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A guided tour through quadratic hedging approaches

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  • Schweizer, Martin

Abstract

This paper gives an overview of results and developments in the area of pricing and hedging contingent claims in an incomplete market by means of a quadratic criterion. We first present the approach of risk-minimization in the case where the underlying discounted price process X is a local martingale. We then discuss the extension to local risk-minimization when X is a semimartingale and explain the relations to the Föllmer-Schweizer decomposition and the minimal martingale measure. Finally we study mean-variance hedging, the variance-optimal martingale measure and the connections to closeness properties of spaces of stochastic integrals.

Suggested Citation

  • Schweizer, Martin, 1999. "A guided tour through quadratic hedging approaches," SFB 373 Discussion Papers 1999,96, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:199996
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
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    5. Rheinländer, Thorsten & Schweizer, Martin, 1997. "On L2-projections on a space of stochastic integrals," SFB 373 Discussion Papers 1997,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    6. Lamberton, Damien & Pham, Huyên & Schweizer, Martin, 1998. "Local risk-minimization under transaction costs," SFB 373 Discussion Papers 1998,18, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    7. Föllmer, H. & Schweizer, M., 1989. "Hedging by Sequential Regression: an Introduction to the Mathematics of Option Trading," ASTIN Bulletin, Cambridge University Press, vol. 19(S1), pages 29-42, November.
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    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Cited by:

    1. Xi Kleisinger-Yu & Vlatka Komaric & Martin Larsson & Markus Regez, 2019. "A multi-factor polynomial framework for long-term electricity forwards with delivery period," Papers 1908.08954, arXiv.org, revised Jun 2020.
    2. S. Cawston & L. Vostrikova, 2010. "$F$-divergence minimal equivalent martingale measures and optimal portfolios for exponential Levy models with a change-point," Papers 1004.3525, arXiv.org, revised Jun 2011.
    3. Moller, Thomas, 2001. "On transformations of actuarial valuation principles," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 281-303, June.
    4. Thomas Møller, 2001. "Risk-minimizing hedging strategies for insurance payment processes," Finance and Stochastics, Springer, vol. 5(4), pages 419-446.
    5. Simon F'ecamp & Joseph Mikael & Xavier Warin, 2019. "Risk management with machine-learning-based algorithms," Papers 1902.05287, arXiv.org, revised Aug 2020.
    6. D. G. Luenberger, 2004. "Pricing a Nontradeable Asset and Its Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 465-487, June.
    7. 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.
    8. Moreau, Ludovic, 2012. "A contribution in stochastic control applied to finance and insurance," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/10711 edited by Bouchard, Bruno.
    9. 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.
    10. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2023. "Deep stochastic optimization in finance," Digital Finance, Springer, vol. 5(1), pages 91-111, March.
    11. Wang, Qizhi & Chidmi, Benaissa, 2009. "Cotton Price Risk Management across Different Countries," 2009 Annual Meeting, January 31-February 3, 2009, Atlanta, Georgia 46762, Southern Agricultural Economics Association.
    12. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    13. Paolo Di Tella & Martin Haubold & Martin Keller-Ressel, 2017. "Semi-Static Variance-Optimal Hedging in Stochastic Volatility Models with Fourier Representation," Papers 1709.05527, arXiv.org.
    14. Arunangshu Biswas & Anindya Goswami & Ludger Overbeck, 2017. "Option Pricing in a Regime Switching Stochastic Volatility Model," Papers 1707.01237, arXiv.org, revised Jan 2018.
    15. 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.
    16. Emmanuel Gobet & Isaque Pimentel & Xavier Warin, 2018. "Option valuation and hedging using asymmetric risk function: asymptotic optimality through fully nonlinear Partial Differential Equations," Working Papers hal-01761234, HAL.
    17. Emmanuel Gobet & Isaque Pimentel & Xavier Warin, 2020. "Option valuation and hedging using an asymmetric risk function: asymptotic optimality through fully nonlinear partial differential equations," Finance and Stochastics, Springer, vol. 24(3), pages 633-675, July.
    18. Tahir Choulli & Jun Deng, 2014. "Structure conditions under progressively added information," Papers 1403.3459, arXiv.org, revised Dec 2018.
    19. Masahiro Fujimoto, 2018. "The Theoretical Price of a Share-Based Payment with Performance Conditions and Implications for the Current Accounting Standards," Papers 1806.05401, arXiv.org.
    20. Saeed Marzban & Erick Delage & Jonathan Yumeng Li, 2020. "Equal Risk Pricing and Hedging of Financial Derivatives with Convex Risk Measures," Papers 2002.02876, arXiv.org, revised Sep 2020.
    21. Zehra Eksi & Damir Filipovi'c, 2020. "Affine Pricing and Hedging of Collateralized Debt Obligations," Papers 2011.10101, arXiv.org.
    22. Ariel Neufeld & Philipp Schmocker, 2022. "Chaotic Hedging with Iterated Integrals and Neural Networks," Papers 2209.10166, arXiv.org, revised Feb 2023.

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    More about this item

    Keywords

    risk-minimization; locally risk-minimizing; mean-variance hedging; minimal martingale measure; variance-optimal martingale measure; Föllmer-Schweizer decomposition; quadratic hedging criteria; incomplete markets;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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