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Portfolio problems stopping at first hitting time with application to default risk

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  • Holger Kraft
  • Mogens Steffensen

Abstract

In this paper a portfolio problem is considered where trading in the risky asset is stopped if a state process hits a predefined barrier. This state process need not to be perfectly correlated with the risky asset. We give a representation result for the value function and provide a verification theorem. As an application, we explicitly solve the problem by assuming that the state process is an arithmetic Brownian motion. Then the result is used as a starting point to solve and analyze a portfolio problem with default risk modeled by the Black-Cox approach. Finally, we discuss how our results can be applied to a portfolio problem with stochastic interest rates and default risk modeled by the approach of Briys and de Varenne. Copyright Springer-Verlag 2006

Suggested Citation

  • Holger Kraft & Mogens Steffensen, 2006. "Portfolio problems stopping at first hitting time with application to default risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 123-150, February.
    • Handle: RePEc:spr:mathme:v:63:y:2006:i:1:p:123-150
    DOI: 10.1007/s00186-005-0026-4
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    References listed on IDEAS

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    1. Briys, Eric & de Varenne, François, 1997. "Valuing Risky Fixed Rate Debt: An Extension," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 32(2), pages 239-248, June.
    2. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    3. Ralf Korn & Holger Kraft, 2003. "Optimal Portfolios With Defaultable Securities A Firm Value Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(08), pages 793-819.
    4. Black, Fischer & Cox, John C, 1976. "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions," Journal of Finance, American Finance Association, vol. 31(2), pages 351-367, May.
    5. Merton, Robert C, 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, American Finance Association, vol. 29(2), pages 449-470, May.
    6. Thaleia Zariphopoulou, 1999. "Optimal investment and consumption models with non-linear stock dynamics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(2), pages 271-296, October.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. Bruno Bouchard & Huyên Pham, 2004. "Wealth-path dependent utility maximization in incomplete markets," Finance and Stochastics, Springer, vol. 8(4), pages 579-603, November.
    9. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    10. repec:dau:papers:123456789/1803 is not listed on IDEAS
    11. Holger Kraft, 2003. "Elasticity approach to portfolio optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(1), pages 159-182, September.
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    Cited by:

    1. Ralf Korn, 2008. "Optimal portfolios: new variations of an old theme," Computational Management Science, Springer, vol. 5(4), pages 289-304, October.
    2. Belkacem Berdjane & Serguei Pergamenshchikov, 2013. "Optimal consumption and investment for markets with random coefficients," Finance and Stochastics, Springer, vol. 17(2), pages 419-446, April.

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    More about this item

    Keywords

    Optimal consumption and investment; Random time horizon; Feynman-Kac representation; Barrier options; G11;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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