Applied Mathematics & Optimization

, Volume 77, Issue 3, pp 405–441 | Cite as

Long-Term Yield in an Affine HJM Framework on \(S_{d}^{+}\)

  • Francesca Biagini
  • Alessandro Gnoatto
  • Maximilian Härtel
Article
  • 45 Downloads

Abstract

We develop the HJM framework for forward rates driven by affine processes on the state space of symmetric positive semidefinite matrices. In this setting we find an explicit representation for the long-term yield in terms of the model parameters. This generalises the results of El Karoui et al. (Rev Deriv Res 1(4):351–369, 1997) and Biagini and Härtel (Int J Theor Appl Financ 17(3):1–24, 2012), where the long-term yield is investigated under no-arbitrage assumptions in a HJM setting using Brownian motions and Lévy processes respectively.

Keywords

HJM Affine process Long-term yield Yield curve Wishart process 

Notes

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. [228087].

References

  1. 1.
    Ahdida, A., Alfonsi, A.: Exact and high order discretization schemes for Wishart processes and their affine extensions. Ann. Appl. Probab. 23(3), 1025–1073 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ang, A., Piazzesi, M.: A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables. J. Monet. Econ. 50(4), 745–787 (2003)CrossRefGoogle Scholar
  3. 3.
    Applebaum, D.: Lévy Processes and Stochastic Calculus, 1st edn. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Bauer, H.: Wahrscheinlichkeitstheorie, 5th edn. Walter de Gruyter, Berlin (2002)MATHGoogle Scholar
  5. 5.
    Benabid, A., Bensusan, H., El Karoui, N.: Wishart stochastic volatility: asymptotic smile and numerical framework. HAL Working Paper (2010). HAL: http://hal.archives-ouvertes.fr/docs/00/45/83/71/PDF/ArticleWishart19Feb2010
  6. 6.
    Biagini, F., Härtel, M.: Behavior of long-term yields in a Lévy term structure. Int. J. Theor. Appl. Financ. 17(3), 1–24 (2014)CrossRefMATHGoogle Scholar
  7. 7.
    Björk, T., Kabanov, Y., Runggaldier, W.: Bond market structure in the presence of marked point processes. Math. Financ. 7(2), 211–223 (1997)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brigo, D., Mercurio, F.: Interest Rate Models: Theory and Practice, 2nd edn. Springer Finance, Heidelberg (2006)MATHGoogle Scholar
  9. 9.
    Bru, M.F.: Wishart processes. J. Theor. Probab. 4(4), 725–751 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Carmona, R., Tehranchi, M.: Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective, 1st edn. Springer Finance, Heidelberg (2006)MATHGoogle Scholar
  11. 11.
    Chiarella, C., Kwon, O.K.: Finite dimensional affine realisations of HJM models in terms of forward rates and yields. Rev. Deriv. Res. 6, 129–155 (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Cuchiero, C.: Affine and polynomial processes. PhD Thesis, ETH Zurich (2011)Google Scholar
  13. 13.
    Cuchiero, C., Filipović, D., Mayerhofer, E., Teichmann, J.: Affine processes on positive semidefinite matrices. Ann. Appl. Probab. 21(2), 397–463 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cuchiero, C., Keller-Ressel, M., Mayerhofer, E., Teichmann, J.: Affine processes on symmetric cones. J. Theor. Probab. (2014). doi: 10.1007/s10959-014-0580-x MATHGoogle Scholar
  15. 15.
    Da Fonseca, J., Grasselli, M.: Riding on the smiles. Quan. Financ. 11(11), 1609–1632 (2011)CrossRefGoogle Scholar
  16. 16.
    Da Fonseca, J., Grasselli, M., Tebaldi, C.: A multifactor volatility Heston model. Quant. Financ. 8(6), 591–604 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Da Fonseca, J., Gnoatto, A., Grasselli, M.: A flexible matrix Libor model with smiles. J. Econ. Dyn. Control 37, 774–793 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Da Fonseca, J., Grasselli, M., Ielpo, F.: Estimating the Wishart affine stochastic correlation model using the empirical characteristic function. Stud. Nonlinear Dyn. Econom. 18(3), 253–289 (2013)MathSciNetMATHGoogle Scholar
  19. 19.
    Diebold, F., Rudebusch, G.D., Aruoba, S.B.: The macroeconomy and the yield curve: a dynamic latent factor approach. J. Econom. 131(1–2), 309–338 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Duffie, D., Kan, R.: A yield-factor model of interest rates. Math. Financ. 6(4), 379–406 (1996)CrossRefMATHGoogle Scholar
  21. 21.
    Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Dybvig, P.H., Ingersoll, J.E., Ross, S.A.: Long forward and zero-coupon rates can never fall. J. Bus. 69(1), 1–25 (1996)CrossRefGoogle Scholar
  23. 23.
    El Karoui, N., Frachot, A., Geman, H.: A note on the behavior of long zero coupon rates in a no arbitrage framework. Rev. Deriv. Res. 1(4), 351–369 (1997)Google Scholar
  24. 24.
    European Central Bank: Long-term interest rates for EU member states (2013). ECB: http://www.ecb.int/stats/money/long/html/index.en.html. Accessed 21 Aug 2015
  25. 25.
    Filipović, D.: Term Structure Models–A Graduate Course, 1st edn. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  26. 26.
    Georgii, H.-O.: Stochastik, 1st edn. Walter de Gruyter, Berlin (2004)MATHGoogle Scholar
  27. 27.
    Gnoatto, A.: The Wishart short-rate model. Int. J. Theor. Appl. Financ. 15(8), 1250056 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Gnoatto, A., Grasselli, M.: The explicit Laplace transform for the Wishart process. J. Appl. Probab. 51(3), 640–656 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Goldammer, V., Schmock, U.: Generalization of the Dybvig-Ingersoll-Ross theorem and asymptotic minimality. Math. Financ. 22(1), 185–213 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Gourieroux, C.: Continuous time Wishart process for stochastic risk. Econom. Rev. 25, 177–217 (2006)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Gourieroux, C., Sufana, R.: Derivative pricing with Wishart multivariate stochastic volatility. J. Bus. Econ. Stat. 28(3), 438–451 (2010)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Grasselli, M., Tebaldi, C.: Solvable affine term structure models. Math. Financ. 18(1), 135–153 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Gürkaynak, R., Sack, B., Swanson, E.: The sensitivity of long-term interest rates to economic news: evidence and implications for macroeconomic news. Am. Econ. Rev. 95(1), 425–436 (2005)CrossRefGoogle Scholar
  34. 34.
    Hansen, L.P., Scheinkman, J.A.: Pricing growth-rate risk. Financ. Stoch. 16(1), 1–15 (2012)CrossRefGoogle Scholar
  35. 35.
    Härtel, M.: The asymptotic behavior of the term structure of interest rates. PhD Thesis, LMU Munich (2016)Google Scholar
  36. 36.
    Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new methodology. Econometrica 60(1), 77–105 (1992)CrossRefMATHGoogle Scholar
  37. 37.
    Hördahl, P., Tristani, O., Vestin, D.: A joint econometric model of macroeconomic and term-structure dynamics. J. Econom. 131(1–2), 405–444 (2006)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Hubalek, F., Klein, I., Teichmann, J.: A general proof of the Dybvig-Ingersoll-Ross-theorem. Math. Financ. 12(4), 447–451 (2002)CrossRefMATHGoogle Scholar
  39. 39.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 1st edn. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  40. 40.
    Kardaras, C., Platen, E.: On the Dybvig-Ingersoll-Ross theorem. Math. Financ. 22(4), 729–740 (2012)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kim, D.H., Wright, J.H.: An arbitrage-free three factor term structure model and the recent behavior of long-term yields and distant-horizon forward rates. Fed. Reserv. Board Financ. Econ. Discuss. Ser. 33, 2005 (2005)Google Scholar
  42. 42.
    Klenke, A.: Wahrscheinlichkeitstheorie, 1st edn. Springer, Berlin (2006)MATHGoogle Scholar
  43. 43.
    Mankiw, N.G., Summers, L.H., Weiss, L.: Do long-term interest rates overreact to short-term interest rates? Brook. Pap. Econ. Activity 1984(1), 223–247 (1984)CrossRefGoogle Scholar
  44. 44.
    Mayerhofer, E.: Affine processes on positive semidefinite \(d \times d\) matrices have jumps of finite variation in dimension \(d > 1\). Stoch. Process. Appl. 122(10), 3445–3459 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Mayerhofer, E., Pfaffel, O., Stelzer, R.: On strong solutions for positive definite jump-diffusions. Stoch. Process. Appl. 121(9), 2072–2086 (2011)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    McCulloch, J.H.: Long forward and zero-coupon rates indeed can never fall. Working Paper # 00(12), Ohio State University (2000)Google Scholar
  47. 47.
    Muhle-Karbe, J., Pfaffel, O., Stelzer, R.: Option pricing in multivariate stochastic volatility models of OU type. SIAM J. Financ. Math. 3(1), 66–94 (2012)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Pfaffel, O.: Wishart Processes. Student Research project supervised by Claudia Klüppelberg and Robert Stelzer, TU Munich (2008)Google Scholar
  49. 49.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005)CrossRefGoogle Scholar
  50. 50.
    Richter, A.: Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models. Stoch. Process. Appl. 124(11), 3578–3611 (2014)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Schulze, K.: Asymptotic maturity behavior of the term structure. Bonn Econ Discussion Papers, University of Bonn (2008)Google Scholar
  52. 52.
    Shiller, R.J.: The volatility of long-term interest rates and expectations models of the term structure. J. Polit. Econ. 87(6), 1190–1219 (1979)CrossRefGoogle Scholar
  53. 53.
    Yao, Y.: Term structure modeling and asymptotic long rate. Insurance 25, 327–336 (1999)MATHGoogle Scholar
  54. 54.
    Yao, Y.: Term structure models: a perspective from the long rate. N. Am. Actuar. J. 3(3), 122–138 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Francesca Biagini
    • 1
    • 2
  • Alessandro Gnoatto
    • 1
  • Maximilian Härtel
    • 3
  1. 1.Department of MathematicsLMU UniversityMunichGermany
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.MEAG Munich ERGO AssetManagement GmbHMunichGermany

Personalised recommendations