High order discretization schemes for the CIR process: Application to affine term structure and Heston models

This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method to get weak second-order schemes that extends the one introduced by Ninomiya and Victoir~\cite{NV}. Combining these both results, this allows to propose a second-order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models. Algorithms are stated in a pseudocode language.

[1]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

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

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

[4]  Dick J. C. van Dijk,et al.  A Comparison of Biased Simulation Schemes for Stochastic Volatility Models , 2008 .

[5]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[6]  D. Talay.  Discrétisation d'une équation différentielle stochastique et calcul approché d'espérances de fonctionnelles de la solution , 1986 .

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

[8]  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 .

[9]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[10]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[11]  Aurélien Alfonsi,et al.  Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model , 2005, Finance Stochastics.

[12]  Leif Andersen.  Simple and efficient simulation of the Heston stochastic volatility model , 2008 .

[13]  K. Singleton,et al.  Specification Analysis of Affine Term Structure Models , 1997 .

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

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

[16]  D. Talay,et al.  Expansion of the global error for numerical schemes solving stochastic differential equations , 1990 .

[17]  A. Diop.  Sur la discrétisation et le comportement à petit bruit d'EDS unidimensionnelles dont les coefficients sont à dérivées singulières , 2003 .

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

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

[20]  D. Brigo,et al.  Interest Rate Models , 2001 .

[21]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

[22]  S. Ninomiya,et al.  Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing , 2006, math/0605361.

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

[24]  Leif Andersen,et al.  Moment Explosions in Stochastic Volatility Models Moment Explosions in the Black–scholes and Exponential Lévy Model Moment Explosions in the Heston Model , 2022 .

[25]  Christian Kahl,et al.  Balanced Milstein Methods for Ordinary SDEs , 2006, Monte Carlo Methods Appl..

[26]  D. Talay,et al.  The law of the Euler scheme for stochastic differential equations , 1996 .