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Modern Finite Difference Approach

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Pricing Derivatives Under Lévy Models

Part of the book series: Pseudo-Differential Operators ((PDO,volume 12))

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Abstract

Modern finite difference schemes usually try to accomplish the following goals: (i) the scheme must be at least of second order of approximation in all independent variables; (ii) it should be unconditionally stable; (iii) it should preserve nonnegativity of the solution. Here we give the main definitions and facts of the modern theory of finite difference schemes using an operator approach to the solution of a parabolic partial differential equations or partial integrodifferentia equations and Padé approximations. We also introduce operator splitting techniques and high-order compact (HOC) schemes. In an appendix, some examples of HOC schemes are provided as applied to pricing American options.

The world is continuous, but the mind is discrete.

David Mumford (ICM 2002 plenary talk, Aug. 21, 2002).

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Notes

  1. 1.

    For sampling barriers discretely, this could result in some problems; see the discussion in [45].

  2. 2.

    An explicit discretization of F k in our case is discussed below.

  3. 3.

    In other words, the order of convergence does not fluctuate significantly with time step change, and is always very close to 2.

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Authors and Affiliations

  1. Department of Finance and Risk Engineering, NYU Tandon School of Engineering, Brooklyn, NY, USA

    Andrey Itkin

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Itkin, A. (2017). Modern Finite Difference Approach. In: Pricing Derivatives Under Lévy Models . Pseudo-Differential Operators, vol 12. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-6792-6_2

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