Let us consider a bond market with a family of stochastic processes describing zero coupon bond prices P ( t , T ) , t ∈ [ 0 , T ], parametrized by the maturity time T > 0 and the short rate process R ( t ) , t ≥ 0. The processes are defined on a probability space ( Ω , F , P ) with a filtration ( F t ) , t ≥ 0. The bond maturing at T pays its owner at time T a nominal value assumed here to be 1, i.e. P ( T , T ) = 1. The discounted value of 1 paid at time t > 0 equals D ( t ) = e − ∫ 0 t R ( s ) d s . The short rate process R is supposed to satisfy, for each T > 0, the condition (1.1) E [ e − ∫ t T R ( s ) d s ∣ F t ] = e − A ( T − t ) − B ( T − t ) R ( t ) , t ∈ [ 0 , T ] , with some deterministic functions A ( · ), B ( · ). Interpreting P as a risk neutral measure, one recognizes in the left side of (1.1) the price at time t of the bond with maturity T, that is P ( t , T ). Thus (1.1) means that the short rate R generates an affine term structure.

The concept of modeling bond prices in the affine fashion was introduced by Filipović [17] and Duffie, Filipović and Schachermeyer [14]. It was motivated by the results of Kawazu and Watanabe [19] on continuous state branching processes with immigration. Further developments on regularity of affine processes are due to Cuchiero, Filipović and Teichmann [11] and Cuchiero and Teichmann [12]. The aforementioned results are settled in the general Markovian setting and the description of affine processes is given in the form of their generators. A class of particular interest are short rates given by stochastic equations. An equation with a solution which generates an affine model is called a generating equation. The precursors of generating equations are two classical equations – one due to Cox, Ingersoll and Ross (CIR) [10] (1.2) d R ( t ) = ( a R ( t ) + b ) d t + c R ( t ) d W ( t ) , with a ∈ R, b ≥ 0, c > 0, and another due to Vasiček [25] (1.3) d R ( t ) = ( a R ( t ) + b ) d t + c d W ( t ) , with a , b , c ∈ R – both driven by a one-dimensional Wiener process W. To make the behavior of the short rate process more realistic and to improve the accuracy of calibration to market data more involved equations are considered in the literature. Into account are taken multidimensional noises, including those with jumps, with possibly correlated variates. Passing to more general types of noise offers more flexibility to the arising bond market which is required from the pricing perspective. Dai and Singleton [13] consider factorial models perturbed by correlated Wiener processes and examine the influence of the correlation structure on the resulting affine model. In the case when W is replaced by a Lévy process, it was shown in Barski and Zabczyk [5] that the generalization of (1.2) must be of the form (1.4) d R ( t ) = ( a R ( t ) + b ) d t + C · ( R ( t − ) ) 1 / α d Z α ( t ) , with a ∈ R, b ≥ 0, C > 0, where Z α is an α-stable process with index α ∈ ( 1 , 2 ]. For a comprehensive study of α-stable processes we refer to Samorodnitsky and Taqqu [24]. It was also shown in [5] that the counterpart of (1.3) in the Lévy setting allows preserving the positivity of R, which clearly lacks in (1.3) like in each Gaussian model. Jiao, Ma and Scotti [18] modify the CIR model by adding an independent α-stable component to the Wiener process. Their α-CIR model reveals better fitting to the European sovereign bond market than the CIR model and the stability index α allows controlling the tail heaviness of the bond prices. Models driven by a multivariate Lévy process with independent coordinates appear, among others, in Duffie and Gârleanu [15], Barndorff-Nielsen and Shephard [3], Barski and Łochowski [4]. Similarly as in [13], it is noticed in [4] that different equations may generate identical affine models. This fact motivated a classification of all generating equations into several classes which are representable by the so-called canonical equations having tractable forms. The case when the coordinates of the multivariate Lévy process are dependent is an unexplored field entered by this paper.

We consider an equation of the form (1.5) d R ( t ) = F ( R ( t ) ) d t + ∑ i = 1 d G i ( R ( t − ) ) d Z i ( t ) , R ( 0 ) = x ≥ 0 , t > 0 , where F , G : = ( G 1 , … , G d ) are deterministic functions and Z = ( Z 1 , … , Z d ) is a multivariate Lévy process and a martingale. As the coordinates of Z may be dependent, the Lévy measure ν of Z is not necessarily concentrated on axes. Its polar decomposition (1.6) ν ( A ) = ∫ S d − 1 ∫ 0 + ∞ 1 A ( r ξ ) γ ξ ( d r ) λ ( d ξ ) , A ∈ B ( R d ) , with a finite measure λ on a unit sphere S d − 1 : = { x ∈ R d : | x | = 1 } called a spherical component of ν and a family of measures { γ ξ ; ξ ∈ S d − 1 } on ( 0 , + ∞ ) called radial components of ν, will play a central role in the sequel. The radial decomposition is known to exist and to be unique for any Lévy measure, see [2], Lemma 2.1 and [22], Proposition 4.2. Let us recall that any α-stable process in R d with index α ∈ ( 1 , 2 ) admits radial decomposition with identical radial measures given by the density (1.7) γ ξ ( d r ) = γ ( d r ) : = 1 r 1 + α d r , r > 0 , ξ ∈ S d − 1 , and arbitrary spherical measure λ. In fact, the radial decomposition can be explicitly determined in the case when ν has a density with respect to the Lebesgue measure, say g. Then the radial measures are of the form γ ξ ( d r ) = g ( r ξ ) r d − 1 1 − ξ 1 2 · 1 − ( ξ 1 2 + ξ 2 2 ) · … · 1 − ( ξ 1 2 + ⋯ + ξ d − 2 2 ) d r , for ξ = ( ξ 1 , … , ξ d ) ∈ S d − 1 , r > 0, and the spherical measure λ is the image of the Lebesgue measure by the polar transformation, see Section 3.1.2 for details. We examine the question which affine models can be generated by equation (1.5) and our analysis of this problem is based on the radial decomposition (1.6).

In Example 2.3 we show that if (1.5) is a generating equation and Z is an R d -valued α-stable process then the resulted affine model is identical with the model generated by (1.4). This means that (1.4) can replace the initial equation, which may be of a complicated form, preserving the bond prices unchanged. In this case we call the initial equation to have the reducibility property. This extends the observations from [13] and [4] to the case with dependent noise coordinates. The main result of this paper, Theorem 3.3, provides conditions for (1.5) to have the reducibility property. We prove that if G is a continuous function for which the limit lim x → 0 + G ( x ) G ( x ) exists and the Laplace exponents associated with the radial measures J γ ξ ( b ) : = ∫ 0 + ∞ ( e − b r − 1 + b r ) γ ξ ( d r ) , b ≥ 0 , ξ ∈ S d − 1 , satisfy the condition (1.8) sup ξ ∈ supp ( λ ) J γ ξ ( b ) ≤ K · inf ξ ∈ supp ( λ ) J γ ξ ( b ) , b ≥ 0 , with some 1 ≤ K < + ∞, then any equation with such G and Z can generate only the same affine model as the one generated by (1.4) with some α ∈ ( 1 , 2 ). Condition (1.8) is shown to be satisfied in the class of tempered stable distributions and in the case when Z is R 2 -valued and its jump measure has a density g such that the functions g _ ( r ) : = inf x = r g ( x ) , g ¯ ( r ) : = sup x = r g ( x ) , r ≥ 0 , satisfy the integrability conditions 0 < ∫ 0 + ∞ min { r 2 , r 3 } g _ ( r ) d r ≤ ∫ 0 + ∞ min { r 2 , r 3 } g ¯ ( r ) d r < + ∞ , and lim ε ↓ 0 ∫ ε 1 r 2 g ¯ ( r ) d r ∫ ε 1 r 2 g _ ( r ) d r < + ∞ and lim ε ↓ 0 ∫ 1 1 / ε r 3 g ¯ ( r ) d r ∫ 1 1 / ε r 3 g _ ( r ) d r < + ∞ . For details and extensions in cases where d > 2, see Section 3.1.2.

The term structure models are calibrated to market data which may contain, for instance, swap or swaption prices or some spot rates. Empirical curves understood as functions of maturities representing market quotes should be well approximated by the resulting model curves. Therefore, understanding which curve shapes can be generated by the model is of prime importance. We examine possible shapes of the simple spot curve defined by F ( x ) = 1 x 1 P ( 0 , x ) − 1 , x ≥ 0 , in the model (1.4) and provide a precise description of normal (increasing), inverse (decreasing) and humped (possesing one local maximum) shapes in terms of the model parameters. Characterization of yield curves x ↦ − 1 x ln P ( 0 , x ) , in affine models is known in the literature, see [20, 21], but it does not imply the shape of simple spot curves. Our characterization in Theorem 4.3 helps to decide if the model can be calibrated to market reference rates like LIBOR or other like SONIA, SOFR, SARON and ESTER. In particular, it helps to decide if the possible simple spot curves produced by the model, which are normal, inverse or humped, fit the shapes of empirical spot curves obtained from these reference rates. Also, comparing (1.4) to the CIR model with a Wiener driving process, we see that the stability index α offers additional fit flexibility.

The paper is organized as follows. In Section 2 we present some basic facts on Lévy processes and the Markovian characterization of generating equations. Section 3 on the reducibility problem contains formulation of the main results including Theorem 3.1 and Theorem 3.3, examples and illustrative analysis of the case when ν has a density, resulting in Theorem 3.8. The proof of Theorem 3.1 and associated auxiliary results are postponed to Section 5. Section 4 is devoted to the description of shapes of simple spot curves.

In this section we specify conditions for the equation (1.5), written now for convenience in the form (3.1) d R ( t ) = ( a R ( t ) + b ) d t + ⟨ G ( R ( t − ) ) , d Z ( t ) ⟩ , R ( 0 ) = x ≥ 0 , t > 0 , to have the reducibility property. The affine form of the above drift is justified by (2.9). This means that (3.1) is supposed to generate the same bond prices as the equation (3.2) d R ( t ) = ( a R ( t ) + b ) d t + C · R ( t − ) 1 / α d Z α ( t ) , with a ∈ R, b ≥ 0, C > 0 and an α-stable real-valued Lévy process Z α with some α ∈ ( 1 , 2 ). Recall that from Example 2.3 we know that each generating equation (3.1) with Z being an α-stable process in R d has the reducibility property.

In (3.1), G : R + ⟶ R d and Z is a Lévy process and martingale in R d , called a Lévy martingale for short. It is characterized by a covariance matrix Q of the Wiener part and a Lévy measure ν with polar decomposition (3.3) ν ( A ) = ∫ S d − 1 ∫ 0 + ∞ 1 A ( r ξ ) γ ξ ( d r ) λ ( d ξ ) , A ∈ B ( R d ) , with a finite spherical measure λ on the unit sphere S d − 1 and some radial measures { γ ξ ; ξ ∈ S d − 1 }. To avoid technical complications we can assume, and we do, the nondegeneracy condition for the radial measures, i.e. (3.4) ξ ∈ supp ( λ ) ⟹ γ ξ ≠ 0 . If (3.4) is not satisfied, one can modify λ by cutting off the part of its support where the radial measures disappear. This operation clearly does not affect (3.3). Since Z is a martingale, it follows from (2.2) that ∫ R d ( y 2 ∧ y ) ν ( d y ) = ∫ S d − 1 ∫ 0 + ∞ ( r ξ 2 ∧ r ξ ) γ ξ ( d r ) λ ( d ξ ) < + ∞ , which means that (3.5) ∫ 0 + ∞ ( r 2 ∧ r ) γ ξ ( d r ) < + ∞ , ξ ∈ supp ( λ ) . If the jump part of Z has infinite variation, then it follows from (2.3) that (3.6) ∫ ∣ y ∣ ≤ 1 y ν ( d y ) = ∫ S d − 1 ∫ 0 1 r γ ξ ( d r ) λ ( d ξ ) = + ∞ . We consider a condition stronger than (3.6), namely, that (3.7) λ ( Γ λ ) > 0 , where Γ λ : = ξ ∈ supp ( λ ) : ∫ 0 1 r γ ξ ( d r ) = + ∞ . Consequently, if we assume that Γ λ is not contained in any proper linear subspace of R d , i.e. (3.8) Linear span ( Γ λ ) = R d , then we obtain that (3.9) G ( 0 ) = 0 . To see this, let us notice that, by (2.11) and (3.3), λ ξ ∈ S d − 1 : ⟨ G ( 0 ) , ξ ⟩ < 0 = 0 which implies that (3.10) ⟨ G ( 0 ) , ξ ⟩ ≥ 0 for any ξ ∈ supp ( λ ) . By (2.12) we have ∫ 0 + ∞ v ν G ( 0 ) ( d v ) = ∫ R d ⟨ G ( 0 ) , y ⟩ ν ( d y ) = ∫ S d − 1 ⟨ G ( 0 ) , ξ ⟩ ∫ 0 + ∞ r γ ξ ( d r ) λ ( d ξ ) < + ∞ , which, in view of (3.7), (3.8) and (3.10) implies (3.9). Obviously, (3.8) also implies that (3.11) Linear span (supp ( λ ) ) = Linear span (supp ( ν ) ) = R d .

Notice that, for example, the measure λ = δ ( 1 , 0 ) + δ ( 0 , 1 ) from Example 2.3 with γ ( 1 , 0 ) ( d r ) = γ ( 0 , 1 ) ( d r ) = r − 1 − α d r, α ∈ ( 1 , 2 ), satisfies (3.7) as well as conditions (3.8) and (3.11).