Efficient and stable numerical solution of the Heston–Cox–Ingersoll–Ross partial differential equation by alternating direction implicit finite difference schemes

This paper concerns the numerical solution of the three-dimensional Heston–Cox–Ingersoll–Ross partial differential equation for the fair values of European-style financial options. We follow the method-of-lines approach and first semidiscretize on suitable nonuniform Cartesian spatial grids by applying finite difference schemes. The main aim of this paper is to investigate various prominent alternating direction implicit schemes for the effective time discretization of the obtained, very large, semidiscrete systems. For this purpose an extensive numerical study is performed, among others, for arbitrary correlation factors, for time-dependent mean-reversion levels, for short and long maturities, for cases where the Feller condition is satisfied and for cases where it is not. Also, the approximation of the hedging quantities Delta and Gamma is discussed and double barrier knock-out call options are considered.

[1]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[2]  ADI FD schemes for the numerical solution of the three-dimensional Heston–Cox–Ingersoll–Ross PDE , 2012 .

[3]  K. I. '. Hout,et al.  ADI finite difference schemes for option pricing in the Heston model with correlation , 2008, 0811.3427.

[4]  K. J. in 't Hout,et al.  Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms , 2012 .

[5]  K. J. in’t Hout,et al.  Stability of the modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term , 2010, Math. Comput. Simul..

[6]  Leif Andersen,et al.  Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing , 2000 .

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

[8]  J. Brandts.  [Review of: W. Hundsdorfer, J.G. Verwer (2003) Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations] , 2006 .

[9]  A. Lipton.  Mathematical methods for foreign exchange , 2001 .

[10]  Bruno Welfert,et al.  Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms , 2009 .

[11]  A Stability Result for the Modified Craig-Sneyd Scheme Applied to 2D and 3D Pure Diffusion Equations , 2010 .

[12]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[13]  Uwe Wystup,et al.  FX Options and Structured Products , 2007 .

[14]  Noelle Foreshaw.  Options… , 2010 .

[15]  Kenneth R. Jackson,et al.  A PDE pricing framework for cross-currency interest rate derivatives , 2010, ICCS.

[16]  S. Shreve.  Stochastic calculus for finance , 2004 .

[17]  Tinne Haentjens,et al.  Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation , 2012 .

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

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

[20]  Bruno Welfert,et al.  Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms , 2007 .

[21]  Andrey Itkin,et al.  Jumps without tears: A new splitting technology for barrier options , 2011 .

[22]  Erik Ekström,et al.  The Black–Scholes equation in stochastic volatility models , 2010 .

[23]  I-Liang Chern,et al.  Interest Rate Modeling , 2019 .

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

[25]  Johan Tysk,et al.  Boundary conditions for the single-factor term structure equation , 2011, 1101.1149.