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Low-Dimensional Partial Integro-differential Equations for High-Dimensional Asian Options

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Inspired by Finance

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Abstract

Asian options on a single asset under a jump-diffusion model can be priced by solving a partial integro-differential equation (PIDE). We consider the more challenging case of an option whose payoff depends on a large number (or even a continuum) of assets. Possible applications include options on a stock basket index and electricity contracts with a delivery period. Both of these can be modeled with an exponential, time-inhomogeneous, Hilbert space valued jump-diffusion process. We derive the corresponding high- or even infinite-dimensional PIDE for Asian option prices in this setting and show how to approximate it with a low-dimensional PIDE. To this end, we employ proper orthogonal decomposition (POD) to reduce the dimension. We generalize the convergence results known for European options to the case of Asian options and give an estimate for the approximation error.

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Authors and Affiliations

  1. Mathematische Statistik M4, Technische Universität München, Boltzmannstr. 3, 85748, Garching, Germany

    Peter Hepperger

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  1. Peter Hepperger
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Correspondence to Peter Hepperger .

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Editors and Affiliations

  1. Labo. Mathématiques U.F.R. des Sciences et Technologie, Université de Franche-Comté, Besancon, France

    Yuri Kabanov

  2. School of Mathematics & Statistics, University of Sydney, Sydney, New South Wales, Australia

    Marek Rutkowski

  3. Depts. of Mathematics and IROM, McCombs School of Business, The University of Texas at Austin, Austin, Texas, USA

    Thaleia Zariphopoulou

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Hepperger, P. (2014). Low-Dimensional Partial Integro-differential Equations for High-Dimensional Asian Options. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_15

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