Stabilized explicit Runge-Kutta methods for multi-asset American options

American derivatives have become very popular instruments in financial markets. However, they are more complicated to price than European options since at each time level we have to determine not only the option value but also whether or not it should be exercised. Several procedures have been proposed to dissolve these difficulties, but they usually involve the solution of nonlinear partial differential equations (PDEs). In the case of multi-dimensional problems, solving these equations is a very challenging task.In this paper we propose Stabilized Explicit Runge-Kutta (SERK) methods to solve this kind of problems. They can easily be applied to many different classes of problems with large dimensions and they have low memory demand. Since these methods are explicit, they do not require algebra routines to solve large nonlinear systems associated to ODEs (as, for example, LAPACK and BLAS packages or multigrid or iterative methods applied together with Newton-type algorithms) and are especially well-suited for the method of lines (MOL) discretizations of parabolic PDEs.

[1]  Desmond J. Higham,et al.  Analysis of stepsize selection schemes for Runge-Kutta codes , 1988 .

[2]  K. Burrage,et al.  Adaptive stepsize based on control theory for stochastic differential equations , 2004 .

[3]  Peter A. Forsyth,et al.  A finite volume approach for contingent claims valuation , 2001 .

[4]  Conall O'Sullivan,et al.  On the acceleration of explicit finite difference methods for option pricing , 2011 .

[5]  J. Martín-Vaquero,et al.  SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems , 2013 .

[6]  Lawrence S. Kroll.  Mathematica--A System for Doing Mathematics by Computer. , 1989 .

[7]  P. Houwen,et al.  On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values , 1979 .

[8]  J. Martín-Vaquero,et al.  Second-order stabilized explicit Runge-Kutta methods for stiff problems , 2009, Comput. Phys. Commun..

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

[10]  D. A. Voss,et al.  A linearly implicit predictor–corrector scheme for pricing American options using a penalty method approach , 2006 .

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

[12]  John Schoenmakers,et al.  Policy iteration for american options: overview , 2006, Monte Carlo Methods Appl..

[13]  P. Boyle,et al.  Numerical Evaluation of Multivariate Contingent Claims , 1989 .

[14]  M. Broadie,et al.  American Capped Call Options on Dividend-Paying Assets , 1995 .

[15]  Kjell Gustafsson,et al.  Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods , 1991, TOMS.

[16]  Willem Hundsdorfer,et al.  RKC time-stepping for advection-diffusion-reaction problems , 2004 .

[17]  Peter A. Forsyth,et al.  A finite element approach to the pricing of discrete lookbacks with stochastic volatility , 1999 .

[18]  Michael D. Marcozzi,et al.  The valuation of foreign currency options under stochastic interest rates , 2003 .

[19]  C. Chiarella,et al.  Evaluation of American strangles , 2002 .

[20]  Aslak Tveito,et al.  Penalty methods for the numerical solution of American multi-asset option problems , 2008 .

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

[22]  Jan G. Verwer,et al.  An Implicit-Explicit Runge-Kutta-Chebyshev Scheme for Diffusion-Reaction Equations , 2004, SIAM J. Sci. Comput..

[23]  Conall O'Sullivan,et al.  Pricing European and American Options under Heston’s Stochastic Volatility Model with Accelerated Explicit Finite Differencing Methods , 2010 .

[24]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[25]  Manuel Torrilhon,et al.  Essentially optimal explicit Runge–Kutta methods with application to hyperbolic–parabolic equations , 2007, Numerische Mathematik.

[26]  V. Thomée,et al.  Incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data , 1989 .

[27]  Eduardo S. Schwartz,et al.  The Valuation of American Put Options , 1977 .

[28]  V. I. LEBEDEV.  A new method for determining the roots of polynomials of least deviation on a segment with weight and subject to additional conditions. Part I , 1993 .

[29]  M. Yousuf,et al.  Smoothing schemes for reaction-diffusion systems with nonsmooth data , 2009 .

[30]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[31]  Abdul-Qayyum M. Khaliq,et al.  Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximants with distinct real poles , 1996, Adv. Comput. Math..

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

[33]  A. Medovikov.  High order explicit methods for parabolic equations , 1998 .

[34]  Linda R. Petzold,et al.  Runge-Kutta-Chebyshev projection method , 2006, J. Comput. Phys..

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

[36]  Assyr Abdulle,et al.  Second order Chebyshev methods based on orthogonal polynomials , 2001, Numerische Mathematik.

[37]  Assyr Abdulle,et al.  Fourth Order Chebyshev Methods with Recurrence Relation , 2001, SIAM J. Sci. Comput..

[38]  L. Shampine,et al.  RKC: an explicit solver for parabolic PDEs , 1998 .

[39]  P. Gremaud,et al.  Super-time-stepping acceleration of explicit schemes for parabolic problems , 1996 .

[40]  Christoph Reisinger,et al.  On the Use of Policy Iteration as an Easy Way of Pricing American Options , 2010, SIAM J. Financial Math..

[41]  Song-Ping Zhu,et al.  An inverse finite element method for pricing American options , 2013 .

[42]  Assyr Abdulle,et al.  S-ROCK: Chebyshev Methods for Stiff Stochastic Differential Equations , 2008, SIAM J. Sci. Comput..

[43]  V. Thomée,et al.  ON RATIONAL APPROXIMATIONS OF SEMIGROUPS , 1979 .

[44]  Kjell Gustafsson,et al.  Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers , 1997, SIAM J. Sci. Comput..

[45]  L. M. Skvortsov.  Explicit stabilized Runge-Kutta methods , 2011 .

[46]  Peter A. Forsyth,et al.  Quadratic Convergence for Valuing American Options Using a Penalty Method , 2001, SIAM J. Sci. Comput..

[47]  Fernando Zapatero,et al.  Monte Carlo Valuation of American Options through Computation of the Optimal Exercise Frontier , 2000, Journal of Financial and Quantitative Analysis.

[48]  A. Abdulle,et al.  S-ROCK methods for stiff Ito SDEs , 2008 .

[49]  Conall O'Sullivan,et al.  PRICING EUROPEAN AND AMERICAN OPTIONS IN THE HESTON MODEL WITH ACCELERATED EXPLICIT FINITE DIFFERENCING METHODS , 2013 .

[50]  P. Forsyth,et al.  Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance , 2007 .

[51]  Conall O'Sullivan,et al.  Pricing Options under Heston's Stochastic Volatility Model via Accelerated Explicit Finite Differencing Methods , 2010 .