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Risk measure pricing and hedging in incomplete markets

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  • Mingxin Xu

Abstract

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.
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Suggested Citation

  • Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January.
    • Handle: RePEc:kap:annfin:v:2:y:2006:i:1:p:51-71
    DOI: 10.1007/s10436-005-0023-x
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    More about this item

    Keywords

    Derivative pricing; Valuation 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; Risk indifference pricing; C60; D46; G13;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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