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On the drawdown of completely asymmetric Lévy processes

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  • Mijatović, Aleksandar
  • Pistorius, Martijn R.
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    Abstract

    The drawdown process Y of a completely asymmetric Lévy process X is equal to X reflected at its running supremum X¯: Y=X¯−X. In this paper we explicitly express in terms of the scale function and the Lévy measure of X the law of the sextuple of the first-passage time of Y over the level a>0, the time G¯τa of the last supremum of X prior to τa, the infimum X¯τa and supremum X¯τa of X at τa and the undershoot a−Yτa− and overshoot Yτa−a of Y at τa. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential Lévy model.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 11 ()
    Pages: 3812-3836

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:11:p:3812-3836

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    Related research

    Keywords: Spectrally one-sided Lévy process; Reflected process; Drawdown; Fluctuation theory; Excursion theory; Sextuple law;

    References

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    1. Olympia Hadjiliadis & Jan Vecer, 2006. "Drawdowns preceding rallies in the Brownian motion model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 403-409.
    2. Foort HAMELINK & Martin HOESLI, 2003. "Maximum Drawdown and the Allocation to Real Estate," FAME Research Paper Series rp87, International Center for Financial Asset Management and Engineering.
    3. Pospisil, Libor & Vecer, Jan & Hadjiliadis, Olympia, 2009. "Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2563-2578, August.
    4. Peter Carr & Liuren Wu, 2002. "The Finite Moment Log Stable Process and Option Pricing," Finance 0207012, EconWPA.
    5. Alexei Chekhlov & Stanislav Uryasev & Michael Zabarankin, 2005. "Drawdown Measure In Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 13-58.
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