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Approximation of forward curve models in commodity markets with arbitrage-free finite-dimensional models

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Abstract

In this paper, we show how to approximate Heath–Jarrow–Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have a finite-dimensional state space. Moreover, we recover a closed-form representation of the forward price dynamics in the approximation models and derive the rate of convergence to the true dynamics uniformly over an interval of time to maturity under certain additional smoothness conditions. In the Markovian case, we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.

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Notes

  1. This is a very useful consequence of the Szőkefalvi-Nagy dilation theorem [32, Theorem I.8.1].

References

  1. Albeverio, S., Ferrario, B.: Some methods of infinite dimensional analysis in hydrodynamics: an introduction. In: da Prato, G., Röckner, M. (eds.) SPDE in Hydrodynamics: Recent Progress and Prospects. Lecture Notes in Math., vol. 1942, pp. 1–50. Springer, Berlin (2005)

    Google Scholar 

  2. Barth, A.: A finite element method for martingale-driven stochastic partial differential equations. Commun. Stoch. Anal. 4, 355–373 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Benth, F.E., Kallsen, J., Meyer-Brandis, T.: A non-Gaussian Ornstein–Uhlenbeck process for electricity spot price modeling and derivatives pricing. Appl. Math. Finance 14, 153–169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benth, F.E., Krühner, P.: Representation of infinite-dimensional forward price models in commodity markets. Commun. Math. Stat. 2, 47–106 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benth, F.E., Krühner, P.: Subordination of Hilbert space valued Lévy processes. Stochastics 87, 458–476 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benth, F.E., Krühner, P.: Derivatives pricing in energy markets: an infinite-dimensional approach. SIAM J. Financ. Math. 6, 825–869 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benth, F.E., Lempa, J.: Optimal portfolios in commodity markets. Finance Stoch. 18, 407–430 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benth, F.E., Šaltytė Benth, J.: Modeling and Pricing in Financial Markets for Weather Derivatives. World Scientific, Singapore (2013)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  10. Björk, T.: On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. Finance 11, 205–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Björk, T., Landén, C.: On the construction of finite dimensional realizations for nonlinear forward rate models. Finance Stoch. 6, 303–331 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Björk, T., Svensson, L.: On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. Finance 11, 205–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Carmona, R., Tehranchi, M.: Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective. Springer, Berlin (2006)

    MATH  Google Scholar 

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

    Google Scholar 

  15. Cuchiero, C., Klein, I., Teichmann, J.: A new perspective on the fundamental theorem of asset pricing for large financial markets. Theory Probab. Appl. 60, 561–579 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dinculeanu, N.: Vector Integration and Stochastic Integration in Banach Spaces. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  18. Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications to finance. Ann. Appl. Probab. 13, 984–1053 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Filipović, D.: Invariant manifolds for weak solutions to stochastic equations. Probab. Theory Relat. Fields 118, 323–341 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Filipović, D.: Consistency Problems for Heath–Jarrow–Morton Interest Rate Models. Lecture Notes in Mathematics, vol. 1760. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  21. Filipović, D., Tappe, S., Teichmann, J.: Jump-diffusions in Hilbert spaces: existence, stability and numerics. Stochastics 82, 475–520 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Filipović, D., Teichmann, J.: Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal. 197, 398–432 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Grecksch, W., Kloeden, P.E.: Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Aust. Math. Soc. 54, 79–85 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new methodology for contingent claim valuation. Econometrica 60, 77–105 (1992)

    Article  MATH  Google Scholar 

  25. Henseler, M., Christoph, C., Seydel, R.C.: A tractable multi-factor dynamic term-structure model for risk management. Working paper (2015), available online at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2225738

  26. Kruse, R.: Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA J. Numer. Anal. 34, 217–251 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kruse, R.: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Lecture Notes in Mathematics, vol. 2093. Springer, Berlin (2014)

    MATH  Google Scholar 

  28. Mishura, Y., Munchak, E.: Rate of convergence of option prices by using the method of pseudomoments. Theory Probab. Math. Stat. 92, 117–133 (2016)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  30. Platen, E., Bruti-Liberati, N.: Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  31. Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)

    MATH  Google Scholar 

  32. Szőkefalvi-Nagy, B., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. North-Holland, Amsterdam (1970)

    MATH  Google Scholar 

  33. Tappe, S.: An alternative approach on the existence of affine realizations for HJM term structure models. Proc. R. Soc. Lond. Ser. A 466, 3033–3060 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tappe, S.: Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures. Int. J. Stoch. Anal. 2012, 236327 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Young, R.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (1980)

    MATH  Google Scholar 

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Acknowledgements

An anonymous referee and the Associate Editor are thanked for their constructive criticisms.

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316, Oslo, Norway

    Fred Espen Benth

  2. Institute for Financial and Actuarial Mathematics, The University of Liverpool, Mathematical Sciences Building, Liverpool, L69 7ZL, UK

    Paul Krühner

Authors
  1. Fred Espen Benth
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  2. Paul Krühner
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Corresponding author

Correspondence to Fred Espen Benth.

Additional information

F.E. Benth acknowledges financial support from the project “Managing Weather Risk in Energy Markets (MAWREM)”, funded by the ENERGIX program of the Norwegian Research Council.

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Benth, F.E., Krühner, P. Approximation of forward curve models in commodity markets with arbitrage-free finite-dimensional models. Finance Stoch 22, 327–366 (2018). https://doi.org/10.1007/s00780-018-0355-9

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