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High order discretization schemes for the CIR process: Application to affine term structure and Heston models


Author: Aurélien Alfonsi
Journal: Math. Comp. 79 (2010), 209-237
MSC (2000): Primary 60H35, 65C30; Secondary 91B70
DOI: https://doi.org/10.1090/S0025-5718-09-02252-2
Published electronically: June 15, 2009
MathSciNet review: 2552224
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Abstract: This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya and Victoir. Combine both these results, this allows us to propose a second order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models.


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  • 1. Alfonsi, A. (2005). On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods and Applications, Vol. 11, No. 4, pp. 355-384. MR 2186814 (2006h:60130)
  • 2. Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance, Vol. 11, No. 3.
  • 3. Andersen, L. and Piterbarg, V. (2007). Moment explosions in stochastic volatility models. Finance and Stochastics, Vol. 11, No. 1, pp. 29-50. MR 2284011 (2008a:65016)
  • 4. Bally, V. and Talay, D. (1996). The law of the Euler scheme for stochastic differential equations I: Convergence rate of the distribution function, Probab. Theory Related Fields, Vol. 104, pp. 43-60. MR 1367666 (96k:60136)
  • 5. Berkaoui, A., Bossy, M. and Diop, A. (2008). Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence. ESAIM Probab. Stat., Vol. 12, pp. 1-11. MR 2367990 (2008i:60115)
  • 6. Bossy, M. and Diop, A. (2004). An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form $ \vert x\vert^a$, $ a$ in [1/2,1). RR-5396, INRIA, Décembre 2004.
  • 7. Brigo, D. and Alfonsi, A. (2005). Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model, Finance Stoch., Vol. 9, No. 1, pp. 29-42. MR 2210926
  • 8. Brigo, D., and Mercurio, F. (2006), Interest Rate Models - Theory and Practice, 2nd edition, with Smile, Inflation and Credit, Springer-Verlag. MR 2255741 (2007d:91002)
  • 9. Broadie, M. and Kaya, Ö. (2003). Exact simulation of stochastic volatility and other affine jump diffusion processes, Working Paper.
  • 10. Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica 53, pp. 385-407. MR 785475
  • 11. Dai, Q. and Singleton, K. (2000). Specification Analysis of Affine Term Structure Models, The Journal of Finance, Vol. LV, No. 5, pp. 1943-1978.
  • 12. Deelstra, G. and Delbaen, F (1998). Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term, Appl. Stochastic Models Data Anal. 14, pp. 77-84. MR 1641781 (99g:60097)
  • 13. Diop, A. (2003). Sur la discrétisation et le comportement à petit bruit d'EDS multidimensionnelles dont les coefficients sont à dérivées singulières, Ph.D. Thesis, INRIA. (available at http://www.inria.fr/rrrt/tu-0785.html)
  • 14. Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering, Springer, Series: Applications of Mathematics , Vol. 53. MR 1999614 (2004g:65005)
  • 15. Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, Vol. 6, No. 2, pp. 327-343.
  • 16. Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edition. Springer, Series : Graduate Texts in Mathematics, Vol. 113. MR 1121940 (92h:60127)
  • 17. Kahl, C. and Schurz, H. (2006). Balanced Milstein Methods for SDE's. Monte Carlo Methods and Applications, Vol 12, No. 2, 2006, pp. 143-170. MR 2237671 (2007b:65015)
  • 18. Lord, R., Koekkoek, R. and van Dijk, D. (2006). A comparison of biased simulation schemes for stochastic volatility models, working paper, Erasmus University Rotterdam, Rabobank International and Robeco Alternative Investments.
  • 19. Ninomiya, S. and Victoir, N. (2008). Weak approximation of stochastic differential equations and application to derivative pricing, Applied Mathematical Finance, Vol. 15, No. 2, pp. 107-121. MR 2409419
  • 20. Strang, G. (1968). On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis, Vol. 5, pp. 506-517. MR 0235754 (38:4057)
  • 21. Talay, D. (1986). Discrétisation d'une équation différentielle stochastique et calcul approché d'espérances de fonctionnelles de la solution. Modélisation Math. Anal. Numér., Vol. 20, No .1, pp. 141-179. MR 844521 (87k:60153)
  • 22. Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Analysis and Applications, Vol. 8 No. 4, pp. 94-120. MR 1091544 (92e:60124)

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Additional Information

Aurélien Alfonsi
Affiliation: CERMICS, MATHFI Project, Ecole des Ponts, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-vallée, France
Email: alfonsi@cermics.enpc.fr

DOI: https://doi.org/10.1090/S0025-5718-09-02252-2
Keywords: Simulation, discretization scheme, squared Bessel process, Cox-Ingersoll-Ross model, Heston model, Affine Term Structure Models (ATSM)
Received by editor(s): October 24, 2008
Received by editor(s) in revised form: December 16, 2008
Published electronically: June 15, 2009
Additional Notes: Most of this work was done when I was at the TU Berlin, thanks to the support of MATHEON. I would like to thank Vlad Bally (Univ. Marne-la-Vallée) and Benjamin Jourdain (Ecole des Ponts) for fruitful comments, and Victor Reutenauer (CALyon) for stimulating discussions on ATSM
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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