Discretizing the Heston Model: An Analysis of the Weak Convergence Rate

In this manuscript we analyze the weak convergence rate of a discretization scheme for the Heston model. Under mild assumptions on the smoothness of the payoff and on the Feller index of the volatility process, respectively, we establish a weak convergence rate of order one. Moreover, under almost minimal assumptions we obtain weak convergence without a rate. These results are accompanied by several numerical examples. Our error analysis relies on a classical technique from Talay & Tubaro, a recent regularity estimate for the Heston PDE by Feehan & Pop and Malliavin calculus.

[1]  Michael Günther,et al.  Structure preserving stochastic integration schemes in interest rate derivative modeling , 2008 .

[2]  F. Utzet,et al.  On the density of log-spot in the Heston volatility model , 2010 .

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

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

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

[6]  A. Kebaier,et al.  Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing , 2005, math/0602529.

[7]  Christian Kahl,et al.  Fast strong approximation Monte Carlo schemes for stochastic volatility models , 2006 .

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

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

[10]  M. Hefter,et al.  Strong Convergence Rates for Cox-Ingersoll-Ross Processes - Full Parameter Range , 2016, 1608.00410.

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

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

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

[14]  Christoph Reisinger,et al.  Exponential integrability properties of Euler discretization schemes for the Cox-Ingersoll-Ross process , 2015, 1601.00919.

[15]  Andreas Neuenkirch,et al.  Multilevel Monte Carlo Quadrature of Discontinuous Payoffs in the Generalized Heston Model Using Malliavin Integration by Parts , 2015, SIAM J. Financial Math..

[16]  Aurélien Alfonsi,et al.  High order discretization schemes for the CIR process: Application to affine term structure and Heston models , 2010, Math. Comput..

[17]  Martin Altmayer.  Quadrature of discontinuous SDE functionals using Malliavin integration by parts , 2015 .

[18]  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.

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

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

[21]  P. Daskalopoulos,et al.  Regularity of the free boundary for the porous medium equation , 1998 .

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

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

[24]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

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

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

[27]  P. Feehan,et al.  A SCHAUDER APPROACH TO DEGENERATE-PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED COEFFICIENTS , 2011, 1112.4824.

[28]  Paul Glasserman,et al.  Gamma expansion of the Heston stochastic volatility model , 2008, Finance Stochastics.