A new radial basis functions method for pricing American options under Merton's jump-diffusion model

References

• Almendral , A. and Oosterlee , C. W. 2005 . Numerical valuation of options with jumps in the underlying . Appl. Numer. Math. , 53 : 1 – 18 .
   Web of Science ®Google Scholar
• Andersen , L. and Andreasen , J. 2000 . Jump diffusion processes: Volatility smile fitting and numerical methods for option pricing . Rev. Derivatives Res. , 4 : 231 – 262 .
   Google Scholar
• Ashi , H. A. , Cummings , L. J. and Matthews , P. C. 2009 . Comparison of methods for evaluating functions of a matrix exponential . J. Appl. Numer. Math. , 59 ( 3–4 ) : 468 – 486 .
   Web of Science ®Google Scholar
• Bellman , R. and Casti , J. 1972 . Differential quadrature: A technique for the rapid solution of non-linear partial differential equations . J. Comput. Phys. , 10 : 40 – 52 .
   Web of Science ®Google Scholar
• Black , F. and Scholes , M. S. 1973 . The pricing of options and corporate liabilities . J. Polit. Economy , 81 : 637 – 659 .
   Web of Science ®Google Scholar
• Briani , M. , Natalini , R. and Russo , G. 2007 . Implicit–explicit numerical schemes for jump diffusion processes . Calcolo , 44 : 33 – 57 .
   Web of Science ®Google Scholar
• Buhmann , M. BD. 2003 . Radial Basis Functions: Theory and Implementation , New York : Cambridge University Press .
   Google Scholar
• Carr , P. and Mayo , A. 2007 . On the numerical evaluation of option prices in jump diffusion processes . Eur. J. Finance , 13 : 353 – 372 .
   Web of Science ®Google Scholar
• Chan , R. T.L. and Hubbert , S. 2010 . A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model . Tech. Rep. , Available at arXiv.org.1011.5650
   Google Scholar
• Cheng , K. An overview of barrier options Working draft, Global derivatives, 2003. Available at http://www.global-derivatives.com.
   Google Scholar
• Cont , R. and Tankov , P. 2004 . Financial Modelling with Jump Processes , Boca Raton , FL : Chapman and Hall/CRC .
   Google Scholar
• Cont , R. and Voltchkova , E. 2005 . A finite difference scheme for option pricing in jump diffusion and exponential Lévy models . SIAM J. Numer. Anal. , 43 : 1596 – 1626 .
   Web of Science ®Google Scholar
• d'Halluin , Y. , Forsyth , P. A. and Vetzal , K. R. 2005 . Robust numerical methods for contingent claims under jump diffusion processes . IMA J. Numer. Anal. , 25 : 87 – 112 .
   Web of Science ®Google Scholar
• Fasshauer , G. E. , Khaliq , Q. M. and Voss , D. A. 2008 . A parallel time stepping approach using meshfree approximations for pricing options with non-smooth payoffs . J. Risk , 10 : 135 – 142 .
   Web of Science ®Google Scholar
• Fasshauer , G. E. , Khaliq , Q. M. and Voss , D. A. 2004 . Using meshfree approximation for multi-asset American option problems . J. Chin. Inst. Eng. , 27 : 563 – 571 .
   Web of Science ®Google Scholar
• Franke , R. 1982 . Scattered data interpolation: Test of some methods . J. Math. Comput. , 38 : 181 – 199 .
   Web of Science ®Google Scholar
• Golberg , M. A. , Chen , C. S. and Karur , S. R. 1996 . Improved multiquadratic approximation for partial differential equations . J. Eng. Anal. Bound. Elem. , 18 : 9 – 17 .
   Web of Science ®Google Scholar
• Han , H. and Wu , X. 2003 . A fast numerical method for the Black–Scholes equation of American options . SIAM J. Numer. Anal. , 41 : 2081 – 2095 .
   Web of Science ®Google Scholar
• Hardy , R. L. 1971 . Multiquadratic equations of topography and other irregular surfaces . J. Geophys. Res. , 176 : 1905 – 1915 .
   Web of Science ®Google Scholar
• Hon , Y. C. and Mao , X. Z. 1997 . A multiquadratic interpolation method for solving initial value problems . J. Sci. Comput. , 12 : 51 – 55 .
   Google Scholar
• Hon , Y. C. and Mao , X. Z. 1999 . A radial basis function method for solving options pricing model . Financ. Eng. , 8 : 31 – 49 .
   Google Scholar
• Ikonen , S. and Toivanen , J. 2004 . Operator splitting methods for American option pricing . Appl. Math. Lett. , 17 : 809 – 814 .
   Web of Science ®Google Scholar
• Kansa , E. J. 1990 . Multiquadratics – a scattered data approximation scheme with application to computational fluid dynamics-I . J. Comput. Math. Appl. , 19 : 127 – 145 .
   Web of Science ®Google Scholar
• Larsson , E. , Åhlander , K. and Hall , A. 2008 . Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform . J. Comput. Appl. Math. , 222 : 175 – 192 .
   Web of Science ®Google Scholar
• Lord , R. , Fang , F. , Bervoets , F. and Oosterlee , C. W. 2008 . A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes . SIAM J. Sci. Comput. , 30 : 1678 – 1705 .
   Web of Science ®Google Scholar
• Madych , W. R. and Nelson , S. A. 1988 . Multivariate interpolation and conditionally positive definite functions I . J. Approx. Theory Appl. , 4 : 77 – 89 .
   Google Scholar
• Merton , R. C. 1976 . Option pricing when the underlying stocks are discontinuous . J. Finan. Econ. , 5 : 125 – 144 .
   Web of Science ®Google Scholar
• Pettersson , U. , Larsson , E. , Marcusson , G. and Persson , J. 2008 . Improved radial basis function methods for multi-dimensional option pricing . J. Comput. Appl. Math. , 222 : 82 – 93 .
   Web of Science ®Google Scholar
• Powell , M. J.D. 1992 . “ The theory of radial basis function approximation in 1990 ” . In Advances in Numerical Analyis , Edited by: Light , W. Vol. 2 , 105 – 210 . Oxford : Clarendon Press .
   Google Scholar
• Rambeerich , N. , Tangman , D. Y. , Gopaul , A. and Bhuruth , M. 2009 . Exponential time integration for fast finite element solutions of some financial engineering problems . J. Comput. Appl. Math. , 224 : 668 – 678 .
   Web of Science ®Google Scholar
• Salmi , S. and Toivanen , J. 2011 . An iterative method for pricing American options under jump diffusion models . Appl. Numer. Math. , 61 : 821 – 831 .
   Web of Science ®Google Scholar
• Santos , G. T. , de Souza , M. C. and Fortes , M. 2009 . Use of radial basis functions for meshless numerical solutions applied to financial engineering barrier options . Pesquisa Operacional [online] , 29 : 419 – 437 .
   Google Scholar
• Sarra , S. A. 2005 . Adaptive radial basis function method for time dependent partial differential equations . Appl. Numer. Math. , 54 : 79 – 94 .
   Web of Science ®Google Scholar
• Tangman , D. Y. 2009 . PDE methods for pricing options PhD thesis, University of Mauritius
   Google Scholar
• Tangman , D. Y. , Gopaul , A. and Bhuruth , M. 2008 . Exponential time integration and Chebychev discretizations schemes for fast pricing of options . Appl. Numer. Math. , 58 : 1309 – 1319 .
   Web of Science ®Google Scholar
• Tangman , D. Y. , Gopaul , A. and Bhuruth , M. 2008 . A fast high-order finite difference algorithm for pricing American options . J. Comput. Appl. Math. , 222 : 17 – 29 .
   Web of Science ®Google Scholar
• Tangman , D. Y. , Gopaul , A. and Bhuruth , M. 2008 . Numerical pricing of options using high-order compact finite difference schemes . J. Comput. Appl. Math. , 218 : 270 – 280 .
   Web of Science ®Google Scholar
• Tangman , D. Y. , Boojhawon , R. , Gopaul , A. and Bhuruth , M. 2011 . “ A general efficient framework for pricing options using exponential time integration schemes ” . In Financial Econometrics Modeling, Derivatives Pricing, Hedge Funds and Term Structure Models , Edited by: Rascalan , R. and Gregoriou , G. N. 70 – 89 . Hampshire : Palgrave-Macmillan .
   Google Scholar
• Wu , Y. L. and Shu , C. 2002 . Development of RBF–DQ method for derivative approximation and its application to simulate natural convection in concentric anuli . Comput. Mech. , 29 : 477 – 485 .
   Web of Science ®Google Scholar