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Variance-optimal hedging for processes with stationary independent increments

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  • Friedrich Hubalek
  • Jan Kallsen
  • Leszek Krawczyk
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    Abstract

    We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace- or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.

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    File URL: http://arxiv.org/pdf/math/0607112
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number math/0607112.

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    Date of creation: Jul 2006
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    Publication status: Published in Annals of Applied Probability 2006, Vol. 16, No. 2, 853-885
    Handle: RePEc:arx:papers:math/0607112

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    Web page: http://arxiv.org/

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    References

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    1. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    2. He, Hua & Pearson, Neil D., 1991. "Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case," Journal of Economic Theory, Elsevier, vol. 54(2), pages 259-304, August.
    3. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
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    Cited by:
    1. Ale\v{s} \v{C}ern\'y & Stephan Denkl & Jan Kallsen, 2013. "Hedging in L\'evy models and the time step equivalent of jumps," Papers 1309.7833, arXiv.org, revised Nov 2013.
    2. Flavio Angelini & Stefano Herzel, 2007. "Explicit formulas for the minimal variance hedging strategy in a martingale case," Quaderni del Dipartimento di Economia, Finanza e Statistica 35/2007, Università di Perugia, Dipartimento Economia, Finanza e Statistica.
    3. Jan Kallsen & Arnd Pauwels, 2011. "Variance-Optimal Hedging for Time-Changed Levy Processes," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(1), pages 1-28.
    4. Ernst Eberlein & Kathrin Glau & Antonis Papapantoleon, 2008. "Analysis of Fourier transform valuation formulas and applications," Papers 0809.3405, arXiv.org, revised Sep 2009.
    5. Ale\v{s} \v{C}ern\'y & Jan Kallsen, 2007. "On the structure of general mean-variance hedging strategies," Papers 0708.1715, arXiv.org.
    6. Mats Brod\'en & Peter Tankov, 2010. "Tracking errors from discrete hedging in exponential L\'evy models," Papers 1003.0709, arXiv.org.
    7. Carmine De Franco & Peter Tankov & Xavier Warin, 2012. "Numerical methods for the quadratic hedging problem in Markov models with jumps," Papers 1206.5393, arXiv.org, revised Dec 2013.
    8. Jan Kallsen & Richard Vierthauer, 2009. "Quadratic hedging in affine stochastic volatility models," Review of Derivatives Research, Springer, vol. 12(1), pages 3-27, April.
    9. Flavio ANGELINI & Stefano HERZEL, 2012. "Delta Hedging in Discrete Time under Stochastic Interest Rate," Quaderni del Dipartimento di Economia, Finanza e Statistica 110/2012, Università di Perugia, Dipartimento Economia, Finanza e Statistica.
    10. Mathieu Rosenbaum & Peter Tankov, 2011. "Asymptotically optimal discretization of hedging strategies with jumps," Papers 1108.5940, arXiv.org, revised Apr 2014.
    11. Mitra, Sovan, 2013. "Operational risk of option hedging," Economic Modelling, Elsevier, vol. 33(C), pages 194-203.
    12. Tankov, Peter & Voltchkova, Ekaterina, 2009. "Asymptotic analysis of hedging errors in models with jumps," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 2004-2027, June.
    13. Rossella Agliardi & Ramazan Gençay, 2012. "Hedging through a Limit Order Book with Varying Liquidity," Working Paper Series 12_12, The Rimini Centre for Economic Analysis.
    14. Jan Kallsen & Johannes Muhle-Karbe & Richard Vierthauer, 2009. "Asymptotic Power Utility-Based Pricing and Hedging," Papers 0912.3362, arXiv.org, revised Jan 2013.
    15. Ales Čern� & Jan Kallsen, 2008. "Mean-Variance Hedging And Optimal Investment In Heston'S Model With Correlation," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 473-492.

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