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Mean‐Variance Hedging for Stochastic Volatility Models

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  • Francesca Biagini
  • Paolo Guasoni
  • Maurizio Pratelli

Abstract

In this paper we discuss the tractability of stochastic volatility models for pricing and hedging options with the mean‐variance hedging approach. We characterize the variance‐optimal measure as the solution of an equation between Doléans exponentials; explicit examples include both models where volatility solves a diffusion equation and models where it follows a jump process. We further discuss the closedness of the space of strategies.

Suggested Citation

  • Francesca Biagini & Paolo Guasoni & Maurizio Pratelli, 2000. "Mean‐Variance Hedging for Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 109-123, April.
    • Handle: RePEc:bla:mathfi:v:10:y:2000:i:2:p:109-123
    DOI: 10.1111/1467-9965.00084
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    Citations

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    Cited by:

    1. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Sep 2023.
    2. Michael Mania & Revaz Tevzadze, 2008. "Backward Stochastic PDEs Related to the Utility Maximization Problem," ICER Working Papers - Applied Mathematics Series 07-2008, ICER - International Centre for Economic Research.
    3. Thorsten Rheinländer, 2005. "An entropy approach to the Stein and Stein model with correlation," Finance and Stochastics, Springer, vol. 9(3), pages 399-413, July.
    4. Chenxu Li, 2016. "Bessel Processes, Stochastic Volatility, And Timer Options," Mathematical Finance, Wiley Blackwell, vol. 26(1), pages 122-148, January.
    5. Abootaleb Shirvani & Frank J. Fabozzi & Stoyan V. Stoyanov, 2020. "Option Pricing in an Investment Risk-Return Setting," Papers 2001.00737, arXiv.org.
    6. David Hobson, 2004. "STOCHASTIC VOLATILITY MODELS, CORRELATION, AND THE q‐OPTIMAL MEASURE," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 537-556, October.
    7. Tahir Choulli & Jun Deng, 2014. "Structure conditions under progressively added information," Papers 1403.3459, arXiv.org, revised Dec 2018.
    8. Tak Siu, 2006. "Option Pricing Under Autoregressive Random Variance Models," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 62-75.
    9. Aleš Černý, 2007. "Optimal Continuous‐Time Hedging With Leptokurtic Returns," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 175-203, April.
    10. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2, July-Dece.
    11. David B. Colwell & Nadima El‐Hassan & Oh Kang Kwon, 2021. "Variance minimizing strategies for stochastic processes with applications to tracking stock indices," International Review of Finance, International Review of Finance Ltd., vol. 21(2), pages 430-446, June.
    12. Vicky Henderson & David Hobson & Sam Howison & Tino Kluge, 2005. "A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation," Review of Derivatives Research, Springer, vol. 8(1), pages 5-25, June.
    13. Ke Du, 2013. "Commodity Derivative Pricing Under the Benchmark Approach," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2013.
    14. Stoyan V. Stoyanov & Yong Shin Kim & Svetlozar T. Rachev & Frank J. Fabozzi, 2017. "Option pricing for Informed Traders," Papers 1711.09445, arXiv.org.
    15. Giulia Di Nunno & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2022. "Option pricing in Sandwiched Volterra Volatility model," Papers 2209.10688, arXiv.org, revised Dec 2023.
    16. Dilip Madan, 2011. "Joint risk-neutral laws and hedging," IISE Transactions, Taylor & Francis Journals, vol. 43(12), pages 840-850.
    17. 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.
    18. Vicky Henderson & David Hobson & Sam Howison & Tino Kluge, 2003. "A Comparison of q-optimal Option Prices in a Stochastic Volatility Model with Correlation," OFRC Working Papers Series 2003mf02, Oxford Financial Research Centre.
    19. Yang Shen, 2020. "Effect of Variance Swap in Hedging Volatility Risk," Risks, MDPI, vol. 8(3), pages 1-34, July.

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