A pure-jump mean-reverting short rate model

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein--Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.


Introduction
Stochastic interest rate models play an important role in the modeling of financial markets. The literature essentially distinguishes between short rate models, forward rate models and market models. In the sequel, we give a brief survey on the different classes of term structure models. For more detailed information, the reader is referred to the respective research articles or the textbooks [7,21] and [26].
Widely applied short rate models are for example the Vasicek model [38], the Hull-White model [29] or the Cox-Ingersoll-Ross (CIR) model [10]. In [38] and [29] the short rate process is modeled by a stochastic differential equation (SDE) of Ornstein-Uhlenbeck (OU) type driven by a Brownian motion (BM). As a consequence, the short rate process is normally distributed in these models and may become arbitrarily negative. Both features embody severe disadvantages with view on real-world market behavior, as the distribution of interest rate data frequently deviates from the normal distribution, while interest rates do not take arbitrarily large negative values in practice. In the recent years, there indeed appeared negative interest rates from time to time, but the negative values usually were small and stayed above some lower bound. However, in [10] the short rate is modeled by a so-called square-root process. This approach leads to a mean-reverting, strictly positive and chi-square distributed short rate process. In [6] the authors propose a time-homogeneous short rate model which is extended by a deterministic shift function in order to allow for negative rates and a perfect fit to the initially observed term structure. A very detailed overview on short rate models and their properties can be found in [7,16] and [21]. In [26] and [33] short rate models in an extended multiple-curve framework are presented. The probably most famous forward rate model is the Heath-Jarrow-Morton (HJM) model proposed in [27]. Therein, the instantaneous forward rate process is modeled directly by an arithmetic SDE driven by a BM. In [4] the HJM model is extended to a jump-diffusion setup where the forward rate process is affected by both diffusion and random jump noise. HJM type models are also treated in [7] and Chapter 7 in [28]. In [12] and [26] HJM forward rate models in an extended multi-curve framework are discussed. The class of the so-called market models was introduced in [5]. For example, the popular LIBOR model belongs to this modeling class. In most cases, market models involve geometric SDEs such that the modeled interest rates usually turn out to be strictly positive. In order to allow the modeled rates also to take small negative values, shifted market model approaches have been proposed recently. Numerous properties of affine LIBOR models are provided in [31]. Market models are also presented in [7]. In [26] LIBOR models in an extended multi-curve framework are discussed. In [17] the authors propose a Lévy forward price model in a multi-curve setup which is able to generate negative interest rates. Term structure models which are driven by Lévy processes have also been proposed in [18][19][20].
In the present paper, we introduce a new pure-jump multi-factor short rate model which is bounded from below by a real-valued function of time which can be chosen arbitrarily. The short rate process is modeled by a deterministic function plus a sum of pure-jump zero-reverting Ornstein-Uhlenbeck processes. It turns out that the short rate is mean-reverting and that the related bond price formula possesses an affine representation. We also provide the dynamics of the related instantaneous forward rate, the latter being of HJM type. We further derive a condition under which the forward rate model can be market-consistently calibrated. The analytical tractability of our model is illustrated by the derivation of a plain-vanilla option price formula with Fourier transform methods. With view on practical applications, we make concrete assumptions on the distribution of the jump noises and show how explicit formulas can be deduced in these cases. We conclude the paper by presenting a post-crisis extension of our short and forward rate model.
The outline of the paper is as follows: In Section 2 we introduce our new purejump multi-factor short rate model which is bounded from below. Section 3 is ded-icated to the derivation of related bond price and forward rate representations. Section 4 is devoted to option pricing. Section 5 contains guidelines for a practical application, while putting a special focus on possible distributional choices for the modeling of the involved jump noises. In Section 6 we consider a post-crisis extension of the proposed short and forward rate model.
2 A pure-jump multi-factor short rate model Let Ω, F, F = (F t ) t∈[0,T ] , Q be a filtered probability space satisfying the usual hypotheses, i.e. F t = F t+ := ∩ s>t F s constitutes a right-continuous filtration and F denotes the sigma-algebra augmented by all Q-null sets (cf. [30,35]). Here, Q is a risk-neutral probability measure and T > 0 denotes a fixed finite time horizon. In this setup, for arbitrary n ∈ N we define the stochastic short rate process r = (r t ) t∈[0,T ] via where µ (t) is a differentiable real-valued deterministic L 1 -function and X k t constitute pure-jump zero-reverting Ornstein-Uhlenbeck (OU) processes satisfying the SDE with deterministic initial values X k 0 := x k ≥ 0, constant mean-reversion velocities λ k > 0 and constant volatility coefficients σ k > 0. Herein, the independent, càdlàg, increasing, pure-jump, compound Poisson Lévy processes L k t are defined by where D k ⊆ R + := ]0, ∞[ ⊂ R denote jump amplitude sets and N k constitute Poisson random measures (PRMs). Note that the processes X k t and L k t always jump simultaneously, while X k t decays exponentially between its jumps due to the dampening linear drift term appearing in (2.2). A typical trajectory of a Lévy-driven OU process is shown in Figure 15.1 in [9]. Further note that the background-driving timehomogeneous Lévy processes L k t are increasing and thus, constitute so-called subordinators. Moreover, for all k ∈ {1, . . . , n} and (s, which constitute (F , Q)-martingale integrators. Herein, the positive and σ-finite Lévy measures ν k satisfy the integrability conditions for an arbitrary constant ̟ ∈ R (cf. [9,17]). For all k ∈ {1, . . . , n} and t ∈ [0, T ] we obtain both being finite entities due to (2.5) (cf. Section 1 in [17]). We remark that the currently proposed multi-factor short rate model (2.1) has been inspired by the electricity spot price model introduced in [2]. Arithmetic multi-factor models of this type have also been investigated in [28] and Section 3.2.2 in [3].
Remark 2.1. (a) Since L k t is increasing and X k t is zero-reverting from above, the function µ (t) is the mean-reversion floor or lower bound of the short rate process r t , i.e. it holds r t ≥ µ (t) Q-a.s. for all t ∈ [0, T ], while r t is mean-reverting from above to µ (t). Also note that the presence of a Brownian motion (BM) as driving noise in one of the processes X 1 t , . . . , X n t would destroy the lower boundedness of r t . In contrast to the presented pure-jump approach, it appears difficult to set up (lower-) bounded processes in arithmetic BM approaches. Moreover, we recall that negative rates have been observed in real-world post-crisis interest rate markets. Such scenarios can easily be captured by our model by choosing, e.g. µ (t) ≡ c, where c < 0 is an arbitrary constant. (In practical applications, it may happen that the floor function µ (t) needs to be readjusted, if interest rates evolve lower than anticipated. This issue has been discussed in [1] in the context of the SABR model.) (b) Our pure-jump model (2.1) is able to generate short rate trajectories which closely resemble those stemming from common Brownian motion approaches, if we allow for small jump sizes only, i.e. D k = ǫ k 1 , ǫ k 2 with small constants 0 < ǫ k 1 < ǫ k 2 . In this context, we emphasize that the well-established pure-jump variance gamma model is likewise able to generate suitable price trajectories, although there is neither any diffusion component involved (cf. Section 2.6.3 in [3], Table 4.5 in [9], Section 5.3.7 in [37]). On top of that, our pure-jump model might even provide more flexibility concerning the modeling of distributional properties than common BM approaches, since we are able to implement tailor-made distributions via an appropriate choice of the Lévy measures ν k which fit the empirical behavior of the rates in a best possible manner. This topic is further discussed in Section 5 below. For instance, (generalized) inverse Gaussian, tempered stable or gamma distributions might embody suitable choices (recall Appendix B.1.2 on p. 151 in [37]). We finally recall that a model of the type (2.1) has been fitted to real market data in [2] (yet in an electricity market context).
For a time partition 0 ≤ t ≤ s ≤ T the solution of (2.2) under Q can be expressed as where we used (2.3). The representation (2.6) implies where 0 ≤ t ≤ T . For all t ∈ [0, T ] we next define the historical filtration F t := σ{L 1 s , . . . , L n s : 0 ≤ s ≤ t}.
where the short rate process r t satisfies (2.1). Both entities are finite due to (2.5).
(Here and in what follows, we omit all proofs which are straightforward.) Taking u = 0 in Proposition 2.2, we find for all t ∈ [0, T ] Note that it is possible to identify the entities E Q X k t and Var Q X k t inside the latter equations due to (2.1). Moreover, suppose that µ (t) →μ for t → ∞ wherẽ µ ∈ R is a finite constant. Then we observe which both constitute finite constants. This limit behavior entirely stands in line with the requirements imposed on short rate models claimed on p. 46 in [7]. In the next step, we investigate the characteristic function of r t which is defined via Then for any u ∈ R and t ∈ [0, T ] the characteristic function of r t can be decomposed as where the characteristic function of X k t is given by u)x k +ρ k (t,u) with deterministic and affine characteristic exponent.

An immediate consequence of Proposition 2.3 is the subsequent affine representation
where we introduced the deterministic functions We emphasize that r t is an affine function of the factors X 1 t , . . . , X n t such that our model turns out to be a special case of the affine short rate models considered in Section 3.3 in [14]. To read more on affine processes we refer to [7,14,16,26] and [31]. We next define the moment generating function of r t via Note that the moment generating function κ rt (v) is well-defined due to (2.5). In the sequel, we derive the time dynamics of the short rate process.
Proposition 2.4. For all t ∈ [0, T ] the short rate process follows the dynamics Remark 2.5. We recall that our model constitutes an extension of the short rate model proposed in [6], whereas we work with multiple pure-jump processes L 1 t , . . . , L n t as driving noises instead of the single Brownian motion W t appearing in Eq. (1) in [6]. Moreover, comparing Eq. (3) in [6] with Eq. (2.1) above, we see that x t and ϕ (t; α) in [6] correspond in our setup to n k=1 X k t and µ (t), respectively.

Bond prices and forward rates
In this section, we derive representations for zero-coupon bond prices, forward rates and the interest rate curve related to the short rate model introduced in Section 2. To begin with, we introduce a bank account with stochastic interest rate r t satisfying with normalized initial capital β 0 = 1. The solution of (3.1) reads as where t ∈ [0, T ]. In this setup, the (zero-coupon) bond price at time t ≤ T with maturity T is given by where t ∈ [0, T ] (cf. [6,7,26]). Note that T ] (similar to, e.g., the CIR model [10]; also see [7,21]). On the other hand, if µ (t) < 0, then we only know that M t,T > 1.
Then the bond price at time t ≤ T with maturity T possesses the affine representation where the factors X k t satisfy (2.7).
Proof. First of all, we put (2.6) into (2.1) and obtain where 0 ≤ t ≤ s ≤ T . We next substitute the latter equation into (3.3), hereafter apply Fubini's theorem and identify the functions B k (·, T ). This procedure yields Taking the independent increment property of the (Q-independent) Lévy processes L 1 , . . . , L n into account, we obtain where t ∈ [0, T ]. The usual expectations appearing here can be handled by the Lévy-Khinchin formula for additive processes (see, e.g., [9,30,36]) which leads us to Putting the latter equations together and identifying the functions A k (·, T ), we end up with the asserted representation (3.6).
Recall that the bond price in (3.6) is the product of exponential affine functions of the factors X 1 t , . . . , X n t (but not of r t ). Also note that for all k ∈ {1, . . . , n} and We remark that the functions B k (t, T ) in (3.5) possess the same structure as the corresponding ones in the Vasicek model (cf. [38], or [7,16,21]). For all t ∈ [0, T ] Eq. (3.6) can be rewritten as which implies P (T, T ) = 1 due to (3.7). Moreover, from (3.5) we infer the time derivatives Hence, the functions B k (t, T ) ≤ 0 are strictly monotone increasing in t. Also note that the formulas found in (3.9) entirely stand in line with those claimed in (4.4)-(4.5) in [31]. From (3.5), (3.7) and (3.9) we deduce the following system of ordinary differential equations (ODEs) where t ∈ [0, T ] and k ∈ {1, . . . , n}. We are now prepared to derive the time dynamics of the bond price process (P (t, T )) t∈[0,T ] .
with B k (t, T ) as in (3.5). Then the bond price satisfies the t-dynamics under Q Recall that it holds ζ k (t, T, z) ≤ 0, since σ k B k (t, T ) z ≤ 0 for all k, t and z. We stress that (3.11) possesses the same structure as the corresponding Eq. (10.9) in [37], whereas the latter stems from a Brownian motion model without jumps. In the next step, we provide the solution of the SDE (3.11). Proposition 3.3. For all t ∈ [0, T ] the solution of (3.11) under Q reads as where the initial value P (0, T ) is deterministic and fulfills P (0, T ) > 0. Furthermore, for all t ∈ [0, T ] let us introduce the discontinuous Doléans-Dade exponential where h k (s, z) is an arbitrary integrable deterministic function (which may also depend on T ). We recall that Ξ k 0 = 1 and that Ξ k t t∈[0,T ] constitutes a local Qmartingale. With definition (3.13) at hand, we can express Eq. (3.12) as follows.
where β t is the bank account process given in (3.2), E denotes the Doleáns-Dade exponential defined in (3.13) and ξ k (s, z) := σ k B k (s, T ) z = log (1 + ζ k (s, T, z)) is a deterministic function.
Moreover, for all 0 ≤ t ≤ T we define the discounted bond pricê such thatP (t, T ) constitutes an F t -adapted (true) martingale under Q, as required by the risk-neutral pricing theory. Plugging (3.14) into (3.15), for all t ∈ [0, T ] we obtain where P (0, T ) is deterministic and ξ k is such as defined in Corollary 3.4. We obtain the following result.
Proposition 3.5. For all t ∈ [0, T ] the discounted bond price satisfies the Q-martingale dynamics dP (t, T ) where the deterministic functions ζ k (t, T, z) are such as defined in (3.10).
With reference to [7], we define the instantaneous forward rate at time t with maturity T via where t ∈ [0, T ] and ∂ T denotes the differential operator with respect to T . Equation Proof. By the definition of B k (t, T ) claimed in (3.5) we find so that the functions B k (t, T ) are strictly monotone decreasing in T . From (3.5) and (3.10) we further deduce whereas Fubini's theorem (see, e.g., Theorem 2.2 in [3]) leads us to (We are able to apply Fubini's theorem here, since the deterministic function ζ k (s,T,z) is measurable and square-integrable with respect to s ∈ [0, T ] and z ∈ D k .) The Leibniz formula for parameter integrals (see Lemma 2.4.1 on p. 13 in [28]) yields where we used (3.10), (3.7) and (3.19). Putting these formulas together, the proof is complete.
Proposition 3.7. For all t ∈ [0, T ] the instantaneous forward rate can be represented as (3.20) where the factor processes X k t satisfy (2.7) and B k (s, T ) is like defined in (3.5).
Replacing T by t in (3.20), we immediately find f (t, t) = r t due to (2.1). This equality stands in line with the usual conventions in interest rate theory (see, e.g., [7,16,21]). In what follows, we illustrate how our forward rate model can be fitted to the initially observed term structure. This procedure is often called market-consistent calibration in the literature. For this purpose, we denote by f M (0, T ) the deterministic initial forward rate. If f (0, T ) = f M (0, T ) and hence, if P (0, T ) = P M (0, T ) for all maturity times T > 0, then the underlying model is called market-consistent.  This object is called continuously-compounded spot rate on p. 60 in [7]. It obviously holds where t ∈ [0, T [. Comparing the exponent in (3.24) with that in (3.8), we infer where A k and B k are such as defined in (3.5). Hence, it turns out that the interest rate curve R (t, T ) can be represented as a sum of affine functions of the pure-jump OU factors X 1 t , . . . , X n t . In this sense, our short rate model possesses an affine term structure (cf. Section 3.2.4 in [7], or [14,16,21]). The latter observation entirely stands in line with (3.8).
where ζ k and B k are such as defined in (3.10) and (3.5), respectively. Proof. We substitute (2.1) and (2.7) into the definition of I t and obtain where B k is like defined in (3.5). An application of Fubini's theorem (see Theorem so that the proof is complete. Recall that the last jump integral in (3.25) constitutes a so-called Volterra integral, as the time parameter t appears both inside the integrand and inside the upper integration bound. Also note that it holds I t = log β t with I 0 = 0 due to (3.2).

Option pricing
In this section, we investigate the evaluation of a plain vanilla option written on the zero-coupon bond price P (·, T ). With reference to the risk-neutral pricing theory, the price at time t ≤ τ of an option with payoff H τ at the maturity time τ reads as where β is the bank account process given in (3.2) and Q denotes a risk-neutral pricing measure (cf. Eq. (3.1) in [7]). We now consider a call option written on the bond price P (·, T ) with maturity time T satisfying T ≥ τ . The payoff of the call option written on P (τ, T ) with deterministic strike price K > 0 and maturity time τ is then given by In what follows, we define the Fourier transform, respectively inverse Fourier transform, of a real-valued deterministic function p (·) ∈ L 1 (R) viâ where the integrated short rate process I t is such as defined in (3.25) and η k (s, z, y) := η k (s, z, T, τ, y) constitute deterministic functions, while a > 1 is an arbitrary real-valued constant. Herein, the functions ζ k and B k are such as defined in (3.10) and (3.5), respectively, while P (0, T ) denotes the deterministic initial bond price.
Proof. We substitute (4.2) and (3.12) into (4.1) and obtain where I t denotes the integrated short rate process defined in (3.25) and is a real-valued stochastic process. For u ∈ R we introduce the deterministic function where a > 1 is a constant real-valued dampening parameter ensuring the integrability of the payoff function. Indeed, it holds q (·) ∈ L 1 (R). With the latter definition at hand, we obtain With reference to [8] (also see [18]), we apply the inverse Fourier transform on the latter equation and hereafter, use Fubini's theorem which leads us to where we have set Z t,τ := I t − I τ + (a + iy) G τ for all 0 ≤ t ≤ τ . By merging the definition of G τ and (3.25) into the definition of Z t,τ we deduce where we identified the deterministic functions θ (τ, y) and η k (s, z, y) defined in (4.4). Hence, since I t is F t -adapted and θ (τ, y) is deterministic. In the derivation of the latter equation, we used the independent increment property under Q of the involved purejump integrals. We next apply the Lévy-Khinchin formula for additive processes (see, e.g., [9,30,36]) and derive where the characteristic exponents ψ k (t, τ, y) are such as defined in (4.4). Putting the latter equations together, we eventually end up with (4.3). The expression for the Fourier transformq (y) is obtained by straightforward calculations using the definition of the function q (u).
Corollary 4.2. In the special case t = 0, the call option price formula (4.3) simplifies to which is deterministic.

Practical applications
In this section, we show how the short rate model introduced in Section 2 can be implemented in practical applications. For this purpose, we now present more detailed expressions in order to prepare our model for a possible calibration of the involved parameters. First of all, let us recall that the increasing compound Poisson processes L k t defined in (2.3) for every k ∈ {1, . . . , n} and t ∈ [0, T ] can be expressed as (cf. Section 5.3.2 in [37]) where N k t constitutes a standard Poisson process under Q with deterministic jump intensity α k > 0. That is, N k t ∼ P oi (α k t) such that for all m ∈ N 0 it holds The strictly positive jump amplitudes of the Lévy process L k t are modeled by the i.i.d. random variables Y k 1 , Y k 2 , . . . which take values in the set D k ⊆ ]0, ∞[. We recall that the random variables Y k 1 , Y k 2 , . . . are independent of the Poisson processes N k t for all combinations of indices k, k ∈ {1, . . . , n}. We further put c k := E Q [Y k 1 ] ∈ D k and recall that the compensated compound Poisson process L k t − c k α k t t∈[0,T ] constitutes an (F t , Q)-martingale for each k which implies due to (2.3) and (2.4). We stress that the Poisson processes N k t shall not be mixed up with the Poisson random measures dN k (t, z).
In the following, we propose a number of probability distributions living on the positive half-line (recall Section B.1.2 in [37]) which constitute suitable candidates for the modeling of the jump size distribution in our new short rate model. As a first example, we propose to work with the gamma distribution and thus, assume that each random variable Y k j is exponentially distributed under Q with parameter ε k > 0 for all j and k. In this case, the related Lévy measure possesses the Lebesgue density where z ∈ D k = ]0, ∞[ and k ∈ {1, . . . , n}. We find c k = 1/ε k and Y k j ∼ Γ (1, ε k ). Hence, following the notation used in Section 5.5.1 in [37], we state that we presently are in a Γ (α k , ε k )-Ornstein-Uhlenbeck process setup (also see Section 8.2 in [31] and Example 15.1 in [9] in this context).
Proposition 5.1. Suppose that the random variables Y k j in (5.1) are exponentially distributed (i.e. Γ (1, ε k )-distributed) under Q with parameters ε k > 0 for all j and k. Then, for u ∈ R and t ∈ [0, T ] the characteristic function of L k t is given by where α k denotes the jump intensity of the standard Poisson process N k t appearing in (5.1).
Proof. Successively applying the definition of the characteristic function, (2.3), the Lévy-Khinchin formula and (5.2), for u ∈ R and t ∈ [0, T ] we obtain We eventually perform the integration and end up with the asserted equality.
An immediate consequence of Proposition 5.1 is the following representation for the moment generating function of L k t being valid for all Proposition 5.2. Assume that the random variables Y k j in (5.1) are exponentially distributed (i.e. Γ (1, ε k )-distributed) under Q with parameters ε k > 0 for all j and k. Then, for all t ∈ [0, T ], k ∈ {1, . . . , n} and x ∈ R the probability density function of L k t under Q takes the form Proof. First, note that it holds due to the definitions of the characteristic function and the Fourier transform claimed in the sequel of (4.2). We next apply the inverse Fourier transform which yields the density function We finally plug the result of Proposition 5.1 into the latter equation which completes the proof.
We stress that Eq. (5.2) can be substituted into the corresponding formulas appearing in the previous Propositions 2.2, 2.3, 3.1, 3.3, 3.7-3.10 and 4.1 yielding more explicit expressions for the involved entities, yet associated with gamma-distributed jump amplitudes in the underlying short rate model. We illustrate this statement by an application of Eq. (5.2) on Proposition 2.3. The precise result reads as follows.
Proposition 5.3. Suppose that the random variables Y k j in (5.1) are exponentially distributed (i.e. Γ (1, ε k )-distributed) under Q with parameters ε k > 0 for all j and k. Let σ k > 0 be the constant volatility coefficients introduced in (2.2). Then, for all t ∈ [0, T ] and v ∈ R with v < min k {ε k /σ k }, k ∈ {1, . . . , n}, the moment generating function under Q of the short rate process r t reads as In this setting, we combine Eq. (5.2) with the definitions of ρ k and Λ k given in Proposition 2.3 and obtain We perform the integration and obtain the formula for ρ k claimed in the proposition. The representation for the moment generating function κ rt (v) finally follows from Proposition 2.3.
Other distributional choices for the random variables Y k j modeling the jump amplitudes would be, for example, the inverse Gaussian distribution (see Section 5.5.2 in [37]), the generalized inverse Gaussian distribution (see Section 5.3.5 in [37]) or the tempered stable distribution (see Section 5.3.6 in [37]). The related formulas for the Lebesgue density of the Lévy measure dν k (z) corresponding to Eq. (5.2) can be found in [37].
Remark 5.4. We recall that the time-homogeneous compound Poisson processes L k t introduced in (2.3) can be simulated according to Algorithms 6.1 and 6.2 in [9]. Further, in our model it is easily possible to calculate the moments of X k t and r t (see the sequel of Proposition 2.2) so that our model can be fitted to any yield curve observed in the market by using the moment estimation method described in Section 7.2.2 in [9]. This procedure is also called moment matching, as the underlying idea is to make the empirical moments match with the theoretical moments of the model by finding a suitable parameter set.

A post-crisis model extension
In this section, we propose a post-crisis extension of the pure-jump lower-bounded short rate model introduced in Section 2. (To read more on post-crisis interest rate models, the reader is referred to [11-14, 17, 22-26, 32-34].) Inspired by the modeling setups presented in [33] and Chapter 2 in [26], for all t ∈ [0, T ] we define the short rate spread under Q by the stochastic process showing a similar structure as (2.1). Herein, µ * (t) ≥ 0 constitutes an integrable realvalued deterministic function and the factors X k t satisfy the SDE (2.2), but presently for indices k ∈ {n + 1, . . . , l} where l ∈ N with l > n. Note that it holds s t ≥ µ * (t) Q-a.s. for all t ∈ [0, T ] such that the short rate spread is bounded from below -similar to the short rate itself [recall Remark 2.1 (a)]. We interpret s t as an additive spread and therefore set for all t ∈ [0, T ] r t := r t + s t (6.1) (cf. [12,33]) where r t denotes the short rate process and r t is called fictitious short rate, similarly to [26]. With reference to p. 46 in [26], we recall that the short rate spread s t not only incorporates credit risks, but also various other risks in the interbank sector which affect the evolution of the LIBOR rates. Let us moreover mention that the short rate r t defined in (2.1) and the short rate spread s t can be 'correlated' by allowing for (at least) one common factor in their respective definitions. More precisely, if the sum in the definition of s t started running from k = n (instead of k = n + 1), then the factor X n t would appear both in the definition of r t and in the definition of s t such that the two latter stochastic processes would no longer be independent.
We next substitute (2.1) as well as the definition of s t into (6.1) and deduce where we introduced the real-valued deterministic function µ (t) := µ (t) + µ * (t). It obviously holds r t ≥ µ (t) Q-a.s. for all t ∈ [0, T ]. In accordance to Section 3.4.1 in [14], Eq. (2.35) in [26] and Section 1 in [33], we define the fictitious bond price in our post-crisis short rate model via where t ∈ [0, T ]. The object P (t, T ) is sometimes called artificial bond price in the literature, as it is not physically traded in the market. Evidently, P (T, T ) = 1. Comparing (6.2) with (2.1) and (6.3) with (3.3), we see that all our single-curve results presented in the previous sections carry over to the currently considered post-crisis case. More precisely, the entities µ (t), r t , P (t, T ) and n emerging in the previous single-curve equations presently have to be replaced by µ (t), r t , P (t, T ) and l, respectively. Moreover, in the present case, we observe Q-a.s. for all t ∈ [0, T ] due to (6.3), the lower boundedness of r t and the monotonicity of conditional expectations.
where P (t, T ) constitutes the bond price defined in (6.3) and P (t, T ) is given in (3.3).
Proof. Note that taking µ * (t) ≥ 0 ∀ t ∈ [0, T ] implies s t ≥ 0 Q-a.s. for all t ∈ [0, T ]. In this case, we deduce r t ≥ r t Q-a.s. for all t ∈ [0, T ] due to (6.1). Hence, we find Q-a.s. for all t ∈ [0, T ]. We next take conditional expectations with respect to F t and Q in the latter inequality, hereafter apply the monotonicity of conditional expectations and finally identify (6.3) and (3.3) in the resulting inequality.
The result obtained in Proposition 6.1 possesses the following economical interpretation: The inequality (6.4) shows that nontraded bonds are cheaper than their nonfictitious counterparts which are physically traded in the market. This feature appears economically reasonable and stands in accordance with the modeling assumptions and statements in [12], Section 2.1.3 in [26] and Section 2.1 in [33]. Moreover, combining (6.3) and (3.18), we obtain (parallel to [12]) where f is sometimes called fictitious/artificial forward rate in the literature. It holds f (t, t) = r t for all t ∈ [0, T ]. With reference to [11] and [12], for all t ∈ [0, T ] we introduce the forward rate spread via g (t, T ) := f (t, T ) − f (t, T ) so that we have not only set up a new pure-jump post-crisis short rate model, but simultaneously a new pure-jump post-crisis forward rate model of HJM-type in the current section. Recall that (6.4) is equivalent to f (t, u) ≤ f (t, u) Q-a.s. for all 0 ≤ t ≤ u ≤ T . From this, we conclude that g (t, T ) ≥ 0 Q-a.s. for all t ∈ [0, T ]. It further holds g (t, t) = s t for all t ∈ [0, T ] due to (6.1).
Furthermore, in the present post-crisis setting, for a time partition 0 ≤ t ≤ T 1 < T 2 we define the (forward) overnight indexed swap (OIS) rate via F (t, T 1 , T 2 ) := 1 δ P (t, T 1 ) P (t, T 2 ) − 1 (cf. Eq. (4.1) in [26]) where P denotes the zero-coupon bond price defined in (3.3) and δ := δ (T 1 , T 2 ) is the year fraction with expiry date T 1 and maturity date T 2 . With reference to [12], for 0 ≤ t ≤ T 1 < T 2 we define the forward LIBOR rate via L (t, T 1 , T 2 ) := 1 δ P (t, T 1 ) where δ is the year fraction and P denotes the bond price introduced in (6.3). Note that the LIBOR rate L (t, T 1 , T 2 ) shall not be mixed up with the Lévy processes L k t defined in (2.3). In a pre-crisis single-curve approach, it holds P (t, T ) = P (t, T ) which implies F (t, T 1 , T 2 ) = L (t, T 1 , T 2 ) Q-a.s. for all t.
We are now prepared to derive the dynamics of the short rate spread s t , the fictitious short rate r t , the bond price P (t, T ), the forward rate f (t, T ), the forward rate spread g (t, T ) and the LIBOR rate L (t, T 1 , T 2 ). The associated computations can be accomplished by similar techniques as presented in Sections 2 and 3 and thus, are not worked out explicitly. We provide as an example two results without proofs in the sequel. For all t ∈ [0, T ] it holds dP (t, T ) P (t−, T ) = r t dt +