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Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management

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  • Lukasz Delong
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    Abstract

    In this paper we investigate novel applications of a new class of equations which we call time-delayed backward stochastic differential equations. Time-delayed BSDEs may arise in finance when we want to find an investment strategy and an investment portfolio which should replicate a liability or meet a target depending on the applied strategy or the past values of the portfolio. In this setting, a managed investment portfolio serves simultaneously as the underlying security on which the liability/target is contingent and as a replicating portfolio for that liability/target. This is usually the case for capital-protected investments and performance-linked pay-offs. We give examples of pricing, hedging and portfolio management problems (asset-liability management problems) which could be investigated in the framework of time-delayed BSDEs. Our motivation comes from life insurance and we focus on participating contracts and variable annuities. We derive the corresponding time-delayed BSDEs and solve them explicitly or at least provide hints how to solve them numerically. We give a financial interpretation of the theoretical fact that a time-delayed BSDE may not have a solution or may have multiple solutions.

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    File URL: http://arxiv.org/pdf/1005.4417
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1005.4417.

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    Date of creation: May 2010
    Date of revision: Jan 2011
    Handle: RePEc:arx:papers:1005.4417

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    Web page: http://arxiv.org/

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    1. Nadine Gatzert & Alexander Kling, 2007. "Analysis of Participating Life Insurance Contracts: A Unification Approach," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 74(3), pages 547-570.
    2. Min Dai & Yue Kuen Kwok & Jianping Zong, 2008. "Guaranteed Minimum Withdrawal Benefit In Variable Annuities," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 595-611.
    3. Marie-Amélie Morlais, 2009. "Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem," Finance and Stochastics, Springer, vol. 13(1), pages 121-150, January.
    4. Ballotta, Laura, 2005. "A Lévy process-based framework for the fair valuation of participating life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 173-196, October.
    5. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71.
    6. Kleinow, Torsten, 2009. "Valuation and hedging of participating life-insurance policies under management discretion," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 78-87, February.
    7. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    8. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    9. Milevsky, Moshe A. & Salisbury, Thomas S., 2006. "Financial valuation of guaranteed minimum withdrawal benefits," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 21-38, February.
    10. Romuald Elie & Nizar Touzi, 2008. "Optimal lifetime consumption and investment under a drawdown constraint," Finance and Stochastics, Springer, vol. 12(3), pages 299-330, July.
    11. Kleinow, Torsten & Willder, Mark, 2007. "The effect of management discretion on hedging and fair valuation of participating policies with maturity guarantees," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 445-458, May.
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