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Merton's portfolio optimization problem in a Black and Scholes market with non‐Gaussian stochastic volatility of Ornstein‐Uhlenbeck type

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  • Fred Espen Benth
  • Kenneth Hvistendahl Karlsen
  • Kristin Reikvam

Abstract

We study Merton's classical portfolio optimization problem for an investor who can trade in a risk‐free bond and a stock. The goal of the investor is to allocate money so that her expected utility from terminal wealth is maximized. The special feature of the problem studied in this paper is the inclusion of stochastic volatility in the dynamics of the risky asset. The model we use is driven by a superposition of non‐Gaussian Ornstein‐Uhlenbeck processes and it was recently proposed and intensively investigated for real market data by Barndorff‐Nielsen and Shephard (2001). Using the dynamic programming method, explicit trading strategies and expressions for the value function via Feynman‐Kac formulas are derived and verified for power utilities. Some numerical examples are also presented.

Suggested Citation

  • Fred Espen Benth & Kenneth Hvistendahl Karlsen & Kristin Reikvam, 2003. "Merton's portfolio optimization problem in a Black and Scholes market with non‐Gaussian stochastic volatility of Ornstein‐Uhlenbeck type," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 215-244, April.
    • Handle: RePEc:bla:mathfi:v:13:y:2003:i:2:p:215-244
    DOI: 10.1111/1467-9965.00015
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    Cited by:

    1. Richter, Anja, 2014. "Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3578-3611.
    2. Francesco C. De Vecchi & Elisa Mastrogiacomo & Mattia Turra & Stefania Ugolini, 2021. "Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries," Mathematics, MDPI, vol. 9(9), pages 1-34, April.
    3. Francesco C. De Vecchi & Elisa Mastrogiacomo & Mattia Turra & Stefania Ugolini, 2021. "Noether theorem in stochastic optimal control problems via contact symmetries," Papers 2102.03172, arXiv.org.
    4. Escobar, Marcos & Ferrando, Sebastian & Rubtsov, Alexey, 2015. "Robust portfolio choice with derivative trading under stochastic volatility," Journal of Banking & Finance, Elsevier, vol. 61(C), pages 142-157.
    5. Daeyung Gim & Hyungbin Park, 2021. "A deep learning algorithm for optimal investment strategies," Papers 2101.12387, arXiv.org.
    6. {L}ukasz Delong & Claudia Kluppelberg, 2008. "Optimal investment and consumption in a Black--Scholes market with L\'evy-driven stochastic coefficients," Papers 0806.2570, arXiv.org.
    7. Indranil SenGupta & William Nganje & Erik Hanson, 2021. "Refinements of Barndorff-Nielsen and Shephard Model: An Analysis of Crude Oil Price with Machine Learning," Annals of Data Science, Springer, vol. 8(1), pages 39-55, March.
    8. Carl Lindberg, 2008. "The estimation of the Barndorff‐Nielsen and Shephard model from daily data based on measures of trading intensity," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 24(4), pages 277-289, July.
    9. Szczepocki Piotr, 2020. "Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process," Statistics in Transition New Series, Polish Statistical Association, vol. 21(2), pages 173-187, June.
    10. Kallsen Jan & Muhle-Karbe Johannes, 2011. "Method of moment estimation in time-changed Lévy models," Statistics & Risk Modeling, De Gruyter, vol. 28(2), pages 169-194, May.
    11. Lancelot F. James, 2005. "Analysis of a Class of Likelihood Based Continuous Time Stochastic Volatility Models including Ornstein-Uhlenbeck Models in Financial Economics," Papers math/0503055, arXiv.org, revised Aug 2005.
    12. Semere Habtemicael & Indranil SenGupta, 2016. "Pricing variance and volatility swaps for Barndorff-Nielsen and Shephard process driven financial markets," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-35, December.
    13. Semere Habtemicael & Indranil Sengupta, 2016. "Pricing Covariance Swaps For Barndorff–Nielsen And Shephard Process Driven Financial Markets," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 11(03), pages 1-32, September.
    14. Marco Piccirilli & Tiziano Vargiolu, 2018. "Optimal Portfolio in Intraday Electricity Markets Modelled by L\'evy-Ornstein-Uhlenbeck Processes," Papers 1807.01979, arXiv.org.
    15. Friedrich Hubalek & Petra Posedel, 2008. "Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models," Papers 0807.3479, arXiv.org.
    16. Wanyang Dai, 2014. "Mean-variance hedging based on an incomplete market with external risk factors of non-Gaussian OU processes," Papers 1410.0991, arXiv.org, revised Aug 2015.
    17. Piotr Szczepocki, 2020. "Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process," Statistics in Transition New Series, Polish Statistical Association, vol. 21(2), pages 173-187, June.

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