LEVY INTEREST RATE MODELS WITH A LONG MEMORY

This article proposes an interest rate model ruled by mean reverting LØvy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein-Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag-Le(cid:31)er function. Based on a representation in term of an in(cid:28)nite dimensional Markov processes, we present the main characteristics of bonds and short-term rates in this setting. Their dynamics under risk neutral and forward measures are studied. Finally, bond options are valued with a discretization scheme and a discrete Fourier’s transform.


Introduction
From the 1980s to nowadays, many interest rate models were proposed in the literature.Their common aim is to explain changes in bond or swap quotes and to capture risks within the interest rates market.Three dominating frameworks coexist: short-term rate, forward rate and the Libor market models.In this last approach proposed by Brace et al. (1997) interest rates are driven by geometric processes.In the forward rate model pioneered by Heath et al. (1992), the term structure of rates is specied through instantaneous forward rates.Mercurio and Moraleda (2000), Falini (2010) propose a forward model with a humped structure of volatilities.Li et al. (2020) develop a forward rate model unifying most of existing Gaussian models for interest rates.Short-term rate models such as promoted by Hull and White (1990), specify a mean reverting dynamic for the instantaneous risk-free rate.The framework developed in this article belongs to this third category.
The family of short-term rate models gathers multiple frameworks and we refer the reader e.g. to Boero and Torricelli (1996) or to Schmidt (2011) for a review.On the other hand, various processes were proposed to explain the interest rate risk.The literature is too vast for being exhaustive.We restrict our attention to contributions related to this work and do not attempt a general overview, referring instead to the Brigo and Mercurio (2006) for detailed accounts on the topic.For instance, Eberlein and Raible (1999) were among the rsts to propose a short-term rate model driven by Lévy processes.In their setting, the short-term rate is ruled by Ornstein-Uhlenbeck processes reverting in an exponential manner to a mean level.Within this framework, Eberlein and Kluge (2005) derive analytical formulae for the prices of caps and oors using bilateral Laplace transforms.In a similar setting, Hainaut and Macgilchrist (2010) propose a pentanomial tree for pricing derivatives.Hainaut (2013) studies the properties of a Gaussian short-term rate with a Markov Switching Multifractal volatility.Moreno and Platania (2015) propose a square-root model replicating economic cyles in the dynamic of interest rates.Hainaut (2016) and Hainaut and Njike Leunga and Hainaut (2020) develop unicurve and multicurve models in which the short-term rate is exposed to self-excitating jumps.Fontana et al. (2021) study a modelling framework exploiting the the self-exciting behavior of continuous-state branching processes with immigration (CBI).
A wide majority of short-term rate models are driven by Markov mean reverting processes.In this setting, the dependency on the past in the dynamics comes in as an exponential decaying function, that we call the memory kernel, in an integral with respect to driving processes.This implies that the inuence of previous variations of interest rates on today's value decreases exponentially with time.In this article, we replace this memory kernel by a Mittag-Leer decaying function.This function is sub-exponential and may be seen as a generalization of the exponential.In this setting, the interest rate remembers its previous occurrences for a longer period than in the exponential framework.
Unfortunately, the interest rate is not a Markov process anymore and bond prices depend on the whole history of rates.Nevertheless, the memory kernel admits a representation as a Laplace Stieltjes integral and the short-term rate can be converted into an innite dimensional Markov process.Approximating this process allows us to infer the dynamic of bond prices and many of their features.
The last part of the article studies the properties of the model under a forward measure and propose a pricing method for bond options.

A rst model with an exponential kernel
In the proposed model, the short-term rate is driven by d ∈ N independent Lévy processes, denoted by (L (j) t ) t≥0 for j = 1, ..., d.These processes are dened on a probability space Ω, endowed with the natural ltration (F t ) t≥0 and the risk measure Q.Each process (L (j) t ) t≥0 has independent and stationary increments.It is fully described by a triplet µ j , σ 2 j , ν l where µ j ∈ R, σ j ∈ R + and ν j (.) is a measure.The moment generating function (mgf ) is denoted by φ (j) t (ω) for j = 1, ..., d and is equal to = exp (tψ j (ω)) .
The function ψ j (ω) is the characteristic exponent of this Lévy process.Without loss of generality, we assume that L (j) 0 = 0.According to the Lévy Itô decomposition, each L (j) t is the sum of three components : a deterministic drift µ j t , a diusion with variance σ 2 j and a jump process, J j (t , z), of intensity ν j (dz) well dened on [−∞, 0) ∪ (0, +∞].This Lévy measure is such that the probability of observing k jumps between [τ 1 , τ 2 ] of a size included in a set B ⊂ R 0 is given by for j = 1, ..., d, are independent Brownian motions on (Ω, F, Q), L (j) t may be split as the sum of a drift, a Brownian motion and a jump process dL (j) where Jj (dt , dz) = J j (dt , dz) − 1 |z|< ν j (dz)dt for any > 0. Without loss of generality, we assume that E dL (j) t = 0.This constraint implies that the drift is equal to: Notice that if the Lévy measure, ν j (.), has no singurality at z = 0, we can set to zero.From Cont and Tankov (2003), for any integrable function f : R → C, the following relation holds This can be proved by approaching f (.) with a stepwise function and by using the property of independence of increments.This property will be useful for developments of following sections.We will also need mean reverting Lévy processes of the form that are also called Ornstein-Uhlenbeck (OU) processes.We assume that Y (j) 0 = 0.The solution of the previous stochastic dierential equation is given by In this OU process, the inuence on Y (j) t of the past sample paths of L (j) t decays in an exponential manner.We call the function e −κj (t−u) , the memory kernel of Y (j) t .A classical approach for modelling the short-term rate, noted (r t ) t≥0 consists to postulate the following dynamic: where ϕ(t) is a continuous and dierentiable function.Such an approach was e.g.developed in Hainaut and Macgilchrist (2010) and approached by a pentanomial tree in order to price interest rate derivatives.As E dL (j) u = 0, the expectation of r t conditionally to the information up to time s is given by This last equation emphasizes that the impact of current values of whereas the autocovariance of r t and r u for t ≥ u ≥ s is given by: This last equation reveals that the autocovariance between the current short rate and the future one decays exponentially.In the next section, we propose a model in which this covariance decreases at a slower pace.Notice that the covariance between r t and r t−∆ for any ∆ > 0 converges to a function of ∆: where the X (j) t are dened in a similar manner to Y (j) t , excepted that the exponential memory kernel is replaced by another decreasing function g j (.): The function g j (.) : R + → R is a continuously decreasing kernel, with an initial value g j (0) = 1 and that admits a representation as a LaplaceStieltjes integral.Contrary to the OU process, X (j) t is in general not Markov and X (j) t |F s may not be reformulated as a function of X (j) s .The interest rate process (r t ) t≥0 is therefore not Markov.This feature makes dicult the evaluation of bond prices at a given time t > 0 since their value depends on the whole sample path of interest rates up to t.
Nevertheless, for some well chosen kernel functions, we will see that the interest rate may be represented as an innite dimensional Markov process.In particular we consider two types of kernel functions, both based on the Mittag-Leer function, denoted by E α (.) where α ∈ [0, 1]: To understand the motivation for working with such functions, we need to review the main properties of the Mittag-Leer function.The Mittag Leer function of order α > 0 plays a fundamental role in the fractional calculus and can be considered as an extension of the exponential function.It is dened as an innite sum .
In this article we assume that α ∈ [0, 1].For this range of values, E α (−t) is an intermediary between the power and the exponential decreasing functions: We refer the reader to the book of Goreno et al. (2014) for a detailed presentation of this function.
Let us recall that a function f : (0, ∞) → ∞ is called completely monotonic if it possesses derivatives f (n) (t) of any order n = 0, 1, 2... and the derivatives are alternating in sign, i.e. (−1) The above property is equivalent to the existence of a representation of the function f in the form of a LaplaceStieltjes integral with non-decreasing density and non-negative measure dγ(.) such that where the derivative of γ α,β (u) is equal to Given that E α (0) = 1, we have that ∞ 0 dγ α,β (u) = 1 and γ α,β (.) is then a measure of probability on R + .In the remainder of this article, E α (−βt) is called the decreasing Mittag-Leer (ML) kernel.When t → 0 the DML behaves at short-term as .
From Haubold et al. (2011), we known that when t → ∞, the DML converges to The function E α (−βt) interpolates for intermediate times t between the decreasing exponential and the inverse power law.The exponential models fast decay for small time t, whereas the asymptotic inverse power law entails a slow decrease at long term.This point is illustrated in the left plot of Figure 1 that compares the ML kernel to the exponential and inverse power functions.The Mittag-Leer function is also related to the fractional calculus.To explain this link, we recall that the Caputo's fractional derivative of order α ∈]0, 1[ for a function h(t) : R + → R, C 1 with respect to t is dened by When α = 1, this derivative corresponds to the rst order derivative.The solution of the fractional dierential equation where the derivative of γ p α,β (u) is given by is a measure of probability on R + .As underlined by Mainardi (2020), when t → 0 the PML behaves at short-term as .
From Erdélyi A et al. (1955), we known that when t → ∞, the PML converges to As a consequence the function E α (−βt α ) interpolates for intermediate times t between the stretched exponential and the negative power law.The stretched exponential models the very fast decay at short-term whereas the asymptotic power law is due to the very slow decay for large time t.The right plot of Figure 1 illustrates this convergence.
Figure 2 presents simulated sample paths of the short-term rate with a one dimensional (d = 1) model ruled by a Brownian motion.These paths are computed for various α with the same random occurrences in order to make comparison feasible.For the ML kernel, trajectories are nearly similar.
An analysis of gures reveals that the sample path is smoother for α = 0.50 than for α = 0.90.This trend becomes visible if we choose a highest value for β.For PML sample paths, the dierence is clearly visible.Decreasing α reduces the volatility of rates and smoothes the sample path.
To conclude this section, we present the rst two moments of the short-term rate and its autocovariance function.The ML and PML kernels dened in Equations ( 13) are continuous decreasing and integrable functions on any bounded interval of R. From Equation (4), the mgf of X (j) t for j = 1, ..., d is then equal to: Deriving the mgf allows us to nd the rst moments of X (j) t conditionally to the initial information.
On the other hand, from the representation of r t , we infer that the conditional expectation of the short-term rate is given by Whereas the conditional variance of r t is equal to the sum of variances of Lévy processes times the integral of the squared kernels Unfortunately the integral of g j (.) 2 does not admit a closed form expression for the ML and PML kernels.Nevertheless, they can be numerically estimated.A direct calculation allows us to infer that the autocovariance is given by Contrary to the exponential case, this covariance function between r t and r t−∆ does not admit any closed form expression when t → ∞.

Empirical motivation
This short section has not for objective to propose a thorough econometric analysis but provides some for k = 1, ..., n obs −1.Under the assumptiont that L t is a Brownian motion without drift and of variance σ 2 , parameters are estimated by log-likelihood maximization.We next consider models with ML and PML kernel functions.In this cases, the variations of the underlying Lévy process are approached by for k = 1, ..., n obs − 1 where g(.) is either the ML or the PML function.Under the assumption of normality, parameters are estimated by log-likelihood maximization.The results of this procedure are reported in Table 1.In terms of log-likelihood, the exponential and ML models achieve the same goodness of t.They share the same estimated volatilities but the parameters of reversion κ and β slightly dier.The best t is obtained with a power Mittag Leer kernel.The parameter α that drives the decay of the memory is small.This means that the process has a longer-term memory than the exponential model.On the other hand, the mean reversion speed is signicantly higher than the ones of other models.We recall to the reader that estimated parameters reect the dynamic of r t under the real measure P and not the risk neutral one.10)) with those of ML and PML models (computed with Equation ( 22)).Parameters used for this exercise are those of Table 1.The interest rate in the ML and PML models being not Markov, the autocorrelation between r t and r t−∆ depends upon t.In the plots, we consider t = 10, 20 and 30 years.We observe that the autocorrelations of the ML kernel are dominated by exponential ones, whatever the time horizon t.On the contrary, the autocorrelations of rates in the PML model is lower at short-term than the exponential ones but globally raise with the time horizon t.This is one important feature of the PML model and a direct consequence of the non-stationarity of increments.

Alternative formulation
Once that the memory is not exponential, the interest rate process (r t ) t≥0 is not Markov anymore.
This feature makes dicult the evaluation of bond prices or interest rate derivatives because their values depend on the whole history of the rate process.Nevertheless, if the memory kernel admits a representation as a LaplaceStieltjes integral, we can convert the short-term rate into an innite dimensional Markov process.Approximating this process allows us to infer the dynamic of bond prices.
As developments are similar in both cases, we denote by γ j (.) the measures γ (j) αj ,βj (.) or γ p(j) αj ,βj (.) of E αj (−β j t) and E αj (−β j t αj ).Using the Fubini's theorem,the processes X (j) t are then rewritten as As this SDE admits the solution the short-term rate is then a sum of integrals of Y (j,ξ) t : Given that γ (j) αj ,βj (.) and γ p(j) αj ,βj (.) are probability measures, the dierential of r t is on the other hand equal to Combining Equations ( 24) and (25) allow us to rewrite the short-term rate as the following sum We will explain in the next section how to approximate the innity of processes by discretizing the measures γ j (ξ).But before we evaluate zero coupon bonds and instantaneous forward rates in this setting.6 Bond prices and forward rates A zero-coupon bond of maturity t delivers an unit cash-ow at expiry.Its price at time s prior to t is denoted by P (s, t) and in absence of arbitrage, is the expected discount factor under the risk neutral measure, P (s, t) = E e − t s ru du | F s .The next proposition presents a semi-closed form expression for this price.Proposition 6.1.Let us respectively dene the functions B (ξ) (s, t) and h j (s, t) as follows for s ≤ t and j = 1, ..., d.The zero coupon bond price is equal to Proof.From Equation ( 19), we infer after a change of the integration order that A direct calculation allows us to rewrite the integrals of g j (.) in this last expression as The zero-coupon bond price becomes then: Finally, the expectation in this last expression is calculated with Equation (4).
Proof.Starting from Equation (26) and permuting the order of integration lead to the following representation of the integral of the short-term rate: The integrals of e −ξ(u−s) Y (j,ξ) s with respect to γ j (.) are equal to: On the other hand, switching the order of integration leads to Therefore the integral of the short-term rate can be rewritten as the following sum As E exp ω t s f (u)dL The integrals in the bond price formula do not admit closed form expressions.Nevertheless, the integrals t s ψ j (−h j (u, t)) du can be numerically computed without any particular diculty whereas ∞ 0 Y (j,ξ) s B (ξ) (s, t) dγ j (ξ) is approached by a discretization scheme developped in the following section.
Let us recall that the initial values of processes Y (j,ξ) t t≥0,j∈{1,...,d} are null, Y (j,ξ) 0 = 0, for all ξ ∈ R + and j = 1, ..., d.Therefore, Equation (30) provides us a way to estimate the function ϕ(.) such that the model perfectly matches the initial term structure of bond prices.More precisely, the integral of ϕ(.) is such that Deriving this last expression leads to the following function: The dynamic of interest rates can also be reformulated in terms of instantaneous forward rates.Let us recall that the instantaneous forward rate, denoted by f (s, t), is such that P (s, t) = exp − t s f (s, u) du and is therefore equal to: The next proposition states that the forward rate is the sum of the expected interest rate under the risk neutral measure and of an adjustment that directly depends upon the memory kernel.
Proposition 6.3.The instantaneous forward rate at time s and of maturity t is given by: where and is equal to Proof.From Equation (30) and ∂B (ξ) (s,t) ∂t = e −ξ(t−s) , we have that If we remember the expression (30) of the characteristic exponent, we immediately infer that As ∞ 0 e −ξ(t−v) dγ j (ξ) = g j (t − v), we obtain Equation (32).The dierential of f (s, t) is obtained by applying the Itô's lemma for Lévy processes.
This last proposition emphasizes that our short-term rate model can be reformulated as a forward rate model in which the forward rate dynamic is independent from processes Y (j,ξ) s ξ≥0 for j = 1, ..., d.

Discretisation scheme
Instead of considering an innity of processes Y (j,ξ) t t≥0,j∈{1,...,d} , we approach the model with a nite number of equivalent processes.This presents several advantages.Firstly, this makes possible implementing our model.Secondly, we can rely on the Itô's calculus in most of developments.This allows us to deduce the dynamic of bond prices both in the approximated and original models.The key step consists to approximate the γ j (.)'s by discrete measures with a nite numbers of atoms.For  this purpose we consider a partition E (n) := {0 < ξ and the mass of corresponding atoms is dened as the measure of intervals of the partition: for k = 0, ..., n − 1.In practice, we choose ξ (n) 0 = 0 and set ξ (n) n to a percentile of the density γ j (z) (e.g.95%).The discrete measure for a partition of size n is dened as follows where δ b (j) k (z) is the Dirac measure located at point b (j) k .We consider that the following assumption holds for the partition E (n) : In this case, for any function f (.) integrable with respect to γ j (.), we have that lim n→∞ 4 shows the dierential of measures dγ(.) of the ML and PML kernels.For the ML kernels, decreasing the α clearly makes fatter the right tail of dγ(.).The left plot also reveals that the construction of the partition requires a particular care for the PML kernel as we have to integrate numerically the measure γ (p) α,β (z) that is not dened for z = 0.The two next propositions respectively present the expressions of integrals needed for calculating the discrete equivalent distributions of γ j (.) in the ML and PML cases.Proposition 7.1.For the ML kernel, the discrete probability mass of atoms is given by m whereas the numerator in the expression of barycenters (33) is This result is a direct consequence of the denition of the ML function.To the best of our knowledge, these expressions does not admit any other representations.In the PML case, barycenters have a closed-form expression.
Proposition 7.2.For the PML kernel, the discrete probability mass of atoms is given by The numerator in the expression of barycenters (33) does not admit a closed form expression but may be approached by: Proof.The expression of the probability mass m (j) k+1 comes from the relation The second result is obtained with an integration by parts in which the integral is approached with the trapezoidal method.
To lighten further developments, we adopt the following notations Y k ) s for k = 1, ..., n and j = 1, ..., d.Let us rst recall that Y (j,k) 0 = 0 and therefore Y (j,k) s is an integral from zero to s with respect to the j th Lévy process Replacing the continuous measures (γ j ) j=1...d by their discrete equivalents leads to the approximation r (n) s of the short-term rate r s : We adopt the following notations: 1 − e −b (j) k (t−s) for k = 1, ..., n and j = 1, ..., d.
The bond price in the discretized model with partitions of size n is denoted by P (n) (s, t) and is equal to Where ϕ (n) (t) is the discretized version of ϕ(t) : As the number of processes in the discretized model is nite, we can apply the Itô's lemma in order to establish the dynamic of P (n) (s, t).
Proposition 7.3.The zero coupon bond price P (n) (s, t) is a geometric Lévy processes solution of the SDE: with the terminal condition Proof.If we remember that dY (j,k) ds, the Itô's lemma gives us the following dierential for P (n) (s, t): The rst and second order partial derivatives of P (n) (s, t) are equal to: Whereas its partial derivative with respect to time is given by where the characteristic exponent is developed as the following sum z1 |z|< ν j (dz) .
Combining Equations ( 45), ( 46), ( 47), ( 48) and (49) allows us to rewrite the dynamic of bond prices as follows: We infer the result if we remember Equation (41) and notice that As by construction lim n→∞ γ (n) j (z) = γ j (z), we immediately infer the dynamic of the bond price in the non-discretized model by considering the limit of Equation (44).
The instantaneous variance of dL (1) t is equal to Of course, we can consider other type of Lévy processes like the variance gamma and the Normal inverse gaussian but this does not fundamentally modify the conclusions drawn in this section.We rst t the curve ϕ(t) We calculate the term structure of expected yields in 5, 10 and 15 years with n =40 atoms.Increasing n does not signicantly modify the results.A sensitivity analysis with respect to the number of atoms is proposed in Section 8.For the parameters, we set β = 1.5, λ 1 = 0.5, σ 1 = 0.01 and η 1 = −0.0002,which are possible realistic values in view of results from Section 4. We remind to the reader that α is the parameter tuning the memory of the model.If α = 1, the memory kernel is exponential and the model forgets in an exponential manner past uctuations of interest rates.Whereas for lower values of α, the model forgets these variations according to a power decaying function.Figures 5and 6 show expected yields computed with the ML and PML models for α = 0.5 and α = 0.9.We use a discretization scheme with 40 atoms that cover the ML and PML density up to their 90% percentiles.
For both models, the range between short and long-term yields is narrowing with the time horizon.In the ML model, the lower is α, the longer is the memory and the quicker is the convergence to a bumped (but nearly at) yield curve.In the PML model, expected yield curves with a low α dominates those with a high α and have a more pronounced curvature.Figures 7 and 8 report the expected standard deviations of future yields calculated with the ML and PML models.In both cases, the term structures of expected standard deviations do not signicantly evolve with the time horizon and is a decreasing function of the maturity.We also notice that standard deviations are respectively directly and inversely proportional to α in the ML and PML models.Proposition 8.3.The moment generating function (mgf) of ln P (t 1 , t 2 ) under the measure F, conditionally to the ltration F s , is given by Proof.The mgf can be rewritten as an expectation under the risk neutral measure using the Bayes' rule E F e ω ln P (t1,t2) |F s = E dF dQ t1 e ω ln P (t1,t2) |F s From Equation (24) and after a change of integration order, the integral of processes Y This allows us to rewrite E F e ω ln P (t1,t2) |F s as follows: wherein the integrand is equal to Combining these three last equations lead to the result.
The expression (64) of the mgf involves integrals of Y (j,ξ) s with respect to ξ ∈ R + .These integrals do not admit analytical formulas and are in practice approached with the discretization scheme introduced in Section 7. Let n be the number of atoms of the discrete approximation of γ j=1,...,d .To lighten future developments, we adopt the following notations for j = 1, ..., d.The discretized version of the mgf of the bond log-return under the forward measure is as follows where ϕ (n) (t), the discretized version of ϕ(t), is detailed in Equation (43).The integrals with respect to times are numerically computed e.g. with a Simpson's rule.Let us denote by f (n) (x), the probability density of ln P (n) (t 1 , t 2 ) conditionally to F s and Υ (n) (iω) = E e i ω ln P (n) (t1,t2) | F s = e iωx f (n) (x)dx , its Fourier's transform.The probability density function can therefore be expressed as the real part of the inverse Fourier's transform: where j = 1 2 1 {j=1} + 1 {j =1} .
The proof of this result is based on the trapezoidal approximation of the integral in Equation (66).
Figure 9 shows the probability density functions of P (n) (5, 10) under the forward measure of numeraire P (n) (0, 5).We consider for this illustration a one factor Lévy model (d = 1) in which the driving process is a jump-diusion such as presented in Equation (54), with the same parameters as those of Section 7. The DFT parameters are x max = 0.2 and M = 2 10 .The plots reveals that variances under F of bond prices displays the same sensitivity to the memory parameter α as under the risk neutral measure.We clearly observe that they are respectively directly and inversely proportional to α, in PML and ML models.(5,10) under the forward measure of numeraire P (n) (0, 5), α =0.5 and 0.9, β = 1.5, λ 1 =0.5 η 1 = -0.002,σ 1 = 0.01.ML and PML kernels with n =40 atoms.
After computation of the log-bond density, the call price is calculated by the following sum: C(s) = P (s, t 1 )E F (P (t 1 , t 2 ) − K) + | F s ≈ P (s, t 1 ) M k=1 where 1 {x k ≥ln K} is an indicator variable equal to one on [ln L, +∞) and zero otherwise.Figure 10 shows call prices on P (n) (5, 10) for various strike prices in the ML and PML kernels.In the PML model, increasing the memory parameter α drives down the option values whatever the strike price.
For the ML kernel, we observe the opposite trend.This is relevant with our previous conclusions about the variance that is the main driving factor of option prices.Figure 11 allows us to conrm the convergence of the call option when the number, n, of atoms in the discretization scheme growths.We recall that the partition ranges from ξ 0 = 0 up to ξ n that Maturity Swap rates (%) Maturity Swap rates (%)

Figure 3
Figure3compares for various ∆ > 0, the asymptotic autocorrelations between r t and r t−∆ in the exponential model (see Equation(10)) with those of ML and PML models (computed with Equation

Figure 3 :
Figure 3: Left and right plots: autocorrelations between r t and r t−∆ for 1D Brownian models with exponential, ML and PML kernels.

u 0 e
which does not depend anymore upon information prior to s.This relation emphasizes that the r t and Y (j,ξ) t ξ∈R + ,j∈{1,...,d} form well an innite dimensional Markov process.If we remember that E(L (j) t ) = 0, we infer an equivalent representation of Equation (20) for the conditional expectation of r t in terms of Y (j,ξ) t : E (r t |F s ) = ϕ(t) + d j=1 ∞ −ξ(t−s) Y (j,ξ) s dγ j (ξ) .
t) and is well dened.Equation (29) emphasizes the dependence of the bond price to the history of Levy processes.A more convenient formulation in terms of Y (j,ξ) s ξ≥0for j = 1, ..., d, is presented in the next proposition.In this alternative valuation formula, the bond price is exclusively calculated with information available at the time of valuation.
ωf (u))du , we conclude that the bond price is well given by Equation (30).
This article proposes an alternative model in which this decay is sub-exponential.For this purpose, we postulate that the risk-free rate, (r t ) t≥0 , is the sum of a deterministic function ϕ(t) : R → R, and of d processes X We abusively call this function the power decreasing Mittag-Leer kernel (PML) .The Laplace transform of E α (−βt α ) is given by s α−1 s α +β where Re(s) > |β| 1 α .By inverting this transform, we can prove as in Goreno and Mainardi (1997) that the PML is the Laplace transform motivating the developments done in this article, in a particular the choice of a ML or PML memory kernel.We have tted univariate Gaussian models (d = 1) with exponential, ML and PML kernels to the Eonia time series from the 13th of March 2000 to the 1st of February 2021.The data set counts n obs =5344 daily observations.As of 1 October 2019 EONIA is calculated with a reformed methodology tracking the euro short-term rate (¿STR).The EONIA is calculated as the euro shortterm rate plus a spread of 8.5 basis points.The dates of observations are denoted by t 0 , t 2 , ..., t n obs −1 whereas the step of time between two successive observations is ∆ t .We rst consider a one dimensional exponential kernel model.Under the assumption that ϕ(t k ) = ϕ(t k + ∆ t ) and Y 0 = 0, we nd that

Table 1 :
Parameter estimates of one dimensional models with exponential, ML and PML kernels.
to the term structure of zero-coupon bond yields, bootstrapped from the ICE swap rates on the 26/2/21 at noon.Since this function is proportional to instantaneous forward rates, we have to interpolate bond yields.For this purpose, we use a Nelson-Siegel (NS) model.Appendix A recalls this model, reports the swap rates curve on the 26/2/21 and parameter estimates.There exist more advanced interpolation methods, as e.g.splines or kriging techniques as detailed in Cousin et al.(2016) but the NS model is suciently accurate for our purpose that is to understand the dynamic of yields over time.

Table 3 :
Nelson-Siegel parameters for the ICE Euro swap rates on the 26/2/21.