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Finance and Stochastics

, Volume 20, Issue 2, pp 267–320 | Cite as

A general HJM framework for multiple yield curve modelling

  • Christa Cuchiero
  • Claudio Fontana
  • Alessandro Gnoatto
Article

Abstract

We propose a general framework for modelling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor’s length. When the driving semimartingale is an affine process, we obtain a flexible and tractable Markovian structure. Finally, we show that the proposed framework allows unifying and extending several recent approaches to multiple yield curve modelling.

Keywords

Multiple yield curves HJM model Semimartingale Forward rate agreement Libor rate Affine processes Multiplicative spreads 

Mathematics Subject Classification (2010)

91G30 91B24 91B70 

JEL Classification

E43 G12 

Notes

Acknowledgements

The research of the second author was partly supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme under grant agreement PIEF-GA-2012-332345.

References

  1. 1.
    Ametrano, F.M., Bianchetti, M.: Everything you always wanted to know about multiple interest rate curve bootstrapping but were afraid to ask (2013). Preprint available at http://ssrn.com/abstract=2219548
  2. 2.
    Bianchetti, M.: Two curves, one price. Risk Magazine, 74–80 (2010) Google Scholar
  3. 3.
    Bielecki, T., Rutkowski, M.: Valuation and hedging of contracts with funding costs and collateralization. SIAM J. Financ. Math. 6, 594–655 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brace, A., Gątarek, D., Musiela, M.: The market model of interest rate dynamics. Math. Finance 7, 127–155 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brigo, D., Mercurio, F.: A deterministic-shift extension of analytically-tractable and time-homogeneous short-rate models. Finance Stoch. 5, 369–387 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bru, M.F.: Wishart processes. J. Theor. Probab. 4, 725–751 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carmona, R.A.H.: A unified approach to dynamic models for fixed income, credit and equity markets. In: Carmona, R.A., et al. (eds.) Paris–Princeton Lectures on Mathematical Finance 2004. Lecture Notes in Math., vol. 1919, pp. 1–50. Springer, Berlin (2007) CrossRefGoogle Scholar
  8. 8.
    Crépey, S., Grbac, Z., Nguyen, H., Skovmand, D.: A Lévy HJM multiple-curve model with application to CVA computation. Quant. Finance 15, 401–419 (2015) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Crépey, S., Grbac, Z., Nguyen, H.N.: A multiple-curve HJM model of interbank risk. Math. Financ. Econ. 6, 155–190 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cuchiero, C., Filipović, D., Mayerhofer, E., Teichmann, J.: Affine processes on positive semidefinite matrices. Ann. Appl. Probab. 21, 397–463 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cuchiero, C., Klein, I., Teichmann, J.: A new perspective on the fundamental theorem of asset pricing for large financial markets. Theory of Probability and its Applications. To appear. Preprint available at http://arxiv.org/abs/1412.7562
  12. 12.
    Cuchiero, C., Teichmann, J.: Path properties and regularity of affine processes on general state spaces. In: Donati-Martin, C., et al. (eds.) Séminaire de Probabilités XLV. Lecture Notes in Mathematics, vol. 2078, pp. 201–244. Springer, Berlin (2013) CrossRefGoogle Scholar
  13. 13.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dörsek, P., Teichmann, J.: Efficient simulation and calibration of general HJM models by splitting schemes. SIAM J. Financ. Math. 4, 575–598 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Douady, R., Jeanblanc, M.: A rating-based model for credit derivatives. Eur. Invest. Rev. 1, 17–29 (2002) Google Scholar
  16. 16.
    Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13, 984–1053 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Eberlein, E., Kluge, W.: Calibration of Lévy term structure models. In: Fu, M., et al. (eds.) Advances in Mathematical Finance: In Honor of D.B. Madan, pp. 147–172. Birkhäuser, Basel (2007) CrossRefGoogle Scholar
  18. 18.
    Eberlein, E., Koval, N.: A cross-currency Lévy market model. Quant. Finance 6, 465–480 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Filipović, D.: Consistency Problems for Heath–Jarrow–Morton Interest Rate Models. Lecture Notes in Mathematics, vol. 1760. Springer, Berlin (2001) CrossRefzbMATHGoogle Scholar
  20. 20.
    Filipović, D., Overbeck, L., Schmidt, T.: Dynamic CDO term structure modeling. Math. Finance 21, 53–71 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Filipović, D., Tappe, S., Teichmann, J.: Jump-diffusions in Hilbert spaces: existence, stability and numerics. Stochastics 82, 475–520 (2010) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Filipović, D., Tappe, S., Teichmann, J.: Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM J. Financ. Math. 1, 523–554 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Filipović, D., Trolle, A.B.: The term structure of interbank risk. J. Financ. Econ. 109, 707–733 (2013) CrossRefGoogle Scholar
  24. 24.
    Fries, C.P.: Curves and term structure models: definition, calibration and application of rate curves and term structure models (2010). Preprint available at http://ssrn.com/abstract=2194907
  25. 25.
    Fujii, M., Shimada, A., Takahashi, A.: A market model of interest rates with dynamic basis spreads in the presence of collateral and multiple currencies. Wilmott 54, 61–73 (2011) CrossRefGoogle Scholar
  26. 26.
    Glau, K., Grbac, Z., Kriens, D.: Martingale property of exponential semimartingales: a note on explicit conditions and applications to asset price and Libor models (2015). Preprint, TU Munich, available at http://arxiv.org/pdf/1506.08127v1.pdf
  27. 27.
    Grasselli, M., Miglietta, G.: A flexible spot multiple-curve model (2014). Preprint available at http://ssrn.com/abstract=2424242
  28. 28.
    Grbac, Z., Papapantoleon, A.: A tractable LIBOR model with default risk. Math. Financ. Econ. 7, 203–227 (2013) CrossRefzbMATHGoogle Scholar
  29. 29.
    Grbac, Z., Papapantoleon, A., Schoenmakers, J., Skovmand, D.: Affine LIBOR models with multiple curves: theory, examples and calibration. SIAM J. Financ. Math. 6, 984–1025 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica 60, 77–105 (1992) CrossRefzbMATHGoogle Scholar
  31. 31.
    Henrard, M.: The irony in the derivatives discounting. Wilmott 30, 92–98 (2007) Google Scholar
  32. 32.
    Henrard, M.: The irony in the derivatives discounting part II: the crisis. Wilmott 2, 301–316 (2010) CrossRefGoogle Scholar
  33. 33.
    Henrard, M.: Interest Rate Modelling in the Multi-Curve Framework. Palgrave Macmillan, Basingstoke (2014) CrossRefGoogle Scholar
  34. 34.
    Jacod, J.: Multivariate point processes: predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 31, 235–253 (1974/1975) Google Scholar
  35. 35.
    Jacod, J., Protter, P.: Discretization of Processes, Stochastic Modelling and Applied Probability, vol. 67. Springer, Berlin (2012) zbMATHGoogle Scholar
  36. 36.
    Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der mathematischen Wissenschaften, vol. 288. Springer, Berlin/Heidelberg/New York (2003) CrossRefzbMATHGoogle Scholar
  37. 37.
    Jarrow, R., Turnbull, S.: Pricing derivatives on financial securities subject to credit risk. J. Finance 50, 53–85 (1995) CrossRefGoogle Scholar
  38. 38.
    Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, London (2009) CrossRefzbMATHGoogle Scholar
  39. 39.
    Kallsen, J., Krühner, P.: On a Heath–Jarrow–Morton approach for stock options. Finance Stoch. 19, 583–615 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kallsen, J., Shiryaev, A.: The cumulant process and Esscher’s change of measure. Finance Stoch. 6, 397–428 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Keller-Ressel, M., Mayerhofer, E.: Exponential moments of affine processes. Ann. Appl. Probab. 25, 714–752 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kenyon, C.: Post-shock short-rate pricing. Risk Magazine, 83–87 (2010) Google Scholar
  43. 43.
    Kijima, M., Tanaka, K., Wong, T.: A multi-quality model of interest rates. Quant. Finance 9, 133–145 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Klein, I., Schmidt, T., Teichmann, J.: When roll-overs do not qualify as numéraire: bond markets beyond short rate paradigms (2013). Preprint available at http://arxiv.org/abs/1310.0032
  45. 45.
    Kluge, W.: Time-inhomogeneous Lévy processes in interest rate and credit risk models. Ph.D. thesis, University of Freiburg (2005). available at https://www.freidok.uni-freiburg.de/data/2090/
  46. 46.
    Koval, N.: Time-inhomogeneous Lévy processes in cross-currency market models. Ph.D. thesis, University of Freiburg (2005). available at https://www.freidok.uni-freiburg.de/data/2041/
  47. 47.
    Krein, M.G., Nudelman, A.A.: The Markov Moment Problem and Extremal Problems: Ideas and Problems of P.L. Čebyšev and A.A. Markov and Their Further Development. Translations of Mathematical Monographs, vol. 50. Am. Math. Soc., Providence (1977) Google Scholar
  48. 48.
    Larsson, M., Ruf, J.: Convergence of local supermartingales and Novikov–Kazamaki type conditions for processes with jumps (2014). Preprint available at http://arxiv.org/abs/1411.6229
  49. 49.
    Levin, K., Zhang, J.X.: Bloomberg volatility cube. Tech. rep., Bloomberg L.P. (2014) Google Scholar
  50. 50.
    Mercurio, F.: Modern Libor market models: using different curves for projecting rates and discounting. Int. J. Theor. Appl. Finance 13, 113–137 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Mercurio, F.: A Libor market model with a stochastic basis. Risk Magazine, 96–101 (2013) Google Scholar
  52. 52.
    Mercurio, F., Xie, Z.: The basis goes stochastic. Risk Magazine, 78–83 (2012) Google Scholar
  53. 53.
    Moreni, N., Pallavicini, A.: Parsimonious HJM modelling for multiple yield-curve dynamics. Quant. Finance 14, 199–210 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Morini, M.: Solving the puzzle in the interest rate market. In: Bianchetti, M., Morini, M. (eds.) Interest Rate Modelling After the Financial Crisis, pp. 61–106. Risk Books, London (2013) Google Scholar
  55. 55.
    Morino, L., Runggaldier, W.J.: On multicurve models for the term structure. In: Dieci, R., et al. (eds.) Nonlinear Economic Dynamics and Financial Modelling: Essays in Honour of Carl Chiarella, pp. 275–290. Springer, Berlin (2014) Google Scholar
  56. 56.
    Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling, 2nd edn. Springer, Berlin (2005) zbMATHGoogle Scholar
  57. 57.
    Pallavicini, A., Tarenghi, M.: Interest rate modelling with multiple yield curves (2010). Preprint available at http://ssrn.com/abstract=1629688
  58. 58.
    Protter, P.E.: Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21, 2nd edn. Springer, Berlin/Heidelberg/New York (2005). Version 2.1, corrected third printing CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Christa Cuchiero
    • 1
  • Claudio Fontana
    • 2
  • Alessandro Gnoatto
    • 3
  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Laboratoire de Probabilités et Modèles AléatoiresUniversité Paris DiderotParisFrance
  3. 3.Department of MathematicsLudwig-Maximilians-Universität MünchenMunichGermany

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