Estimation of stable CARMA models with an application to electricity spot prices

Abstract

We discuss theoretical properties and estimation of continuous-time ARMA (CARMA) processes, which are driven by a stable Lévy process. Such processes are very useful in a continuous-time linear stationary setup: they have a similar structure as the widely used ARMA models, and provide all advantages of a continuous-time model. As an application we consider data from a deregulated electricity market. Here we t a CARMA(2,1) model to spot prices from the Singapore New Electricity Market. The quality of the estimates is assessed in a simulation study. The continuous-time modelling aims at a new pricing methodology for energy derivatives.

Representation of Infinite-Dimensional Forward Price Models in Commodity Markets

“…More generally, we obtain a spot price dynamics in terms of so-called Lévy semistationary processes, which have gained recent attention in power markets (see Garcia et al. [28] for the special case of CARMA processes, and Barndorff-Nielsen et al [4]). These processes are also natural in modelling the temperature dynamics, and thus are relevant to weather forward prices.…”

“…In the traditional set-up, one is modelling the logarithmic spot price dynamics by a finite sum of Ornstein-Uhlenbeck processes, each driven by a Lévy process. However, several papers also advocate to model the spot price dynamics directly by such a finite sum of Ornstein-Uhlenbeck processes (see, e.g., Benth et al [10] and Garcia et al [28] for the case of power). In a very simplistic setting, one starts out with a model…”

“…Note that we are free to choose the covariance kernel q and the θ , which means that we can have many different forward curve models resulting in the same CARMA spot model. CARMA-processes have been applied to temperature models (see Benth et al [9] and Härdle and Lopez-Cabrera [30]), spot power prices (see Garcia et al [28]) and the prices of oil (see Paschke and Prokopczuk [39]). For a detailed analysis of CARMA processes, we refer the reader to Brockwell [16].…”

Pricing of Forwards and Other Derivatives in Cointegrated Commodity Markets

“…We consider the cases p = 2 and p = 3, which seems to be of most practical interest (see e.g. Garcia, Klüppelberg and Müller [23] and Pasche and Prokopczuk [35] for a CARMA(2,1) model for power spot and crude oil futures prices, respectively, and Benth andŠaltytė Benth [4] and Härdle and Lopez-Cabrera [25] for CARMA(3,0) models applied to temperature dynamics.) Let us first consider the matrix…”

“…Recall that Garcia, Klüppelberg and Müller [23], Benth et al [7] and Paschke and Prokopczuk [35] showed empirically that power and oil prices follow a CARMA(2,1) dynamics. In Fig.…”