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

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

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  • [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.

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