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Numeraire-invariant quadratic hedging and mean–variance portfolio allocation

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  • Černý, Aleš
  • Czichowsky, Christoph
  • Kallsen, Jan

Abstract

The paper investigates quadratic hedging in a general semimartingale market that does not necessarily contain a risk-free asset. An equivalence result for hedging with and without numeraire change is established. This permits direct computation of the optimal strategy without choosing a reference asset and/or performing a numeraire change. New explicit expressions for optimal strategies are obtained, featuring the use of oblique projections that provide unified treatment of the case with and without a risk-free asset. The main result advances our understanding of the efficient frontier formation in the most general case where a risk-free asset may not be present. Several illustrations of the numeraire-invariant approach are given.

Suggested Citation

  • Černý, Aleš & Czichowsky, Christoph & Kallsen, Jan, 2021. "Numeraire-invariant quadratic hedging and mean–variance portfolio allocation," LSE Research Online Documents on Economics 112612, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:112612
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    File URL: http://eprints.lse.ac.uk/112612/
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    References listed on IDEAS

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    3. Yao, Haixiang & Li, Zhongfei & Chen, Shumin, 2014. "Continuous-time mean–variance portfolio selection with only risky assets," Economic Modelling, Elsevier, vol. 36(C), pages 244-251.
    4. Sara Biagini & Aleš Černý, 2020. "Convex duality and Orlicz spaces in expected utility maximization," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 85-127, January.
    5. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    6. Duan Li & Wan‐Lung Ng, 2000. "Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation," Mathematical Finance, Wiley Blackwell, vol. 10(3), pages 387-406, July.
    7. Takuji Arai, 2005. "An extension of mean-variance hedging to the discontinuous case," Finance and Stochastics, Springer, vol. 9(1), pages 129-139, January.
    8. Alev{s} v{C}ern'y & Johannes Ruf, 2020. "Simplified calculus for semimartingales: Multiplicative compensators and changes of measure," Papers 2006.12765, arXiv.org, revised May 2023.
    9. Dimitris Bertsimas & Leonid Kogan & Andrew W. Lo, 2001. "Hedging Derivative Securities and Incomplete Markets: An (epsilon)-Arbitrage Approach," Operations Research, INFORMS, vol. 49(3), pages 372-397, June.
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    More about this item

    Keywords

    mean–variance portfolio selection; quadratic hedging; numeraire change; oblique projection; opportunity-neutral measure; mean–variance hedging;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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