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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Giles, Multilevel Monte Carlo path simulation,, Operations Research, 56 (2008), 607.  doi: 10.1287/opre.1070.0496.  Google Scholar

[13]

P. Glasserman, Monte Carlo Methods in Financial Engineering,, Springer, (2004).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations,, preprint, ().   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[22]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $2^{nd}$ edition, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Giles, Multilevel Monte Carlo path simulation,, Operations Research, 56 (2008), 607.  doi: 10.1287/opre.1070.0496.  Google Scholar

[13]

P. Glasserman, Monte Carlo Methods in Financial Engineering,, Springer, (2004).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Hutzenthaler, A. Jentzen and X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations,, preprint, ().   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[22]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, $2^{nd}$ edition, (1991).  doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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