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 [18–20].

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 pure-jump multi-factor short rate model which is bounded from below. Section 3 is dedicated 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.

Let Ω,F,F=Ftt∈0,T,Q be a filtered probability space satisfying the usual hypotheses, i.e. Ft=Ft+:=∩s>tFs t}}{\mathcal{F}_{s}}$]]> constitutes a right-continuous filtration and F denotes the sigma-algebra augmented by all Q -null sets (cf. [30, 35]). Here, Q is a risk-neutral probability measure and T>0 0$]]> denotes a fixed finite time horizon. In this setup, for arbitrary n∈N we define the stochastic short rate process r=rtt∈0,T via (2.1) rt:=μt+ ∑k=1nXtk where μt is a differentiable real-valued deterministic L1 -function and Xtk constitute pure-jump zero-reverting Ornstein–Uhlenbeck (OU) processes satisfying the SDE (2.2) dXtk=−λkXtkdt+σkdLtk with deterministic initial values X0k:=xk≥0 , constant mean-reversion velocities λk>0 0$]]> and constant volatility coefficients σk>0 0$]]>. Herein, the independent, càdlàg, increasing, pure-jump, compound Poisson Lévy processes Ltk are defined by (2.3) Ltk:=∫0t∫DkzdNks,z where Dk⊆R+:=0,∞⊂R denote jump amplitude sets and Nk constitute Poisson random measures (PRMs). Note that the processes Xtk and Ltk always jump simultaneously, while Xtk 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 time-homogeneous Lévy processes Ltk are increasing and thus, constitute so-called subordinators. Moreover, for all k∈1,…,n and s,z∈0,T×Dk we define the Q -compensated PRMs (2.4) dN˜kQs,z:=dNks,z−dνkzds which constitute F,Q -martingale integrators. Herein, the positive and σ-finite Lévy measures νk satisfy the integrability conditions (2.5) ∫Dk1∧zdνkz<∞,∫z>1eϖzdνkz<∞ 1}}{e^{\varpi z}}d{\nu _{k}}\left(z\right)<\infty \]]]> for an arbitrary constant ϖ∈R (cf. [9, 17]). For all k∈1,…,n and t∈0,T we obtain EQ[Ltk]=t∫Dkzdνkz,VarQ[Ltk]=t∫Dkz2dνkz 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].

For a time partition 0≤t≤s≤T the solution of (2.2) under Q can be expressed as (2.6) Xsk=Xtke−λks−t+σk∫ts∫Dke−λks−uzdNku,z where we used (2.3). The representation (2.6) implies (2.7) Xtk=xke−λkt+σk∫0t∫Dke−λkt−szdNks,z where 0≤t≤T . For all t∈0,T we next define the historical filtration Ft:=σ{Ls1,…,Lsn:0≤s≤t}.

(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 EQrt=μt+ ∑k=1nxke−λkt+σk1−e−λktλk∫Dkzdνkz,VarQrt= ∑k=1nσk21−e−2λkt2λk∫Dkz2dνkz. Note that it is possible to identify the entities EQXtk and VarQXtk inside the latter equations due to (2.1). Moreover, suppose that μt→μ˜ for t→∞ where μ˜∈R is a finite constant. Then we observe limt→∞EQrt=μ˜+ ∑k=1nσkλk∫Dkzdνkz,limt→∞VarQrt= ∑k=1nσk22λk∫Dkz2dνkz 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 rt which is defined via Φrtu:=EQ[eiurt] where u∈R and t∈0,T .

An immediate consequence of Proposition 2.3 is the subsequent affine representation (2.8) Φrtu= ∏k=1neψkt,uxk+ϕkt,u where we introduced the deterministic functions ϕkt,u:=ρkt,u+iuμt/n. We emphasize that rt is an affine function of the factors Xt1,…,Xtn 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 rt via (2.9) κrtv:=EQevrt which implies the well-known equalities Φrtu=κrtiu and κrtv=Φrt−iv . Note that the moment generating function κrtv 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 (2.10) drt=μ′t− ∑k=1nλkXtkdt+ ∑k=1nσk∫DkzdNkt,z.

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 rt satisfying (3.1) dβt=rtβtdt with normalized initial capital β0=1 . The solution of (3.1) reads as (3.2) βt=exp∫0trsds where t∈0,T . In this setup, the (zero-coupon) bond price at time t≤T with maturity T is given by (3.3) Pt,T:=βtEQβT−1|Ft=EQexp−∫tTrsds|Ft where t∈0,T (cf. [6, 7, 26]). Note that Pt,T>0Q 0\hspace{2.5pt}\mathbb{Q}$]]>-a.s. ∀t∈0,T by construction. Since rt≥μtQ -a.s. ∀t∈0,T [recall Remark 2.1 (a)], we observe (3.4) Pt,T≤Mt,T:=exp−∫tTμsds Q -a.s. ∀t∈0,T due to (3.3) and the monotonicity of conditional expectations. The upper bound Mt,T appearing in (3.4) is deterministic and strictly positive for all 0≤t≤T . If μt≥0 , then it holds Pt,T≤1Q -a.s. ∀t∈0,T (similar to, e.g., the CIR model [10]; also see [7, 21]). On the other hand, if μt<0 , then we only know that Mt,T>1 1$]]>.

Recall that the bond price in (3.6) is the product of exponential affine functions of the factors Xt1,…,Xtn (but not of rt ). Also note that for all k∈1,…,n and t∈0,T it holds (3.7) Akt,t=Bkt,t=0. We remark that the functions Bkt,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 (3.8) Pt,T=exp ∑k=1nAkt,T+Bkt,TXtk which implies PT,T=1 due to (3.7). Moreover, from (3.5) we infer the time derivatives (3.9) Ak′t,T=μtn−∫Dk[eσkBkt,Tz−1]dνkz,Bk′t,T=e−λkT−t>0 0\]]]> where Ak′:=∂tAk and Bk′:=∂tBk . Hence, the functions Bkt,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) Akt,T=−∫tTAk′s,Tds,Bk′t,T=1+λkBkt,T,AkT,T=BkT,T=0 where t∈0,T and k∈1,…,n . We are now prepared to derive the time dynamics of the bond price process Pt,Tt∈0,T .

Recall that it holds ζkt,T,z≤0 , since σkBkt,Tz≤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).

Furthermore, for all t∈0,T let us introduce the discontinuous Doléans-Dade exponential (3.13) Ξtk:=E(hk∗N˜kQ)t:=exp{∫0t∫Dkhks,zdN˜kQs,z−∫0t∫Dk[ehks,z−1−hks,z]dνkzds} where hks,z is an arbitrary integrable deterministic function (which may also depend on T). We recall that Ξ0k=1 and that Ξtkt∈0,T constitutes a local Q -martingale. With definition (3.13) at hand, we can express Eq. (3.12) as follows.

Moreover, for all 0≤t≤T we define the discounted bond price (3.15) Pˆt,T:=Pt,Tβt where Pˆ0,T=P0,T . From (3.3) we deduce Pˆt,T=EQ(βT−1|Ft) such that Pˆt,T constitutes an Ft -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 (3.16) Pˆt,T=P0,T ∏k=1nE(ξk∗N˜kQ)t where P0,T is deterministic and ξk is such as defined in Corollary 3.4. We obtain the following result.

With reference to [7], we define the instantaneous forward rate at time t with maturity T via (3.17) ft,T:=−∂TlogPt,T where t∈0,T and ∂T denotes the differential operator with respect to T. Equation (3.17) is equivalent to the representation (3.18) Pt,T=exp−∫tTft,udu.

Replacing T by t in (3.20), we immediately find ft,t=rt 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 fM0,T the deterministic initial forward rate. If f0,T=fM0,T and hence, if P0,T=PM0,T for all maturity times T>0 0$]]>, then the underlying model is called market-consistent.

Note that the floor function μt for all t∈0,T can be obtained from (3.22) by replacing T by t therein. Moreover, we define the interest rate curve at time t<T with maturity T via (3.23) Rt,T:=logPt,Tt−T. This object is called continuously-compounded spot rate on p. 60 in [7]. It obviously holds (3.24) Pt,T=e−T−tRt,T where t∈0,T . Comparing the exponent in (3.24) with that in (3.8), we infer Rt,T=1t−T ∑k=1n[Akt,T+Bkt,TXtk] where Ak and Bk are such as defined in (3.5). Hence, it turns out that the interest rate curve Rt,T can be represented as a sum of affine functions of the pure-jump OU factors Xt1,…,Xtn . 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).

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 It=logβt with I0=0 due to (3.2).

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 Hτ at the maturity time τ reads as (4.1) Ct=βtEQ(βτ−1Hτ|Ft)=EQ(e−∫tτrsdsHτ|Ft) 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 Pτ,T with deterministic strike price K>0 0$]]> and maturity time τ is then given by (4.2) Hτ=Pτ,T−K+:=max0,Pτ,T−K. In what follows, we define the Fourier transform, respectively inverse Fourier transform, of a real-valued deterministic function p·∈L1R via pˆy:=12π∫Rpue−iyudu,pu=∫Rpˆyeiyudy. Proposition 4.1.