Abstract
Risk management by applying operational flexibility is becoming a key issue for production companies. This paper discusses how a power portfolio can be hedged through its own production assets. In particular we model operational flexibility of a hydro pump storage plant and show how to dispatch it to hedge against adverse movements in the portfolio. Moreover, we present how volume risk, which is not hedgeable with standard contracts from power exchanges, can be managed by an intelligent dispatch policy. Despite the incompleteness of the market we quantify the value of this operational flexibility in the framework of coherent risk measures.
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Notes
i.e. non-risk-neutral.
Evaluated in this context means to decide on the technical investment or its zero-coupon equivalent. Additionally the optimal level of operations can be determined.
Ramp-up times are within 10 s.
Spot prices are determined for each hour of the next day. This is done on hourly auctions.
Subadditivity is probably the most important property to be a good risk measure for portfolios.
See Rockafellar and Uryasev (2002) for the mathematical assumptions.
i.e. we assume that the strategy of a single power generation company will not influence spot markets.
In our case for every scenario j, ω j corresponds to a joint path of the stochastic values, spot price S t , demand D t and inflow I t , over all periods t=1,...,T.
A more rigorous mathematical analysis is presented in Rockafellar and Uryasev (2000).
All marginal production costs exclude costs of electricity and neglect Swiss water taxes.
Monthly futures contracts have the highest liquidity in the EEX market.
Using Monte-Carlo simulation techniques.
The risk constraint is no longer a binding restriction.
Note that ΔV is always negative. A smaller end level of water leads to a higher volume as more water can now be used to produce electricity.
Assuming once again non-degeneracy. For degenerated optimal points directional derivatives have to be applied. This additional step will not change the methodology we suggest.
Any coherent or convex risk measure could have been used.
References
Acerbi C, Tasche D (2002) On the coherence of expected shortfall. In: Szegö G (ed) “Beyond VaR” (Special Issue). Journal of Banking & Finance 26:1505–1518
Arrow KJ (1971) Essays on the theory of risk-bearing. Markham, Chicago
Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent risk measures. Math Financ 9:203–228
Bertsimas D, Lauprete GJ, Samarov A (2000) Shortfall as a risk measure: properties, optimization and applications. Working paper, MIT
Borenstein S, Bushnell J (1999) An empirical analysis of the potential for market power in California’s electricity industry. J Ind Econ 47(3):285–323
Burger M, Klar B, Müller A, Schindlmayr G (2004) A spot market model for pricing derivatives in electricity markets. Quantitative Finance 4:109–122
Carmona R, Dayanik S (2004) Optimal multiple-stopping of linear diffusions and swing options. Preprint, Princeton University
Clewlow L, Strickland C (2001) Energy derivatives – pricing and risk management. Lacima Publications, London
Delbaen F (2000) Coherent risk measures, lecture notes at Cattedra Galileiana. Scuola Normale di Pisa, Pisa
Deng S, Johnson B, Sogomonian A (2001) Exotic electricity options and the valuation of electricity generation and transmission assets. Decis Support Syst 30(3):383–392
Eberlein E, Stahl G (2003) Both sides of the fence: a statistical and regulatory view of electricity risk. Energy Power Risk Manag 8(6):34–38
Eydeland A, Geman H (2000) Fundamentals of electricity derivatives in energy modeling and the management of uncertainty. Risk Books, New York
Eydeland A, Wolyniec K (2002) Energy and power risk management, Wiley
Fleten S-E, Wallace SW, Ziemba WT (2002) Hedging electricity portfolios via stochastic programming. Decision making under uncertainty: energy and power. Springer, Berlin, Heidelberg New York, pp 71–94
Föllmer H, Schied A (2002) Convex measures of risk and trading constraints. Finance Stoch 6(4):429–447
Frey R, McNeil A (2002) VaR and expected shortfall in credit portfolios: conceptual and practical insights. In: Szegö G (ed) “Beyond VaR” (Special Issue). J Bank Financ 26:1317–1334
Geman H (2001) Spot and derivatives trading in deregulated European electricity markets. Rev Econ Soc 8
Gröwe-Kuska N, Römisch W (2002) Stochastic unit commitment in hydrthermal power production planning. Preprint 02-3, Institute for Mathematics, Humboldt-University Berlin
Güssow J (2001) Power systems operations and trading in competitive energy markets. PhD thesis, University of St. Gallen, Switzerland
Hinz J (2003) Optimizing a portfolio of power-producing plants. Bernoulli 9(4):659–669
Hinz J, von Grafenstein L, Verschuere M, Wilhelm M (2004) Pricing electricity risk by interest rate methods, to appear in quantitative finance
Jaillet P, Ronn EI, Tompaidis S (2004) Valuation of commodity-based swing options. Manag Sci 50:909–921
Kamat R, Oren S (2002) Exotic options for interruptible electricity supply contracts. Oper Res 50(5):835–850
Kholodnyi VA (2004) Valuation and hedging of European contingent claims on power with spikes: a non-Markovian approach. J Eng Math 49:233–252
Ku A (2003) Risk and flexibility in electricity. Risk Books, London
Lüthi H-J, Doege J (2005) Convex risk measures for portfolio optimization and concepts of flexibility. Math Program, Series B 104(2–3):541–559
Markowitz HM (1952) Portfolio selection. J Finance 7:77–91
Mas-Colell A, Whinston M, Green J (1995) Microeconomic theory. Oxford University Press, New York
Pilipovic D (1997) Energy risk: valuing and managing energy derivatives. McGraw-Hill
Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):21–41
Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26:1443–1471
Rudin W (1976) Principles of mathematical analysis. McGraw-Hill, Singapore
Schwartz ES, Lucia JJ (2002) Electricity prices and power derivatives. Evidence from the Nordic power exchange. Rev Deriv Res 5(1):5–50
Stoft S (2002) Power system economics. IEEE, Wiley-Interscience, New York
Thompson AC (1995) Valuation of path-dependent contingent claims with multiple exercise decisions over time: the case of take-or-pay. J Financ Quant Anal 30(2):271–293
Unger G (2002) Hedging strategy and electricity contract engineering, Ph D thesis, Swiss Federal Institute of Technology – ETH Zürich, http://www.ifor.math.ethz.ch/publications/diss unger
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This research project is gratefully supported by the Swiss Innovation Promotion Agency KTI/CTI, Berne (CH).
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Doege, J., Schiltknecht, P. & Lüthi, HJ. Risk management of power portfolios and valuation of flexibility. OR Spectrum 28, 267–287 (2006). https://doi.org/10.1007/s00291-005-0005-4
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DOI: https://doi.org/10.1007/s00291-005-0005-4