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On the optimal wealth process in a log-normal market: Applications to risk management

Author

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  • Phillip Monin

    (Office of Financial Research, U.S. Department of the Treasury, 717 14th St NW, Washington, DC 20005, USA)

  • Thaleia Zariphopoulou

    (Department of Mathematics and IROM, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, TX 78712, USA)

Abstract

Using a stochastic representation of the optimal wealth process in the classical Merton problem, we calculate its cumulative distribution and density functions and provide bounds and monotonicity results for these quantities under general risk preferences. We also show that the optimal wealth and portfolio processes for different utility functions are related through a deterministic transformation and appropriately modified initial conditions. We analyze the value at risk (VaR) and expected shortfall (ES) of the optimal wealth process and show how each can be used to infer a constant relative risk aversion (CRRA) investor's risk aversion coefficient. Drawing analogies to the option greeks, we study the sensitivities of the optimal wealth process with respect to the cumulative excess stock return, time, and market parameters. We conclude with a study of how sensitivities of the excess return on the optimal wealth process relate to the portfolio's beta.

Suggested Citation

  • Phillip Monin & Thaleia Zariphopoulou, 2014. "On the optimal wealth process in a log-normal market: Applications to risk management," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(02), pages 1-37.
    • Handle: RePEc:wsi:jfexxx:v:01:y:2014:i:02:n:s2345768614500135
    DOI: 10.1142/S2345768614500135
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    Cited by:

    1. Ljudmila A. Bordag, 2019. "Portfolio optimization in the case of an exponential utility function and in the presence of an illiquid asset," Papers 1910.07417, arXiv.org, revised May 2020.
    2. Wai Mun Fong, 2018. "Synthetic growth stocks," Journal of Asset Management, Palgrave Macmillan, vol. 19(3), pages 162-168, May.

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