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Pseudoparabolic and Fractional Equations of Option Pricing

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Book cover Pricing Derivatives Under Lévy Models

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

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Abstract

We propose a new, unified approach to solving the jump–diffusion partial integrodifferential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finite difference methods, which for multidimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudodifferential operator.

I became an atheist because, as a graduate student studying quantum physics, life seemed to be reducible to second-order differential equations. Mathematics, chemistry and physics had it all. And I didn’t see any need to go beyond that.

Francis Collins.

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Notes

  1. 1.

    In general, the CF of a jump–diffusion process in not known, unless we are talking about a Lévy process. What is known is the so-called symbol of the generator.

  2. 2.

    For the cosine transform, see [25, 26, 28] and references therein. For the modern FFT technique, see [16, 36, 38] and references therein.

  3. 3.

    For some models, it can be computed analytically, so in what follows, we do not take such models into account.

  4. 4.

    A second-order approximation could in principle be constructed as well, but that would result in a massive calculation for the coefficients. Therefore, this approach has not been further elaborated.

  5. 5.

    This equation arises naturally at some step of the splitting procedure if splitting is organized by separating diffusion from jumps.

  6. 6.

    In other words, the PIDE grid is a superset of the corresponding PDE grid.

  7. 7.

    We recall that a standard Brownian motion already has paths of infinite variation. Therefore, the Lévy process in Eq. (5.2) has infinite variation, since it contains a continuous martingale component. However, here we refer to the infinite variation that comes from the jumps.

  8. 8.

    See http://www.netlib.org/scalapack/scalapack_home.html.

  9. 9.

    We use 2N instead of N because in order to avoid undesirable wraparound errors, a common technique is to embed a discretization Toeplitz matrix in a circulant matrix. This requires doubling the initial vector of unknowns.

  10. 10.

    We keep the same notation for the put option value, which is C(x, τ), since this comparison means basically to compare two numerical methods rather than provide some deep economic meaning.

  11. 11.

    It actually uses 4N points, as it discussed earlier.

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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). Pseudoparabolic and Fractional Equations of Option Pricing. 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_5

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