An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams--Bashforth (AB) method, while the rest is treated implicitly using the Crank--Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. F...

[1]  S. Ikonen,et al.  Efficient numerical methods for pricing American options under stochastic volatility , 2008 .

[2]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[3]  G. Meyer,et al.  The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines , 2008 .

[4]  Younhee Lee,et al.  A Second-Order Tridiagonal Method for American Options under Jump-Diffusion Models , 2011, SIAM J. Sci. Comput..

[5]  Jari Toivanen,et al.  COMPONENTWISE SPLITTING METHODS FOR PRICING AMERICAN OPTIONS UNDER STOCHASTIC VOLATILITY , 2007 .

[6]  Jari Toivanen.  A Componentwise Splitting Method for Pricing American Options Under the Bates Model , 2010 .

[7]  Xiao-Qing Jin,et al.  Quadratic finite element and preconditioning methods for options pricing in the SVCJ model , 2014 .

[8]  Muddun Bhuruth,et al.  High-order computational methods for option valuation under multifactor models , 2013, Eur. J. Oper. Res..

[9]  Gabriel Wittum,et al.  On multigrid for anisotropic equations and variational inequalities “Pricing multi-dimensional European and American options” , 2004 .

[10]  Cornelis W. Oosterlee,et al.  Numerical valuation of options with jumps in the underlying , 2005 .

[11]  M. Giles,et al.  Convergence analysis of Crank-Nicolson and Rannacher time-marching , 2006 .

[12]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[13]  Jari Toivanen,et al.  Operator splitting methods for pricing American options under stochastic volatility , 2009, Numerische Mathematik.

[14]  D. Duffie,et al.  Transform Analysis and Asset Pricing for Affine Jump-Diffusions , 1999 .

[15]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options , 1998 .

[16]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[17]  Rama Cont,et al.  A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models , 2005, SIAM J. Numer. Anal..

[18]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[19]  O. Pironneau,et al.  Computational Methods for Option Pricing (Frontiers in Applied Mathematics) (Frontiers in Applied Mathematics 30) , 2005 .

[20]  A. George.  Nested Dissection of a Regular Finite Element Mesh , 1973 .

[21]  Jonas Persson,et al.  Space-time adaptive finite difference method for European multi-asset options , 2007, Comput. Math. Appl..

[22]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[23]  Kazufumi Ito,et al.  Lagrange Multiplier Approach with Optimized Finite Difference Stencils for Pricing American Options under Stochastic Volatility , 2009, SIAM J. Sci. Comput..

[24]  Willem Hundsdorfer,et al.  Stability of implicit-explicit linear multistep methods , 1997 .

[25]  Jari Toivanen,et al.  IMEX schemes for pricing options under jump-diffusion models , 2014 .

[26]  Jari Toivanen,et al.  An Iterative Method for Pricing American Options Under Jump-Diffusion Models , 2011 .

[27]  Jari Toivanen,et al.  A Projected Algebraic Multigrid Method for Linear Complementarity Problems , 2011 .

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

[29]  Jonas Persson,et al.  A highly accurate adaptive finite difference solver for the Black–Scholes equation , 2009, Int. J. Comput. Math..

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

[31]  Vadim Linetsky,et al.  Pricing Options in Jump-Diffusion Models: An Extrapolation Approach , 2008, Oper. Res..

[32]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[33]  Luca Vincenzo Ballestra,et al.  The evaluation of American options in a stochastic volatility model with jumps: An efficient finite element approach , 2010, Comput. Math. Appl..

[34]  Jari Toivanen,et al.  Operator splitting methods for American option pricing , 2004, Appl. Math. Lett..

[35]  Nicholas G. Polson,et al.  The Impact of Jumps in Volatility and Returns , 2000 .

[36]  Younhee Lee,et al.  A Second-order Finite Difference Method for Option Pricing Under Jump-diffusion Models , 2011, SIAM J. Numer. Anal..

[37]  Jari Toivanen,et al.  Comparison and survey of finite difference methods for pricing American options under finite activity jump-diffusion models , 2012, Int. J. Comput. Math..

[38]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

[39]  Kevin Parrott,et al.  Multigrid for American option pricing with stochastic volatility , 1999 .

[40]  G. Papanicolaou,et al.  Derivatives in Financial Markets with Stochastic Volatility , 2000 .

[41]  R. Rannacher.  Finite element solution of diffusion problems with irregular data , 1984 .

[42]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[43]  Xiao-Qing Jin,et al.  Quadratic Finite Element and Preconditioning for Options Pricing in the SVCJ Model , 2011 .

[44]  E. Miglio,et al.  A finite element discretization method for option pricing with the Bates model , 2011 .

[45]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[46]  Jonas Persson,et al.  Pricing European multi-asset options using a space-time adaptive FD-method , 2007 .

[47]  Curt Randall,et al.  Pricing Financial Instruments: The Finite Difference Method , 2000 .

[48]  A. Brandt,et al.  Multigrid Algorithms for the Solution of Linear Complementarity Problems Arising from Free Boundary Problems , 1983 .

[49]  Peter A. Forsyth,et al.  Penalty methods for American options with stochastic volatility , 1998 .

[50]  Cornelis W. Oosterlee,et al.  On multigrid for linear complementarity problems with application to American-style options. , 2003 .

[51]  D. Bartuschat.  Algebraic Multigrid , 2007 .

[52]  Jonas Persson,et al.  Pricing American options using a space-time adaptive finite difference method , 2010, Math. Comput. Simul..

[53]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[54]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[55]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[56]  Lina von Sydow,et al.  Iterative Methods for Pricing American Options under the Bates Model , 2013, ICCS.