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Dupire'S Equation For Bubbles

Author

Listed:
  • ERIK EKSTRÖM

    (Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden)

  • JOHAN TYSK

    (Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden)

Abstract

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

Suggested Citation

  • Erik Ekström & Johan Tysk, 2012. "Dupire'S Equation For Bubbles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(06), pages 1-12.
    • Handle: RePEc:wsi:ijtafx:v:15:y:2012:i:06:n:s0219024912500410
    DOI: 10.1142/S0219024912500410
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    References listed on IDEAS

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    1. Amel Bentata & Marc Yor, 2008. "From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon," Papers 0806.0239, arXiv.org.
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    Cited by:

    1. Petteri Piiroinen & Lassi Roininen & Tobias Schoden & Martin Simon, 2018. "Asset Price Bubbles: An Option-based Indicator," Papers 1805.07403, arXiv.org, revised Jul 2018.
    2. Francesca Biagini & Lukas Gonon & Andrea Mazzon & Thilo Meyer-Brandis, 2022. "Detecting asset price bubbles using deep learning," Papers 2210.01726, arXiv.org, revised Dec 2022.
    3. P. Carr & A. Itkin, 2021. "An Expanded Local Variance Gamma Model," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 949-987, April.
    4. Andrey Itkin, 2020. "Geometric Local Variance Gamma Model," World Scientific Book Chapters, in: Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models, chapter 6, pages 137-173, World Scientific Publishing Co. Pte. Ltd..

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