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High order discretization schemes for the CIR process: application to affine term structure and heston models. (English) Zbl 1198.60030

Summary: 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 S. Ninomiya and N. Victoir [Appl. Math. Finance 15, No. 2, 107–121 (2008; Zbl 1134.91524)]. 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.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
91B70 Stochastic models in economics

Citations:

Zbl 1134.91524
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References:

[1] Aurélien Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods Appl. 11 (2005), no. 4, 355 – 384. · Zbl 1100.65007
[2] Andersen, L. (2008). Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance, Vol. 11, No. 3.
[3] Leif B. G. Andersen and Vladimir V. Piterbarg, Moment explosions in stochastic volatility models, Finance Stoch. 11 (2007), no. 1, 29 – 50. · Zbl 1142.65004
[4] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields 104 (1996), no. 1, 43 – 60. · Zbl 0838.60051
[5] Abdel Berkaoui, Mireille Bossy, and Awa Diop, Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence, ESAIM Probab. Stat. 12 (2008), 1 – 11. · Zbl 1183.65004
[6] Bossy, M. and Diop, A. (2004). An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form \( | x|^a\), \( a\) in [1/2,1). RR-5396, INRIA, Décembre 2004.
[7] Damiano Brigo and Aurélien Alfonsi, Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model, Finance Stoch. 9 (2005), no. 1, 29 – 42. · Zbl 1065.60085
[8] Damiano Brigo and Fabio Mercurio, Interest rate models — theory and practice, 2nd ed., Springer Finance, Springer-Verlag, Berlin, 2006. With smile, inflation and credit. · Zbl 1109.91023
[9] Broadie, M. and Kaya, Ö. (2003). Exact simulation of stochastic volatility and other affine jump diffusion processes, Working Paper. · Zbl 1167.91363
[10] John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385 – 407. · Zbl 1274.91447
[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] G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Appl. Stochastic Models Data Anal. 14 (1998), no. 1, 77 – 84. , https://doi.org/10.1002/(SICI)1099-0747(199803)14:13.0.CO;2-2 · Zbl 0915.60064
[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] Paul Glasserman, Monte Carlo methods in financial engineering, Applications of Mathematics (New York), vol. 53, Springer-Verlag, New York, 2004. Stochastic Modelling and Applied Probability. · Zbl 1038.91045
[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. · Zbl 1384.35131
[16] Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. · Zbl 0734.60060
[17] Christian Kahl and Henri Schurz, Balanced Milstein methods for ordinary SDEs, Monte Carlo Methods Appl. 12 (2006), no. 2, 143 – 170. · Zbl 1105.65009
[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. · Zbl 1198.91240
[19] Syoiti Ninomiya and Nicolas Victoir, Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance 15 (2008), no. 1-2, 107 – 121. · Zbl 1134.91524
[20] Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506 – 517. · Zbl 0184.38503
[21] Denis Talay, Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 1, 141 – 179 (French, with English summary). · Zbl 0662.65129
[22] Denis Talay and Luciano Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990), no. 4, 483 – 509 (1991). · Zbl 0718.60058
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