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Optimal investment in derivative securities

Author

Listed:
  • Dilip B. Madan

    (Robert H. Smith School of Business, Van Munching Hall, University of Maryland, College Park, MD. 20742, USA Manuscript)

  • Xing Jin

    (Robert H. Smith School of Business, Van Munching Hall, University of Maryland, College Park, MD. 20742, USA Manuscript)

  • Peter Carr

    (Banc of America Securities, 9 West 57th Street, 40th floor, New York, N.Y. 10019, USA)

Abstract

We consider the problem of optimal investment in a risky asset, and in derivatives written on the price process of this asset, when the underlying asset price process is a pure jump Lévy process. The duality approach of Karatzas and Shreve is used to derive the optimal consumption and investment plans. In our economy, the optimal derivative payoff can be constructed from dynamic trading in the risky asset and in European options of all strikes. Specific closed forms illustrate the optimal derivative contracts when the utility function is in the HARA class and when the statistical and risk-neutral price processes are in the variance gamma (VG) class. In this case, we observe that the optimal derivative contract pays a function of the price relatives continuously through time.

Suggested Citation

  • Dilip B. Madan & Xing Jin & Peter Carr, 2001. "Optimal investment in derivative securities," Finance and Stochastics, Springer, vol. 5(1), pages 33-59.
    • Handle: RePEc:spr:finsto:v:5:y:2001:i:1:p:33-59
    Note: received: November 1999; final version received: February 2000
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    Citations

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    Cited by:

    1. Topaloglou, Nikolas & Vladimirou, Hercules & Zenios, Stavros A., 2011. "Optimizing international portfolios with options and forwards," Journal of Banking & Finance, Elsevier, vol. 35(12), pages 3188-3201.
    2. Libo Yin & Liyan Han, 2013. "Options strategies for international portfolios with overall risk management via multi-stage stochastic programming," Annals of Operations Research, Springer, vol. 206(1), pages 557-576, July.
    3. Pietro Siorpaes, 2015. "Optimal investment and price dependence in a semi-static market," Finance and Stochastics, Springer, vol. 19(1), pages 161-187, January.
    4. Jakša Cvitanić & Vassilis Polimenis & Fernando Zapatero, 2008. "Optimal portfolio allocation with higher moments," Annals of Finance, Springer, vol. 4(1), pages 1-28, January.
    5. Bo, Lijun & Tang, Dan & Wang, Yongjin, 2017. "Optimal investment of variance-swaps in jump-diffusion market with regime-switching," Journal of Economic Dynamics and Control, Elsevier, vol. 83(C), pages 175-197.
    6. Liu, Jun & Pan, Jun, 2003. "Dynamic derivative strategies," Journal of Financial Economics, Elsevier, vol. 69(3), pages 401-430, September.
    7. Yacine Aït-Sahalia & Thomas Robert Hurd, 2016. "Portfolio Choice in Markets with Contagion," Journal of Financial Econometrics, Oxford University Press, vol. 14(1), pages 1-28.
    8. Jun Deng & Huifeng Pan & Shuyu Zhang & Bin Zou, 2021. "Optimal Bitcoin trading with inverse futures," Annals of Operations Research, Springer, vol. 304(1), pages 139-163, September.
    9. Ajay Khanna & Dilip Madan, 2004. "Understanding option prices," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 55-63.
    10. Robert Kohn & Oana Papazoglu-Statescu, 2006. "On the equivalence of the static and dynamic asset allocation problems," Quantitative Finance, Taylor & Francis Journals, vol. 6(2), pages 173-183.
    11. Aït-Sahalia, Yacine & Matthys, Felix, 2019. "Robust consumption and portfolio policies when asset prices can jump," Journal of Economic Theory, Elsevier, vol. 179(C), pages 1-56.
    12. Xue, Xiaole & Wei, Pengyu & Weng, Chengguo, 2019. "Derivatives trading for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 40-53.

    More about this item

    Keywords

    Lévy process; market completeness; stochastic duality; option pricing; variance gamma model;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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