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Symmetric martingales and symmetric smiles

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  • Tehranchi, Michael R.

Abstract

A local martingale X is called arithmetically symmetric if the conditional distribution of XT-Xt is symmetric given , for all 0 T- t) for all 0 =0. The notion of a geometrically symmetric martingale is also defined and characterized as the Doléans-Dade exponential of an arithmetically symmetric local martingale. As an application of these results, we show that a market model of the implied volatility surface that is initially flat and that remains symmetric for all future times must be the Black-Scholes model.

Suggested Citation

  • Tehranchi, Michael R., 2009. "Symmetric martingales and symmetric smiles," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3785-3797, October.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:10:p:3785-3797
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    References listed on IDEAS

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    Cited by:

    1. Erhan Bayraktar & Sergey Nadtochiy, 2013. "Weak reflection principle for L\'evy processes," Papers 1308.2250, arXiv.org, revised Oct 2015.
    2. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, January.
    3. Julien Claisse & Gaoyue Guo & Pierre Henry-Labordere, 2015. "Some Results on Skorokhod Embedding and Robust Hedging with Local Time," Papers 1511.07230, arXiv.org, revised Oct 2017.
    4. Molchanov, Ilga & Schmutz, Michael & Stucki, Kaspar, 2012. "Invariance properties of random vectors and stochastic processes based on the zonoid concept," DES - Working Papers. Statistics and Econometrics. WS ws122014, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Thorsten Rheinlander & Michael Schmutz, 2012. "Quasi self-dual exponential L\'evy processes," Papers 1201.5132, arXiv.org.
    6. Ilya Molchanov & Michael Schmutz, 2009. "Exchangeability type properties of asset prices," Papers 0901.4914, arXiv.org, revised Apr 2011.
    7. Fajardo, José, 2015. "Barrier style contracts under Lévy processes: An alternative approach," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 179-187.
    8. Thorsten Rheinlander & Michael Schmutz, 2012. "Self-dual continuous processes," Papers 1201.6516, arXiv.org.
    9. Jan Vecer, 2013. "Asian options on the harmonic average," Quantitative Finance, Taylor & Francis Journals, vol. 14(8), pages 1315-1322, September.
    10. Luciano Campi & Ismail Laachir & Claude Martini, 2014. "Change of numeraire in the two-marginals martingale transport problem," Papers 1406.6951, arXiv.org, revised Mar 2016.
    11. Thorsten Rheinländer & Jenny Sexton, 2011. "Hedging Derivatives," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8062, January.
    12. Rheinländer, Thorsten & Schmutz, Michael, 2013. "Self-dual continuous processes," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1765-1779.
    13. Fajardo, José & Mordecki, Ernesto, 2010. "Market symmetry in time-changed Brownian models," Finance Research Letters, Elsevier, vol. 7(1), pages 53-59, March.

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