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Worst-Case Optimal Investment in Incomplete Markets

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  • Sascha Desmettre
  • Sebastian Merkel
  • Annalena Mickel
  • Alexander Steinicke

Abstract

We study and solve the worst-case optimal portfolio problem as pioneered by Korn and Wilmott (2002) of an investor with logarithmic preferences facing the possibility of a market crash with stochastic market coefficients by enhancing the martingale approach developed by Seifried in 2010. With the help of backward stochastic differential equations (BSDEs), we are able to characterize the resulting indifference optimal strategies in a fairly general setting. We also deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. We therefore solve the corresponding BSDEs via solving their associated PDEs using a utility crash-exposure transformation. Our approach is subsequently demonstrated for Heston's stochastic volatility model, Bates' stochastic volatility model including jumps, and Kim-Omberg's model for a stochastic excess return.

Suggested Citation

  • Sascha Desmettre & Sebastian Merkel & Annalena Mickel & Alexander Steinicke, 2023. "Worst-Case Optimal Investment in Incomplete Markets," Papers 2311.10021, arXiv.org, revised Dec 2024.
    • Handle: RePEc:arx:papers:2311.10021
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    References listed on IDEAS

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    1. George Chacko & Luis M. Viceira, 2005. "Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets," The Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1369-1402.
    2. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    3. Alexander Schied, 2005. "Optimal Investments for Robust Utility Functionals in Complete Market Models," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 750-764, August.
    4. Kim, Tong Suk & Omberg, Edward, 1996. "Dynamic Nonmyopic Portfolio Behavior," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 141-161.
    5. Briand, Ph. & Delyon, B. & Hu, Y. & Pardoux, E. & Stoica, L., 2003. "Lp solutions of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 109-129, November.
    6. Holger Kraft, 2005. "Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility," Quantitative Finance, Taylor & Francis Journals, vol. 5(3), pages 303-313.
    7. Tina Engler & Ralf Korn, 2014. "Worst-Case Portfolio Optimization under Stochastic Interest Rate Risk," Risks, MDPI, vol. 2(4), pages 1-20, December.
    8. Belak, Christoph & Christensen, Sören & Menkens, Olaf, 2014. "Worst-case optimal investment with a random number of crashes," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 140-148.
    9. Engler, Tina & Korn, Ralf, 2014. "Worst-case portfolio optimization under stochastic interest rate risk," EconStor Open Access Articles and Book Chapters, ZBW - Leibniz Information Centre for Economics, vol. 2(4), pages 469-488.
    10. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    11. 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, January.
    12. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    13. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    14. Ralf Korn & Olaf Menkens, 2005. "Worst-Case Scenario Portfolio Optimization: a New Stochastic Control Approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(1), pages 123-140, September.
    15. Briand, Philippe & Confortola, Fulvia, 2008. "BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 118(5), pages 818-838, May.
    16. Branger, Nicole & Schlag, Christian & Schneider, Eva, 2008. "Optimal portfolios when volatility can jump," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 1087-1097, June.
    17. Frank Thomas Seifried, 2010. "Optimal Investment for Worst-Case Crash Scenarios: A Martingale Approach," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 559-579, August.
    18. Denis Talay & Ziyu Zheng, 2002. "Worst case model risk management," Finance and Stochastics, Springer, vol. 6(4), pages 517-537.
    19. Ralf Korn & Paul Wilmott, 2002. "Optimal Portfolios Under The Threat Of A Crash," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 171-187.
    20. Jan Kallsen & Johannes Muhle-Karbe, 2010. "Utility Maximization In Affine Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(03), pages 459-477.
    21. Jun Liu, 2007. "Portfolio Selection in Stochastic Environments," The Review of Financial Studies, Society for Financial Studies, vol. 20(1), pages 1-39, January.
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