Pricing American options under multi-state regime switching with an efficient L- stable method

An efficient second-order method based on exponential time differencing approach for solving American options under multi-state regime switching is developed and analysed for stability and convergence. The method is seen to be strongly stable (L-stable) in each regime. The implicit predictor–corrector nature of the method makes it highly efficient in solving nonlinear systems of partial differential equations arising from multi-state regime switching model. Stability and convergence of the method are examined. The impact of regime switching on option prices for different jump rates and volatility is illustrated. A general framework for multi-state regime switching in multi-asset American option has been provided. Numerical experiments are performed on one and two assets to demonstrate the performance of the method with convex as well as non-convex payoffs. The method is compared with some of the existing methods available in the literature and is found to be reliable, accurate and efficient.

[1]  Abdul Q. M. Khaliq,et al.  New Numerical Scheme for Pricing American Option with Regime-Switching , 2009 .

[2]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[3]  Kok Lay Teo,et al.  A Robust Numerical Scheme For Pricing American Options Under Regime Switching Based On Penalty Method , 2014 .

[4]  Jingtang Ma,et al.  Convergence rates of trinomial tree methods for option pricing under regime-switching models , 2015, Appl. Math. Lett..

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

[6]  Hao Zhou,et al.  Term Structure of Interest Rates with Regime Shifts , 2001 .

[7]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[8]  Hailiang Yang,et al.  Option pricing with regime switching by trinomial tree method , 2010, J. Comput. Appl. Math..

[9]  J. M. Keiser,et al.  A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , 1998 .

[10]  Phelim P. Boyle,et al.  Pricing exotic options under regime switching , 2007 .

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

[12]  R. Liu,et al.  Regime-Switching Recombining Tree For Option Pricing , 2010 .

[13]  Emil M. Constantinescu,et al.  Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA) , 2013, SIAM J. Sci. Comput..

[14]  Sam D. Howison,et al.  The Effect of Nonsmooth Payoffs on the Penalty Approximation of American Options , 2010, SIAM J. Financial Math..

[15]  Y. Huang,et al.  Methods for Pricing American Options under Regime Switching , 2011, SIAM J. Sci. Comput..

[16]  Andrew L. Rukhin,et al.  Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach , 2001, Technometrics.

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

[18]  Robert J. Elliott,et al.  Option pricing and Esscher transform under regime switching , 2005 .

[19]  Bruce A. Wade,et al.  An ETD Crank‐Nicolson method for reaction‐diffusion systems , 2012 .

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

[21]  Mary R. Hardy,et al.  A Regime-Switching Model of Long-Term Stock Returns , 2001 .

[22]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[23]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[24]  R. H. Liu,et al.  A lattice method for option pricing with two underlying assets in the regime-switching model , 2013, J. Comput. Appl. Math..

[25]  Robert J. Elliott,et al.  American options with regime switching , 2002 .

[26]  Qiang Du,et al.  Analysis and Applications of the Exponential Time Differencing Schemes and Their Contour Integration Modifications , 2005 .

[27]  Shuhua Zhang,et al.  A front-fixing finite element method for the valuation of American options with regime switching , 2012, Int. J. Comput. Math..

[28]  A. Tveito,et al.  Penalty and front-fixing methods for the numerical solution of American option problems , 2002 .

[29]  Qing Zhang,et al.  Continuous-Time Markov Chains and Applications , 1998 .

[30]  Abdul-Qayyum M. Khaliq,et al.  The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost , 2012, Int. J. Comput. Math..

[31]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[32]  Svetlana Boyarchenko,et al.  American Options in Regime-Switching Models , 2006, SIAM J. Control. Optim..

[33]  Peter A. Forsyth,et al.  Implications of a regime-switching model on natural gas storage valuation and optimal operation , 2010 .

[34]  B. Kleefeld,et al.  Solving complex PDE systems for pricing American options with regime‐switching by efficient exponential time differencing schemes , 2013 .

[35]  SANTTU SALMI,et al.  An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps , 2014, SIAM J. Sci. Comput..

[36]  Hongtao Yang,et al.  A Numerical Analysis of American Options with Regime Switching , 2010, J. Sci. Comput..