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

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December 2016, 21(10): 3359-3377. doi: 10.3934/dcdsb.2016101

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, United Kingdom, United Kingdom

Received December 2015 Revised August 2016 Published November 2016

Abstract
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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.

Keywords: exponential integrability, stochastic volatility model., Cox--Ingersoll--Ross process, explicit Euler scheme, implicit Euler scheme.

Mathematics Subject Classification: Primary: 60H35; Secondary: 65C3.

Citation: Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101

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. 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. 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. 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. 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. 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, (5396).
[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.
[8]	J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates,, Econometrica, 53 (1985), 385. 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, ().
[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. 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. doi: 10.1098/rspa.2011.0505.
[12]	M. Giles, Multilevel Monte Carlo path simulation,, Operations Research, 56 (2008), 607. 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. 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. 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. 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. 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. 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, ().
[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, ().
[21]	M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients,, Mem. Amer. Math. Soc., 236 (2015). doi: 10.1090/memo/1112.
[22]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $2^{nd}$ edition, (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), (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).
[25]	R. Lord, R. Koekkoek and D. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models,, Quantitative Finance, 10 (2010), 177. 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. 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). doi: 10.1142/S0219024914500459.

show all references

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. 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. 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. 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. 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. 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, (5396).
[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.
[8]	J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates,, Econometrica, 53 (1985), 385. 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, ().
[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. 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. doi: 10.1098/rspa.2011.0505.
[12]	M. Giles, Multilevel Monte Carlo path simulation,, Operations Research, 56 (2008), 607. 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. 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. 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. 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. 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. 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, ().
[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, ().
[21]	M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients,, Mem. Amer. Math. Soc., 236 (2015). doi: 10.1090/memo/1112.
[22]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $2^{nd}$ edition, (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), (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).
[25]	R. Lord, R. Koekkoek and D. Van Dijk, A comparison of biased simulation schemes for stochastic volatility models,, Quantitative Finance, 10 (2010), 177. 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. 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). doi: 10.1142/S0219024914500459.

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