On the speed towards the mean for continuous time autoregressive moving average processes with applications to energy markets

- The concept of half life is extended to Levy-driven continuous time autoregressive moving average processes
- The dynamics of Malaysian temperatures are modeled using a continuous time autoregressive model with stochastic volatility
- Forward prices on temperature become constant when time to maturity tends to infinity
- Convergence in time to maturity is at an exponential rate given by the eigenvalues of the model temperature model

We extend the concept of half life of an Ornstein-Uhlenbeck process to Lévy-driven continuous-time autoregressive moving average processes with stochastic volatility. The half life becomes state dependent, and we analyze its properties in terms of the characteristics of the process. An empirical example based on daily temperatures observed in Petaling Jaya, Malaysia, is presented, where the proposed model is estimated and the distribution of the half life is simulated. The stationarity of the dynamics yield futures prices which asymptotically tend to constant at an exponential rate when time to maturity goes to infinity. The rate is characterized by the eigenvalues of the dynamics. An alternative description of this convergence can be given in terms of our concept of half life.