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

We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler 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.

[1]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[2]  C. Reisinger,et al.  Convergence of an Euler discretisation scheme for the Heston stochastic-local volatility model with CIR interest rates , 2015 .

[3]  M. Hutzenthaler,et al.  Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries , 2014, 1403.6385.

[4]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[5]  D. Higham,et al.  Convergence of Monte Carlo Simulations involving the Mean-Reverting Square Root Process , 2005 .

[6]  Mireille Bossy,et al.  Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence , 2006 .

[7]  Patrick Cheridito,et al.  Market price of risk speci-fications for a ne models: theory and evidence , 2004 .

[8]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[9]  M. Hutzenthaler,et al.  Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients , 2012, 1203.5809.

[10]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[11]  L. Szpruch,et al.  An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  D. Dijk,et al.  A comparison of biased simulation schemes for stochastic volatility models , 2008 .

[13]  F. Delbaen,et al.  Convergence of discretized stochastic (interest rate) processes with stochastic drift term , 1998 .

[14]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[15]  Cornelis W. Oosterlee,et al.  The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation , 2013 .

[16]  Mireille Bossy,et al.  An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient function of the form |x|^a, a in [1/2,1) , 2006 .

[17]  P. Kloeden,et al.  Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  Cornelis W. Oosterlee,et al.  On the Heston Model with Stochastic Interest Rates , 2010, SIAM J. Financial Math..

[19]  Rehez Ahlip,et al.  Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates , 2013 .

[20]  Andrew M. Stuart,et al.  Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations , 2002, SIAM J. Numer. Anal..

[21]  Andreas Neuenkirch,et al.  First order strong approximations of scalar SDEs defined in a domain , 2014, Numerische Mathematik.

[22]  Aurélien Alfonsi,et al.  Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process , 2013 .

[23]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[24]  Aurélien Alfonsi,et al.  On the discretization schemes for the CIR (and Bessel squared) processes , 2005, Monte Carlo Methods Appl..

[25]  Vladimir V. Piterbarg,et al.  Moment explosions in stochastic volatility models , 2005, Finance and Stochastics.

[26]  S. Liberty,et al.  Existence of Solutions of a Riccati Differential System from a General Cumulant Control Problem , 2011 .