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Optimal portfolios for exponential Lévy processes

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  • Jan Kallsen

Abstract

We consider the problem of maximizing the expected utility from consumption or terminal wealth in a market where logarithmic securities prices follow a Lévy process. More specifically, we give explicit solutions for power, logarithmic and exponential utility in terms of the Lévy-Khintchine triplet. In the first two cases, a constant fraction of current wealth should be invested in each of the securities, as is well-known for related discrete-time models and for Brownian motion. The situation is different for exponential utility. Copyright Springer-Verlag Berlin Heidelberg 2000

Suggested Citation

  • Jan Kallsen, 2000. "Optimal portfolios for exponential Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(3), pages 357-374, August.
    • Handle: RePEc:spr:mathme:v:51:y:2000:i:3:p:357-374
    DOI: 10.1007/s001860000048
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    Citations

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    Cited by:

    1. Thorsten Rheinländer & Jenny Sexton, 2011. "Hedging Derivatives," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8062.
    2. Küchler Uwe & Tappe Stefan, 2009. "Option pricing in bilateral Gamma stock models," Statistics & Risk Modeling, De Gruyter, vol. 27(4), pages 281-307, December.
    3. Martin Herdegen & Sebastian Herrmann, 2017. "Strict Local Martingales and Optimal Investment in a Black-Scholes Model with a Bubble," Papers 1711.06679, arXiv.org.
    4. Alev{s} v{C}ern'y & Johannes Ruf, 2019. "Simplified stochastic calculus with applications in Economics and Finance," Papers 1912.03651, arXiv.org, revised Jan 2021.
    5. João Guerra & Manuel Guerra & Zachary Polaski, 2019. "Market Timing with Option-Implied Distributions in an Exponentially Tempered Stable Lévy Market," Working Papers REM 2019/74, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    6. Černý, Aleš & Ruf, Johannes, 2020. "Simplified stochastic calculus with applications in economics and finance," LSE Research Online Documents on Economics 108156, London School of Economics and Political Science, LSE Library.
    7. Laura Pasin & Tiziano Vargiolu, 2010. "Optimal Portfolio for CRRA Utility Functions when Risky Assets are Exponential Additive Processes," Economic Notes, Banca Monte dei Paschi di Siena SpA, vol. 39(1‐2), pages 65-90, February.
    8. S. Cawston & L. Vostrikova, 2010. "$F$-divergence minimal equivalent martingale measures and optimal portfolios for exponential Levy models with a change-point," Papers 1004.3525, arXiv.org, revised Jun 2011.
    9. Fred Espen Benth & Jan Kallsen & Thilo Meyer-Brandis, 2007. "A Non-Gaussian Ornstein-Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(2), pages 153-169.
    10. Mauricio Junca & Rafael Serrano, 2014. "Utility maximization in pure-jump models driven by marked point processes and nonlinear wealth dynamics," Papers 1411.1103, arXiv.org, revised Sep 2015.
    11. Robert Jarrow, 2018. "An Equilibrium Capital Asset Pricing Model in Markets with Price Jumps and Price Bubbles," Quarterly Journal of Finance (QJF), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 1-33, June.
    12. Jérôme Detemple, 2014. "Portfolio Selection: A Review," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 1-21, April.
    13. Ariel Neufeld & Marcel Nutz, 2015. "Robust Utility Maximization with L\'evy Processes," Papers 1502.05920, arXiv.org, revised Mar 2016.
    14. Zongxia Liang & Ming Ma, 2020. "Robust consumption‐investment problem under CRRA and CARA utilities with time‐varying confidence sets," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1035-1072, July.
    15. Alev{s} v{C}ern'y & Johannes Ruf, 2020. "Simplified stochastic calculus via semimartingale representations," Papers 2006.11914, arXiv.org, revised Jan 2022.
    16. Milan Kumar Das & Anindya Goswami & Nimit Rana, 2016. "Risk Sensitive Portfolio Optimization in a Jump Diffusion Model with Regimes," Papers 1603.09149, arXiv.org, revised Jan 2018.
    17. Černý, Aleš & Ruf, Johannes, 2021. "Simplified stochastic calculus with applications in Economics and Finance," European Journal of Operational Research, Elsevier, vol. 293(2), pages 547-560.
    18. Rafael Serrano & Camilo Castillo, 2018. "ALM for insurers with multiple underwriting lines and portfolio constraints: a Lagrangian duality approach," Papers 1810.08466, arXiv.org, revised Aug 2021.
    19. Marco Piccirilli & Tiziano Vargiolu, 2018. "Optimal Portfolio in Intraday Electricity Markets Modelled by L\'evy-Ornstein-Uhlenbeck Processes," Papers 1807.01979, arXiv.org.
    20. Paweł Kliber, 2019. "Continuous and jump changes in prices processes in the selected stock markets," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 54, pages 333-344.
    21. 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.
    22. Buccioli, Alice & Kokholm, Thomas & Nicolosi, Marco, 2019. "Expected shortfall and portfolio management in contagious markets," Journal of Banking & Finance, Elsevier, vol. 102(C), pages 100-115.
    23. Le Courtois, Olivier & Menoncin, Francesco, 2015. "Portfolio optimisation with jumps: Illustration with a pension accumulation scheme," Journal of Banking & Finance, Elsevier, vol. 60(C), pages 127-137.

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