EXPONENTIAL-POLYNOMIAL FAMILIES AND THE TERM STRUCTURE OF INTEREST RATES ´ DAMIR FILIPOVIC ¨ ¨ DEPARTMENT OF MATHEMATICS, ETH, RAMISTRASSE 101, CH-8092 ZURICH, SWITZERLAND. E-MAIL: [email protected]

Abstract. Exponential-polynomial families like the Nelson–Siegel or Svensson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with inter-temporal modelling. We characterize the consistent Itˆ o 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 in particular diffusion processes. We show that there is a strong limitation on their choice. Bounded exponential-polynomial families should rather not be used for modelling the term structure of interest rates. Keywords: consistent Itˆ o process, diffusion process, exponential-polynomial family, 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 furthermore by the central banks as 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 parameterized families of smooth curves {F ( . , z)}z∈Z , where Z ⊂ 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 7→ F (x, z) to the observed data is attained, where x ≥ 0 denotes time to maturity. In that sense z represents the current state of the economy taking values in the state space Z. Examples are the Nelson–Siegel (1987) family with curve shape [0, ∞) 3 x 7→ FN S (x, z) = z1 + (z2 + z3 x)e−z4 x ,

z ∈ Z ⊂ R4

and the Svensson (1994) family, an extension of Nelson–Siegel, [0, ∞) 3 x 7→ FS (x, z) = z1 + (z2 + z3 x)e−z5 x + z4 xe−z6 x ,

z ∈ Z ⊂ R6 .

Table 1 gives an overview of the fitting procedures used by some selected central banks. It is taken from a discussion paper of the Deutsche Bundesbank, see Schich (1996). Despite the flexibility and low number of parameters of FN S and FS , their choice is somewhat arbitrary. We shall discuss them from an inter-temporal point of view: A lot of cross-sectional data, i.e. daily estimations of z, is available. Therefore it would be natural to ask for the stochastic evolution of the parameter z over time. But then there exist economic constraints based on no arbitrage considerations. 1

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central bank Finland France Norway Spain Sweden Switzerland UK USA

curve fitting procedure Svensson Nelson–Siegel, Svensson Svensson, cubic splines Nelson–Siegel, Svensson Svensson Svensson Nelson–Siegel, Svensson, cubic splines Nelson–Siegel, Svensson, smoothing splines

Table 1. Overview of forward rate curve fitting procedures used by some selected central banks

Following Bj¨ ork and Christensen (1997), instead of FN S and FS we consider general exponential-polynomial families containing curves of the form [0, ∞) 3 x 7→ F (x, z) =

ni K ³X X i=1

´ zi,µ xµ e−zi,ni +1 x ,

z ∈ Z ⊂ RN ,

µ=0

i.e. linear combinations of exponential functions exp(−zi,ni +1 x) over some polynomials of degree ni ∈ N0 . Obviously FN S and FS are of this type. We replace then z by an Itˆ o process Z = (Zt )t≥0 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–Morton framework with deterministic volatility structure, Bj¨ ork and Christensen (1997) showed that the exponential-polynomial families are in a certain sense to large to carry an interest rate model. This result has been generalized for the Nelson–Siegel family in Filipovi´c (1998), including stochastic volatility structure. Expanding the methods used in 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 Itˆ o processes consistent with a given parameterized family of forward rate curves. Consistent Itˆo processes provide an arbitrage free interest rate model when driving the parameterized family. They are characterized in terms of their drift and diffusion coefficients. By solving an inverse problem we get the main result for Itˆ o processes consistent with bounded exponential-polynomial families, stated in section 3. As it is shown 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 reads 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 e-consistent with the Nelson– Siegel family. Furthermore we identify the diffusion process e-consistent with the

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Svensson family, which contains just one non deterministic component. The corresponding short rate model is shown to be essentially the Vasicek model. We conclude that bounded exponential-polynomial families, in particular FN S and FS , should rather not be used for modelling the term structure of interest rates. ˆ processes 2. Consistent Ito For the stochastic background and notations we refer to Revuz and Yor (1994) and Jacod and Shiryaev (1987). Let (Ω, F, (Ft )0≤t 0 | |Xt − Yt | = 0}. Define T (n, p) := inf{t > S(n, p) | |Xt − Yt | > 0} and by induction S(n, p + 1) := inf{t > S(n, p) | |Xt − Yt | = 0 and supS(n,p)≤s≤t |Xs − Ys | > 2−n }. Then by continuity S we have limp→∞ S(n, p) = ∞ for all n ∈ N and it follows that {X = Y } = n,p∈N [S(n, p), T (n, p)]. Now proceed as in Jacod and Shiryaev (1987), Lemma I.1.31, to find the sequences (Sn ) and (Tn ) with the desired properties. From above we have 1{X=Y } (γ X − γ Y )2 = 0, dt ⊗ dP-a.s. For any 0 ≤ t < ∞ R T ∧t therefore Snn∧t (γsX − γsY ) dWs = 0, P-a.s. Hence Z Tn ∧t 0 = (X − Y )Tn ∧t − (X − Y )Sn ∧t = (βsX − βsY ) ds, P-a.s. Sn ∧t

We conclude Z t XZ X Y 1{Xs =Ys } (βs − βs ) ds = 0

n∈N

Tn ∧t

Sn ∧t

(βsX − βsY ) ds = 0,

for 0 ≤ t < ∞, P-a.s.

Using the same arguments as in the proof of Proposition 3.2 in Filipovi´c (1998), we derive the desired result. Secondly there are listed two results in matrix algebra. Lemma 4.2. Let γ = (γi,j ) be a N × d-matrix and define the symmetric nonnegPd ative definite N × N -matrix α := γγ ∗ , i.e. αi,j = αj,i = λ=1 γi,λ γj,λ . Let I and J denote two arbitrary subsets of {1, . . . , N }. Define XX αI,J = αJ,I := αi,j . j∈J i∈I

√ √ Then it holds that αI,I ≥ 0 and |αI,J | ≤ αI,I αJ,J . P Proof. For 1 ≤ λ ≤ d define γI,λ := i∈I γi,λ . Then by definition αI,J =

d XXX j∈J i∈I λ=1

γi,λ γj,λ =

d ³X X λ=1

i∈I

γi,λ

´³ X j∈J

´ γj,λ =

d X λ=1

γI,λ γJ,λ .

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Hence αI,I =

d X

2

(γI,λ ) ≥ 0

λ=1

and by Schwarz inequality v v u d u d uX uX √ √ 2 2 |αI,J | ≤ t (γI,λ ) t (γJ,λ ) = αI,I αJ,J . λ=1

λ=1

Lemma 4.3. Let α = (αi,j ) be a n×n-matrix, n ∈ N, which is diagonally dominant from the right, i.e. |αi,i | ≥

|αi,i | >

n X

|αi,j |

j=1 j6=i n X

|αi,j |,

j=i+1

³ set

´ · · · := 0 ,

n X j=n+1

for all 1 ≤ i ≤ n. Then α is regular. Proof. The proof is a slight modification of an argument given in Schwarz (1986, Theorem 1.5). Pn Gaussian elimination: by assumption |α1,1 | > j=2 |α1,j | ≥ 0, in particular α1,1 6= 0. If n = 1 we are done. If n > 1, the elimination step αi,1 (1) αi,j := αi,j − α1,j , 2 ≤ i, j ≤ n, α1,1 (1)

leads to the (n − 1) × (n − 1)-matrix α(1) = (αi,j )2≤i,j≤n . We show that α(1) is diagonal dominant from the right. If αi,1 = 0, there is nothing to prove for the i-th row. Let αi,1 6= 0, for some 2 ≤ i ≤ n. We have ¯ ¯ ¯ αi,1 ¯ (1) ¯ |α1,j |, 2 ≤ j ≤ n. ¯ |αi,j | ≥ |αi,j | − ¯ α1,1 ¯ Therefore ¯ ¯ n ¯ n n n ¯ n X X X ¯ ¯ X ¯ αi,1 ¯ X αi,1 (1) (1) ¯ ¯ ¯ ¯ |αi,j | ≤ |αi,j | = |αi,j | + ¯ |α1,j | ¯αi,j − α1,1 α1,j ¯ ≤ ¯ α 1,1 j=i+1 j=2 j=2 j=2 j=2 j6=i n X

j6=i

j6=i

¯ n ¯ ´ ¯ αi,1 ¯ ³ X ¯ ¯ = |αi,j | − |αi,1 | + ¯ |α1,j | − |α1,i | α1,1 ¯ j=2 j=1 j6=i

¯ ¯ ´ ¯ αi,1 ¯ ³ ¯ ¯ |α1,1 | − |α1,i | < |αi,i | − |αi,1 | + ¯ α1,1 ¯ ¯ ¯ ¯ αi,1 ¯ ¯ ¯ |α1,i | ≤ |α(1) |. = |αi,i | − ¯ i,i α1,1 ¯ Proceed inductively to α(2) , . . . , α(n−1) .

j6=i

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5. The case EP B(1, n) We will treat the case K = 1 separately, since it represents a key step in the proof of the general EP B(K, n) case. For simplicity we shall skip the index i = 1 and write n = n1 ∈ N0 , p = p1 , bj = b1,j , ai,j = a1,i;1,j , etc. In particular we use the notation of section 2 with N = n + 2. Lemma 5.1. Let n ∈ N0 and Z be as above. If Z is consistent with EP B(1, n), then necessarily n+1

Zti = Z0i e−Z0

t

n+1

Ztn = Z0n e−Z0

n+1

+ Z0i+1 te−Z0

t

t



Z

Ztn+1 = Z0n+1 + 

t

bn+1 ds + s 0

d Z X j=1

 t

σsn+1,j dWsj  1Ω0 ,

0

for 0 ≤ i ≤ n − 1 and 0 ≤ t < ∞, P-a.s., where Ω0 := {p(Z0 ) = 0}. Consequently, if Z is consistent with EP B(1, n), then {p(Z) = 0} = [0, ∞) × Ω0 . Hence {Z n+1 6= Z0n+1 } ⊂ {p(Z) = 0}. Therefore we may state Corollary 5.2. If Z is consistent with EP B(1, n), then Z is as in the lemma and n+1

F (x, Z) = p(x, Z)e−Z0

x

.

Hence the corresponding interest rate model is quasi deterministic, i.e. all randomness remains F0 -measurable. Proof of Lemma 5.1. Let n ∈ N0 and let Z be an Itˆ o process consistent with EP B(1, n). Fix a point (t, ω) in [0, ∞) × Ω. For simplicity write zi for Zti (ω), i ai,j for ai,j t (ω) and bi for bt (ω). We are going to expand equation (1) in the point z = (z0 , . . . , zn+1 ). The terms appearing are ¶ µ ∂ ∂ (11) F (x, z) = p(x, z) − zn+1 p(x, z) e−zn+1 x ∂x ∂x ( 0≤i≤n xi e−zn+1 x , ∂ F (x, z) = (12) −z x ∂zi −xp(x, z)e n+1 , i = n + 1  0, 0 ≤ i, j ≤ n ∂ 2 F (x, z) ∂ 2 F (x, z)  i+1 −zn+1 x = = −x e (13) , 0 ≤ i ≤ n, j = n + 1  ∂zi ∂zj ∂zj ∂zi  2 x p(x, z)e−zn+1 x , i = j = n + 1. Finally for m ∈ N0 we need the relation ( Z x m! −qm (x)e−zn+1 x + zm+1 , zn+1 6= 0 m −zn+1 η n+1 η e dη = m+1 x 0 zn+1 = 0, m+1 , Pm xm−k m! where qm (x) = k=0 (m−k)! is a polynomial in x of order m. z k+1

(14)

n+1

Let’s suppose first that zn+1 6= 0. Thus, subtracting of (1) we get a null equation of the form

∂ ∂x F (x, Z)

q1 (x)e−zn+1 x + q2 (x)e−2zn+1 x = 0,

from both sides (15)

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which has to hold simultaneously for all x ≥ 0. The appearing polynomials q1 and q2 depend on the zi ’s, bi ’s and ai,j ’s. Equality (15) implies q1 = q2 = 0. We have to distinguish the two cases p(z) 6= 0 and p(z) = 0. Let’s suppose first the former is true, then there exists an index i ∈ {0, . . . , n} such that zi 6= 0. Set m := max{i | zi 6= 0}. With regard to (12), (13) and (14) it follows that deg q2 = 2m + 2. In particular q2 (x) = an+1,n+1

2 zm x2m+2 + . . . , zn+1

where . . . denotes terms of lower order in x. Hence an+1,n+1 = 0. But the matrix a has to be nonnegative definite, so necessarily an+1,j = aj,n+1 = 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 ⊗ 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 am,m 2m q2 (x) = x + ... . 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 get that the diffusion matrix a is equal to zero and hence q2 = 0 is fulfilled. Using once more Lemma 4.1 we may assume bi = 0, for m + 1 ≤ i ≤ n. With regard to (11) and (12), q1 reduces therefore to q1 (x) = −bn+1 zm xm+1 + . . . . It follows bn+1 = 0 and it remains m

q1 (x) = (bm + zn+1 zm )x +

m−1 X

(bi − zi+1 + zn+1 zi )xi

i=0

= (bn + zn+1 zn )xn +

n−1 X

(bi − zi+1 + zn+1 zi )xi .

i=0

If p(z) = 0, i.e. z0 = · · · = zn = 0, we may assume ai,j = aj,i = bi = 0, 0 ≤ j ≤ n + 1, for all i ≤ n. But this means that q1 = q2 = 0, independently of the choice of bn+1 and an+1,n+1 . It remains the case where zn+1 = 0. By the boundedness assumption z ∈ Z it follows that z1 = · · · = zn = 0, see (5). 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 (1) reduces to 0 = b0 − a0,0 x and therefore b0 = a0,0 = 0. Summarizing all cases we conclude bi = −zn+1 zi + zi+1 ,

0≤i≤n−1

bn = −zn+1 zn ai,j = 0,

for (i, j) 6= (n + 1, n + 1).

Whereas bn+1 and an+1,n+1 are arbitrary real, resp. nonnegative real, numbers whenever p(z) = 0. Otherwise bn+1 = an+1,n+1 = 0.

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The rest of the proof is analogous to the proof of Proposition 4.1 in Filipovi´c (1998). 6. The general case EP B(K, n) Using again the notation of section 3 we give the proof of the main result for the case K ≥ 2. Proof of Theorem 3.2. Let K ≥ 2, n = (n1 , . . . , nK ) ∈ NK 0 , and let Z be consistent with EP B(K, n). As in the proof of Lemma 5.1 we fix a point (t, ω) in [0, ∞) × Ω and use the shorthand notation zi,µ for Zti,µ (ω), ai,µ;j,ν for ai,µ;j,ν (ω) and bi,µ for t i,µ bt (ω), etc. Since we are characterizing a and b up to a dt ⊗ dP-nullset, we assume that (t, ω) is chosen outside of an exceptional dt ⊗ dP-nullset. In particular the lemmas from section 4 shall apply each time we use them. The strategy is the same as for the case K = 1. Thus expanding equation (1) in the point z = (z1,0 , . . . , zK,nK +1 ) to get a linear combination of (ideally) linearly independent exponential functions over the ring of polynomials K X

qi (x)e−zi,ni +1 x +

i=1

X

qi,j (x)e−(zi,ni +1 +zj,nj +1 )x = 0.

(16)

1≤i≤j≤K

Concluding that all polynomials qi and qi,j have to be zero. The main difference to the case K = 1 consists in the possibly multiple occurrence of the cases i) zi,ni +1 = zj,nj +1 , for i 6= j, ii) 2zi,ni +1 = zj,nj +1 + zk,nk +1 , iii) 2zi,ni +1 = zj,nj +1 , iv) zi,ni +1 = zj,nj +1 + zk,nk +1 , for some indices 1 ≤ i, j, k ≤ K. It turns out that the lemmas in section 4 and the boundedness assumption z ∈ Z are good enough to settle these four cases. Let’s suppose first that pi (z) 6= 0, for all i ∈ {1, . . . , K}. To settle case i), let ∼ denote the equivalence relation defined in (3). After re-parameterization if necessary we may assume that ˜ {1, . . . , K}/∼ = {[1], . . . , [K]} ˜ ˜ and z1,n1 +1 < · · · < zK,n ˜ ˜ +1 for some integer K ≤ K. Write I := {1, . . . , K}. In K view of Lemma 4.1 we may assume aj,nj +1;j,nj +1 = ai,ni +1;i,ni +1 and bj,nj +1 = bi,ni +1 for all j ∈ [i], i ∈ I.

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Claim 1. ai,ni +1;i,ni +1 = 0, for all i ∈ I. Expression (16) takes the form X X q˜i (x)e−zi,ni +1 x + q˜i,j (x)e−(zi,ni +1 +zj,nj +1 )x = 0, i∈I

i,j∈I i≤j

for some polynomials q˜i and q˜i,j . Taking into account cases ii), iii) and iv) this representation may not be unique. However if for an index i ∈ I there exist no j, k ∈ I such that 2zi,ni +1 = zj,nj +1 + zk,nk +1 or 2zi,ni +1 = zj,nj +1 (in particular zi,ni +1 6= 0) then we have P 2 j∈I[i],µm zj,µm 2µm +2 q˜i,i (x) = ai,ni +1;i,ni +1 x + ..., zi,ni +1

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where µm := max{ν | ν ≤ nj and zj,ν 6= 0 for some j ∈ [i]} ∈ N0 . Hence ai,ni +1;i,ni +1 = 0 and Claim 1 is proved in this case. If zi,ni +1 = 0 then by Lemma 4.1 also ai,ni +1;i,ni +1 = 0 and we are done again. Split I into two disjoint subsets I1 and I2 , where I1 := {i ∈ I | zi,ni +1 6= 0 and there exist j, k ∈ I, such that 2zi,ni +1 = zj,nj +1 + zk,nk +1 or 2zi,ni +1 = zj,nj +1 } I2 := I \ I1 . ˜ Observe that zK,n ˜ ˜ +1 > 0 implies K ∈ I2 and z1,n1 +1 < 0 implies 1 ∈ I2 . Since at K least one of these events has to happen, the set I2 is not empty. We have shown above that ai,ni +1;i,ni +1 = 0, for i ∈ I2 . If I1 is not empty, we will show that for each i ∈ I1 , the parameter zi,ni +1 can be written as a linear combination of zj,nj +1 ’s with j ∈ I2 . From which it follows by Lemma 4.1 and Lemma 4.2 that ai,ni +1;i,ni +1 = 0 for all i ∈ I1 . We proceed as follows. Write I1 = {i1 , . . . , ir } with i1 < · · · < ir . For each ik ∈ I1 take one linear equation of the form   zi1 ,ni1 +1   ..   .  ¡ ¢  ∗, . . . , ∗, 2, ∗, . . . , ∗  zik ,nik +1  = αk ,   ..   . zir ,nir +1 where ∗ stands for 0 or −1, but at most one −1 on each side of 2. The αk on the right hand side is either 0 or zi,ni +1 or zi,ni +1 + zj,nj +1 for some indices i, j ∈ I2 . Obviously zir ,nir +1 > 0 implies αr 6= 0 and zi1 ,ni1 +1 < 0 implies α1 6= 0. Hence we get the system of linear equations      zi1 ,ni1 +1 2 ∗ ... ∗ α1      . .  . . . .. . . ..   .. ∗   ..     =  , . .    .  .. . . . . . ∗   ..   ..  . ∗ ... ∗ 2 αr zir ,nir +1 where the vector on the right hand side is not zero. The matrix on the left hand side is regular which is seen from Lemma 4.3. Therefore Claim 1 is proved. Claim 2. aj,nj +1;k,ν = ak,ν;j,nj +1 = 0, for 0 ≤ ν ≤ nk , for all 1 ≤ j, k ≤ K. In view of (17), this follows immediately from Claim 1 and Lemma 4.2. Analogous to the notation introduced in (4) we set X b[i],µ := bj,µ j∈I[i],µ

σ[i],µ;λ :=

X

σj,µ;λ

j∈I[i],µ

a[i],µ;k,ν :=

X

j∈I[i],µ

aj,µ;k,ν ,

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for 0 ≤ µ ≤ n[i] , 0 ≤ ν ≤ nk , 1 ≤ k ≤ K, 1 ≤ λ ≤ d, i ∈ I, and X X a[i],µ;[k],ν := aj,µ;l,ν , l∈I[k],ν j∈I[i],µ

for 0 ≤ µ ≤ n[i] , 0 ≤ ν ≤ n[k] , i, k ∈ I. Claim 3. If z[i],µ = 0, for two indices i ∈ I and µ ∈ {0, . . . , n[i] }, then b[i],µ = a[i],µ;[i],µ = a[i],µ;k,ν = ak,ν;[i],µ = 0, for all 0 ≤ ν ≤ nk , 1 ≤ k ≤ K. Pd 2 Notice that a[i],µ;[i],µ = λ=1 σ[i],µ;λ . Hence Claim 3 follows by Lemma 4.1 and Lemma 4.2. Claim 4. bi,ni +1 = 0, for all i ∈ I such that p[i] (z) 6= 0. Suppose first that zi,ni +1 6= 0, for all i ∈ I. Let i ∈ I such that p[i] (z) 6= 0, and let’s assume there exist no j, k ∈ I with zi,ni +1 = zj,nj +1 + zk,nk +1 . How does the polynomial q˜i look like? With regard to (17), Claim 2, Lemma 4.1 and equalities (11) to (14) the contributing terms are ∧nj ∧nj ´ ³ µm ´´ ³³ µm X X ∂ −zj,nj +1 x µ−1 − zi,ni +1 pj (x, z)e = zj,µ x zj,µ xµ e−zi,ni +1 x ∂x µ=1 µ=0

X

nj +1

µ=0

∧nj ∧nj ³³ µm ´ ³ µm ´´ X X ∂ µ bj,µ F (x, z) = bj,µ x − bi,ni +1 zj,µ xµ+1 e−zi,ni +1 x ∂zj,µ µ=0 µ=0

(18) and 1³ X ∂ −2 aj,µ;k,ν F (x, z) 2 µ=0 ∂zj,µ nj

∧nj ³ µm X =− aj,µ;k,ν µ=0

nk ! nk +1 zk,n k +1

Z 0

x

´ ∂ F (η, z) dη ∂zk,ν

³ ´ ´ xµ e−zi,ni +1 x − polynomial e−(zi,ni +1 +zk,nk +1 )x , in x

(19)

for 0 ≤ ν ≤ nk , for all 1 ≤ k ≤ K and j ∈ [i]. We have used the integer µm := max{λ | λ ≤ nl and zl,λ 6= 0 for some l ∈ [i]}. Define µ ˜m := max{λ | λ ≤ n[i] and z[i],λ 6= 0} ∈ N0 . Obviously µ ˜m ≤ µm . Furthermore a[i],µ;k,ν = 0, for all µ ˜m < µ ≤ n[i] , by Claim 3. Thus summing up the above expressions over j ∈ [i] we get q˜i (x) = −bi,ni +1 z[i],˜µm xµ˜m +1 + . . . , from which eventually bi,ni +1 = 0. To proceed, we split I into two disjoint subsets J1 and J2 , where J1 := {i ∈ I | there exist j, k ∈ I, such that zi,ni +1 = zj,nj +1 + zk,nk +1 and zj,nj +1 > 0 and zk,nk +1 > 0} J2 := I \ J1 . Notice that in any case 1 ∈ J2 . We have shown above that for each i ∈ J2 such that zi,ni +1 is not the sum of two other zj,nj +1 ’s it follows bi,ni +1 = 0. We will show

EXPONENTIAL-POLYNOMIAL FAMILIES

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now that bi,ni +1 = 0, for all i ∈ J2 . Let i ∈ J2 and assume there exist j, k ∈ I with zi,ni +1 = zj,nj +1 + zk,nk +1 . Then necessarily one of the summands is strictly less than zero. Without loss of generality zj,nj +1 < 0. By the boundedness assumption z ∈ Z we have p[j] (z) = 0, see (5). Thus a[j],µ;[j],µ = 0 by Claim 3 and therefore a[j],µ;k,ν = 0, for all 0 ≤ µ ≤ n[j] , 0 ≤ ν ≤ nk , 1 ≤ k ≤ K. The contributing terms to the polynomial in front of e−zi,ni +1 x , i.e. q˜i + q˜j,k + . . . , are those in (18) and (19) and also Z ³ ∂F (x, z) Z x ∂F (η, z) 1 ∂F (x, z) x ∂F (η, z) ´ − 2 al,µ;m,ν dη + dη 2 ∂zl,µ ∂zm,ν ∂zm,ν ∂zl,µ 0 0 Z x Z x ³ ´ µ −zj,nj +1 x ν −zk,nk +1 η ν −zk,nk +1 x = −al,µ;m,ν x e η e dη+x e η µ e−zj,nj +1 η dη , 0

0

(20) for 0 ≤ µ ≤ nl , 0 ≤ ν ≤ nm , l ∈ [j], m ∈ [k]. Fix µ, m and ν. Summing the right hand side of (20) over l ∈ I[j],µ gives zero. Hence the terms in (20) actually don’t contribute to the meant polynomial. Arguing in this way for all j, k ∈ I with the property zi,ni +1 = zj,nj +1 + zk,nk +1 it finally follows as before that bi,ni +1 = 0. If J1 is not empty, we show that for each i ∈ J1 , the parameter zi,ni +1 can be written as a linear combination of zj,nj +1 ’s with j ∈ J2 . Concluding from Lemma 4.1 that bi,ni +1 = 0, for all i ∈ J1 . We proceed as follows. Write J1 = {i1 , . . . , ir0 } with i1 < · · · < ir0 . For each ik ∈ J1 take one linear equation of the form   zi1 ,ni1 +1 ..     .  ¡ ¢ 0  ∗, . . . , ∗, 1, 0, . . . , 0   zik ,nik +1  = αk ,   ..   . zir0 ,ni 0 +1 r

where ∗ stands for 0 or −1, but at most two of them are −1. The αk0 on the right hand side is either 0 or zi,ni +1 or zi,ni +1 + zj,nj +1 for some indices i, j ∈ J2 with zi,ni +1 > 0 and zj,nj +1 > 0. Obviously α10 is of the latter form. Hence we get the system of linear equations      zi1 ,ni1 +1 1 0 ... 0 zi,ni +1 + zj,nj +1 ..    .   α20 ∗ . . . . . . ..     .    = , . . .    . ..  .. . . . . . 0   ..   0 αr0 zir0 ,ni 0 +1 ∗ ... ∗ 1 r

for some i, j ∈ J2 . On the left hand side stands a lower-triangular matrix, which is therefore regular. Hence Claim 4 is proved in case where zi,ni +1 6= 0, for all i ∈ I. Assume now that there is an index i ∈ I with zi,ni +1 = 0. Then i ∈ J2 . We have to make sure that also in this case bj,nj +1 = 0, for all j ∈ J2 . Clearly bi,ni +1 is zero by Lemma 4.1. The problem is that zj,nj +1 = zi,ni +1 + zj,nj +1 for all j ∈ J2 . But following the lines above it is enough to show a[i],µ;[i],µ = 0, for all 0 ≤ µ ≤ n[i] . From the boundedness assumption z ∈ Z we know that p[i] (z) = z[i],0 , see (5). Hence a[i],µ;[i],µ = 0, for 1 ≤ µ ≤ n[i] . Suppose there is no pair of indices j, k ∈ I \ {i} with zj,nj +1 + zk,nk +1 = 0. Summing up the contributing terms in

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(18) and (19) over j ∈ [i] we get the polynomial in front of e0 , i.e. q˜i (x) + q˜i,i (x) = −a[i],0;[i],0 x + . . . ,

(21)

hence a[i],0;[i],0 = 0. If there exist a pair of indices j, k ∈ I \ {i} with zj,nj +1 + zk,nk +1 = 0, then one of these summands is strictly less than zero. Arguing as before, the polynomial in front of e0 remains of the form (21) and again a[i],0;[i],0 = 0. Thus Claim 4 is proved. If there is an index i ∈ {1, . . . , K} with pi (z) = 0, then by Lemma 4.1 we may assume ai,µ;i,µ = bi,µ = 0, for all 0 ≤ µ ≤ ni . In view of Lemma 4.2 it is seen that therefore none of the terms including the index i do appear in (16). In particular ai,ni +1;i,ni +1 and bi,ni +1 can be chosen arbitrarily without affecting equation (16). This means on the other hand that w.l.o.g. we may skip i and proceed, after a re-parameterization if necessary, with the new index set {1, . . . , K − 1} as above. This all has to hold for dt ⊗ dP-a.e. (t, ω). Hence (9) and (10) are proved. A closer look to the proof of (9), i.e. Claim 1, shows that the boundedness assumption z ∈ Z was not explicitly used there. Whence Remark 3.3. Now let v ≥ 0 be a rational number and let Tv := inf{t > v | pi (Zt ) = 0 or p[i] (Zt ) = 0} denote the debut of the optional set [v, ∞[∩Ai . By (9) and (10) and the continuity of Z we have that Z i,ni +1 is P-a.s. constant on [v, v + Tv ], hence P-a.s. constant on every such interval [v, v + Tv ]. Since every open interval where pi (Zt ) 6= 0 or p[i] (Zt ) 6= 0 is covered by a countable union of intervals [v, v + Tv ] and by continuity of Z the first part of the theorem is proved. Let τ be a stopping time with [τ ] ∈ D0 and P(τ < ∞) > 0. Define the stopping time τ 0 (ω) := inf{t ≥ τ | (t, ω) 6∈ D0 }. By the continuity of Z we conclude that τ < τ 0 on {τ < ∞}. Choose a point (t, ω) in [τ, τ 0 [, and use shorthand notation as above. By definition of D0 we can exclude the cases zi,ni +1 = zj,nj +1 or ˜ = K, 2zi,ni +1 = zj,nj +1 , for some indices 1 ≤ i, j ≤ K with i 6= j. In particular K hence I = {1, . . . , K}. If there is an index i ∈ I with pi (z) = 0 then argued as above ai,µ;i,µ = bi,µ = 0, for all 0 ≤ µ ≤ ni , and we may skip the index i. Hence we assume now that there is a K 0 ≤ K such that pi (z) 6= 0 and thus zi,ni +1 ≥ 0 by the boundedness assumption z ∈ Z, for all 1 ≤ i ≤ K 0 . Let I 0 := {1, . . . , K 0 }. Split I 0 into two disjoint subsets I10 and I20 , where I10 := {i ∈ I 0 | zi,ni +1 > 0 and there exist j, k ∈ I 0 , such that 2zi,ni +1 = zj,nj +1 + zk,nk +1 } I20 := I 0 \ I10 . Hence zi,ni +1 = 0 for i ∈ I 0 implies i ∈ I20 . We have shown in the proof of Claim 4 that in this case ai,µ;i,µ = 0, for all 0 ≤ µ ≤ ni . The same follows for i ∈ I20 with zi,ni +1 > 0, as it was demonstrated for the case K = 1. Now let i ∈ I10 and let l, m be in I 0 , l ≤ m, such that 2zi,ni +1 = zl,nl +1 + zm,nm +1 . Thus the polynomial in front of e−2zi,ni +1 x is qi,i + ql,m + . . . , and among the contributing terms are also those in (20). If l or m is in I20 , those are all zero. Write I10 = {i1 , . . . , ir00 } with zi1 ,ni1 +1 < · · · < zir00 ,ni 00 +1 . Then necessarily l ∈ I20 in the above representation r

−2z

x

for zi1 ,ni1 +1 . Thus the polynomial in front of e i1 ,ni1 +1 is qi1 ,i1 . It follows ai1 ,µ;i1 ,µ = 0, for all 0 ≤ µ ≤ ni1 , as it was demonstrated for the case K = 1. Proceeding inductively for i2 , . . . , ir00 , we derive eventually that ai,µ;i,µ = 0, for all 0 ≤ µ ≤ ni and i ∈ I 0 .

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17

In view of (10) we have that bi,ni +1 = 0, for all i ∈ I 0 . Thus (16) reduces to 0

K X

qi (x)e−zi,ni +1 x = 0,

i=1

with ni

qi (x) = (bi,ni + zi,ni +1 zi,ni )x

+

nX i −1

(bi,µ − zi,µ+1 + zi,ni +1 zi,µ )xµ .

µ=0

We conclude that for all 1 ≤ i ≤ K (in particular if pi (z) = 0) bi,µ = zi,µ+1 − zi,ni +1 zi,µ ,

0 ≤ µ ≤ ni − 1

bi,ni = −zi,ni +1 zi,ni ai,µ;i,µ = 0,

0 ≤ µ ≤ ni .

By continuity this holds pathwise on the semi open interval [τ (ω), τ 0 (ω)[ for almost every ω. Therefore Zτ + . is of the claimed form on [0, τ 0 − τ [. Now replace D0 by D and proceed as above. By (8) we have τ < τ 0 on {τ < ∞}, and since D ⊂ D0 , all the above results remain valid. In addition pi (z) = p[i] (z) 6= 0 and thus ai,ni +1;i,ni +1 = bi,ni +1 = 0, for all 1 ≤ i ≤ K, by the first part of the i +1 theorem. Hence up to evanescence Zτi,n = Zτi,ni +1 on [0, τ 0 −τ [, for all 1 ≤ i ≤ K. +. But this again implies τ 0 = ∞ by the continuity of Z. ˆ processes 7. e-consistent Ito An Itˆ o process Z is by definition consistent with a family {F ( . , z)}z∈Z if and only if P 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)}z∈Z if there exists an equivalent martingale measure Q. Then obviously consistency implies econsistency, and e-consistency implies the absence of arbitrage opportunities, as it is well known. In case where the filtration is generated by the Brownian motion W , i.e. (Ft ) = (FtW ), we can give the following stronger result: W Proposition 7.1. Let K ∈ N and n = (n1 , . . . , nK ) ∈ NK 0 . If (Ft ) = (Ft ), then any Itˆ o process Z, which is e-consistent with EP B(K, n), is of the form as stated in Theorem 3.2.

Proof. Let Z be an e-consistent Itˆ o process under P, and let Q be an equivalent martingale measure. Since (Ft ) = (FtW ), we know that all P-martingales have the representation property relative to W . By Girsanov’s theorem it follows therefore that Z remains an Itˆ o process under Q, which is consistent with EP B(K, n). The i,µ drift coefficients b change under Q into ˜bi,µ . Whereas bi,µ = ˜bi,µ on {ai,µ;i,µ = 0}, dt ⊗ dP-a.s. The diffusion matrix a remains the same. Therefore and since the measures dt ⊗ dQ and dt ⊗ dP are equivalent on [0, ∞) × Ω, the Itˆ o process Z is of the form as stated in Theorem 3.2. Notice that in this case the expression quasi deterministic, i.e. F0 -measurable, in Corollary 3.5 and Corollary 5.2 can be replaced by purely deterministic.

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8. The diffusion case The main result from section 3 reads much clearer for diffusion processes. In all applications the generic Itˆ o process Z on (Ω, F, (Ft )0≤t 0. Hence it’s immediate from Theorem 8.2 that there is no non trivial e-consistent diffusion. This result has already been obtained in Filipovi´c (1998) for e-consistent Itˆo processes. 9.2. The Svensson family. FS (x, z) = z1 + (z2 + z3 x)e−z5 x + z4 xe−z6 x By Theorem 8.1 and Theorem 8.2 there remain the two choices i) 2z6 = z5 > 0 ii) 2z5 = z6 > 0 We shall identify the e-consistent diffusion process in both cases . Let thus the diffusion Z = (Z 1 , . . . , Z 6 ) be e-consistent with the Svensson family. By Theorem 8.1 thus Z 5 ≡ z05 and Z 6 ≡ z06 for some initial values z05 , z06 > 0. Let Q be an equivalent martingale measure. Under Q the diffusion Z transforms into a

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consistent one. Now we proceed as in the proof of Theorem 3.2. The expansion (16) reads as follows Q1 (x) + Q2 (x)e−z5 x + Q3 (x)e−z6 x + Q4 (x)e−2z5 x + Q5 (x)e−(z5 +z6 )x + Q6 (x)e−2z6 x = 0, for some polynomials Q1 . . . , Q6 . Explicitly Q1 (x) = −a1,1 x + . . . Q2 (x) = −a1,3 x2 + . . . Q3 (x) = −a1,4 x2 + . . . a3,3 2 Q4 (x) = x + ... z5 a4,4 2 Q6 (x) = x + ... z6 where . . . denotes terms of lower order in x. Hence a1,1 = 0 in any case. By the usual arguments (the matrix a is nonnegative definite) the degree of Q2 and Q3 reduces to at most 1. Thus in both cases i) and ii) it follows a3,3 = a4,4 = 0. It remains Q1 (x) = b1 Q2 (x) = (b3 + z3 z5 )x + b2 − z3 −

a2,2 + z2 z5 z5

Q3 (x) = (b4 + z4 z6 )x − z4 a2,2 Q4 (x) = z5

(24)

while Q5 = Q6 = 0. Since in case i) it must hold that Q4 = 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 = z4 z5 .

(25)

Hence a possibility for a non deterministic consistent diffusion Z. We derive from (24) and (25) b1 = 0 b2 = z3 + z4 − z5 z2 b3 = −z5 z3 b4 = −2z5 z4 . Therefore the dynamics of Z 1 , Z 3 , . . . , Z 6 are deterministic Zt1 ≡ z01 Zt3 = z03 e−z0 t 5

−2z05 t

Zt4 = z04 e

(26) ,

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21

˜ the Girsanov transform of W we while Zt5 ≡ z05 and Zt6 ≡ 2z05 . Denoting by W have under the equivalent martingale measure Q Z t³ d Z t ´ X 2 2 5 2 ˜ λ, Zt = z0 + σ 2,λ (s) dW (27) Φ(s) − z0 Zs ds + s 0

λ=1

0

where Φ(t) and σ 2,λ (t) are some deterministic functions in t with Φ(t) := z03 e−z0 t + z04 e−2z0 t 5

5

and d ³ X

´2 5 σ 2,λ (t) = z04 z05 e−2z0 t .

λ=1

By L´evy’s characterization theorem, see Revuz and Yor (1994, Theorem (3.6), Chap. IV), the real valued process d Z t X σ 2,λ (s) ∗ ˜ λ , 0 ≤ t < ∞, p Wt := dW s 5s 4 5 −z z0 z0 e 0 λ=1 0 is a (Ft )-Brownian motion under Q. Hence the corresponding short rates rt = FS (0, Zt ) = z01 + Zt2 satisfy ³ ´ drt = φ(t) − z05 rt dt + σ ˜ (t) dWt∗ , p 5 where φ(t) := Φ(t) + z01 z05 and σ ˜ (t) := z04 z05 e−z0 t . This is just the generalized Vasicek model. It can be easily given a closed form solution for rt , see Musiela and Rutkowski (1997, p. 293). Summarizing case ii) we have found a non trivial e-consistent diffusion process, which is identified by (26) and (27). Actually Φ has to be replaced by a predictable ˜ due to the change of measure. Nevertheless this is just a one factor process Φ model. The corresponding interest rate model is essentially a 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 seen that after all just one of them (z2 ) can be chosen to be non deterministic. 10. Conclusions Bounded exponential-polynomial families like the Nelson–Siegel or Svensson family may be well suited for daily estimations of the forward rate curve. They are rather not to be used for inter-temporal interest rate modelling by diffusion processes. This is due to the facts that • the exponents have to be kept constant • and moreover this choice is very restricted whenever you want to exclude arbitrage possibilities. It is 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, but still a very limited one, since all parameters but one have to be kept either constant or deterministic. Acknowledgement. I am grateful to T. Bj¨ ork for pointing out this extension of my earlier work. Also I would like to thank B.J. Christensen for his interest in

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these results and for discussion. Finally I thank my Ph.D. advisor F. Delbaen for his helpful comments. Support from Credit Suisse is gratefully acknowledged. References [1] Bj¨ ork T., Christensen B.J. (1997): “Interest rate dynamics and consistent forward rate curves”. Working paper, Stockholm School of Economics, submitted. [2] Filipovi´c D. (1998): “A note on the Nelson–Siegel family”, to appear in Mathematical Finance. [3] Jacod J., Shiryaev A.N. (1987): “Limit Theorems for Stochastic Processes”. Grundlehren der mathematischen Wissenschaften, 288, Springer, Berlin-Heidelberg-New York. [4] Musiela M., Rutkowski M. (1997): “Martingale Methods in Financial Modelling”. Applications of Mathematics, 36, Springer, Berlin-Heidelberg. [5] Nelson C., Siegel A. (1987): ”Parsimonious modeling of yield curves”. Journal of Business, 60, 473-489. [6] Revuz D., Yor M. (1994): “Continuous Martingales and Brownian Motion”. Second Edition, Grundlehren der mathematischen Wissenschaften, 293, Springer, Berlin-HeidelbergNew York. [7] Schich S.T. (1996): “Alternative Spezifikationen der deutschen Zinsstrukturkurve und ihr Informationsgehalt hinsichtlich der Inflation”. Diskussionspapier 8/96, Volkwirtschaftliche Forschungsgruppe der Deutschen Bundesbank. [8] Schwarz H.R. (1986): “Numerische Mathematik”. Second Edition, Teubner, Stuttgart. [9] Svensson L.E.O. (1994): “Estimating and interpreting forward interest rates: Sweden 19921994”. IMF Working Paper No. 114, September.