Advertisement

Review of Derivatives Research

, Volume 22, Issue 1, pp 77–167 | Cite as

Pricing and risk of swing contracts in natural gas markets

  • Hendrik Kohrs
  • Hermann Mühlichen
  • Benjamin R. AuerEmail author
  • Frank Schuhmacher
Article
  • 141 Downloads

Abstract

Motivated by the growing importance of swing contracts in natural gas markets, this article extends the literature on commodity price modelling as well as valuation methods and sensitivity analysis for swing options. While most previous studies focused on simple price models, we face the challenge of deriving option properties under more realistic commodity price dynamics. We begin by formulating a multi-factor price forward curve model with parametric volatility functions, which can capture uncertainty in both yearly seasonality and time-to-maturity effects, and propose a two-step calibration procedure to fit such models to empirical data. We then show how results from the literature can be combined to obtain swing option values and sensitivities in such a general framework. In this context, we also provide new theoretical results and a first numerical approach to efficiently estimate swing options’ gammas. For options’ deltas, we expand upon existing studies by including a larger variety of contract specifications and by focusing on a multidimensional variant of the Longstaff–Schwartz algorithm as an alternative option valuation method. With these contributions, we supply important tools for swing option sellers and buyers relying on accurate option value and risk estimates to maintain their business models, hedge option-related risks and adequately represent swing options in financial reporting.

Keywords

Natural gas Forward curve dynamics Swing options Delta Gamma Simulation 

JEL Classification

G12 G13 C63 C65 

Notes

Acknowledgements

We thank the Verbundnetz Gas AG for supporting our research by supplying forward price data and by contributing important ideas for the design of the estimation methods developed in our article. We are also indebted to Max von Renesse, Ralf Wunderlich, Xaver Muschik, the editor, and an anonymous reviewer for valuable comments and suggestions. Generous financial support was provided by the Wissenschaftsförderung der Sparkassen-Finanzgruppe e.V.

Supplementary material

References

  1. Alexandrov, A. (1939). Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningrad State University Annals, Mathematics Series, 37, 3–35.Google Scholar
  2. Anupindi, R., & Bassok, Y. (1999). Supply contracts with quantity commitments and stochastic demands. In S. Tayur, G. Ram, & M. Magazine (Eds.), Quantitative models for supply change management (pp. 197–232). Boston: Kluwer Academic Publishers.Google Scholar
  3. Asmussen, S., & Glynn, P. (2007). Stochastic simulation: Algorithms and analysis (57th ed.). New York: Springer.Google Scholar
  4. Balvers, R., Wu, Y., & Gilliland, E. (2000). Mean reversion across national stock markets and parametric contrarian investment strategies. Journal of Finance, 55(2), 745–772.Google Scholar
  5. Barbieri, A., & Garman, M. (1996). Putting a price on swings. Energy Power Risk Management, 1(6), 17–19.Google Scholar
  6. Bardou, O., Bouthemy, S., & Pagés, G. (2010). When are swing options bang-bang and how to use it. International Journal of Theoretical and Applied Finance, 13(6), 867–899.Google Scholar
  7. Barrera-Esteve, C., Bergeret, F., Dossal, C., Gobet, E., Meziou, A., Munos, R., et al. (2006). Numerical methods for the pricing of swing options: A stochastic control approach. Methodology and Computing in Applied Probability, 8(4), 517–540.Google Scholar
  8. Bellman, R. (1952). On the theory of dynamic programming. Proceedings of the National Academy of Sciences, 38(8), 716–719.Google Scholar
  9. Bellman, R. (1957). Dynamic programming. Princeton, NJ: Princeton University Press.Google Scholar
  10. Benmenzer, G., Gobet, E., & Jérusalem, T. (2007). Arbitrage free cointegrated models in gas and oil future markets. Working Paper, Cornell University.Google Scholar
  11. Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3(1–2), 167–179.Google Scholar
  12. Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.Google Scholar
  13. Bonnans, J., Cen, Z., & Christel, T. (2012). Sensitivity analysis of energy contracts by stochastic programming techniques. In R. Carmona, P. Del Moral, P. Hu, & N. Oudjane (Eds.), Numerical methods in finance (pp. 447–471). Berlin, Heidelberg: Springer.Google Scholar
  14. Bonnans, J., & Shapiro, A. (2000). Perturbation analysis of optimization problems. New York: Springer.Google Scholar
  15. Boogert, A., & De Jong, C. (2011). Gas storage valuation using a multi-factor price process. Journal of Energy Markets, 4(4), 29–52.Google Scholar
  16. Boyd, S., & Vandenberghe, L. (2009). Convex optimization (7th ed.). Cambridge, MA: Cambridge University Press.Google Scholar
  17. Boyle, R., Broadie, M., & Glasserman, P. (1997). Monte Carlo methods for security pricing. Journal of Economic Dynamics and Control, 21(8–9), 1267–1321.Google Scholar
  18. Breslin, J., Clewlow, L., Kwok, C., & Strickland, C. (2008). Gaining from complexity: MFMC models. Energy Risk, April, 60–64.Google Scholar
  19. Brezis, H. (2010). Functional analysis, Sobolev spaces and partial differential equations. New York: Springer.Google Scholar
  20. Broadie, M., & Glasserman, P. (1996). Estimating security price derivatives using simulation. Management Science, 42(2), 269–285.Google Scholar
  21. Broadie, M., & Glasserman, P. (1997). Pricing American-style securities using simulation. Journal of Economic Dynamics and Control, 21(8–9), 1323–1352.Google Scholar
  22. Brown, S., & Yücel, M. (2008). What drives natural gas prices? Energy Journal, 29(2), 45–60.Google Scholar
  23. Carmona, R., & Touzi, N. (2008). Optimal multiple stopping and valuation of swing options. Mathematical Finance, 18(2), 239–268.Google Scholar
  24. Chen, N., & Liu, Y. (2014). American option sensitivities estimation via a generalized infinitesimal perturbation analysis approach. Operations Research, 62(3), 616–632.Google Scholar
  25. Chiarella, C., Clewlow, L., & Kang, B. (2009). Modelling and estimating the forward price curve in the energy market. Working Paper 260, Quantitative Finance Research Centre, University of Technology Sydney.Google Scholar
  26. Clément, E., Lamberton, D., & Protter, P. (2002). An analysis of a least squares regression method for American option pricing. Finance and Stochastics, 6(4), 449–471.Google Scholar
  27. Clewlow, L., & Strickland, C. (1999a). A multi-factor model for energy derivatives. QFRC Research Paper Series 28, University of Technology Sydney.Google Scholar
  28. Clewlow, L., & Strickland, C., (1999b). Valuing energy options in a one factor model fitted to forward prices. QFRC Research Paper Series 10, University of Technology Sydney.Google Scholar
  29. Clewlow, L., & Strickland, C. (2000). Energy derivatives: Pricing and risk management. London: Lacima Publications.Google Scholar
  30. Cortazar, G., & Schwartz, E. (1994). The valuation of commodity contingent claims. Journal of Derivatives, 1(4), 27–39.Google Scholar
  31. Dahlgren, M. (2005). A continuous time model to price commodity-based swing options. Review of Derivatives Research, 8(1), 27–47.Google Scholar
  32. De Jong, C., (2006). The nature of power spikes: A regime-switch approach. Studies in Nonlinear Dynamics and Econometrics, 10(3), Article 3.Google Scholar
  33. De Maeseneire, J. (2010). Estimating forward price curve in energy markets with MCMC method. Research Paper, University of Waterloo.Google Scholar
  34. Deaton, A., & Laroque, G. (1992). On the behaviour of commodity prices. Review of Economic Studies, 59(1), 1–23.Google Scholar
  35. Efron, B., & Tibshirani, R. (1993). An introduction to the bootstrap. Boca Raton: Chapman & Hall.Google Scholar
  36. Eydeland, A., & Wolyniec, K. (2003). Energy and power risk management: New developments in modeling, pricing and hedging (Vol. 206). Hoboken, NJ: Wiley.Google Scholar
  37. Garman, M., & Barbieri, A. (1997). Ups and downs of swing. Energy Power Risk Management, 2(1).Google Scholar
  38. Geman, H. (2005). Commodities and commodity derivatives: Modelling and pricing for agricultures, metals and energy. Chichester: Wiley.Google Scholar
  39. Gibson, R., & Schwartz, E. (1990). Stochastic convenience yield and the pricing of oil contingent claims. Journal of Finance, 45(3), 959–976.Google Scholar
  40. Glasserman, P. (2003). Monte Carlo methods in financial engineering. New York: Springer.Google Scholar
  41. Glasserman, P., & Yu, B. (2004). Number of paths versus number of basis functions in American option pricing. Annals of Applied Probability, 14(4), 2090–2119.Google Scholar
  42. Hambly, B., Howison, S., & Kluge, T. (2009). Modelling spikes and pricing swing options in electricity markets. Quantitative Finance, 9(8), 937–949.Google Scholar
  43. Haugh, M., & Kogan, L. (2004). Pricing American options: A duality approach. Operations Research, 52(2), 258–270.Google Scholar
  44. Hayashi, T., & Mykland, P. (2005). Evaluating hedging error: An asymptotic approach. Mathematical Finance, 15(1), 309–343.Google Scholar
  45. Hedestig, J. (2014). Pricing and hedging of swing options in the European electricity and gas markets. Student Paper, Lund University.Google Scholar
  46. Hubbard, R., & Weiner, R. (1986). Regulation and long-term contracting US natural gas markets. Journal of Industrial Economics, 35(1), 71–79.Google Scholar
  47. Hull, J., & White, A. (1994). Numerical procedures for implementing term structure models i: Single factor models. Journal of Derivatives, 2(1), 7–16.Google Scholar
  48. Ibáñez, A., & Zapatero, F. (2004). Monte Carlo valuation of American options through computation of the optimal exercise frontier. Journal of Financial and Quantitative Analysis, 39(2), 253–275.Google Scholar
  49. Jaillet, J., Ronn, E., & Tompaidis, S. (2004). Valuation of commodity-based swing options. Management Science, 50(7), 909–921.Google Scholar
  50. Jamishidian, F. (1991). Commodity option valuation in the Gaussian futures term structure model. Review of Futures Markets, 10(2), 324–346.Google Scholar
  51. Jin, Y., & Jorion, P. (2006). Firm value and hedging: Evidence from US oil and gas producers. Journal of Finance, 61(2), 893–919.Google Scholar
  52. Johnson, R., & Wichern, D. (2007). Applied multivariate statistical analysis (6th ed.). Upper Saddle River, NJ: Prentice Hall.Google Scholar
  53. Joskow, P. (1985). Vertical integration and long-term contracts: The case of coal-burning electric generating plants. Journal of Law, Economics, & Organization, 1(1), 33–80.Google Scholar
  54. Joskow, P. (1987). Contract duration and relationship-specific investments: Empirical evidence from coal markets. American Economic Review, 77(1), 168–185.Google Scholar
  55. Kiesel, R., Gernhard, J., & Stoll, S. (2010). Valuation of commodity-based swing options. Journal of Energy Markets, 3(3), 91–112.Google Scholar
  56. Kiesel, R., Schindlmayr, G., & Börger, R. (2009). A two-factor model for the electricity forward market. Quantitative Finance, 9(3), 279–287.Google Scholar
  57. Klenke, A. (2007). Probability theory: A comprehensive course. London: Springer.Google Scholar
  58. Kohler, M. (2011). A review of regression-based Monte Carlo methods for pricing American options. In L. Devroye, B. Karasözen, M. Kohler, & R. Korn (Eds.), Recent developments in applied probability and statistics (pp. 37–58). Heidelberg: Physica.Google Scholar
  59. Kolb, R., & Overdahl, J. (Eds.). (2010). Financial derivatives: Pricing and risk management. Hoboken, NJ: Wiley.Google Scholar
  60. Kovacevic, R., Pflug, G., & Vespucci, M. (2013). Handbook of risk management in energy production and trading. New York: Springer.Google Scholar
  61. L’Ecuyer, P., (2007). Variance reduction’s greatest hits. In Proceedings of the 2007 European simulation and modeling conference (pp. 5–12). European Multidisciplinary Society for Modelling and Simulation Technology, Ostend.Google Scholar
  62. L’Ecuyer, P., & Perron, G. (1994). On the convergence rates of IPA and FDC derivative estimators. Operations Research, 42(4), 643–656.Google Scholar
  63. Lioui, A., & Poncet, P. (2005). Dynamic asset allocation with forwards and futures. New York: Springer.Google Scholar
  64. Løland, A., & Lindqvist, O., (2008). Valuation of commodity-based swing options: A survey. Note SAMBA/38/80, Norwegian Computing Center.Google Scholar
  65. Longstaff, F., & Schwartz, E. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies, 14(1), 113–147.Google Scholar
  66. MacAvoy, P. (2000). The natural gas market: Sixty years of regulation and deregulation. New Haven, London: Yale University Press.Google Scholar
  67. Maciejowska, K., & Weron, R., (2013). Forecasting of daily electricity spot prices by incorporating intra-day relationships: Evidence from the UK power market. In IEEE conference proceedings, 10th international conference in the European energy market (EEM).Google Scholar
  68. Manoliu, M., & Tompaidis, S. (2002). Energy futures prices: Term structure models with Kalman filter estimation. Applied Mathematical Finance, 9(1), 21–43.Google Scholar
  69. McLeish, D. (2011). Monte Carlo simulation and finance. Hoboken, NJ: Wiley.Google Scholar
  70. Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.Google Scholar
  71. Musiela, M., & Rutkowski, M. (2004). Martingale methods in financial modelling (2nd ed.). Berlin: Springer.Google Scholar
  72. Pascucci, A. (2011). PDE and martingale methods in option pricing. Milano: Springer, Bocconi University Press.Google Scholar
  73. Pflug, G., & Broussev, N. (2009). Electricity swing options: Behavioral models and pricing. European Journal of Operational Research, 197(3), 1041–1050.Google Scholar
  74. Pilipovic, D. (2007). Energy risk: Valuing and managing energy derivatives (2nd ed.). New York: McGraw-Hill.Google Scholar
  75. Pilipovic, D., & Wengler, J. (1998). Getting into the swing. Energy Power Risk Management, 2(10).Google Scholar
  76. Pinar, M. (2007). Robust scenario optimization based on downside-risk measure for mulit-period portfolio selection. OR Spectrum, 29(2), 295–309.Google Scholar
  77. Poterba, J. (1988). Mean reversion in stock prices: Evidence and implications. Journal of Financial Economics, 22(1), 27–59.Google Scholar
  78. Rockafellar, R., & Wets, R. (2009). Variational analysis (Vol. 317). Berlin: Springer.Google Scholar
  79. Rodríguez, R. (2008). Real option valuation of free destination in long-term liquefied natural gas supplies. Energy Economics, 30(4), 1909–1932.Google Scholar
  80. Rudin, W. (1964). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill.Google Scholar
  81. Schwartz, E. (1997). The stochastic behaviour of commodity prices: Implications for valuation and hedging. Journal of Finance, 52(3), 923–973.Google Scholar
  82. Schwartz, E., & Smith, J. (2000). Short-term variations and long-term dynamics in commodity prices. Management Science, 46(7), 893–911.Google Scholar
  83. Smith, C., & Zimmerman, J. (1976). Valuing employee stock option plans using option pricing models. Journal of Accounting Research, 14(2), 357–364.Google Scholar
  84. Thompson, A. (1995). Valuation of path-dependent contingent claims with multiple exercise decisions over time: The case of take-or-pay. Journal of Financial and Quantitative Analysis, 30(2), 271–293.Google Scholar
  85. van der Hoek, J., & Elliott, R. (2006). Binomial models in finance. New York: Springer.Google Scholar
  86. Vinga, S., (2004). Convolution integrals of normal distribution functions. Supplementary material to Vinga and Almeida (2004) “Rényi continuous entropy of DNA sequences”.Google Scholar
  87. Wahab, M., & Lee, C. (2011). Pricing swing options with regime switching. Annals of Operations Research, 185(1), 139–160.Google Scholar
  88. Wahab, M., Yin, Z., & Edirisinghe, N. (2010). Pricing swing options in the electricity markets under regime-switching uncertainty. Quantitative Finance, 10(9), 975–994.Google Scholar
  89. Warin, X. (2012). Hedging swing contract on gas markets. Working Paper, Cornell University.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Hendrik Kohrs
    • 1
  • Hermann Mühlichen
    • 2
  • Benjamin R. Auer
    • 1
    • 3
    Email author
  • Frank Schuhmacher
    • 1
  1. 1.Department of FinanceUniversity of LeipzigLeipzigGermany
  2. 2.Risk Management/Quantitative AnalysisVNG Handel & Vertrieb GmbHLeipzigGermany
  3. 3.Research Network Area Macro, Money and International FinanceCESifo MunichMunichGermany

Personalised recommendations