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Splitting Methods in Communication, Imaging, Science, and Engineering pp 541–575Cite as

Application of Operator Splitting Methods in Finance

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Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems.

Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g., given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems.

Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black–Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps.

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Notes

  1. 1.

    That is, the order for fixed nonstiff ODE systems.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020, Antwerp, Belgium

    Karel in’t Hout

  2. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, 94305, USA

    Jari Toivanen

  3. Department of Mathematical Information Technology, FI-40014 University of Jyväskylä, Jyväskylä, Finland

    Jari Toivanen

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  1. Karel in’t Hout
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  2. Jari Toivanen
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Correspondence to Karel in’t Hout .

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Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

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Hout, K., Toivanen, J. (2016). Application of Operator Splitting Methods in Finance. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_16

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