Valuation and pricing of electricity delivery contracts: the producer’s view. (English) Zbl 1427.91183
Summary: This paper analyzes the valuation and pricing of physical electricity delivery contracts from the viewpoint of a producer with given capacities for production and fuel-storage. Using stochastic optimization problems in discrete time with general state space, the dual problems of production problems are used to derive no-arbitrage conditions for fuel and electricity prices as well as superhedging values and prices of bilaterally traded electricity delivery contracts. In particular we take the perspective of an electricity producer, who serves contractual deliveries but avoids unacceptable losses. The resulting no-arbitrage conditions, stochastic discount factors and superhedging prices account for typical frictions like limitation of storage and production capacity and for the fact that it is possible to produce electricity from fuel, but not to produce fuel from electricity. Similarities, but also substantial differences to purely financial results can be demonstrated in this way. Furthermore, using acceptability measures, we analyze capital requirements and acceptability prices for delivery contracts, when the producer accepts some risk.
MSC:
| 91B74 | Economic models of real-world systems (e.g., electricity markets, etc.) |
| 91B24 | Microeconomic theory (price theory and economic markets) |
References:
| [1] | Alonso-Ayuso, A., Escudero, L. F., & Pizarro-Romero, C. (2009). Introduction to stochastic programming. Madrid: Ciencias Experimentales Y Tecnologia: Universidad Rey Juan Carlos. |
| [2] | Analui, B., & Kovacevic, R. (2014). Medium term hydroelectric production planning – a multistage stochastic optimization problem. Civil Engineering Infrastructures Journal (CEIJ), 47(1), 139-152. |
| [3] | Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203-228. · Zbl 0980.91042 · doi:10.1111/1467-9965.00068 |
| [4] | Bernardo, A., & Ledoit, O. (2000). Gain, loss, and asset pricing. Journal of Political Economy, 108, 144-172. · doi:10.1086/262114 |
| [5] | Bertocchi, M., Consigli, G., & Dempster, M. A. (Eds.). (2011). Stochastic optimization methods in finance and energy., International series in operations research and management science Berlin: Springer. |
| [6] | Björk, (2009). Arbitrage theory in continuous time (3rd ed.). Oxford: Oxford University Press. · Zbl 1140.91038 |
| [7] | Bot, R. I., Grad, S.-M., & Wanka, G. (2009). Duality in vector space optimization. Dordrecht: Springer. · Zbl 1177.90355 · doi:10.1007/978-3-642-02886-1 |
| [8] | Carmona, R. (2009). Indifference pricing: Theory and applications. Princeton: Princeton University Press. · Zbl 1155.91008 |
| [9] | Carmona, R., & Touzi, N. (2008). Optimal multiple stopping and valuation of swing options. Mathematical Finance, 18(2), 239-268. · Zbl 1133.91499 · doi:10.1111/j.1467-9965.2007.00331.x |
| [10] | Cochrane, J. H., & Saá-Roquejo, J. (2000). Beyond arbitrage; good deal asset price bounds in incomplete markets. Journal of Political Economy, 108, 79-119. · doi:10.1086/262112 |
| [11] | Delbaen, F., & Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Mathematische Annalen, 300, 463-520. · Zbl 0865.90014 · doi:10.1007/BF01450498 |
| [12] | Delbaen, F., & Schachermayer, W. (1998). The fundamental theorem for unbounded processes. Mathematische Annalen, 312, 212-250. · Zbl 0917.60048 · doi:10.1007/s002080050220 |
| [13] | Deng, S.-J., Johnson, B., & Sogomonian, A. (2001). Exotic electricity options and the valuation of electricity generation and transmission assets. Decision Support Systems, 30, 383-392. · doi:10.1016/S0167-9236(00)00112-3 |
| [14] | Dupacova, J., Gröwe-Kuska, N., & Römisch, W. (2003). Scenario reduction in stochastic programming: An approach using probability metrics. Mathematical Programming, Series A, 95, 493-511. · Zbl 1023.90043 · doi:10.1007/s10107-002-0331-0 |
| [15] | Eichhorn, A., Römisch, W., & Wegner, I. (2004). Polyhedral risk measures in electricity portfolio optimization. Proceedings in Applied Mathematics and Mechanics, 4(1), 7-10. · Zbl 1354.91172 · doi:10.1002/pamm.200410002 |
| [16] | Eydeland, A., & Wolyniec, K. (2003). Energy and power risk management. New York: Wiley. |
| [17] | Flåm, S . D. (2008). Option pricing by mathematical programming. Optimization: A Journal of Mathematical Programming and Operations Research, 57(1), 165-182. · Zbl 1151.91507 · doi:10.1080/02331930701779054 |
| [18] | Fritelli, M., & Rosazza, G. E. (2002). Putting order in risk measures. Journal of Banking and Finance, 2, 1473-1486. · doi:10.1016/S0378-4266(02)00270-4 |
| [19] | Gollmer, R., Nowak, W., Römisch, W., & Schultz, R. (2000). Unit commitment in power generation, a basic model and some extensions. Annals of Operations Research, 96, 167-189. · Zbl 0997.90534 · doi:10.1023/A:1018947401538 |
| [20] | Gross, P., & Pflug, G. C. (2016). Behavioral pricing of energy swing options by stochastic bilevel optimization. Energy Systems,. https://doi.org/10.1007/s12667-016-0190-z. · doi:10.1007/s12667-016-0190-z |
| [21] | Haarbrücker, G., & Kuhn, D. (2009). Valuation of electricity swing options by multistage stochastic programming. Automatica, 45, 889-899. · Zbl 1177.90299 · doi:10.1016/j.automatica.2008.11.022 |
| [22] | Hansen, L. P., & Jagannathan, R. (1991). Implications of security market data for models of dynamic economics. Journal of Political Economy, 99, 225-262. · doi:10.1086/261749 |
| [23] | Hao, T. (2008). Option pricing and hedging bounds in incomplete markets. Journal of Derivatives and Hedge Funds, 14(2), 89. · doi:10.1057/jdhf.2008.9 |
| [24] | Heitsch H., & Römisch W. (2010) Stability and scenario trees for multistage stochastic programs. In G. Infanger (Ed.), Stochastic programming. International series in operations research & management science (Vol. 150). New York: Springer. · Zbl 1251.90297 |
| [25] | Jouini, E., Schachermayer, W., & Touzi, N. (2008). Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18(2), 269-292. · Zbl 1133.91360 · doi:10.1111/j.1467-9965.2007.00332.x |
| [26] | King, A. J. (2002). Duality and martingales: A stochastic programming perspective on contingent claims. Mathematical Programming, Series B, 91, 543-562. · Zbl 1074.91018 · doi:10.1007/s101070100257 |
| [27] | Kovacevic, R., & Paraschiv, F. (2014). Medium term planning for thermal electricity production. Operations Research Spectrum, 36(3), 723-759. · Zbl 1305.90319 · doi:10.1007/s00291-013-0340-9 |
| [28] | Kovacevic, R. M. (2012). Conditional risk and acceptability mappings as banach lattice valued mappings. Statistics & Risk Modeling, 29(1), 1-18. · Zbl 1238.91084 · doi:10.1524/strm.2012.1041 |
| [29] | Kovacevic, R. M., & Pflug, G. C. (2014). Electricity swing option pricing by stochastic bilevel optimization: A survey and new approaches. European Journal of Operational Research, 237(2), 389-403. · Zbl 1304.91218 · doi:10.1016/j.ejor.2013.12.029 |
| [30] | Kovacevic, R. M., Pflug, G. C., & Vespucci, M. (Eds.). (2013). Handbook of risk management in energy production and trading., International series in operations research and management science Berlin: Springer. |
| [31] | Kovacevic, R. M., & Pichler, A. (2015). Tree approximation for discrete time stochastic processes – a process distance approach. Annals of OR, 235(1), 395-421. · Zbl 1332.90182 · doi:10.1007/s10479-015-1994-2 |
| [32] | Lofberg, J. (2005). Yalmip: A toolbox for modeling and optimization in MATLAB. In: 2004 IEEE international symposium on computer aided control systems design. https://doi.org/10.1109/CACSD.2004.1393890. |
| [33] | Luenberger, D. G. (1969). Optimization by vector space methods. New York: Wiley. · Zbl 0176.12701 |
| [34] | Nowotarski, J., & Weron, R. (2018). Recent advances in electricity price forecasting: A review of probabilistic forecasting. Renewable and Sustainable Energy Reviews, 81(1), 1548-1568. · doi:10.1016/j.rser.2017.05.234 |
| [35] | Pagès, G., & Printems, J. (2003). Optimal quadratic quantization for numerics: The Gaussian case. Monte Carlo Methods and Applications, 9(2), 135-165. · Zbl 1029.65012 · doi:10.1515/156939603322663321 |
| [36] | Pansera, J. (2012). Discrete-time local risk minimization of payment processes and applications to equity linked life-insurance contracts. Insurance: Mathematics and Economics, 50, 1-11. · Zbl 1235.91104 |
| [37] | Pennanen, T. (2011a). Dual representation of superhedging costs in illiquid markets. Mathematics and Financial Economics, 5(4), 233-248. · Zbl 1275.91063 · doi:10.1007/s11579-012-0061-x |
| [38] | Pennanen, T. (2011b). Superhedging in illiquid markets. Mathematical Finance, 21(3), 519-540. · Zbl 1229.91322 · doi:10.1111/j.1467-9965.2010.00437.x |
| [39] | Pennanen, T. (2012). Introduction to convex optimization in financial markets. Mathematical Programming, Series B, 134(1), 157-186. · Zbl 1254.91766 · doi:10.1007/s10107-012-0573-4 |
| [40] | Pfaff, B. (2008). VAR, SVAR and SVEC models: Implementation within R package vars. Journal of Statistical Software, 27(4), 1-32. · doi:10.18637/jss.v027.i04 |
| [41] | Pflug, G. C. (2006). Subdifferential representations of risk measures. Mathematical Programming, 108, 339-354. · Zbl 1138.91505 · doi:10.1007/s10107-006-0714-8 |
| [42] | Pflug, G. C., & Pichler, A. (2014). Multistage stochastic optimization., Springer series in operations research and financial engineering Berlin: Springer. · Zbl 1317.90220 · doi:10.1007/978-3-319-08843-3 |
| [43] | Pflug, G. C., & Pichler, A. (2015). Dynamic generation of scenario trees. Computational Optimization and Applications, 62(3), 641-668. · Zbl 1337.90047 · doi:10.1007/s10589-015-9758-0 |
| [44] | Pflug, G. C., & Römisch, W. (2007). Modeling, measuring and managing risk. Singapore: World Scientific. · Zbl 1153.91023 |
| [45] | Philpott, A., & Schultz, R. (2006). Unit commitment in electricity pool markets. Mathematical Programming, 108, 313-337. · Zbl 1130.90358 · doi:10.1007/s10107-006-0713-9 |
| [46] | Rudloff, B. (2007). Convex hedging in incomplete markets. Applied Mathematical Finance, 14(5), 437-452. · Zbl 1151.91537 · doi:10.1080/13504860701352206 |
| [47] | Sagastizabal, C. (2012). Divide to conquer: Decomposition methods for energy optimization. Mathematical Programming, Series B, 134(1), 187-222. · Zbl 1254.90238 · doi:10.1007/s10107-012-0570-7 |
| [48] | Sen, S., Yu, L., & Genc, T. (2006). A stochastic programming approach to power portfolio optimization. Operations Research, 54(1), 55-72. · Zbl 1167.90612 · doi:10.1287/opre.1050.0264 |
| [49] | Takriti, S., Birge, J., & Long, E. (1996). A stochastic model for the unit commitment problem. IEEE Transactions on Power Systems, 11, 1497-1508. · doi:10.1109/59.535691 |
| [50] | Thompson, A. (1995). Valuation of path-dependent contingent claims with multiple exercise decisions over time. Journal of Financial and Quantitative Analysis, 30(2), 271-293. · doi:10.2307/2331121 |
| [51] | Vayanos, P., Wiesemann, W., & Kuhn, D. (2011). Hedging electricity swing options in incomplete markets. In Preprints of the 18th IFAC World Congress Milano, pp. 846-853. |
| [52] | Wallace, S. W., & Fleten, S. E. (2003). Stochastic programming. Vol. 10 of Handbooks in operations research and management science. Ch. Stochastic programming models in energy (pp. 637-677). Elsevier: Amsterdam. · Zbl 1115.90001 |
| [53] | Zephyr, L., & Anderson, C. L. (2018). Stochastic dynamic programming approach to managing power system uncertainty with distributed storage. Computational Management Science, 15(1), 87-110. · Zbl 1397.90227 · doi:10.1007/s10287-017-0297-2 |
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