IDEAS home Printed from https://ideas.repec.org/p/uts/rpaper/377.html
   My bibliography  Save this paper

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
    as

    Download full text from publisher

    File URL: https://www.uts.edu.au/sites/default/files/QFR-rp377.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. Peter Carr & Travis Fisher & Johannes Ruf, 2012. "Why are quadratic normal volatility models analytically tractable?," Papers 1202.6187, arXiv.org, revised Mar 2013.
    4. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Cyril Grunspan & Joris van der Hoeven, 2017. "Effective asymptotic analysis for finance," Working Papers hal-01573621, HAL.
    2. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Craddock, Mark & Grasselli, Martino, 2020. "Lie symmetry methods for local volatility models," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3802-3841.
    2. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    3. Yukihiro Tsuzuki, 2024. "Boundary conditions at infinity for Black-Scholes equations," Papers 2401.05549, arXiv.org, revised Mar 2024.
    4. Martino Grasselli, 2017. "The 4/2 Stochastic Volatility Model: A Unified Approach For The Heston And The 3/2 Model," Mathematical Finance, Wiley Blackwell, vol. 27(4), pages 1013-1034, October.
    5. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.
    6. Jan Baldeaux & Eckhard Platen, 2012. "Computing Functionals of Multidimensional Diffusions via Monte Carlo Methods," Papers 1204.1126, arXiv.org.
    7. Umut Cetin & Kasper Larsen, 2020. "Uniqueness in Cauchy problems for diffusive real-valued strict local martingales," Papers 2007.15041, arXiv.org, revised May 2022.
    8. Fisher, Travis & Pulido, Sergio & Ruf, Johannes, 2019. "Financial models with defaultable numéraires," LSE Research Online Documents on Economics 84973, London School of Economics and Political Science, LSE Library.
    9. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching To Nonaffine Stochastic Volatility: A Closed-Form Expansion For The Inverse Gamma Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-37, August.
    10. Kim, Seong-Tae & Kim, Jeong-Hoon, 2020. "Stochastic elasticity of vol-of-vol and pricing of variance swaps," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 420-440.
    11. Travis Fisher & Sergio Pulido & Johannes Ruf, 2019. "Financial Models with Defaultable Numéraires," Post-Print hal-01240736, HAL.
    12. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    13. Martin Herdegen & Martin Schweizer, 2018. "Semi‐efficient valuations and put‐call parity," Mathematical Finance, Wiley Blackwell, vol. 28(4), pages 1061-1106, October.
    14. Yukihiro Tsuzuki, 2023. "Pitman's Theorem, Black-Scholes Equation, and Derivative Pricing for Fundraisers," Papers 2303.13956, arXiv.org.
    15. Paolo Guasoni & Miklós Rásonyi, 2015. "Fragility of arbitrage and bubbles in local martingale diffusion models," Finance and Stochastics, Springer, vol. 19(2), pages 215-231, April.
    16. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching to nonaffine stochastic volatility: a closed-form expansion for the Inverse Gamma model," Post-Print hal-02909113, HAL.
    17. Jos'e Da Fonseca & Claude Martini, 2014. "The $\alpha$-Hypergeometric Stochastic Volatility Model," Papers 1409.5142, arXiv.org.
    18. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2018. "Most-Likely-Path In Asian Option Pricing Under Local Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(05), pages 1-32, August.
    19. Alexander Lipton, 2024. "Hydrodynamics of Markets:Hidden Links Between Physics and Finance," Papers 2403.09761, arXiv.org.
    20. Martin HERDEGEN & Martin SCHWEIZER, 2016. "Economically Consistent Valuations and Put-Call Parity," Swiss Finance Institute Research Paper Series 16-02, Swiss Finance Institute.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:uts:rpaper:377. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Duncan Ford (email available below). General contact details of provider: https://edirc.repec.org/data/qfutsau.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.