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Monte Carlo Euler approximations of HJM term structure financial models

Abstract

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

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References

  1. Barth, A.: A finite element method for martingale-driven stochastic partial differential equations. Commun. Stochast. Anal. 4, 355–375 (2010)

    MathSciNet  Google Scholar 

  2. Baxter, M., Rennie, A.: Financial Calculus: An Introduction to Derivate Pricing. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  3. Björck, Å., Dahlquist, G.: Numerical Methods. Prentice-Hall, Englewood Cliffs (1974)

    Google Scholar 

  4. Björk, T.: Arbitrage Theory in Continuous Time, 2nd edn. Oxford University Press Inc., Oxford (2004)

    MATH  Book  Google Scholar 

  5. Boyle, P., Broadie, M., Glasserman, P.: Option Pricing, Interest Rates and Risk Management, pp. 185–238. Cambridge Univ. Press, Cambridge (2001)

    Google Scholar 

  6. Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1–49 (1998). Cambridge Univ. Press, Cambridge

    MathSciNet  Article  Google Scholar 

  7. Carverhill, A.: A note on the models of hull and white for pricing options on the term structure. J. Fixed Income 5, 89–96 (1995)

    Article  Google Scholar 

  8. Carverhill, A., Pang, K.: Efficient and flexible bond option valuation in the Heath, Jarrow and Morton framework. J. Fixed Income 5, 70–77 (1995)

    Article  Google Scholar 

  9. Dekker, K., Verwer, J.G.: Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  10. Duffie, D.: Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton (1996)

    Google Scholar 

  11. Durett, R.: Probability: Theory and Examples. Duxbury Press, N. Scituate (1994)

    Google Scholar 

  12. Dörsek, P., Teichmann, J.: Efficient simulation and calibration of general HJM models by splitting schemes arXiv:1112.5330 (2011)

  13. Fishman, G.S.: Monte Carlo Concepts, Algorithms and Applications. Springer Series in Operations Research. Springer, New York (1996)

    MATH  Book  Google Scholar 

  14. Hammersley, J.M., Morton, K.W.: A new Monte Carlo technique: antithetic variates. Proc. Camb. Philos. Soc. 52, 449–475 (1956)

    MathSciNet  MATH  Article  Google Scholar 

  15. Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a discrete time approximation. J. Financ. Quant. Anal. 25, 419–440 (1990)

    Article  Google Scholar 

  16. 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)

    MATH  Article  Google Scholar 

  17. Hull, J.: Options, Futures and Other Derivatives. Prentice Hall, New York (1993)

    Google Scholar 

  18. Krivko, M., Tretyakov, M.V.: Numerical integration of Heath-Jarrow-Morton model of interest rates arXiv:1109.2557 (2011)

  19. Metropolis, N., Ulam, S.: The Monte Carlo method. J. Am. Stat. Assoc. 44, 335–341 (1949)

    MathSciNet  MATH  Article  Google Scholar 

  20. Moon, K.-S., Szepessy, A., Tempone, R., Zouraris, G.E.: Convergence rates for adaptive weak approximation of stochastic differential equations. Stoch. Anal. Appl. 23, 511–558 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  21. Press, W.H.: Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  22. Rebonato, R.: Interest-Rate Option Models. Wiley, New York (1996)

    Google Scholar 

  23. Shreve, S.E.: Stochastic Calculus and Finance. Lecture Notes. Carnegie-Mellon University, Pittsburg (1996)

    Google Scholar 

  24. Shreve, S.E.: Stochastic Calculus for Finance II: Continuous-Time Models. Springer, New York (2010)

    Google Scholar 

  25. Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of stochastic differential equations. Commun. Pure Appl. Math. 54, 1169–1214 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  26. Tempone, R.: Numerical complexity analysis of weak approximation of stochastic differential equations. Ph.D. Dissertation, KTH, Stockholm, Sweden (2002)

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Acknowledgements

This work has been partially supported by: The Swedish National Network in Applied Mathematics (NTM) ‘Numerical approximation of stochastic differential equations’ (NADA, KTH), The EU-TMR project HCL # ERBFMRXCT960033, UdelaR and UdeM in Uruguay, The Swedish Research Council for Engineering Science (TFR) Grant#222-148, The VR project ‘Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar’ (NADA, KTH), the European Union’s Seventh Framework Programme (FP7-REGPOT-2009-1) under grant agreement no. 245749 ‘Archimedes Center for Modeling, Analysis and Computation’ (University of Crete, Greece), The University of Crete (Sabbatical Leave of the fourth author), and The King Abdullah University of Science and Technology (KAUST). The third author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

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Correspondence to R. Tempone.

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Communicated by Desmond Higham.

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Björk, T., Szepessy, A., Tempone, R. et al. Monte Carlo Euler approximations of HJM term structure financial models. Bit Numer Math 53, 341–383 (2013). https://doi.org/10.1007/s10543-012-0410-4

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  • DOI: https://doi.org/10.1007/s10543-012-0410-4

Keywords

  • HJM model
  • Option price
  • Bond market
  • Stochastic differential equations
  • Monte Carlo methods
  • A priori error estimates
  • A posteriori error estimates

Mathematics Subject Classification

  • 65C05
  • 65C30
  • 65C20
  • 91B28
  • 91B70