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Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model

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  • Jingtang Ma
  • Wenyuan Li
  • Harry Zheng

Abstract

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution of the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yarri utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.

Suggested Citation

  • Jingtang Ma & Wenyuan Li & Harry Zheng, 2017. "Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model," Papers 1710.10487, arXiv.org.
    • Handle: RePEc:arx:papers:1710.10487
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    References listed on IDEAS

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    1. Ma, Jingtang & Li, Wenyuan & Zheng, Harry, 2017. "Dual control Monte-Carlo method for tight bounds of value function in regime switching utility maximization," European Journal of Operational Research, Elsevier, vol. 262(3), pages 851-862.
    2. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    3. 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.
    4. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    5. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    6. 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.
    7. 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.
    8. Bian, Baojun & Zheng, Harry, 2015. "Turnpike property and convergence rate for an investment model with general utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 51(C), pages 28-49.
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