Modelling Electricity Prices: A Time Change Approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump CoxIngersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a timeinhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the timechanged JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss-Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.