Modeling, Pricing and Risk Management of Power Derivatives

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1 Diss. ETH No Modeling, Pricing and Risk Management of Power Derivatives A dissertation submitted to ETH Zurich for the degree of Doctor of Sciences presented by Martina Wilhelm Dipl. Math. ETH born March 3, 1978 citizen of Safenwil AG accepted on the recommendation of Prof. Dr. Hans Jakob Lüthi, examiner Prof. Dr. Rajna Gibson Brandon, co examiner Dr. Juri Hinz, co examiner 2007

2 Abstract Deregulation of energy markets has necessitated the adoption of risk management techniques in the power industry. The launched liberalization and therewith the uncertainty involved in gas, fuel and electrical power prices requires an efficient management of production facilities and financial contracts. Thereby derivatives build essential instruments to exchange volume as well as price risks. However, the valuation of financial derivatives in power markets has proven to be particularly challenging. The fact that electrical power is not physically storable eliminates the direct application of the no arbitrage methodology from financial mathematics. Consequently, new approaches are required to value even the simplest derivative products in electricity markets. That is, modeling arbitrage free electricity price dynamics turns out to be the crucial step towards fair pricing of financial products in power markets. In this work we propose an approach, which converts an electricity market into a virtual base market consisting of zero bonds and an additional risky asset. Using this structure, interest rate theory as well as the change of numeraire technique are applied to elaborate risk neutral price dynamics in electricity markets. As a result, explicit formulas are obtained for European type derivatives such as spread, cap, floor and collar options. Through the performed historical calibration it can be seen that in the proposed model, contract volatilities close to maturity increase significantly. For valuing swing type derivatives, which possess no closed form solutions, an algorithm based on finite element methods is proposed. Thereby the reduction of multiple stopping time problems to a cascade of single stopping time problems is utilized. The obtained numerical results for different swing options all show a smooth and stable behavior. This allows an interpretation of the optimal exercise boundary and an analysis

3 iv of the dependence of swing option prices on initial spot prices and on the number of exercise rights. A comparison of the finite element algorithm to Monte Carlo methods demonstrates the strengths of the developed numerical procedure.

4 Zusammenfassung Die Deregulierung der Strommärkte bedingt die Einführung von Risikomanagement Methoden in der Energieindustrie. Die lancierte Liberalisierung und die damit verbundenen Unsicherheiten in Gas, Öl und Strompreisen verlangen einen adäquaten Einsatz von Produktionsstätten und finanziellen Verträgen. Dabei bilden Derivate wesentlich Instrumentarien zum Austausch von Nachfrageunsicherheiten und finanziellen Risiken. Die Bewertung von derivativen Finanzinstrumenten im Strommärkten erweist sich jedoch als äussert anspruchsvoll. Denn die Nichtspeicherbarkeit von Elektrizität schliesst eine direkte Anwendung der arbitragefreien Bewertungstheorie der Finanzmathematik aus. Folglich sind neue Ansätze erforderlich um selbst die einfachsten derivativen Produkte in Strommärkten zu bewerten. Das heisst, die arbitragefreie Preisdynamik ist eine der zentralen Fragen zur Bestimmung von fairen Optionspreisen in Strommärkten. In der vorliegenden Arbeit wird ein Ansatz hergeleitet, der einen Strommarkt in einen virtuellen Basismarkt, bestehend aus Nullkuponanleihen und einer risikobehafteten Anlage überführt. Diese Struktur ermöglicht die Anwendung der Geldmarkttheorie und der Change of Nummeraire Technik zur Bestimmung von arbitragefreien Preisprozessen im Strommarkt. Als Ergebnis konnten explizite Formeln für Europäische Optionen wie Caps, Floors, Collars und Spreads hergeleitet werden. Durch die historische Kalibrierung wurde aufgezeigt, dass in diesem Modell die Volatilitäten der obengenannten Optionen kurz vor der Verfallsfrist deutlich ansteigen. Zur Bewertung von Swing Optionen, welche keine geschlossenen Formeln besitzen, wird ein Algorithmus basierend auf Finiten Elementen erarbeitet. Dabei wird verwendet, dass multiple stopping time problems auf

5 vi eine Sequenz von single stopping time problems reduzierbar sind. Die numerischen Resultate für verschiedene Swing Optionen weisen ein glattes, stabiles Verhalten auf. Dies ermöglicht eine Interpretation der optimalen Ausübungszeiten, sowie eine Analyse des Optionspreises in Bezug auf den initialen Strompreis und auf die Anzahl der Ausübungsrechte. Ein Vergleich des Finite Elemente Algorithmus zu Monte Carlo Methoden zeigt die Stärken des in dieser Arbeit entwickelten, numerischen Verfahrens auf.

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