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Optimal Investment with Correlated Stochastic Volatility Factors

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  • Maxim Bichuch
  • Jean-Pierre Fouque

Abstract

The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the situation with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. Our proposed approximation requires to solve numerically two linear equations in lower dimension instead of a fully non-linear HJB equation. A rigorous accuracy result is derived by constructing sub- and super- solutions so that their difference is at the desired order of accuracy. We illustrate our result with a particular model for which we have explicit formulas for the approximation. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.

Suggested Citation

  • Maxim Bichuch & Jean-Pierre Fouque, 2019. "Optimal Investment with Correlated Stochastic Volatility Factors," Papers 1908.07626, arXiv.org, revised Nov 2022.
    • Handle: RePEc:arx:papers:1908.07626
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    References listed on IDEAS

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    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584.
    2. Thaleia Zariphopoulou, 1999. "Optimal investment and consumption models with non-linear stock dynamics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(2), pages 271-296, October.
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    4. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    5. Levon Avanesyan & Mykhaylo Shkolnikov & Ronnie Sircar, 2018. "Construction of Forward Performance Processes in Stochastic Factor Models and an Extension of Widder's Theorem," Papers 1805.04535, arXiv.org.
    6. Jean-Pierre Fouque & George Papanicolaou & K. Ronnie Sircar, 2000. "Mean-Reverting Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 101-142.
    7. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    8. Jean-Pierre Fouque & Ruimeng Hu, 2017. "Optimal Portfolio under Fractional Stochastic Environment," Papers 1703.06969, arXiv.org, revised Dec 2017.
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