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Forward Curve Modelling by Ambit Fields

  • Ole E. Barndorff-Nielsen
  • Fred Espen Benth
  • Almut E. D. Veraart
Chapter
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Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 88)

Abstract

The final chapter of the book is devoted to forward pricing. We apply ambit fields in order to extend the so-called Heath-Jarrow-Morton approach to forward price modelling. Indeed, we state general ambit field models with drift, where the spatial dimension is the delivery time of the forward contract. The ambit sets will have a simple form in our setting, and we derive explicit no-arbitrage conditions for the drift in both arithmetic and geometric (i.e., exponentiated) ambit specifications. Basic properties such as the temporal and spatial correlation are studied, as well as the implied spot price dynamics. The chapter ends with a detailed study of spread option pricing, where we show that in the Gaussian case we recover a general version of the classical Margrabe formula, while in the Lévy case we can provide a Fourier-based expression for the price.

References

  1. Albeverio, S., Lytvynov, E. & Mahning, A. (2004), ‘A model of the term structure of interest rates based on Lévy fields’, Stochastic Processes and their Applications 114, 251–263.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Audet, N., Heiskanen, P., Keppo, J. & Vehviläinen, I. (2004), Modeling electricity forward curve dynamics in the Nordic Market, in D. W. Bunn, ed., ‘Modelling prices in competitive electricity markets’, John Wiley & Sons, pp. 251–265.Google Scholar
  3. Barndorff-Nielsen, O. E., Benth, F. E. & Veraart, A. E. D. (2013a), ‘Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes’, Bernoulli 19(3), 803–845.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Barndorff-Nielsen, O. E., Benth, F. E. & Veraart, A. E. D. (2014c), ‘Modelling electricity futures by ambit fields’, Advances in Applied Probability 46(3), 719–745.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Barndorff-Nielsen, O. E., Benth, F. E. & Veraart, A. E. D. (2015a), Cross-commodity modelling by multivariate ambit fields, in M. Ludkovski, R. Sircar & R. Aïd, eds, ‘Commodities, Energy and Environmental Finance’, Vol. 74 of Fields Institute Communications, Springer, New York, pp. 109–148.Google Scholar
  6. Barndorff-Nielsen, O. E., Benth, F. E. & Veraart, A. E. D. (2015b), ‘Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency’, Banach Center Publications 104, 25–60.Google Scholar
  7. Barndorff-Nielsen, O. E. & Shephard, N. (2001), ‘Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics’, Journal of the Royal Statistical Society. Series B. Statistical Methodology 63(2), 167–241.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Barndorff-Nielsen, O. E. & Shiryaev, A. (2015), Change of time and change of measure, Vol. 21 of Advanced Series on Statistical Science and Applied Probability, second edn, World Scientific.Google Scholar
  9. Benth, F. E., Šaltytė Benth, J. & Koekebakker, S. (2008), Stochastic Modelling of Electricity and Related Markets, Vol. 11 of Advanced Series on Statistical Science and Applied Probability, World Scientific.zbMATHGoogle Scholar
  10. Bjerksund, P., Rasmussen, H. & Stensland, G. (2010), Valuation and risk management in the Nordic electricity market, in P. M. P. E. Bjørndal, M. Bjørndal & M. Rønnqvist, eds, ‘Energy, Natural Resources and Environmental Economics’, Springer Verlag, pp. 167–185.Google Scholar
  11. Black, F. & Scholes, M. (1973), ‘The pricing of options and corporate liabilities’, Journal of Political Economy 81(3), 637–654.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Caldeira, J. & Torrent, H. (2017), ‘Forecasting the US term structure of interest rates using nonparametric functional data analysis’, Journal of Forecasting 36(1), 56–73.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Carmona, R. & Durrleman, V. (2003), ‘Pricing and hedging spread options’, SIAM Review 45(4), 627–685.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Goldstein, R. S. (2000), ‘The term structure of interest rates as a random field’, The Review of Financial Studies 13(2), 365–384.CrossRefGoogle Scholar
  15. Goutis, C. & Casella, G. (1999), ‘Explaining the saddlepoint approximation’, The American Statistician 53(3), 216–224.MathSciNetGoogle Scholar
  16. Heath, D., Jarrow, R. & Morton, A. (1992), ‘Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation’, Econometrica 60(1), 77–105.zbMATHCrossRefGoogle Scholar
  17. Hurd, T. R. & Zhou, Z. (2010), ‘A Fourier transform method for spread option pricing’, SIAM Journal on Financial Mathematics 1(1), 142–157.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Kallsen, J. & Tankov, P. (2006), ‘Characterization of dependence of multidimensional Lévy processes using Lévy copulas’, Journal of Multivariate Analysis 97(7), 1551–1572.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Kennedy, D. P. (1994), ‘The term structure of interest rates as a Gaussian random field’, Mathematical Finance 4(3), 247–258.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Kennedy, D. P. (1997), ‘Characterizing Gaussian models of the term structure of interest rates’, Mathematical Finance 7(2), 107–116.zbMATHCrossRefGoogle Scholar
  21. Kimmel, R. L. (2004), ‘Modeling the term structure of interest rates: A new approach’, Journal of Financial Economics 72, 143–183.CrossRefGoogle Scholar
  22. Koekebakker, S. & Ollmar, F. (2005), ‘Forward curve dynamics in the Nordic electricity market’, Managerial Finance 31(6), 73–94.CrossRefGoogle Scholar
  23. Margrabe, W. (1978), ‘The value of an option to exchange one asset for another’, The Journal of Finance 33(1), 177–186.CrossRefGoogle Scholar
  24. Samuelson, P. (1965), ‘Proof that properly anticipated prices fluctuate randomly’, Industrial Management Review 6, 41–44.Google Scholar
  25. Santa-Clara, P. & Sornette, D. (2001), ‘The dynamics of the forward interest rate curve with stochastic string shocks’, The Review of Financial Studies 14(1), 149–185.CrossRefGoogle Scholar
  26. Walsh, J. (1986), An introduction to stochastic partial differential equations, in R. Carmona, H. Kesten & J. Walsh, eds, ‘Lecture Notes in Mathematics 1180’, Ecole d’Eté de Probabilités de Saint-Flour XIV (1984), Springer.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Fred Espen Benth
    • 2
  • Almut E. D. Veraart
    • 3
  1. 1.Department of MathematicsUniversity of AarhusAarhusDenmark
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.Department of MathematicsImperial College LondonLondonUK

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