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Heath-Jarrow-Morton Type Models

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Stochastic Models for Prices Dynamics in Energy and Commodity Markets

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Abstract

In this main chapter of the book, infinite-dimensional stochastic processes are defined for the forward dynamics using a Hilbert space as state space for the term structures. Arithmetic and geometric models are introduced, where the noise driver is a Wiener process or a Lévy process and the context is cross-commodity markets. Moreover, we also allow for a class of stochastic volatility models in the forward dynamics. Drift conditions are derived ensuring a risk-neutral dynamics, i.e., a no-arbitrage dynamics under a pricing measure. We furthermore study swap prices (forward with delivery period) and finite factor models in this HJM-framework. To include seasonality and modeling under the market probability require a study of measure change, where one may apply the Girsanov and Esscher transform in our context. To have available data and the initial forward curve, a smoothing approach based on a combination of parametric curves (i.e., the Nelson-Siegel model) the the interpolation technique kriging is proposed and applied in an empirical example.

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Notes

  1. 1.

    To simplify the exposition, we let \(d=1\) here.

  2. 2.

    The data are downloaded from finance.yahoo.com. We do not make any distinction between forward and futures here, but treat them as the same asset class.

References

  1. Andersen, A., Koekebakker, S., and Westgaard, S. (2010). Modeling electricity forward prices using the multivariate normal inverse Gaussian distribution. Journal of Energy Markets, 3(3), pp. 3–25.

    Article  Google Scholar 

  2. Audet, N., Heiskanen, P., Keppo, J., and Vehvilainen, I. (2002). Modelling of electricity forward curve dynamics in the Nordic electricity market. In: Modelling Prices in Competitive Electricity Markets, D. W. Bunn (editor), pp. 251–265, John Wiley & Sons, Chichester.

    Google Scholar 

  3. Barndorff-Nielsen, O. E., Benth, F. E., and Veraart, A. (2014). Modelling electricity futures by ambit fields. Advances in Applied Probability, 46(3), pp. 719–745.

    Article  MathSciNet  MATH  Google Scholar 

  4. Barndorff-Nielsen, O. E., Benth, F. E., and Veraart, A. (2018). Ambit Stochastics, Springer-Verlag, Cham.

    Book  MATH  Google Scholar 

  5. Barndorff-Nielsen, O. E., and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in economics. Journal of the Royal Statistical Society, Series B, 63(2), pp. 167–241 (with discussion).

    Google Scholar 

  6. Barndorff-Nielsen, O. E., and Stelzer, R. (2011). Multivariate supOU processes. Annals of Applied Probability, 21(1), pp. 140–182.

    Article  MathSciNet  MATH  Google Scholar 

  7. Barth, A., and Benth, F. E. (2014). The forward dynamics in energy markets – infinite-dimensional modelling and simulation. Stochastics, 86(6), pp. 932–966.

    Article  MathSciNet  MATH  Google Scholar 

  8. Benth, F. E. (2015). Kriging smooth futures curves. Energy Risk, February, pp. 64–69.

    Google Scholar 

  9. Benth, F. E., Šaltytė Benth, J., and Koekebakker, S. (2008). Stochastic Modelling of Electricity and Related Markets. World Scientific, Singapore.

    Book  MATH  Google Scholar 

  10. 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. Journal of Banking & Finance, 32(10), pp. 2006–2021.

    Article  Google Scholar 

  11. Benth, F. E., and Harang, F. A. (2021). Infinite dimensional pathwise Volterra processes driven by Gaussian noise – probabilistic properties and applications. Electronic Journal of Probability, 26, article no. 114.

    Google Scholar 

  12. Benth, F. E., and Koekebakker, S. (2008). Stochastic modeling of financial electricity contracts. Energy Economics, 30(3), pp. 1116–1157.

    Article  Google Scholar 

  13. Benth, F. E., Koekebakker, S., and Ollmar, F. (2007). Extracting and applying smooth forward curves from average-based commodity contracts with seasonal variation. Journal of Derivatives, 15, pp. 52–66.

    Article  Google Scholar 

  14. Benth, F. E., and Krühner, P. (2014). Representation of infinite dimensional forward price models in commodity markets. Communications in Mathematics and Statistics, 2(1), pp. 47–106.

    Article  MathSciNet  MATH  Google Scholar 

  15. Benth, F. E., and Krühner, P. (2018) Approximation of forward curve models in commodity markets with arbitrage-free finite-dimensional models. Finance & Stochastics, 22(2), pp. 327–366.

    Article  MathSciNet  MATH  Google Scholar 

  16. Benth, F. E., and Krühner, P. (2015). Derivatives pricing in energy markets: an infinite dimensional approach. SIAM Journal of Financial Mathematics, 6(1), pp. 825–869.

    Article  MathSciNet  MATH  Google Scholar 

  17. Benth, F. E., and Lempa, J. (2014). Optimal portfolios in commodity markets. Finance & Stochastics, 18(2), pp. 407–430.

    Article  MathSciNet  MATH  Google Scholar 

  18. Benth, F. E., and Paraschiv, F. (2018). A structural model for electricity forward prices. Journal of Banking & Finance, 95, pp. 203–216.

    Article  Google Scholar 

  19. Benth, F. E., Rüdiger, B., and Süss, A. (2018). Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility. Stochastic Processes and their Applications, 128, pp. 461–486.

    Article  MathSciNet  MATH  Google Scholar 

  20. Benth, F. E., and Šaltytė Benth (2004). The normal inverse Gaussian distribution and spot price modelling in energy markets. International Journal of Theoretical and Applied Finance, 7(2), pp. 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  21. Benth, F. E., and Šaltytė Benth (2012). Modelling and Pricing in Financial Markets for Weather Derivatives. World Scientific, Singapore.

    Book  MATH  Google Scholar 

  22. Benth, F. E., Schroers, D., and Veraart, A. (2020). A weak law of large numbers for realized covariation in a Hilbert space setting. Stochastic Processes and their Applications, 145, pp. 241–268.

    Article  MATH  Google Scholar 

  23. Benth, F. E., and Sgarra, C. (2021). A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets. E-print, available at ssrn:abstract=3835053.

    Google Scholar 

  24. Benth, F. E., and Simonsen, I. C. (2018). The Heston stochastic volatility model in Hilbert space. Stochastic Analysis and Applications, 36(4), pp. 733–750.

    Article  MathSciNet  MATH  Google Scholar 

  25. Bessembinder, H., and Lemon, M. L. (2002). Equilibrium pricing and optimal hedging in electricity forward markets. Journal of Finance, 57(3), pp. 1347–1382.

    Article  Google Scholar 

  26. Björk, T. (2009). Arbitrage Theory in Continuous Time. Oxford University Press, Oxford.

    MATH  Google Scholar 

  27. Björk T., and Landen C. (2002). On the term structure of futures and forward prices. In: Mathematical Finance – Bachelier Congress 2000, Geman H., Madan D., Pliska S.R., Vorst T. (eds.), Springer Finance. Springer-Verlag, Berlin Heidelberg, pp. 111–149.

    Chapter  Google Scholar 

  28. Callegaro, G., Mazzoran, A., and Sgarra, C. (2022). A self-exciting modeling framework for forward prices in power markets. Applied Stochastic Models in Business and Industry, 38(1), pp. 27–48.

    Article  MathSciNet  Google Scholar 

  29. Carmona, R. (2015). Financialization of the commodities markets: a non-technical introduction. In Commodities, Energy and Environmental Finance, R. Aid, M. Ludkovski and R. Sircar (eds.), Springer-Verlag, New York, pp. 3–37.

    Chapter  Google Scholar 

  30. Clewlow, L., and Strickland, C. (2000). Energy Derivatives. Pricing and Risk Management. Lacima Publications, London.

    Google Scholar 

  31. Cox, S., Karbach, S., and Khedher, A. (2022). Affine pure-jump processes on positive Hilbert-Schmidt operators. Stochastic Processes and their Applications, 151, pp. 191–229.

    Article  MathSciNet  MATH  Google Scholar 

  32. Cox, S., Karbach, S., and Khedher, A. (2022). An infinite-dimensional affine stochastic volatility model. Mathematical Finance, 32(3), pp. 878–906.

    Article  MathSciNet  MATH  Google Scholar 

  33. Cressie, N. A. C. (1993). Statistics for Spatial Data. John Wiley & Sons, New York.

    Book  MATH  Google Scholar 

  34. Cressie, N. A. C., and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. John Wiley & Sons, New York.

    MATH  Google Scholar 

  35. Da Prato, G., and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  36. Di Persio, L., and Perin, I. (2015). An ambit stochastic approach to pricing electricity forward contracts: The case of the German energy market. Journal of Probability and Statistics, 2015 (2015).

    Google Scholar 

  37. Diebold, F. X., and Rudebusch, G. D. (2013). Yield Curve Modeling and Forecasting: The Dynamic Nelson-Siegel Approach. Princeton University Press, Princeton.

    Book  MATH  Google Scholar 

  38. Duffie, D. (1992). Dynamic Asset Pricing Theory. Princeton University Press, Princeton.

    MATH  Google Scholar 

  39. Esscher, F. (1932). On the probability function in collective theory of risk. Skandinavisk Aktuarietidsskrift, 15, pp. 175–195.

    MATH  Google Scholar 

  40. Eydeland, A., and Wolyniec, K. (2003). Energy Risk and Power Risk Management – New Developments in Modeling, Pricing, and Hedging. Wiley-Finance. John Wiley & Sons, Hoboken, New Jersey.

    Google Scholar 

  41. Filipović, D., Pelger, M., and Ye, Ye (2022). Shrinking the term structure. Swiss Finance Institute Research Paper No. 61. Available at SSRN: https://ssrn.com/abstract=4182649 or http://dx.doi.org/10.2139/ssrn.4182649

  42. Frestad, D., Benth, F. E., and Koekebakker, S. (2010). Modeling term structure dynamics in the Nordic electricity swap market. Energy Journal, 31(2), pp. 53–86.

    Article  Google Scholar 

  43. Geman, H. (2005). Commodities and Commodity Derivatives. Wiley-Finance. John Wiley & Sons, Chichester.

    Google Scholar 

  44. Geman, H., and Vašíček, O. (2001). Forwards and futures on non storable commodities: the case of electricity. RISK, August issue.

    Google Scholar 

  45. Gerber, H. U., and Shiu, E. S. W. (1994). Option pricing by Esscher transform. Transaction of the Society of Actuaries, 46, pp. 91–191 (with discussion).

    Google Scholar 

  46. Goutte, S., Oudjane, N., and Russo, F. (2013). Variance optimal hedging for discrete time processes with independent increments. Application to electricity markets. Journal of Computational Finance, 17(2), pp. 71–111.

    Article  Google Scholar 

  47. 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(1), pp. 77–105.

    Article  MATH  Google Scholar 

  48. Hinderks, W. J., and Wagner, A. (2020). Factor models in the German electricity market: Stylized facts, seasonality, and calibration. Energy Economics, 85, article 104351.

    Google Scholar 

  49. Jacod, J., and Shiryaev, A. (2003). Limit Theorems for Stochastic Processes, 2nd Ed. Springer-Verlag, Berlin Heidelberg.

    Book  MATH  Google Scholar 

  50. Jamshidian, F. (1992). Commodity option valuation in the Gaussian futures term structure model. Review of Futures Markets, 10, pp. 61–86.

    Google Scholar 

  51. Kiesel, R., Börger, R., and Schindlmayr, G. (2009). A two-factor model for the electricity forward market. Quantitative Finance, 9(3), pp. 279–287.

    Article  MathSciNet  MATH  Google Scholar 

  52. Kiesel, R., Paraschiv, F., and Sætherø, A. (2019). On the construction of hourly price forward curves for electricity prices. Computational Management Finance, 16(1-2), pp. 345–369.

    MathSciNet  MATH  Google Scholar 

  53. Kristoufek, L. (2014). Leverage effect in energy futures. Energy Economics, 45, pp. 1–9.

    Article  Google Scholar 

  54. Latini, L., Piccirilli, M., and Vargiolu, T. (2019). Mean-reverting no-arbitrage additive models for forward curves in energy markets. Energy Economics, 79, pp. 157–170.

    Article  Google Scholar 

  55. Nelson, C., and Siegel, A. F. (1987). Parsimonious modeling of yield curves. Journal of Business, 60, pp. 473–489.

    Article  Google Scholar 

  56. Nomikos, N. K., and Andriosopoulos, K. (2012). Modelling energy spot prices: empirical evidence from NYMEX. Energy Economics, 34(4), pp. 1153–1169.

    Article  Google Scholar 

  57. Peszat, S., and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  58. Piccirilli, M., Schmeck, M. D:, and Vargiolu, T. (2021). Capturing the power options smile by an additive two-factor model for overlap** futures prices. Energy Economics, 95, article 105006.

    Google Scholar 

  59. Quiao, H., and Wu, J.-L. (2019). On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete and Continuous Dynamical Systems – B, 24(4), pp. 1449–1467.

    Article  MathSciNet  MATH  Google Scholar 

  60. Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  61. Schmidt, T., Tappe, S., and Yu, W. (2020). Infinite dimensional affine processes. Stochastic Processes and their Applications, 130(12), pp. 7131–7169.

    Article  MathSciNet  MATH  Google Scholar 

  62. Svensson, L. E. O. (1994). Estimating and interpreting forward interest rates: Sweden 1992–4. NBER Working Paper Series, no 4871, September.

    Google Scholar 

  63. Veraart, A. E. D., and Veraart, L. A. M. (2014). Risk premiums in energy markets, Journal of Energy Markets, 6(4), pp. 91–132.

    Article  MathSciNet  Google Scholar 

  64. Weron, R. (2006). Modeling and Forecasting Electricity Loads and Prices – A Statistical Approach. John Wiley & Sons, Chichester.

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, Oslo, Norway

    Fred Espen Benth

  2. Institute for Statistics and Mathematics, Vienna University of Economics and Business, Vienna, Austria

    Paul Krühner

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  1. Fred Espen Benth
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  2. Paul Krühner
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Benth, F.E., Krühner, P. (2023). Heath-Jarrow-Morton Type Models. In: Stochastic Models for Prices Dynamics in Energy and Commodity Markets. Springer Finance. Springer, Cham. https://doi.org/10.1007/978-3-031-40367-5_6

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