A pure-jump mean-reverting short rate model

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 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 $\left(\Omega,\mathbb{F},\mathcal{F}=\left(\mathcal{F}_{t}\right)_{t\in\left[0,T\right]},\mathbb{Q}\right)$ be a filtered probability space satisfying the usual hypotheses, i.e. $\mathcal{F}_{t}=\mathcal{F}_{t+}:=\cap_{s>t}\mathcal{F}_{s}$ constitutes a right-continuous filtration and $\mathbb{F}$ denotes the sigma-algebra augmented by all $\mathbb{Q}$-null sets (cf. [30, 35]). Here, $\mathbb{Q}$ is a risk-neutral probability measure and $T>0$ denotes a fixed finite time horizon. In this setup, for arbitrary $n\in\mathbb{N}$ we define the stochastic short rate process $r=\left(r_{t}\right)_{t\in\left[0,T\right]}$ via

where $\mu\left(t\right)$ is a differentiable real-valued deterministic $\mathcal{L}^{1}$-function and $X_{t}^{k}$ constitute pure-jump zero-reverting Ornstein–Uhlenbeck (OU) processes satisfying the SDE

with deterministic initial values $X_{0}^{k}:=x_{k}\geq 0$, constant mean-reversion velocities $\lambda_{k}>0$ and constant volatility coefficients $\sigma_{k}>0$. Herein, the independent, càdlàg, increasing, pure-jump, compound Poisson Lévy processes $L_{t}^{k}$ are defined by

where $D_{k}\subseteq\mathbb{R}^{+}:=\left]0,\infty\right[\subset\mathbb{R}$ denote jump amplitude sets and $N_{k}$ constitute Poisson random measures (PRMs). Note that the processes $X_{t}^{k}$ and $L_{t}^{k}$ always jump simultaneously, while $X_{t}^{k}$ 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 $L_{t}^{k}$ are increasing and thus, constitute so-called subordinators. Moreover, for all $k\in\left\{1,\dots,n\right\}$ and $\left(s,z\right)\in\left[0,T\right]\times D_{k}$ we define the $\mathbb{Q}$-compensated PRMs

which constitute $\left(\mathcal{F},\mathbb{Q}\right)$-martingale integrators. Herein, the positive and $\sigma$-finite Lévy measures $\nu_{k}$ satisfy the integrability conditions

for an arbitrary constant $\varpi\in\mathbb{R}$ (cf. [9, 17]). For all $k\in\left\{1,\dots,n\right\}$ and $t\in\left[0,T\right]$ 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].

For a time partition $0\leq t\leq s\leq T$ the solution of (2.2) under $\mathbb{Q}$ can be expressed as

where we used (2.3). The representation (2.6) implies

where $0\leq t\leq T$. For all $t\in\left[0,T\right]$ we next define the historical filtration

(Here and in what follows, we omit all proofs which are straightforward.) Taking $u=0$ in Proposition 2.2, we find for all $t\in\left[0,T\right]$

Note that it is possible to identify the entities $\mathbb{E}_{\mathbb{Q}}\left[X_{t}^{k}\right]$ and $\mathbh{Var}_{\mathbb{Q}}\left[X_{t}^{k}\right]$ inside the latter equations due to (2.1). Moreover, suppose that $\mu\left(t\right)\rightarrow\tilde{\mu}$ for $t\rightarrow\infty$ where $\tilde{\mu}\in\mathbb{R}$ 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

where $u\in\mathbb{R}$ and $t\in\left[0,T\right]$.

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 $X_{t}^{1},\dots,X_{t}^{n}$ 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

which implies the well-known equalities $\Phi_{r_{t}}\left(u\right)=\kappa_{r_{t}}\left(iu\right)$ and $\kappa_{r_{t}}\left(v\right)=\Phi_{r_{t}}\left(-iv\right)$. Note that the moment generating function $\kappa_{r_{t}}\left(v\right)$ is well-defined due to (2.5). In the sequel, we derive the time dynamics of the short rate process.

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 $\beta_{0}=1$. The solution of (3.1) reads as

where $t\in\left[0,T\right]$. In this setup, the (zero-coupon) bond price at time $t\leq T$ with maturity $T$ is given by

where $t\in\left[0,T\right]$ (cf. [6, 7, 26]). Note that $P\left(t,T\right)>0\ \mathbb{Q}$-a.s. $\forall\ t\in\left[0,T\right]$ by construction. Since $r_{t}\geq\mu\left(t\right)\ \mathbb{Q}$-a.s. $\forall\ t\in\left[0,T\right]$ [recall Remark 2.1 (a)], we observe

$\mathbb{Q}$-a.s. $\forall\ t\in\left[0,T\right]$ due to (3.3) and the monotonicity of conditional expectations. The upper bound $M_{t,T}$ appearing in (3.4) is deterministic and strictly positive for all $0\leq t\leq T$. If $\mu\left(t\right)\geq 0$, then it holds $P\left(t,T\right)\leq 1\ \mathbb{Q}$-a.s. $\forall\ t\in\left[0,T\right]$ (similar to, e.g., the CIR model [10]; also see [7, 21]). On the other hand, if $\mu\left(t\right)<0$, then we only know that $M_{t,T}>1$.

Recall that the bond price in (3.6) is the product of exponential affine functions of the factors $X_{t}^{1},\dots,X_{t}^{n}$ (but not of $r_{t}$). Also note that for all $k\in\left\{1,\dots,n\right\}$ and $t\in\left[0,T\right]$ it holds

We remark that the functions $B_{k}\left(t,T\right)$ in (3.5) possess the same structure as the corresponding ones in the Vasicek model (cf. [38], or [7, 16, 21]). For all $t\in\left[0,T\right]$ Eq. (3.6) can be rewritten as

which implies $P\left(T,T\right)=1$ due to (3.7). Moreover, from (3.5) we infer the time derivatives

where $A_{k}^{\prime}:=\partial_{t}A_{k}$ and $B_{k}^{\prime}:=\partial_{t}B_{k}$. Hence, the functions $B_{k}\left(t,T\right)\leq 0$ 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 $t\in\left[0,T\right]$ and $k\in\left\{1,\dots,n\right\}$. We are now prepared to derive the time dynamics of the bond price process $\left(P\left(t,T\right)\right)_{t\in\left[0,T\right]}$.

Recall that it holds $\zeta_{k}\left(t,T,z\right)\leq 0$, since $\sigma_{k}\ B_{k}\left(t,T\right)\ z\leq 0$ 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 $t\in\left[0,T\right]$ let us introduce the discontinuous Doléans-Dade exponential

where $h_{k}\left(s,z\right)$ is an arbitrary integrable deterministic function (which may also depend on $T$). We recall that $\Xi_{0}^{k}=1$ and that $\left(\Xi_{t}^{k}\right)_{t\in\left[0,T\right]}$ constitutes a local $\mathbb{Q}$-martingale. With definition (3) at hand, we can express Eq. (3.3) as follows.

Moreover, for all $0\leq t\leq T$ we define the discounted bond price

where $\hat{P}\left(0,T\right)=P\left(0,T\right)$. From (3.3) we deduce $\hat{P}\left(t,T\right)=\mathbb{E}_{\mathbb{Q}}(\beta_{T}^{-1}|\ \mathcal{F}_{t})$ such that $\hat{P}\left(t,T\right)$ constitutes an $\mathcal{F}_{t}$-adapted (true) martingale under $\mathbb{Q}$, as required by the risk-neutral pricing theory. Plugging (3.14) into (3.15), for all $t\in\left[0,T\right]$ we obtain

where $P\left(0,T\right)$ is deterministic and $\xi_{k}$ 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

where $t\in\left[0,T\right]$ and $\partial_{T}$ denotes the differential operator with respect to $T$. Equation (3.17) is equivalent to the representation

Replacing $T$ by $t$ in (3.20), we immediately find $f\left(t,t\right)=r_{t}$ 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 $f^{M}\left(0,T\right)$ the deterministic initial forward rate. If $f\left(0,T\right)=f^{M}\left(0,T\right)$ and hence, if $P\left(0,T\right)=P^{M}\left(0,T\right)$ for all maturity times $T>0$, then the underlying model is called market-consistent.

Note that the floor function $\mu\left(t\right)$ for all $t\in\left[0,T\right]$ 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

This object is called continuously-compounded spot rate on p. 60 in [7]. It obviously holds

where $t\in\left[0,T\right[$. 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 $R\left(t,T\right)$ can be represented as a sum of affine functions of the pure-jump OU factors $X_{t}^{1},\dots,X_{t}^{n}$. 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).