Laplace transformation method for the Black-Scholes equation

In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thom\'ee (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various options. Existence and uniqueness properties of the Laplace transformed Black-Scholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.

[1]  Christoph Schwab,et al.  Fast deterministic pricing of options on Lévy driven assets , 2002 .

[2]  M. Fu,et al.  Pricing Continuous Asian Options: A Comparison of Monte Carlo and Laplace Transform Inversion Methods , 1998 .

[3]  Vidar Thomée,et al.  Time discretization via Laplace transformation of an integro-differential equation of parabolic type , 2006, Numerische Mathematik.

[4]  Christoph Schwab,et al.  Fast Numerical Solution of Parabolic Integrodifferential Equations with Applications in Finance , 2005, SIAM J. Sci. Comput..

[5]  A. M. Cohen.  Numerical Methods for Laplace Transform Inversion , 2007 .

[6]  Antoon Pelsser,et al.  Pricing double barrier options using Laplace transforms , 2000, Finance Stochastics.

[7]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.

[8]  Jinwoo Lee,et al.  A Parallel Method for Backward Parabolic Problems Based on the Laplace Transformation , 2006, SIAM J. Numer. Anal..

[9]  Ivan P. Gavrilyuk,et al.  fMathematik in den Naturwissenschaften Leipzig H-matrix approximation for the operator exponential with applications , 2000 .

[10]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[11]  Pricing Multi-Asset Options with Sparse Grids and Fourth Order Finite Differences , 2006 .

[12]  Radha Panini,et al.  Pricing perpetual options using Mellin transforms , 2005, Appl. Math. Lett..

[13]  C. Schwab,et al.  NUMERICAL SOLUTION OF PARABOLIC EQUATIONS IN HIGH DIMENSIONS , 2004 .

[14]  Almerico Murli,et al.  Algorithm 682: Talbot's method of the Laplace inversion problems , 1990, TOMS.

[15]  T. J. I'a. Bromwich,et al.  Normal Coordinates in Dynamical Systems , 1917 .

[16]  Vidar Thomée,et al.  A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature , 2005 .

[17]  C. Palencia,et al.  On the numerical inversion of the Laplace transform of certain holomorphic mappings , 2004 .

[18]  Lloyd N. Trefethen,et al.  Parabolic and hyperbolic contours for computing the Bromwich integral , 2007, Math. Comput..

[19]  William T. Weeks,et al.  Numerical Inversion of Laplace Transforms Using Laguerre Functions , 1966, JACM.

[20]  Stig Larsson,et al.  Resolvent estimates for elliptic finite element operators in one dimension , 1994 .

[21]  J. A. C. Weideman,et al.  Algorithms for Parameter Selection in the Weeks Method for Inverting the Laplace Transform , 1999, SIAM J. Sci. Comput..

[22]  Michael Griebel,et al.  A Domain Decomposition Method Using Sparse Grids , 1994 .

[23]  Gabriel Wittum,et al.  Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems , 2007, SIAM J. Sci. Comput..

[24]  V. Thomée,et al.  Time discretization of an evolution equation via Laplace transforms , 2004 .

[25]  RAUL KANGRO,et al.  Far Field Boundary Conditions for Black-Scholes Equations , 2000, SIAM J. Numer. Anal..

[26]  Dongwoo Sheen,et al.  An accurate numerical inversionof Laplace transforms based on the location of their poles , 2004 .

[27]  Kenny S. Crump,et al.  Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation , 1976, J. ACM.

[28]  Cornelis W. Oosterlee,et al.  On coordinate transformation and grid stretching for sparse grid pricing of basket options , 2008 .

[29]  D. I. Cruz-Báez,et al.  A different approach for pricing European options , 2005 .

[30]  Dongwoo Sheen,et al.  A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature , 2003 .

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

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

[33]  Dongwoo Sheen,et al.  A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature , 2000, Math. Comput..

[34]  Ivan P. Gavrilyuk,et al.  Data-sparse approximation to a class of operator-valued functions , 2004, Math. Comput..

[35]  Ghada Alobaidi,et al.  Laplace transforms and American options , 2000 .

[36]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .