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Mean–Variance Hedging And Optimal Investment In Heston'S Model With Correlation

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  • Aleš Černý
  • Jan Kallsen

Abstract

This paper solves the mean–variance hedging problem in Heston's model with a stochastic opportunity set moving systematically with the volatility of stock returns. We allow for correlation between stock returns and their volatility (so‐called leverage effect). Our contribution is threefold: using a new concept of opportunity‐neutral measure we present a simplified strategy for computing a candidate solution in the correlated case. We then go on to show that this candidate generates the true variance‐optimal martingale measure; this step seems to be partially missing in the literature. Finally, we derive formulas for the hedging strategy and the hedging error.

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  • Aleš Černý & Jan Kallsen, 2008. "Mean–Variance Hedging And Optimal Investment In Heston'S Model With Correlation," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 473-492, July.
    • Handle: RePEc:bla:mathfi:v:18:y:2008:i:3:p:473-492
    DOI: 10.1111/j.1467-9965.2008.00342.x
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    References listed on IDEAS

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    4. Alev{s} v{C}ern'y & Jan Kallsen, 2007. "On the Structure of General Mean-Variance Hedging Strategies," Papers 0708.1715, arXiv.org, revised Jul 2017.
    5. Friedrich Hubalek & Jan Kallsen & Leszek Krawczyk, 2006. "Variance-optimal hedging for processes with stationary independent increments," Papers math/0607112, arXiv.org.
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    1. Aleš Černý & Jan Kallsen, 2008. "A Counterexample Concerning The Variance‐Optimal Martingale Measure," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 305-316, April.
    2. Alev{s} v{C}ern'y & Jan Kallsen, 2007. "On the Structure of General Mean-Variance Hedging Strategies," Papers 0708.1715, arXiv.org, revised Jul 2017.
    3. Liao Wang & Johannes Wissel, 2013. "Mean-variance hedging with oil futures," Finance and Stochastics, Springer, vol. 17(4), pages 641-683, October.
    4. Alev{s} v{C}ern'y & Christoph Czichowsky, 2022. "The law of one price in quadratic hedging and mean-variance portfolio selection," Papers 2210.15613, arXiv.org, revised Sep 2024.
    5. Černý, Aleš & Maccheroni, Fabio & Marinacci, Massimo & Rustichini, Aldo, 2012. "On the computation of optimal monotone mean–variance portfolios via truncated quadratic utility," Journal of Mathematical Economics, Elsevier, vol. 48(6), pages 386-395.
    6. Chiu, Mei Choi & Wong, Hoi Ying, 2014. "Mean–variance asset–liability management with asset correlation risk and insurance liabilities," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 300-310.
    7. Jan Kallsen & Arnd Pauwels, 2011. "Variance-Optimal Hedging for Time-Changed Levy Processes," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(1), pages 1-28.
    8. 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.
    9. Flavio ANGELINI & Stefano HERZEL, 2012. "Delta Hedging in Discrete Time under Stochastic Interest Rate," Quaderni del Dipartimento di Economia, Finanza e Statistica 110/2012, Università di Perugia, Dipartimento Economia.
    10. Marcos Escobar & Daniela Neykova & Rudi Zagst, 2015. "Portfolio Optimization In Affine Models With Markov Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-46.
    11. Aiqin Ma & Cuiyun Zhang & Yubing Wang, 2023. "Optimal Consumption and Investment Problem under 4/2-CIR Stochastic Hybrid Model," Mathematics, MDPI, vol. 11(17), pages 1-19, August.
    12. Dorival Le~ao & Alberto Ohashi & Vinicius Siqueira, 2013. "A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility," Papers 1308.1704, arXiv.org, revised Aug 2013.
    13. Yang Shen, 2020. "Effect of Variance Swap in Hedging Volatility Risk," Risks, MDPI, vol. 8(3), pages 1-34, July.

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