Modelling electricity prices: a time change approach

Abstract

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 Cox-Ingersoll-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 time-inhomogeneous 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 time-changed 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.

Keywords:

• Electricity spot prices
• Electricity futures and futures options
• Spikes
• Mean-reversion
• Seasonality
• Stochastic time change
• Laplace transform

JEL Classifications:

• G12
• G13

Acknowledgements

We thank Prof. Ehud Ronn for helpful discussions and suggestions. We are also very grateful to two anonymous referees and the editor whose suggestions helped to improve the quality of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Lingfei Li acknowledges research support from Hong Kong Research Grant Council ECS [grant number 24200214].