Fast calibration of two-factor models for energy option pricing

TL;DR

This paper introduces analytical and numerical methods for faster market calibration of two-factor energy commodity models within the Black framework, significantly reducing computation time and enabling efficient option pricing.

Contribution

It presents a straightforward Lyapunov-based approach for variance calculation in multi-factor models, enhancing calibration speed and extending applicability to higher-dimensional models.

Findings

Analytical method speeds up calibration by 14 times.

Lyapunov approach simplifies variance derivation.

Open-source Python implementation available.

Abstract

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.