Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process

Andrei  Cozma; Christoph  Reisinger

doi:10.3934/dcdsb.2016101

Article Contents

2016, Volume 21, Issue 10: 3359-3377. Doi: 10.3934/dcdsb.2016101

This issue Previous Article Optimal control of a perturbed sweeping process via discrete approximations Next Article Dynamics of a networked connectivity model of epidemics

Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process

Andrei Cozma^1, and
Christoph Reisinger^1,

1.
Mathematical Institute, University of Oxford, Oxford OX2 6GG

Received: November 30, 2015

Revised: July 31, 2016

Published: November 2016

Abstract Related Papers Cited by

Abstract

We study exponential integrability properties of the Cox--Ingersoll--Ross (CIR) process and its Euler--Maruyama discretizations with various types of truncation and reflection at $0$. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of stochastic differential equations arising in finance. We prove that both implicit and explicit Euler--Maruyama discretizations for the CIR process preserve the exponential integrability of the exact solution for a wide range of parameters, and find lower bounds on the explosion time.

Keywords:

Mathematics Subject Classification: Primary: 60H35; Secondary: 65C30.

Citation:

Related Papers

Cited by

References

[1]	R. Ahlip and M. Rutkowski, Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates, Quantitative Finance, 13 (2013), 955-966.doi: 10.1080/14697688.2013.769688.
[2]	A. Alfonsi, On the discretization schemes for the CIR (and Bessel squared) processes, Monte Carlo Methods and Applications, 11 (2005), 355-384.doi: 10.1515/156939605777438569.
[3]	A. Alfonsi, Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process, Statistics and Probability Letters, 83 (2013), 602-607.doi: 10.1016/j.spl.2012.10.034.
[4]	L. Andersen and V. Piterbarg, Moment explosions in stochastic volatility models, Finance and Stochastics, 11 (2007), 29-50.doi: 10.1007/s00780-006-0011-7.
[5]	A. Berkaoui, M. Bossy and A. Diop, Euler scheme for SDEs with non-Lipschitz diffusion coefficient: Strong convergence, ESAIM: Probability and Statistics, 12 (2008), 1-11.doi: 10.1051/ps:2007030.
[6]	M. Bossy and A. Diop, An Efficient Discretization Scheme for one Dimensional SDEs with a Diffusion Coefficient Function of the Form $\|x\|^{\alpha}$, $\alpha \in [1/2,1)$, INRIA Research Report 5396, 2007.
[7]	P. Cheridito, D. Filipović and R. L. Kimmel, Market price of risk specifications for affine models: Theory and evidence, Journal of Financial Economics, 83 (2007), 123-170.
[8]	J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.doi: 10.2307/1911242.
[9]	A. Cozma, M. Mariapragassam and C. Reisinger, Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets, preprint, arXiv:1501.06084.
[10]	G. Deelstra and F. Delbaen, Convergence of discretized stochastic (interest rate) processes with stochastic drift term, Applied Stochastic Models and Data Analysis, 14 (1998), 77-84.doi: 10.1002/(SICI)1099-0747(199803)14:1<77::AID-ASM338>3.0.CO;2-2.
[11]	S. Dereich, A. Neuenkirch and L. Szpruch, An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process, Proceedings of the Royal Society of London A, 468 (2012), 1105-1115.doi: 10.1098/rspa.2011.0505.
[12]	M. Giles, Multilevel Monte Carlo path simulation, Operations Research, 56 (2008), 607-617.doi: 10.1287/opre.1070.0496.
[13]	P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004.
[14]	L. A. Grzelak and C. W. Oosterlee, On the Heston model with stochastic interest rates, SIAM Journal on Financial Mathematics, 2 (2011), 255-286.doi: 10.1137/090756119.
[15]	S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.doi: 10.1093/rfs/6.2.327.
[16]	D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002), 1041-1063.doi: 10.1137/S0036142901389530.
[17]	D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, The Journal of Computational Finance, 8 (2005), 35-62.doi: 10.21314/JCF.2005.136.
[18]	M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society of London A, 467 (2011), 1563-1576.doi: 10.1098/rspa.2010.0348.
[19]	M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations, preprint, arXiv:1309.7657.
[20]	M. Hutzenthaler, A. Jentzen and M. Noll, Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries, preprint, arXiv:1403.6385.
[21]	M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015), v+99 pp.doi: 10.1090/memo/1112.
[22]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, $2^{nd}$ edition, Springer, 1991.doi: 10.1007/978-1-4612-0949-2.
[23]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992.doi: 10.1007/978-3-662-12616-5.
[24]	S. R. Liberty and L. Mou, Existence of solutions of a Riccati differential system from a general cumulant control problem, International Journal of Differential Equations, (2011), 13 pages.
[25]	R. Lord, R. Koekkoek and D. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models, Quantitative Finance, 10 (2010), 177-194.doi: 10.1080/14697680802392496.
[26]	A. Neuenkirch and L. Szpruch, First order strong approximations of scalar SDEs defined in a domain, Numerische Mathematik, 128 (2014), 103-136.doi: 10.1007/s00211-014-0606-4.
[27]	A. Van der Stoep, L. A. Grzelak and C. W. Oosterlee, The Heston stochastic-local volatility model: Efficient Monte Carlo simulation, International Journal of Theoretical and Applied Finance, 17 (2014), 1450045, 30 pp.doi: 10.1142/S0219024914500459.