A general method is presented to compute the variance of a linear stochastic process through a matrix Lyapunov differential equation. This approach, adopted from control theory, is alternative and easier with respect to the classical arguments found in quantitative finance literature and can be readily applied to high-dimensional models. Both analytical and numerical methods to solve the Lyapunov equation are discussed and compared in terms of computational efficiency. A practical application is presented, where numerical and analytical solutions for the variance of a two-factor mean-reverting model are embedded into the Black pricing framework and market calibration of model parameters is performed. It is shown that the availability of an analytical formula for the variance makes the calibration 14 times faster , thus proving the practical value of tractable and general methods to derive it .
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 discussed and 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 .
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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
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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
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