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  market calibration is the computation of the variance of the asset. For energy commodities , it is common to adopt multi-factor linear models, whose variance obeys a matrix Lyapunov differential equation. In this paper, both analytical and numerical methods to derive the variance through a Lyapunov equation are discussedand compared in terms of computational efficiency. The Lyapunov approach illustrated herein is more straightforward than ad-hoc derivations found in the quantitative finance literature and can be readily extended by a practitioner to higher-dimensional models. A practical 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 then calibrated against listed options. In particular, the analytical and numerical method are compared, showing that the former makes the calibration 14 times faster. A Python implementation of the proposed procedures is available as open-source software on GitHub.
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 multi-factor linear models, whose variance obeys a matrix Lyapunov differential equation. In this paper, analytical and numerical 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 method 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.
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quantitative finance
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