Abstract
We give a short introduction to energy markets, describing how they function and what products are traded. Next we survey some of the popular models that have been proposed in the literature. We extend the analysis of one of these models to include for stochastic volatility effects. In particular, we analyse a mean reverting stochastic spot price dynamics with a stochastic mean level modelled as an Ornstein–Uhlenbeck process. We include in this dynamics a stochastic volatility model of the Barndorff-Nielsen and Shephard type. Some properties of the dynamics are derived and discussed in relation to energy markets. Moreover, we derive a semi-analytical expression for the forward price based on such a spot dynamics. In the last part of these lecture notes we consider a cross-commodity spot price model including jumps. A Margrabe formula for options on the spread is derived, along with an analysis of the dependency risk under an Esscher measure transform. An empirical example demonstrates that the Esscher transform may increase the tail dependency in the bivariate jump part of the spot model.
Financial support from the project “Energy Markets: Modeling, Optimization and Simulation” (EMMOS), funded by the Norwegian Research Council under grant 205328/v30, is greatly acknowledged. An anonymous referee is thanked for careful reading of the paper and the several suggestions which improved the presentation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
We have selected this rather old period of data for illustration only, since it was a period where prices had a very apparent seasonality and spike pattern.
References
O. Barndorff-Nielsen, N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in economics. J. R. Stat. Soc. B 63(2), 167–241 (2001) (with discussion)
O.E. Barndorff-Nielsen, F.E. Benth, A. Veraart, Modelling energy spot prices by Lévy semistationary processes. Bernoulli (2010) (to appear)
F.E. Benth, The stochastic volatility model of Barndorff-Nielsen and Shephard in commodity markets. Math. Financ. 21(4), 595–625 (2011)
F.E. Benth, S. Koekebakker, Stochastic modeling of financial electricity contracts. Energy Econ. 30(3), 1116–1157 (2008)
F.E. Benth, T. Meyer-Brandis, The information premium for non-storable commodities. J. Energy Mark. 2(3), 111–140 (2009)
F.E. Benth, C. Sgarra, The risk premium and the Esscher transform in power markets. Stoch. Anal. Appl. 30, 20–43 (2012)
F.E. Benth, J. Šaltytė Benth, J. Koekebakker, Stochastic Modelling of Electricity and Related Markets (World Scientific, Singapore, 2008)
F.E. Benth, A. Cartea, R. Kiesel, Pricing forward contracts in power markets by the certainty equivalence principle: Explaining the sign of the market risk premium. J. Bank. Financ. 32(10), 2006–2021 (2008)
F.E. Benth, G. Di Nunno, A. Khedher, Lévy model robustness and sensitivity, in QP-PQ: Quantum Probability and White Noise Analysis, vol. 25, ed. by H. Ouerdiane, A. Barhoumi. Proceedings of the 29th Conference in Hammamet, Tunisia, 13–18 October 2008 (World Scientific, Singapore, 2010), pp. 153–184
F.E. Benth, C. Klüppelberg, G. Müller, L. Vos, Futures pricing in electricity markets based on stable CARMA spot models. (2011) (submitted)
F.E. Benth, J. Lempa, T.K. Nilsen, On optimal exercise of swing options in electricity markets. J. Energy Mark. 4(4), 3–28 (2012)
F.E. Benth, G. Di Nunno, A. Khedher, Computations of Greeks in multi-factor models with applications to power and commodity markets. J. Energy Mark. 5(4), 3–31 (2013)
D. Brigo, F. Mercurio, Interest Rate Models – Theory and Practice (Springer, Berlin, 2001)
R. Carmona, V. Durrleman, Pricing and hedging spread options. SIAM Rev. 45, 627–685 (2003)
A. Eydeland, K. Wolyniec, Energy and Power Risk Management (Wiley, New York, 2003)
G.B. Folland, Real Analysis – Modern Techniques and their Applications (Wiley, New York, 1984)
H. Geman, Commodities and Commodity Derivatives (Wiley-Finance, Chichester, 2005)
D. Heath, R. Jarrow, A. Morton, Bond pricing and the term structure of interest rates: A new methodology. Econometrica 60, 77–105 (1992)
S. Hikspoors, S. Jaimungal, Asymptotic pricing of commodity derivatives for stochastic volatility spot models. Appl. Math. Financ. 15(5, 6), 449–467 (2008)
N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland/Kodansha, 1981)
J. Lucia, E.S. Schwartz, Electricity prices and power derivatives: Evidence from the Nordic power exchange. Rev. Deriv. Res. 5(1), 5–50 (2002)
W. Margrabe, The value of an option to exchange one asset for another. J. Financ. 33, 177–186 (1978)
R.B. Nelsen, An Introduction to Copulas, 2nd edn. (Springer, Berlin, 2010)
K. Sato, Lévy Processes and Infinite Divisibility (Cambridge University Press, Cambridge, 1999)
E.S. Schwartz, The stochastic behaviour of commodity prices: Implications for valuation and hedging. J. Financ. LII(3), 923–973 (1997)
E.S. Schwartz, J.E. Smith, Short-term variations and long-term dynamics in commodity prices. Manag. Sci. 46(7), 893–911 (2000)
A.B. Trolle, E.S. Schwartz, Unspanned stochastic volatility and the pricing of commodity derivatives. Rev. Financ. Stud. 22(11), 4423–4461 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Benth, F.E. (2013). Stochastic Volatility and Dependency in Energy Markets: Multi-Factor Modelling. In: Paris-Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Mathematics, vol 2081. Springer, Cham. https://doi.org/10.1007/978-3-319-00413-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-00413-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00412-9
Online ISBN: 978-3-319-00413-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)