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Risk Measure Pricing and Hedging in Incomplete Markets


  • Mingxin Xu

    (University of North Carolina at Charlotte)


This article attempts to extend the complete market option pricing theory to incomplete markets. Instead of eliminating the risk by a perfect hedging portfolio, partial hedging will be adopted and some residual risk at expiration will be tolerated. The risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. We will give the definition of risk-efficient options and confirm that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures proposed in Carr et al. (2001) will be discussed. Examples using shortfall risk measure and average VaR will be discussed.

Suggested Citation

  • Mingxin Xu, 2004. "Risk Measure Pricing and Hedging in Incomplete Markets," Finance 0406004, EconWPA, revised 07 Mar 2006.
  • Handle: RePEc:wpa:wuwpfi:0406004
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    References listed on IDEAS

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    2. Marco Frittelli, 2000. "Introduction to a theory of value coherent with the no-arbitrage principle," Finance and Stochastics, Springer, vol. 4(3), pages 275-297.
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    Cited by:

    1. Michail Anthropelos & Gordan Žitković, 2010. "Partial equilibria with convex capital requirements: existence, uniqueness and stability," Annals of Finance, Springer, vol. 6(1), pages 107-135, January.
    2. Guo, Ivan & Zhu, Song-Ping, 2017. "Equal risk pricing under convex trading constraints," Journal of Economic Dynamics and Control, Elsevier, vol. 76(C), pages 136-151.
    3. repec:spr:decfin:v:41:y:2018:i:1:d:10.1007_s10203-018-0207-2 is not listed on IDEAS
    4. repec:eee:pacfin:v:45:y:2017:i:c:p:186-210 is not listed on IDEAS
    5. Hans Buhler & Lukas Gonon & Josef Teichmann & Ben Wood, 2018. "Deep Hedging," Papers 1802.03042,
    6. Michail Anthropelos, 2012. "The Effect of Market Power on Risk-Sharing," Papers 1206.0384,, revised May 2016.
    7. François, Pascal & Gauthier, Geneviève & Godin, Frédéric, 2014. "Optimal hedging when the underlying asset follows a regime-switching Markov process," European Journal of Operational Research, Elsevier, vol. 237(1), pages 312-322.
    8. Wayne King Ming Chan, 2015. "RAROC-Based Contingent Claim Valuation," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 21, June.
    9. Takuji Arai, 2015. "Good deal bounds with convex constraints," Papers 1506.00396,
    10. Takuji Arai & Masaaki Fukasawa, 2011. "Convex risk measures for good deal bounds," Papers 1108.1273,
    11. Pascal François & Geneviève Gauthier & Frédéric Godin, 2012. "Optimal Hedging when the Underlying Asset Follows a Regime-switching Markov Process," Cahiers de recherche 1234, CIRPEE.
    12. Michail Anthropelos & Nikolaos E. Frangos & Stylianos Z. Xanthopoulos & Athanasios N. Yannacopoulos, 2008. "On contingent claims pricing in incomplete markets: A risk sharing approach," Papers 0809.4781,, revised Feb 2012.
    13. Takuji Arai, 2016. "Good deal bounds with convex constraints: --- examples and proofs ---," Keio-IES Discussion Paper Series 2016-017, Institute for Economics Studies, Keio University.
    14. Wurth, A.M., 2009. "Pricing and hedging in incomplete financial markets," Other publications TiSEM 45e60d16-cf9e-4740-bb05-0, Tilburg University, School of Economics and Management.
    15. Mustafa Pınar, 2011. "Gain–loss based convex risk limits in discrete-time trading," Computational Management Science, Springer, vol. 8(3), pages 299-321, August.
    16. Ravi Kashyap, 2016. "Securities Lending Strategies, Valuation of Term Loans using Option Theory," Papers 1609.01274,, revised Nov 2016.
    17. Michail Anthropelos & Gordan Zitkovic, 2009. "Partial Equilibria with Convex Capital Requirements: Existence, Uniqueness and Stability," Papers 0901.3318,
    18. repec:wsi:ijtafx:v:20:y:2017:i:02:n:s021902491750011x is not listed on IDEAS

    More about this item


    Pricing and Hedging; Incomplete Markets; Dynamic Shortfall Risk; Average Value at Risk; Utility Indifference Pricing; Convex Measure of Risk; Coherent Risk Measure; Risk-Efficient Options; Semimartingale Models;

    JEL classification:

    • G - Financial Economics

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