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Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

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  • Michael B. Giles
  • Christoph Reisinger
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    Abstract

    In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.

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    File URL: http://arxiv.org/pdf/1204.1442
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    Paper provided by arXiv.org in its series Papers with number 1204.1442.

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    Date of creation: Apr 2012
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    Handle: RePEc:arx:papers:1204.1442

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    Web page: http://arxiv.org/

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    1. Robert A. Jarrow, 2009. "The Term Structure of Interest Rates," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 69-96, November.
    2. Merton, Robert C., 1973. "On the pricing of corporate debt: the risk structure of interest rates," Working papers 684-73., Massachusetts Institute of Technology (MIT), Sloan School of Management.
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