Lévy Processes and Path Integral Methods with Applications in the Energy Markets

Abstract

The objective of this thesis was to explore methods for valuation of derivatives in energy markets. One aim was to determine whether the Normal inverse Gaussian distributions would be better suited for modelling energy prices than normal distributions. Another aim was to develop working implementations of Path Integral methods for valuing derivatives, based on some one-factor model of the underlying spot price. Energy prices are known to display properties like mean-reversion, periodicity, volatility clustering and extreme jumps. Periodicity and trend are modelled as a deterministic function of time, while mean-reversion effects are modelled with auto-regressive dynamics. It is established that the Normal inverse Gaussian distributions are superior to the normal distributions for modelling the residuals of an auto-regressive energy price model. Volatility clustering and spike behaviour are not reproduced with the models considered here. After calibrating a model to fit real energy data, valuation of derivatives is achieved by propagating probability densities forward in time, applying the Path Integral methodology. It is shown how this can be implemented for European options and barrier options, under the assumptions of a deterministic mean function, mean-reversion dynamics and Normal inverse Gaussian distributed residuals. The Path Integral methods developed compares favourably to Monte Carlo simulations in terms of execution time. The derivative values obtained by Path Integrals are sometimes outside of the Monte Carlo confidence intervals, and the relative error may thus be too large for practical applications. Improvements of the implementations, with a view to minimizing errors, can be subject to further research.