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Modelling Electricity Futures by Ambit Fields

Part of: Mathematical finance Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Ole E. Barndorff-Nielsen ,
Fred Espen Benth  and
Almut E. D. Veraart
Ole E. Barndorff-Nielsen*
Affiliation:
Aarhus University
Fred Espen Benth*
Affiliation:
University of Oslo
Almut E. D. Veraart*
Affiliation:
Imperial College London
*
∗ Postal address: Thiele Center, Department of Mathematics, and CREATES, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark. Email address: oebn@imf.au.dk
∗∗ Postal address: Centre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, N-0316 Oslo, Norway. Email address: fredb@math.uio.no
∗∗∗ Postal address: Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK. Email address: a.veraart@imperial.ac.uk
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Abstract

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In this paper we propose a new modelling framework for electricity futures markets based on so-called ambit fields. The new model can capture many of the stylised facts observed in electricity futures and is highly analytically tractable. We discuss martingale conditions, option pricing, and change of measure within the new model class. Also, we study the corresponding model for the spot price, which is implied by the new futures model, and show that, under certain regularity conditions, the implied spot price can be represented in law as a volatility modulated Volterra process.

Keywords

Electricity marketfutures pricerandom fieldambit fieldLévy baseSamuelson effectstochastic volatility

MSC classification

Primary: 91G99: None of the above, but in this section
Secondary: 60G51: Processes with independent increments; L'evy processes 60G57: Random measures 60G60: Random fields
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 3 , September 2014 , pp. 719 - 745
Copyright
© Applied Probability Trust 

References

Albeverio, S., Lytvynov, E. and Mahnig, A. (2004). “A model of the term structure of interest rates based on Lévy fields.” Stoch. Process. Appl. 114, 251–263.Google Scholar
Andresen, A., Koekebakker, S. and Westgaard, S. (2010). “Modeling electricity forward prices using the multivariate normal inverse Gaussian distribution.” J. Energy Markets 3, 1–23.CrossRefGoogle Scholar
Audet, N., Heiskanen, P., Keppo, J. and Vehviläinen, I. (2004). “Modelling electricity forward curve dynamics in the Nordic market.“In Modelling Prices in Competitive Electricity Markets, ed. Bunn, D. W., John Wiley, Chichester, pp. 251–266.Google Scholar
Barlow, M. T. (2002). “A diffusion model for electricity prices.” Math. Finance 12, 287–298.CrossRefGoogle Scholar
Barlow, M., Gusev, Y. and Lai, M. (2004). “Calibration of multifactor models in electricity markets.” Internat. J. Theoret. Appl. Finance 7, 101–120.Google Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2004). “Lévy-based spatial-temporal modelling, with applications to turbulence.” Uspekhi Mat. NAUK 59, 63–90.Google Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2007). “Ambit processes: with applications to turbulence and tumour growth.“In Stochastic Analysis and Applications (Abel Symp. 2), Springer, Berlin, pp. 93–124.Google Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2008). “A stochastic differential equation framework for the timewise dynamics of turbulent velocities.” Theory Prob. Appl. 52, 372–388.Google Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2008). “Time change and universality in turbulence.” Res. Rep. 2007-8, Thiele Centre for Applied Mathematics in Natural Science, Aarhus University.Google Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2008). “Time change, volatility, and turbulence.“In Mathematical Control Theory and Finance, Springer, Berlin, pp. 29–53.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Schmiegel, J. (2009). “Brownian semistationary processes and volatility/intermittency.“In Advanced Financial Modelling (Radon Ser. Comput. Appl. Math. 8).“De Gruyter, Berlin, pp. 1–25.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). “Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics.” J. R. Statist. Soc. B 63, 167–241.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shiryaev, A. (2010). “Change of Time and Change of Measure (Adv. Ser. Statist. Sci. Appl. Prob. 13).” World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Benth, F. E. and Veraart, A. E. D. (2011). “Ambit processes and stochastic partial differential equations.“In Advanced Mathematical Methods for Finance, Springer, Heidelberg, pp. 35–74.Google Scholar
Barndorff-Nielsen, O. E., Benth, F. E. and Veraart, A. E. D. (2014). “Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency.” Banach Center Publications. To appear.Google Scholar
Barndorff-Nielsen, O. E., Benth, F. E. and Veraart, A. E. D. (2013). “Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes.” Bernoulli 19, 803–845.Google Scholar
Barth, A. and Benth, F. E. (2014). “The forward dynamics in energy markets – infinite-dimensional modelling and simulation.” To appear in Stochastics.CrossRefGoogle Scholar
Benth, F. E. (2011). “The stochastic volatility model of Barndorff-Nielsen and Shephard in commodity markets.” Math. Finance 21, 595–625.Google Scholar
Benth, F. E. and Koekebakker, S. (2008). “Stochastic modelling of financial electricity contracts.” Energy Economics 30, 1116–1157.Google Scholar
Benth, F. E., Cartea, A. and Kiesel, R. (2008). “Pricing forward contracts in power markets by the certainty equivalence principle: explaining the sign of the market risk premium.” J. Banking Finance 32, 2006–2021.Google Scholar
Benth, F. E., Kallsen, J. and Meyer-Brandis, T. (2007). “A non-Gaussian Ornstein–Uhlenbeck process for electricity spot price modeling and derivatives pricing.” Appl. Math. Finance 14, 153–169.Google Scholar
Benth, F. E., Šaltytė Benth, J. and Koekebakker, S. (2008). “Stochastic Modelling of Electricity and Related Markets (Adv. Ser. Statist. Sci. Appl. Prob. 11).” World Scientific, Hackensack, NJ.Google Scholar
Bernhardt, C., Klüppelberg, C. and Meyer-Brandis, T. (2008). “Estimating high quantiles for electricity prices by stable linear models.” J. Energy Markets 1, 3–19.Google Scholar
Bichteler, K. (2002). “Stochastic Integration with Jumps (Encyclopedia Math. Appl. 89).” Cambridge University Press.Google Scholar
Bjerksund, P., Rasmussen, H. and Stensland, G. (2010). “Valuation and risk management in the Norwegian electricity market.” In Energy, Natural Resources and Environmental Economics, eds Bjørndal, E., et al. Springer, Berlin, pp. 167–185.Google Scholar
Brockwell, P. J. (2001). “Continuous-time ARMA processes.“In Stochastic Processes: Theory and Methods (Handbook Statist. 19), North-Holland, Amsterdam, pp. 249–276.Google Scholar
Brockwell, P. J. (2001). “Lévy-driven CARMA processes.” Ann. Ins. Statist. Math. 53, 113–124.CrossRefGoogle Scholar
Carmona, R. A. and Tehranchi, M. R. (2006). “Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective.” Springer, Berlin.Google Scholar
Carr, P. and Madan, D. (1998). “Option valuation using the fast Fourier transform.” J. Computational Finance 2, 61–73.CrossRefGoogle Scholar
Cartea, A. and Figueroa, M. G. (2005). “Pricing in electricity markets: a mean reverting Jump diffusion model with seasonality.” Appl. Math. Finance 12, 313–335.Google Scholar
Chung, K. L. (2001). “A Course in Probability Theory, 3rd edn.” Academic Press, San Diego, CA.Google Scholar
Clewlow, L. and Strickland, C. (2000). “Energy Derivatives: Pricing and Risk Management.” Lacima, Houston, TX.Google Scholar
Da Prato, G. and Zabczyk, J. (1992). “Stochastic Equations in Infinite Dimensions.” Cambridge University Press.Google Scholar
Delbaen, F. and Schachermayer, W. (1994). “A general version of the fundamental theorem of asset pricing.” Math. Ann. 300, 463–520.Google Scholar
Diko, P., Lawford, S. and Limpens, V. (2006). “Risk premia in electricity forward prices.” Studies Nonlinear Dynamics Econometrics 10, Article 7.Google Scholar
Eydeland, A. and Wolyniec, K. (2003). “Energy and Power Risk Management.” John Wiley, Hoboken, NJ.Google Scholar
Folland, G. B. (1984). “Real Analysis. Modern Techniques and their Applications.” John Wiley, New York.Google Scholar
Frestad, D., Benth, F. E. and Koekebakker, S. (2010). “Modeling term structure dynamics in the Nordic electricity swap market.” Energy J.. 31, 53–86.Google Scholar
Garcı´a, I., Klüppelberg, C. and Müller, G. (2011). “Estimation of stable CARMA models with an application to electricity spot prices.” Statist. Modelling 11, 447–470.Google Scholar
Geman, H. (2005). “Commodities and Commodity Derivatives.” John Wiley, Chichester.Google Scholar
Geman, H. and Roncoroni, A. (2006). “Understanding the fine structure of electricity prices.” J. Business 79, 1225–1261.Google Scholar
Geman, H. and Vasicek, O. (2001). Forwards and futures on non-storable commodities: the case of electricity. RISK, August, 2001.Google Scholar
Goldstein, R. S. (2000). “The term structure of interest rates as a random field.” Rev. Financial Studies 13, 365–384.Google Scholar
Hambly, B., Howison, S. and Kluge, T. (2009). “Modelling spikes and pricing swing options in electricity markets.” Quant. Finance 9, 937–949.Google Scholar
Heath, D., Jarrow, R. and Morton, A. (1992). “Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation.” Econometrica 60, 77–105.Google Scholar
Hikspoors, S. and Jaimungal, S. (2008). “Asymptotic pricing of commodity derivatives using stochastic volatility spot models.” Appl. Math. Finance 15, 449–477.CrossRefGoogle Scholar
Hinz, J., von Grafenstein, L., Verschuere, M. and Wilhelm, M. (2005). “Pricing electricity risk by interest rate methods.” Quant. Finance 5, 49–60.Google Scholar
Kallenberg, O. (2002). “Foundations of Modern Probability, 2nd edn. ” Springer, New York.CrossRefGoogle Scholar
Kallsen, J. and Shiryaev, A. N. (2002). “The cumulant process and Esscher's change of measure.” Finance Stoch. 6, 397–428.CrossRefGoogle Scholar
Kennedy, D. P. (1994). “The term structure of interest rates as a Gaussian random field.” Math. Finance 4, 247–258.CrossRefGoogle Scholar
Kennedy, D. P. (1997). “Characterizing Gaussian models of the term structure of interest rates.” Math. Finance 7, 107–118.Google Scholar
Kiesel, R., Schindlmayr, G. and Börger, R. H. (2009). “A two-factor model for the electricity forward market.” Quant. Finance 9, 279–287.CrossRefGoogle Scholar
Kimmel, R. L. (2004). “Modeling the term structure of interest rates: a new approach.” J. Financial Econom. 72, 143–183.Google Scholar
Koekebakker, S. and Ollmar, F. (2005). “Forward curve dynamics in the Nordic electricity market.” Managerial Finance 31, 73–94.CrossRefGoogle Scholar
Lucia, J. J. and Schwartz, E. S. (2002). “Electricity prices and power derivatives: evidence from the Nordic power exchange.” Rev. Derivatives Res. 5, 5–50.Google Scholar
Rajput, B. S. and Rosiński, J. (1989). “Spectral representations of infinitely divisible processes.” Prob. Theory Relat. Fields 82, 451–487.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). “Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance.” Chapman & Hall, New York.Google Scholar
Samuelson, P. (1965). “Proof that properly anticipated prices fluctuate randomly.” Industrial Manag. Rev. 6, 41–49.Google Scholar
Santa-Clara, P. and Sornette, D. (2001). “The dynamics of the forward interest rate curve with stochastic string shocks.” Rev. Financial Studies 14, 149–185.Google Scholar
Sato, K.-I. (1999). “Lévy Processes and Infinitely Divisible Distributions.” Cambridge University Press.Google Scholar
Sato, K.-I. (2004). “Stochastic integrals in additive processes and application to semi-Lévy processes.” Osaka J. Math. 41, 211–236.Google Scholar
Shiryaev, A. N. (1999). “Essentials of Stochastic Finance. Facts, Models, Theory (Adv. Ser. Statist. Sci. Appl. Prob. 3).” World Scientific, River Edge, NJ.Google Scholar
Vedel Jensen, E. B., Jónsdóttir, K. Y., Schmiegel, J. and Barndorff-Nielsen, O. E. (2006). “Spatio-temporal modelling – with a view to biological growth.“In Statistical Methods for Spatio-Temporal Systems, eds Finkenstädt, B. et al., Chapman & Hall/CRC, Boca Raton, FL, pp. 47–76.Google Scholar
Veraart, A. E. D. and Veraart, L. A. M. (2013). “Risk premiums in energy markets.” J. Energy Markets 6, 1–42.CrossRefGoogle Scholar
Veraart, A. E. D. and Veraart, L. A. M. (2014). “Modelling electricity day-ahead prices by multivariate Lévy semistationary processes.“In Quantitative Energy Finance, eds Benth, F. E. et al., Springer, New York, pp. 157–188.Google Scholar
Walsh, J. B. (1986). “An introduction to stochastic partial differential equations.“In Ecole d'Été de Probabilités de Saint-Flour, XIV—1984 (Lecture Notes Math. 1180), Springer, Berlin, pp. 265–439.Google Scholar
Weron, R. (2006). “Modeling and Forecasting Electricity Loads and Prices. A Statistical Approach.” John Wiley, Chichester.Google Scholar