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Energy markets: modelling, optimization and simulation

Awarded: NOK 6.5 mill.

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Project Period:

2011 - 2015

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The EMMOS project has focused on several topics of analysis on energy markets. Forward and futures contracts are liquidly traded in today's energy markets, and require sophisticated mathematical models in order to be used efficiently in risk management. In the EMMOS project we have proposed and analyzed a new class of stochastic processes called Ambit fields, which embraces many of the existing stochastic dynamical models for the forward price curve evolution in energy markets. The models are generalizations of the more common hyperbolic stochastic partial differential equations with Wiener noise, and require advanced tools from stochastics and functional analysis to be studied. This activity has involved collaborations with researchers from Århus, Denmark, and Imperial College, London. Linda Vos defended her PhD thesis in October 2012 on multivariate models for energy markets as part of this activity. Methods for simulating ambit fields and subclasses have been developed using two novel approaches. The first applies Fourier techniques and the second recast the problem to simulating a stochastic partial differential equation. In both cases we obtain an iterative scheme which has was not known for simulating these fields. Even more, we can control the numerical error. Such schemes are crucial for risk management in the market, as well as for efficient pricing of financial contracts and estimation of these models to data. This work has resulted in a PhD thesis defended in October 2013 by Heidar Ejyolfsson. A big effort has been invested in analysis of so-called swing options, financial contracts with flexibility. We have developed new theory for stochastic control to analyze these derivatives in the various cases relevant to energy (in particular power). Numerical methods have also been developed for practical pricing and optimal management of the contracts. This work is part of the PhD thesis of Marcus Eriksson, under joint supervision of the former EMMOS post doc Jukka Lempa, EMMOS research associate professor Ruediger Kiesel and project manager Fred Espen Benth. We have collaborated with researchers from Agder University. Marcus Eriksson defended his thesis in November 2014. Optimal management of portfolios of forward and futures contracts is an important theme in the EMMOS project. New mathematics have been developed to analyze such problems, involving infinite dimenisional stochastic analysis. This research has been performed by former EMMOS post doc Jukka Lempa in collaboration with Fred Espen Benth, leading to a published scientific paper in a prestigous journal. Based on functional analysis, we have devised methods to statistically detect forward-looking information in power markets. Power is a non-storable commodity, and this impacts the forward and futures market. Information about future events like planned shut-downs of power plants or policy changes can make a direct impact on the for ward/futures markets but not the current day-ahead prices. We apply the theory to analyze the market in case of the German "Atom Moratorium", where Germany decided to shut down some of their nuclear power plants in response to the tsunami hitting the Fukushima power plant in April 2011 in Japan. Based on this work, Richard Biegler-Koenig, jointly supervised by Ruediger Kiesel and Fred Espen Benth, successfully defended his thesis at the University of Essen in April 2013. His thesis was awarded a price for the best PhD thesis at the University of Duisburg-Essen. Benth has for several years been a Pauli Fellow at the Wolfgang Pauli Institute in Vienna, organizing a special topic program on energy and finance with people from Austria, Italy, UK and France. A conference was jointly organized by the Institute and the EMMOS project, as well as a two-day workshop and an intensive course. EMMOS PhD student Solanilla Blanco has with Benth analysed the impact of memory in spot prices on the shape of forward curves. It turens out that the forward curves can be related to factors-shapes, similar to those from a principal component analysis. EMMOS post doc Ortiz-Latorre has with Benth developed a mathematical framework for modelling the stochastic risk premium in electricity forward markets. Developments in measure theory leads to a class of pricing probabilities that leads to a risk premium depending on price spikes in the spot. In October 2014 a two-day workshop was organized as a meeting place between industry and academia. In September 2014 an international conference was organized in the Academy in Oslo, attracting more than 50 international participants.

The project aims at developing infinite dimensional stochastic processes for modelling and analysis of the term structures (futures markets) in energy. Investment strategies for optimal management of risk will be derived, based on trading in energy future s and options. The study of these investment decisions can be formalized as stochastic control problems, and we will be concerned with both the theoretical analysis and the practical implementation. This involves development of theory in mathematics and n umerical analysis. On the theoretical side, new methods for stochastic control in infinite dimensions, including dynamic programming principles and Hilbert-space valued Hamilton-Jacobi-Bellman (HJB) equations. On the numerical side, we will focus on Mon te Carlo based simulation methods of infinite dimensional stochastic processes, including stochastic partial differential equations, and numerical methods for Hilbert-space valued HJB equations. For the latter, we will use finite-dimensional approximation s combined with numerical schemes for partial differential equations. The theoretical and numerical developments will have practical problems and solutions in view. The formulation of investment problems and interpretation of results will be done in clos e collaboration with industry. Since in general the mathematical solution will live in function (Hilbert) spaces, approximations must be found for practical use. It is part of the project to develop such approximations, in the sense of risk management dec ision rules in terms investments in existing market assets like energy futures and options. The results of the project will help industry's ability to manage their risk exposure optimally.

Publications from Cristin

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