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Original Articles

Asynchronous time-parallel method based on Laplace transform

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Pages 179-194 | Received 12 Nov 2018, Accepted 24 Feb 2020, Published online: 09 Mar 2020

References

  • J. Abate and P.P. Valkó, Multi-precision Laplace transform inversion, Int. J. Numer. Methods Eng. 60 (2004), pp. 979–993. doi: 10.1002/nme.995
  • J. Abate and W. Whitt, A unified framework for numerically inverting Laplace transforms, INFORMS J. Comput. 18 (2006), pp. 408–421. doi: 10.1287/ijoc.1050.0137
  • G. Barles and H.M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Financ. Stoch. 2 (1998), pp. 369–397. doi: 10.1007/s007800050046
  • G.M. Baudet, Asynchronous iterative methods for multiprocessors, J. ACM 25 (1978), pp. 226–244. doi: 10.1145/322063.322067
  • D.P. Bertsekas, Distributed asynchronous computation of fixed points, Math. Program. 27 (1983), pp. 107–120. doi: 10.1007/BF02591967
  • D.P. Bertsekas, J.N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Upper Saddle River, NJ1989.
  • F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81 (1973), pp. 637–654. doi: 10.1086/260062
  • P.P. Boyle and T. Vorst, Option replication in discrete time with transaction costs, J. Financ. 47 (1992), pp. 271–293. doi: 10.1111/j.1540-6261.1992.tb03986.x
  • D. Chazan and W. Miranker, Chaotic relaxation, Linear Algebra Appl. 2 (1969), pp. 199–222. doi: 10.1016/0024-3795(69)90028-7
  • A.M. Cohen, Numerical Methods for Laplace Transform Inversion, Springer, New York, NY2007.
  • J.C. Cox, S.A. Ross, and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ. 7 (1979), pp. 229–263. doi: 10.1016/0304-405X(79)90015-1
  • D. Crann, A.J. Davies, C.H. Lai, and S.H. Leong, Time domain decomposition for European options in financial modelling, Contemp. Math. 218 (1998), pp. 486–491. doi: 10.1090/conm/218/03047
  • B. Davies and B. Martin, Numerical inversion of the Laplace transform: a survey and comparison of methods, J. Comput. Phys. 33 (1979), pp. 1–32. doi: 10.1016/0021-9991(79)90025-1
  • J.C. Duan, The GARCH option pricing model, Math. Financ. 5 (1995), pp. 13–32. doi: 10.1111/j.1467-9965.1995.tb00099.x
  • D.G. Duffy, On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications, ACM Trans. Math. Softw. 19 (1993), pp. 333–359. doi: 10.1145/155743.155788
  • D. El Baz, P. Spitéri, J.C. Miellou, and D. Gazen, Asynchronous iterative algorithms with flexible communication for nonlinear network flow problems, J. Parallel Distrib. Comput. 38 (1996), pp. 1–15. doi: 10.1006/jpdc.1996.0124
  • M.N. El Tarazi, Some convergence results for asynchronous algorithms, Numer. Math. 39 (1982), pp. 325–340. (in French). doi: 10.1007/BF01407866
  • A. Frommer and D.B. Szyld, Asynchronous two-stage iterative methods, Numer. Math. 69 (1994), pp. 141–153. doi: 10.1007/s002110050085
  • A. Frommer and D.B. Szyld, On asynchronous iterations, J. Comput. Appl. Math. 123 (2000), pp. 201–216. doi: 10.1016/S0377-0427(00)00409-X
  • M.J. Gander, 50 years of time parallel time integration, in Multiple Shooting and Time Domain Decomposition Methods, T. Carraro, M. Geiger, S. Körkel, and R. Rannacher, eds., Springer, Cham, 2015, pp. 69–113.
  • J.M. Harrison and D.M. Kreps, Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory 20 (1979), pp. 381–408. doi: 10.1016/0022-0531(79)90043-7
  • J. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Financ. 42 (1987), pp. 281–300. doi: 10.1111/j.1540-6261.1987.tb02568.x
  • K.L. Kuhlman, Review of inverse Laplace transform algorithms for Laplace-space numerical approaches, Numer. Algorithms 63 (2013), pp. 339–355. doi: 10.1007/s11075-012-9625-3
  • A. Kuznetsov, On the convergence of the Gaver–Stehfest algorithm, SIAM J. Numer. Anal. 51 (2013), pp. 2984–2998. doi: 10.1137/13091974X
  • C.H. Lai, On transformation methods and the induced parallel properties for the temporal domain, in Substructuring Techniques and Domain Decomposition Methods, F. Magoulès, ed., chap. 3, Saxe-Coburg Publications, Stirlingshire, 2010, pp. 45–70.
  • C.H. Lai, A.K. Parrott, S. Rout, and M.E. Honnor, A distributed algorithm for European options with nonlinear volatility, Comput. Math. Appl. 49 (2005), pp. 885–894. doi: 10.1016/j.camwa.2004.03.014
  • H. Lee, J. Lee, and D. Sheen, Laplace transform method for parabolic problems with time-dependent coefficients, SIAM J. Numer. Anal. 51 (2013), pp. 112–125. doi: 10.1137/110824000
  • J. Ma and Z. Zhou, Convergence analysis of iterative Laplace transform methods for the coupled PDEs from regime-switching option pricing, J. Sci. Comput. 74 (2018), pp. 49–69. doi: 10.1007/s10915-017-0423-x
  • F. Magoulès and A.K. Cheik Ahamed, Alinea: an advanced linear algebra library for massively parallel computations on graphics processing units, Int. J. High Perform. Comput. Appl. 29 (2015), pp. 284–310. doi: 10.1177/1094342015576774
  • F. Magoulès and G. Gbikpi-Benissan, JACK: an asynchronous communication kernel library for iterative algorithms, J. Supercomput. 73 (2017), pp. 3468–3487. doi: 10.1007/s11227-016-1702-2
  • F. Magoulès and G. Gbikpi-Benissan, Asynchronous Parareal time discretization for partial differential equations, SIAM J. Sci. Comput. 40 (2018), pp. C704–C725. doi: 10.1137/17M1149225
  • F. Magoulès and G. Gbikpi-Benissan, Distributed convergence detection based on global residual error under asynchronous iterations, IEEE Trans. Parallel Distrib. Syst. 29 (2018), pp. 819–829. doi: 10.1109/TPDS.2017.2780856
  • F. Magoulès and G. Gbikpi-Benissan, JACK2: an MPI-based communication library with non-blocking synchronization for asynchronous iterations, Adv. Eng. Softw. 119 (2018), pp. 116–133. doi: 10.1016/j.advengsoft.2018.01.009
  • F. Magoulès, G. Gbikpi-Benissan and Q. Zou, Asynchronous iterations of Parareal algorithm for option pricing models, Mathematics 6 (2018), pp. 1–18. doi: 10.3390/math6040045
  • F. Magoulès, D.B. Szyld, and C. Venet, Asynchronous optimized Schwarz methods with and without overlap, Numer. Math. 137 (2017), pp. 199–227. doi: 10.1007/s00211-017-0872-z
  • F. Magoulès and C. Venet, Asynchronous iterative sub-structuring methods, Math. Comput. Simul.145 (2018), pp. 34–49. doi: 10.1016/j.matcom.2016.05.009
  • R.C. Merton, Theory of rational option pricing, Bell J. Econ. 4 (1973), pp. 141–183. doi: 10.2307/3003143
  • R.C. Merton, On the pricing of corporate debt: the risk structure of interest rates, J. Financ. 29 (1974), pp. 449–470.
  • J.C. Miellou, Algorithmes de relaxation chaotique à retards, ESAIM: M2AN 9 (1975), pp. 55–82. (in French).
  • J.C. Miellou, D. El Baz, and P. Spitéri, A new class of asynchronous iterative algorithms with order intervals, Math. Comput. 67 (1998), pp. 237–255. doi: 10.1090/S0025-5718-98-00885-0
  • D. Sheen, I.H. Sloan, and V. Thomée, A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature, Math. Comput. 69 (2000), pp. 177–195. doi: 10.1090/S0025-5718-99-01098-4
  • H. Stehfest, Numerical inversion of Laplace transforms, Commun. ACM 13 (1970), pp. 47–49. doi: 10.1145/361953.361969
  • A. Talbot, The accurate numerical inversion of Laplace transforms, IMA J. Appl. Math. 23 (1979), pp. 97–120. doi: 10.1093/imamat/23.1.97
  • S.G. Walker, A Laplace transform inversion method for probability distribution functions, Stat. Comput. 27 (2017), pp. 439–448. doi: 10.1007/s11222-016-9631-8
  • Z. Zhou, J. Ma, and X. Gao, Convergence of iterative Laplace transform methods for a system of fractional PDEs and PIDEs arising in option pricing, East Asian J. Appl. Math. 8 (2018), pp. 782–808. doi: 10.4208/eajam.130218.290618
  • Z. Zhou, J. Ma, and H.W. Sun, Fast Laplace transform methods for free-boundary problems of fractional diffusion equations, J. Sci. Comput. 74 (2018), pp. 49–69. doi: 10.1007/s10915-017-0423-x

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