Fast calibration of two-factor models for energy option pricing

Energy companies need efficient procedures to perform market calibration of stochastic models for commodities. If the Black framework is chosen for option pricing, the bottleneck of the market calibration is the computation of the variance of the asset. Energy commodities are commonly represented by multifactor linear models, whose variance obeys a matrix Lyapunov differential equation. In this article, analytical and methods to derive the variance are discussed: the Lyapunov approach is shown to be more straightforward than ad hoc derivations found in the literature and can be readily extended to higher dimensional models. A case study is presented, where the variance of a two-factor mean-reverting model is embedded into the Black formulae and the model parameters are calibrated against listed options. The analytical and numerical methods are compared, showing that the former makes the calibration 14 times faster. A Python implementation of the proposed methods is available as open-source software on GitHub.