Exact simulation of normal tempered stable processes of OU type with applications

We study the Ornstein-Uhlenbeck process having a symmetric normal tempered stable stationary law and represent its transition distribution in terms of the sum of independent laws. In addition, we write the background driving Levy process as the sum of two independent Levy components. Accordingly, we can design two alternate algorithms for the simulation of the skeleton of the Ornstein-Uhlenbeck process. The solution based on the transition law turns out to be faster since it is based on a lower number of computational steps, as confirmed by extensive numerical experiments. We also calculate the characteristic function of the transition density which is instrumental for the application of the FFT-based method of Carr and Madan (J Comput Finance 2:61-73, 1999) to the pricing of a strip of call options written on markets whose price evolution is modeled by such an Ornstein-Uhlenbeck dynamics. This setting is indeed common for spot prices in the energy field. Finally, we show how to extend the range of applications to future markets.Peer reviewe

Pricing Energy Derivatives in Markets Driven by Tempered Stable and CGMY Processes of Ornstein–Uhlenbeck Type

In this study, we consider the pricing of energy derivatives when the evolution of spot prices follows a tempered stable or a CGMY-driven Ornstein–Uhlenbeck process. To this end, we first calculate the characteristic function of the transition law of such processes in closed form. This result is instrumental for the derivation of nonarbitrage conditions such that the spot dynamics is consistent with the forward curve. Moreover, we also conceive efficient algorithms for the exact simulation of the skeleton of such processes and propose a novel procedure when they coincide with compound Poisson processes of Ornstein–Uhlenbeck type. We illustrate the applicability of the theoretical findings and the simulation algorithms in the context of pricing different contracts, namely strips of daily call options, Asian options with European style and swing options

Pricing Energy Derivatives in Markets Driven by Tempered Stable and CGMY Processes of Ornstein–Uhlenbeck Type

In this study, we consider the pricing of energy derivatives when the evolution of spot prices follows a tempered stable or a CGMY-driven Ornstein–Uhlenbeck process. To this end, we first calculate the characteristic function of the transition law of such processes in closed form. This result is instrumental for the derivation of nonarbitrage conditions such that the spot dynamics is consistent with the forward curve. Moreover, we also conceive efficient algorithms for the exact simulation of the skeleton of such processes and propose a novel procedure when they coincide with compound Poisson processes of Ornstein–Uhlenbeck type. We illustrate the applicability of the theoretical findings and the simulation algorithms in the context of pricing different contracts, namely strips of daily call options, Asian options with European style and swing options

Modelling energy spot prices by volatility modulated Levy-driven Volterra processes

This paper introduces the class of volatility modulated L\'{e}vy-driven Volterra (VMLV) processes and their important subclass of L\'{e}vy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of VMLV processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ476 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Statistics, pricing and model risk

In dieser Arbeit entwickeln wir neue Methoden und Verfahren, um aktuelle Modellierungsverfahren zu ergänzen und zu verbessern und um so ein tieferes Verständnis von Energiemärkten zu gewinnen. Wir untersuchen verschiedene Aspekte der stochastischen Modellierung von Energiemärkten: wir analysieren stochastische Eigenschaften von Elektrizitätsmärkten, betrachten Bewertungsmethoden für verschiedene Energie-verwandte Finanzinstrumente, entwerfen ein neues Speichermodell und untersuchen Modellrisiko. Im Zuge dessen wenden wir Methoden der verschiedenen Bereiche der angewandten Mathematik, von statistischen und ökonometrischen Techniken über einen auf partielle Differenzialgleichung basierenden Ansatz bis hin zu Algorithmen der numerischen Analysis, an. Wir modifizieren und erweitern diese Methoden, um sie auf unsere Problemstellung anwenden zu können. Die Resultate der Arbeit sind theoretischer und praktischer Natur. Folgende Ergebnisse der Arbeit seien besonders hervorgehoben: 1. Ein kritischer Vergleich der Eigenschaften und der Schätzverfahren von drei kürzlich veröffentlichten und weitverbreiteten stochastischen Elektrizitätspreismodellen zeigt, dass keines der Modelle eines der anderen übertrifft. Wichtiger bei der Modellierung von Elektrizitätspreisen ist, dass additive Modelle aufgrund ihrer analytischen Lenkbarkeit effizienter sind und die Elektrizitätspreise als Summe verschiedener stochastischer Prozesse verantwortlich für unterschiedliche Preisschwankungsausschläge und mean-reversion Kräfte darstellen. 2. Eine auf integro-partielle Differenzialgleichung (integro-PDE) basierende Methode wird implementiert, um die Dynamiken des Elektrizitätsforwardpreises für ein regime-switching Elektrizitätspreismodell zu finden, die für Hedging von grundlegender Bedeutung sind. 3. Ein neuer Ansatz der Speicherbewertung wird entwickelt, um aktuelle stochastische Methoden der optimalen Steuerung zu ergänzen und eine optimale Speichersteuerung zu finden. Die Hauptneuheit ist der Speicherstandsprozess, der als beschränkte Diffusion dargestellt wird. Hierzu können wir Formeln für die Übergangswahrscheinlichkeitsdichten herleiten, die eine große Variabilität weiterer Anwendungen in der Bepreisung und Speicherbewertung erlauben. 4. Wir wenden eine detaillierte Untersuchung der verschiedenen Risikoquellen bei der Elektrizitätspreismodellierung auf das Beispiel eines Gaskraftwerkes an and finden, dass das Risiko von Preisspikes bei weitem die wichtigste Quelle des Modellrisikos ist.In this thesis we develop new methods and procedures to complement and improve current modelling frameworks and to provide a deeper and better understanding of energy markets. We investigate various aspects of stochastic modelling of energy markets: we analyse statistical properties of power markets, study pricing methods for different financial energy-related instruments, design a new storage model and examine model risk. In doing so we apply a wide range of methods from different branches of applied mathematics ranging from statistical and econometric techniques to a partial differential equations based approach and algorithms from numerical analysis. We modify and extend these methods to make them applicable to our problem setting. The study reveals results of both theoretical and practical importance. In particular, there are the main findings of this thesis: 1. A critical comparison of the properties and estimation procedures of three recently proposed and widely used stochastic power price models shows that none of the models outperforms each other, as all of them have some drawbacks. The more important issue when modelling power prices is that it is more efficient to use additive models (due to their analytical tractability) which present a power price as a sum of various stochastic process responsible for different price fluctuation magnitudes and mean-reversion forces. 2. An integro-partial differential equation (integro-PDE) based method is implemented to find the power forward price dynamics for a regime-switching power price model which is a critical issue for hedging purposes. 3. A new approach to storage value modelling is developed to complement current stochastic optimal control methods on finding an optimal storage policy. The main novelty is that the storage level process is represented as a bounded diffusion for which we are able to derive the transition probability density formula which in turn allows for a great variability of further applications to pricing and value storage. 4. A detailed investigation of various sources of risks when modelling power price is applied to the example of a gas-fired power plant and finds that spike risk is by far the most important source of model risk