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.

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 [

Widely applied short rate models are for example the Vasicek model [

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

Let

For a time partition

(Here and in what follows, we omit all proofs which are straightforward.) Taking

An immediate consequence of Proposition

We recall that our model constitutes an extension of the short rate model proposed in [

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

First of all, we put (

Recall that the bond price in (

Recall that it holds

Furthermore, for all

Moreover, for all

With reference to [

By the definition of

We substitute (

Replacing

In what follows, we illustrate how our forward rate model can be fitted to the initially observed term structure. This procedure is often called

Note that the floor function

We substitute (

Recall that the last jump integral in (

In this section, we investigate the evaluation of a plain vanilla option written on the zero-coupon bond price

We substitute (

In this section, we show how the short rate model introduced in Section

In the following, we propose a number of probability distributions living on the positive half-line (recall Section B.1.2 in [

Successively applying the definition of the characteristic function, (

An immediate consequence of Proposition

First, note that it holds

We stress that Eq. (

For each

Other distributional choices for the random variables

We recall that the time-homogeneous compound Poisson processes

In this section, we propose a post-crisis extension of the pure-jump lower-bounded short rate model introduced in Section

We next substitute (

Note that taking

The result obtained in Proposition

Furthermore, in the present post-crisis setting, for a time partition

We are now prepared to derive the dynamics of the short rate spread