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Strict local martingales and bubbles

Author

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  • Kardaras, Constantinos
  • Kreher, Dörte
  • Nikeghbali, Ashkan

Abstract

This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term” apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given.

Suggested Citation

  • Kardaras, Constantinos & Kreher, Dörte & Nikeghbali, Ashkan, 2015. "Strict local martingales and bubbles," LSE Research Online Documents on Economics 64967, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:64967
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    File URL: http://eprints.lse.ac.uk/64967/
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    References listed on IDEAS

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    1. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    2. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2012. "Strict local martingale deflators and valuing American call-type options," Finance and Stochastics, Springer, vol. 16(2), pages 275-291, April.
    3. Hardy Hulley, 2010. "The Economic Plausibility of Strict Local Martingales in Financial Modelling," Research Paper Series 279, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    5. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    6. Patrick Cheridito & Ashkan Nikeghbali & Eckhard Platen, 2012. "Processes of Class Sigma, Last Passage Times, and Drawdowns," Published Paper Series 2012-4, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    7. Pal, Soumik & Protter, Philip, 2010. "Analysis of continuous strict local martingales via h-transforms," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1424-1443, August.
    8. 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.
    9. Soumik Pal & Philip Protter, 2007. "Analysis of continuous strict local martingales via h-transforms," Papers 0711.1136, arXiv.org, revised Jun 2010.
    10. Ju-Yi Yen & Marc Yor, 2011. "Call Option Prices Based on Bessel Processes," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 329-347, June.
    11. Robert A. Jarrow & Philip Protter, 2009. "Forward And Futures Prices With Bubbles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(07), pages 901-924.
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    Citations

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    Cited by:

    1. Kreher, Dörte & Nikeghbali, Ashkan, 2015. "A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 94-101.
    2. Robert A. Jarrow, 2015. "Asset Price Bubbles," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 201-218, December.
    3. Petteri Piiroinen & Lassi Roininen & Martin Simon, 2019. "Brexit Risk Implied by the SABR Martingale Defect in the EUR-GBP Smile," Papers 1912.05773, arXiv.org, revised Mar 2020.
    4. Cetin, Umut, 2018. "Diffusion transformations, Black-Scholes equation and optimal stopping," LSE Research Online Documents on Economics 87261, London School of Economics and Political Science, LSE Library.
    5. Johannes Muhle-Karbe & Marcel Nutz, 2018. "A risk-neutral equilibrium leading to uncertain volatility pricing," Finance and Stochastics, Springer, vol. 22(2), pages 281-295, April.
    6. Antoine Jacquier & Martin Keller-Ressel, 2015. "Implied volatility in strict local martingale models," Papers 1508.04351, arXiv.org.
    7. Petteri Piiroinen & Lassi Roininen & Tobias Schoden & Martin Simon, 2018. "Asset Price Bubbles: An Option-based Indicator," Papers 1805.07403, arXiv.org, revised Jul 2018.
    8. Christian Furrer, 2022. "Scaled insurance cash flows: representation and computation via change of measure techniques," Finance and Stochastics, Springer, vol. 26(2), pages 359-382, April.
    9. Martin Herdegen & Dorte Kreher, 2021. "Bubbles in discrete time models," Papers 2104.12740, arXiv.org, revised Jul 2022.
    10. Christa Cuchiero & Guido Gazzani & Janka Moller & Sara Svaluto-Ferro, 2023. "Joint calibration to SPX and VIX options with signature-based models," Papers 2301.13235, arXiv.org.
    11. Martin Herdegen & Dörte Kreher, 2022. "Bubbles in discrete-time models," Finance and Stochastics, Springer, vol. 26(4), pages 899-925, October.
    12. Peter Imkeller & Nicolas Perkowski, 2015. "The existence of dominating local martingale measures," Finance and Stochastics, Springer, vol. 19(4), pages 685-717, October.
    13. Li, Xue-Mei, 2017. "Strict local martingales: Examples," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 65-68.
    14. Umut Cetin & Kasper Larsen, 2020. "Uniqueness in Cauchy problems for diffusive real-valued strict local martingales," Papers 2007.15041, arXiv.org, revised May 2022.
    15. Thomas Krabichler & Josef Teichmann, 2020. "A constraint-based notion of illiquidity," Papers 2004.12394, arXiv.org.

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    More about this item

    Keywords

    Strict local martingales; bubbles;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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