguishability)determined and satisfies

Z1t = Z1

0

Z2t = Z2

0 +∫ t

0

(Z3

s − Z2s Z4

s

)ds

Z3t = Z3

0 −∫ t

0Z3

s Z4s ds

Z4t = Z4

0 +∫ t

0b4

s ds+d∑

j=1

∫ t

0σ 4 j

s dWjs


for some progressively measurable processesb4 andσ 4 j , j = 1, . . . ,d, being compatiblewith equation (2.2) and vanishing outside the set{(t, ω) | Z2

t (ω) = Z3t (ω) = 0}.

Note thatZ2 andZ3 satisfy (path-wise) a system of linear ODEs with continuous coeffi-cients. Hence they are indistinguishable from zero onÄ0. So the statement of the theoremis proved onÄ0. It remains to prove it onÄ1 := Ä \Ä0.

Introduce the stopping timeτ := inf{s > 0 | Z2s = Z3

s = 0}. We have just argued thatÄ0 ⊂ {τ = 0}. By continuity of Z the reverse inclusion also holds, henceÄ0 = {τ = 0}.The stopped processZτ =: Y satisfies (path-wise) the following system of linear stochasticintegral equations

Y1t = Z1

0(4.2)

Y2t = Z2

0 +∫ t∧τ

0

(Y3

s − Y2s Z4

0

)ds

Y3t = Z3

0 +∫ t∧τ

0Y3

s Z40 ds

Y4t = Z4

0, for 0≤ t <∞.

We have used the fact thatb4 = b41[τ,∞] andσ 4 = σ 41[τ,∞] . Then the last equation followsfrom an elementary property of the stopped stochastic integral:

d∑j=1

∫ t∧τ

0σ 4 j

s dWjs =

d∑j=1

∫ t

0

(σ 4 j

s 1[τ,∞])

1[0,τ ] dWjs =

d∑j=1

∫ t

0σ 4 j

s 1[τ ] dWjs = 0,

by continuity ofW.The system (4.2) has the unique solution for 0≤ t <∞:

Y1t = Z1

0

Y2t = Z2

0e−Z40(t∧τ) + Z3

0(t ∧ τ)e−Z40(t∧τ)

Y3t = Z3

0e−Z40(t∧τ)

Y4t = Z4

0.

SinceZ = Y on the stochastic interval [0, τ ] and sinceYt 6= 0, ∀t < ∞, P-a.s. onÄ1, itfollows by the continuity ofZ, thatÄ1 = {τ > 0} = {τ = ∞}. Inserting this in the abovesolution, the theorem is proved also onÄ1.

5. E-CONSISTENT ITO PROCESSES

Note that by definitionZ is consistent if and only ifP is a martingale measure for thediscounted bond price processes. We could generalize this definition and call a state spaceprocessZ e-consistentif there exists an equivalent martingale measureQ. Then obvi-ously consistency implies e-consistency, and e-consistency implies the absence of arbitrageopportunities, as it is well known.

In the case where the filtration is generated by the Brownian motionW (that is,(Ft ) =(FW

t )), we can give the following stronger result:


PROPOSITION5.1. If (Ft ) = (FWt ), then any e-consistent Ito process Z is of the form

as stated in Theorem 4.1. In particular the corresponding interest rate model is purelydeterministic.

Proof. Let Z be an e-consistent Itô process underP, and letQ be an equivalent martin-gale measure. Since(Ft ) = (FW

t ), we know that allP-martingales have the representationproperty relative toW. By Girsanov’s theorem it follows therefore thatZ remains an Itôprocess underQ. In particular,Z is a consistent state space Itô process with respect to thestochastic basis(Ä,F, (FW

t )0≤t<∞,Q). HenceZ is of the form as stated in Theorem 4.1.SinceFW

0 consists of sets of measure 0 or 1, the processesZ1, Z2, Z3 are (after changingthe Zi

0’s on a set of measure 0) purely deterministic and therefore also(r (t, . ))0≤t<∞; seeRemark 4.2.

REFERENCES

BIS (1999): “Zero-Coupon Yield Curves: Technical Documentation.” Bank for International Settle-ments, Basle, March 1999.

BJORK, T., and B. J. CHRISTENSEN(1999): Interest Rate Dynamics and Consistent Forward RateCurves,Math. Finance, this issue, 323–348.

DELBAEN, F., and W. SCHACHERMAYER (1994): A General Version of the Fundamental Theorem ofAsset Pricing,Math. Annalen300, 463–520.

FILIPOVIC, D. (1998): Exponential-Polynomial Families and the Term Structure of Interest Rates,unpublished manuscript.

HEATH, D., R. JARROW, and A. MORTON (1992): Bond Pricing and the Term Structure of InterestRates: A New Methodology for Contingent Claims Valuation,Econometrica60, 77–105.

JACOD, J., and A. N. SHIRYAEV (1987):Limit Theorems for Stochastic Processes. Berlin: Springer.

NELSON, C., and A. SIEGEL (1987): Parsimonious Modeling of Yield Curves,J. of Business60,473–489.

REVUZ, D., and M. YOR (1994): Continuous Martingales and Brownian Motion, 2nd ed. Berlin:Springer.

a note on the nelson–siegel family

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