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Lie Symmetry Methods for Local Volatility Models

Author

Listed:
  • Mark Craddock

    (School of Mathematical and Physical Sciences, University of Technology Sydney)

  • Martino Grasselli

    (Department of Mathematics, University of Padova)

Abstract

We investigate PDEs of the form ut = 1/2 s^2 (t, x)u_xx - g(x)u which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can be used to obtain standard integral transforms of fundamental solutions of the PDE. We detail explicit computations in the separable volatility case when s(t, x) = h(t)(a + ßx + ?x^2), g = 0, corresponding to the so called Quadratic Normal Volatility Model. We also consider choices of g for which we can obtain exact fundamental solutions that are also positive and continuous probability densities.

Suggested Citation

  • Mark Craddock & Martino Grasselli, 2016. "Lie Symmetry Methods for Local Volatility Models," Research Paper Series 377, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:377
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    File URL: https://www.uts.edu.au/sites/default/files/QFR-rp377.pdf
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    References listed on IDEAS

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    1. Andrey Itkin, 2013. "New solvable stochastic volatility models for pricing volatility derivatives," Review of Derivatives Research, Springer, vol. 16(2), pages 111-134, July.
    2. Mark Craddock & Eckhard Platen, 2003. "Symmetry Group Methods for Fundamental Solutions and Characteristic Functions," Research Paper Series 90, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    4. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
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    Cited by:

    1. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.
    2. Cyril Grunspan & Joris van der Hoeven, 2017. "Effective asymptotic analysis for finance," Working Papers hal-01573621, HAL.
    3. Mark Craddock, 2017. "Integral Transform and Lie Symmetry Methods for Scalar and Multi-Dimensional Diffusions," Research Paper Series 380, Quantitative Finance Research Centre, University of Technology, Sydney.

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