Polynomial Maps of Polynomial Processes for Energy Markets

Date
2024-12-05
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Abstract
The empirical evidence reveals that energy prices are different from other commodity prices, exhibiting seasonality, mean reversion and extremely high volatility. These features arise from the interaction between supply and demand, with each subject to seasonal variations, and in particular from constraints on storage, transmission or transportation, perhaps exacerbated by sudden changes due to unforeseen infrastructure breakdowns. Pricing energy assets and energy derivatives is a significant challenge. In this thesis, we present a modeling framework based on the mathematical foundations for polynomial diffusions, involving polynomial maps of polynomial processes (PMPP models) for various energy prices such as oil, natural gas and electricity prices. Under the PMPP framework, underlying factors are modeled by polynomial processes such as the Ornstein-Uhlenbeck process, geometric Brownian motion, inhomogeneous geometric Brownian motion and etc. Energy prices are generated by polynomial maps of these underlying processes. The polynomial maps can be determined either by monotone polynomial maps or regression, revealing the relationship between underlying factors and energy prices. We will show that PMPP models are able to describe the dynamics of spot and forward prices in energy markets, providing the advantage of cheap and convenient computation of forward prices because of the property of polynomial processes that the conditional expectations of polynomial functions of future states, conditional on current states, are given by polynomials of the current states. We also introduce a potential extension of the PMPP framework where the underlying factor process and the map may not be polynomial. Under this extended framework, the PMPP model will still reserve a quasi-polynomial property whereby conditional expectation of polynomial functions of the future state are given by functions of the current state in a space spanned by a finite set of basis functions. The optimal PMPP model is determined by the statistical criteria. The maximum likelihood estimation and Kalman filters are applied for model calibration. Preliminary results indicate that the PMPP models are able to capture the behavior of energy prices well and can be useful for risk management purposes.
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Keywords
Energy Markets, Polynomial Processes, Polynomial Maps, Gas Prices, Oil Prices, Power Prices, Forward Prices, Volatility Analysis
Citation
Sun, Z. (2024). Polynomial maps of polynomial processes for energy markets (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.