Interest Rate Models

  • Ernst Eberlein
  • Jan Kallsen
Part of the Springer Finance book series (FINANCE)


From an abstract point of view interest rate markets fall into the reach of the general theory which is discussed in Chaps.  9 13. They differ from equity markets just in the kind of assets, namely bonds, swaps, caps and other interest rate products. On closer observation, however, one faces new phenomena. For example, the natural choice of a numeraire is not obvious in markets where interest rates change randomly over time.


  1. 26.
    T. Bielecki, M. Rutkowski, Credit Risk: Modelling, Valuation and Hedging (Springer, Berlin, 2002)zbMATHGoogle Scholar
  2. 29.
    T. Björk, Arbitrage Theory in Continuous Time, 3rd edn. (Oxford Univ. Press, Oxford, 2009)zbMATHGoogle Scholar
  3. 30.
    T. Björk, An overview of interest rate theory, in Handbook of Financial Time Series, ed. by T. Andersen, R. Davis, J.-P. Kreiß, T. Mikosch (Springer, Berlin, 2009), pp. 615–651zbMATHCrossRefGoogle Scholar
  4. 31.
    T. Björk, B. Christensen, Interest rate dynamics and consistent forward rate curves. Math. Finance 9(4), 323–348 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 32.
    T. Björk, L. Svensson, On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. Finance 11(2), 205–243 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 33.
    T. Björk, G. Di Masi, Y. Kabanov, W. Runggaldier, Towards a general theory of bond markets. Finance Stochast. 1(2), 141–174 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 34.
    T. Björk, Y. Kabanov, W. Runggaldier, Bond market structure in the presence of marked point processes. Math. Finance 7(2), 211–239 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 39.
    A. Brace, D. Ga̧tarek, M. Musiela, The market model of interest rate dynamics. Math. Finance 7(2), 127–155 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 42.
    D. Brigo, F. Mercurio, Interest Rate Models—Theory and Practice, 2nd edn. (Springer, Berlin, 2006)zbMATHGoogle Scholar
  10. 44.
    D. Brody, L. Hughston, E. Mackie, Rational term structure models with geometric Lévy martingales. Stochastics 84(5–6), 719–740 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 46.
    R. Carmona, M. Tehranchi, Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective (Springer, Berlin, 2006)zbMATHGoogle Scholar
  12. 62.
    S. Crépey, T. Bielecki, Counterparty Risk and Funding (CRC Press, Boca Raton, 2014)zbMATHGoogle Scholar
  13. 83.
    E. Eberlein, W. Kluge, Exact pricing formulae for caps and swaptions in a Lévy term structure model. J. Comput. Finance 9(2), 99–125 (2006)CrossRefGoogle Scholar
  14. 84.
    E. Eberlein, W. Kluge, Valuation of floating range notes in Lévy term-structure models. Math. Finance 16(2), 237–254 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 85.
    E. Eberlein, W. Kluge, Calibration of Lévy term structure models, in Advances in Mathematical Finance (Birkhäuser, Boston, 2007), pp. 147–172zbMATHGoogle Scholar
  16. 86.
    E. Eberlein, F. Özkan, The defaultable Lévy term structure: Ratings and restructuring. Math. Finance 13(2), 277–300 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 87.
    E. Eberlein, F. Özkan, The Lévy LIBOR model. Finance Stochast. 9(3), 327–348 (2005)zbMATHCrossRefGoogle Scholar
  18. 89.
    E. Eberlein, S. Raible, Term structure models driven by general Lévy processes. Math. Finance 9(1), 31–53 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 94.
    E. Eberlein, J. Jacod, S. Raible, Lévy term structure models: No-arbitrage and completeness. Finance Stochast. 9(1), 67–88 (2005)zbMATHCrossRefGoogle Scholar
  20. 103.
    D. Filipović, Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, volume 1760 of Lecture Notes in Mathematics (Springer, Berlin, 2001)zbMATHCrossRefGoogle Scholar
  21. 105.
    D. Filipović, Term-Structure Models (Springer, Berlin, 2009)zbMATHCrossRefGoogle Scholar
  22. 108.
    D. Filipović, S. Tappe, Existence of Lévy term structure models. Finance Stochast. 12(1), 83–115 (2008)zbMATHCrossRefGoogle Scholar
  23. 109.
    D. Filipović, J. Teichmann, Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal. 197(2), 398–432 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 110.
    D. Filipović, M. Larsson, A. Trolle, Linear-rational term structure models. J. Finance 72(2), 655–704 (2017)CrossRefGoogle Scholar
  25. 111.
    D. Filipović, M. Larsson, A. Trolle, On the relation between linearity-generating processes and linear-rational models. arXiv preprint arXiv:1806.03153 (2018)Google Scholar
  26. 113.
    B. Flesaker, L. Hughston, Positive interest. Risk 9(1), 46–49 (1996)Google Scholar
  27. 124.
    K. Glau, Z. Grbac, A. Papapantoleon, A unified view of LIBOR models, in Advanced Modelling in Mathematical Finance, vol. 189, ed. by J. Kallsen, A. Papapantoleon (Springer, Cham, 2016), pp. 423–452CrossRefGoogle Scholar
  28. 129.
    Z. Grbac, W. Runggaldier, Interest Rate Modeling: Post-Crisis Challenges and Approaches (Springer, Cham, 2015)zbMATHCrossRefGoogle Scholar
  29. 131.
    P. Harms, D. Stefanovits, J. Teichmann, M. Wüthrich, Consistent recalibration of yield curve models. Math. Finance 28(3), 757–799 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 140.
    D. Heath, R. Jarrow, A. Morton, Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 60, 77–105 (1992)zbMATHCrossRefGoogle Scholar
  31. 156.
    F. Jamshidian, An exact bond option formula. J. Finance 44(1), 205–209 (1989)CrossRefGoogle Scholar
  32. 157.
    F. Jamshidian, Libor and swap market models and measures. Finance Stochast. 1(4), 293–330 (1997)zbMATHCrossRefGoogle Scholar
  33. 158.
    F. Jamshidian, Libor market model with semimartingales. Technical report, Working Paper, NetAnalytic Ltd, 1999Google Scholar
  34. 159.
    R. Jarrow, D. Madan, Option pricing using the term structure of interest rates to hedge systematic discontinuities in asset returns. Math. Finance 5(4), 311–336 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 190.
    M. Keller-Ressel, A. Papapantoleon, J. Teichmann, The affine LIBOR models. Math. Finance 23(4), 627–658 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 193.
    W. Kluge, Time-Inhomogeneous Lévy Processes in Interest Rate and Credit Risk Models. PhD thesis, University of Freiburg, 2005Google Scholar
  37. 221.
    K. Miltersen, K. Sandmann, D. Sondermann, Closed form solutions for term structure derivatives with log-normal interest rates. J. Finance 52(1), 409–430 (1997)CrossRefGoogle Scholar
  38. 223.
    M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, 2nd edn. Stochastic Modelling and Applied Probability (Springer, Berlin, 2005)zbMATHCrossRefGoogle Scholar
  39. 224.
    T. Nguyen, F. Seifried, The affine rational potential model (2015). PreprintGoogle Scholar
  40. 229.
    A. Papapantoleon, Old and new approaches to LIBOR modeling. Statistica Neerlandica 64(3), 257–275 (2010)MathSciNetCrossRefGoogle Scholar
  41. 247.
    C. Rogers, The potential approach to the term structure of interest rates and foreign exchange rates. Math. Finance 7(2), 157–176 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 255.
    M. Rutkowski, A note on the Flesaker-Hughston model of the term structure of interest rates. Appl. Math. Finance 4(3), 151–163 (1997)zbMATHCrossRefGoogle Scholar
  43. 262.
    P. Schönbucher, Credit Derivatives Pricing Models: Models, Pricing and Implementation (Wiley, New York, 2003)Google Scholar
  44. 274.
    H. Shirakawa, Interest rate option pricing with Poisson-Gaussian forward rate curve processes. Math. Finance 1(4), 77–94 (1991)zbMATHCrossRefGoogle Scholar
  45. 279.
    S. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (Springer, New York, 2004)zbMATHCrossRefGoogle Scholar
  46. 296.
    R. Zagst, Interest-Rate Management (Springer, Berlin, 2002)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ernst Eberlein
    • 1
  • Jan Kallsen
    • 2
  1. 1.Department of Mathematical StochasticsUniversity of FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsKiel UniversityKielGermany

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