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PAPERS

Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing

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Pages 107-121 | Received 24 Apr 2006, Accepted 05 Feb 2007, Published online: 17 Mar 2008

References

  • Black , F. and Scholes , M. 1973 . The Pricing of Options and Corporate Liabilities . Journal of Political Economy , 81 : 637 – 59 .
  • Bossy , Mireille and Diop , Awa . 2004 . “ An efficient discretisation scheme for 1‐dimensional sides with a diffusion coefficient function of the form |x| a , a ∊ [1/2, 1) ” . In RR 5396, INRIA December
  • Butcher , J. C. 1987 . The Numerical Analysis of Ordinary Differential Equations , Chichester : John Wiley & Sons .
  • Castell , F. 1993 . Asymptotic expansion of stochastic flows . Probability theory and related fields , 96 (2) : 225 – 239 .
  • Feller , W. 1951 . Two singular diffusion problems . Annals of Mathematics , 54 : 173 – 182 .
  • Glasserman , P. 2004 . Monte Carlo Methods in Financial Engineering (New York: Springer‐Verlag)
  • Heston , S. L. 1993 . A Closed‐Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options . The Review of Financial Studies , 6 : 327 – 343 .
  • Ikeda , N. and Watanabe , S. 1981 . Stochastic differential equations and diffusion processes (North Holland/Kodansha)
  • Kloeden , P. E. and Platen , E. 1999 . Numerical Solution of Stochastic Differential Equations (Berlin: Springer)
  • Kusuoka , S. 2001 . “ Approximation of Expectation of Diffusion Process and Mathematical Finance ” . In Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 Edited by: Sunada , T . 147 – 165 . 31
  • Kusuoka , S. 2003 . Malliavin Calculus Revisited . Jounal of Mathematical Sciences The University of Tokyo , 10 : 261 – 277 .
  • Kusuoka , S. 2005 . Kusuoka Scheme and Gaussian type approximation . Presentation at “Mathematical Finance Seminar in Graduate School of Mathematical Sciences The University of Tokyo (1/June/2005)” ,
  • Kusuoka , S. and Ninomiya , S. 2004 . “ A new simulation method of diffusion processes applied to Finance ” . In Stochastic processes and application to mathematical finance, Proceedings of the Ritsumeikan International Symposium Edited by: Akahori , J , Ogawa , S and Watanabe , S . 233 – 253 . (Singapore: World Scientific)
  • Lapeyre , B. and Temam , E. 2001 . Competitive Monte Carlo methods for the pricing of Asian options . Journal of Computational Finance , 5 : 39 – 59 .
  • Lapeyre , B. , Pardoux , E. and Sentis , R. 1998 . Méthodes de Monte‐Carlo pour les équations de transport et de diffusion (Mathematics and Applications 29). (Berlin: Springer‐Verlag)
  • Liu , X. Q. and Li , C. W. 2000 . Weak approximation and extrapolations of stochastic differential equations with jumps . SIAM Journal on Numerical Analysis , 37 : 1747 – 1767 .
  • Lyons , T. and Victoir , N. 2004 . Cubature on Wiener Space . Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , 460 : 169 – 198 .
  • Ninomiya , S. 2003a . A new simulation scheme of diffusion processes: Application of the Kusuoka approximation to Finance Problems . Mathematics and Computers in Simulation , 62 ( 3–6 ) : 479 – 486 .
  • Ninomiya , S. 2003b . A partial sampling method applied to the Kusuoka approximation . Monte Carlo Methods and Applications , 9 : 27 – 38 .
  • Ninomiya , S. and Tezuka , S. 1996 . Toward real‐time pricing of complex financial derivatives . Applied Mathematical Finance , 3 : 1 – 20 .
  • Paskov , S. H. 1997 . “ New methodologies for Valuing Derivatives ” . In Mathematics of Derivative Securities Edited by: Pliska , S and Dempster , M . 545 – 582 . (Cambridge University Press)
  • Stroud , A. H. 1971 . Approximate calculation of multiple integrals (Englewood Cliffs, New Jersey: Prentice Hall)
  • Talay , D. 1990 . Second‐order discretization schemes of stochastic differential systems for the computation of the invariant law . Stochastics and Stochastics Reports , 29 : 13 – 36 .
  • Talay , D. 1995 . “ Simulation of Stochastic Differential Systems ” . In Probabilistic Methods in Applied Physics, LNP 451 Edited by: Kree , P and Wedig , W . 54 – 96 . (Springer‐Verlag)
  • Talay , D. and Tubaro , L. 1990 . Expansion of the global error for numerical schemes solving Stochastic Differential Equations . Stochastic Analysis and Applications , 8 : 483 – 509 .
  • Wong , E. and Zakai , M. 1965 . On the relation between ordinary and stochastic differential equations . Intern. J. Engng. Sci. , 3 : 213 – 229 .

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