On non-negative modeling with CARMA processes

Abstract

Two stationary and non-negative processes that are based on continuoustime autoregressive moving average (CARMA) processes are discussed. First, we consider a generalization of Cox-Ingersoll-Ross (CIR) processes. Next, we consider CARMA processes driven by compound Poisson processes with exponential jumps which are generalizations of Ornstein-Uhlenbeck (OU) processes driven by the same noise. The way in which the two processes generalize CIR and OU processes and the relation between them will be discussed. Furthermore, the stationary distribution, the autocorrelation function, and pricing of zero-coupon bonds are considered.

The Laplace transform of the integrated Volterra Wishart process

“…where X is the d × m Volterra process as in (3.1), Q ∈ S d + and ξ : [0, T ] → R is an input curve used to match today's yield curve and/or control the negativity level of the short rate. The model replicates the asymmetrical distribution of interest rates, allows for rich auto-correlation structures, and the possibility to account for long range dependence, see for instance Benth and Rohde (2018); Corcuera et al (2013).…”

“…We consider a quadratic short rate model of the form $$\begin{equation*} r_t = \operatorname{tr}{\left(X_t^\top Q X_t \right)} + \xi (t), \quad t\le T, \end{equation*}$$where X is the $$d\times m$$ Volterra process as in (42), $$Q\in {\mathbb {S}}^d_+$$ and $$\xi :[0,T]\rightarrow {\mathbb {R}}$$ is an input curve used to match today's yield curve and/or control the negativity level of the short rate. The model replicates the asymmetrical distribution of interest rates, allows for rich auto‐correlation structures, and the possibility to account for long range dependence, see for instance Benth and Rohde (2019), Corcuera et al. (2013).…”

“…Volterra Wishart processes have been recently studied in Cuchiero and Teichmann (2019); Yue and Huang (2018). Applications of certain quadratic Gaussian processes can be found in Benth and Rohde (2018); Corcuera et al (2013); Harms and Stefanovits (2019); Kleptsyna et al (2002). Gaussian stochastic volatility models have been treated in Gulisashvili (2018); Gulisashvili et al (2019).…”

Pricing of Commodity and Energy Derivatives for Polynomial Processes

“…The CIR stochastic dynamics is an example of a polynomial process. Benth and Rohde [28] extended the study of Bensoussan and Brouste [27] in two different directions. Considering a finite sum of squared CARMA processes, they defined a so-called CIR-CARMA model on the one hand.…”

“…We recall that factor models with Ornstein-Uhlenbeck dynamics driven by jump processes are relevant for power price and wind speed modelling. In particular, Ornstein-Uhlenbeck processes with exponential jump processes leading to invariant Γ-distributions are applied (recall discussion from [4,26,28] in Section 4, say). In addition, we have CIR-processes as a model for wind speeds as we recall from [27].…”