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Extending pricing rules with general risk functions


  • Balbás, Alejandro
  • Balbás, Raquel
  • Garrido, José


The paper addresses pricing issues in imperfect and/or incomplete markets if the risk level of the hedging strategy is measured by a general risk function. Convex Optimization Theory is used in order to extend pricing rules for a wide family of risk functions, including Deviation Measures, Expectation Bounded Risk Measures and Coherent Measures of Risk. Necessary and sufficient optimality conditions are provided in a very general setting. For imperfect markets the extended pricing rules reduce the bid-ask spread. The findings are particularized so as to study with more detail some concrete examples, including the Conditional Value at Risk and some properties of the Standard Deviation. Applications dealing with the valuation of volatility linked derivatives are discussed.

Suggested Citation

  • Balbás, Alejandro & Balbás, Raquel & Garrido, José, 2010. "Extending pricing rules with general risk functions," European Journal of Operational Research, Elsevier, vol. 201(1), pages 23-33, February.
  • Handle: RePEc:eee:ejores:v:201:y:2010:i:1:p:23-33

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    References listed on IDEAS

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    6. Jeremy Staum, 2004. "Fundamental Theorems of Asset Pricing for Good Deal Bounds," Mathematical Finance, Wiley Blackwell, vol. 14(2), pages 141-161.
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    8. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    9. Renata Mansini & Włodzimierz Ogryczak & M. Speranza, 2007. "Conditional value at risk and related linear programming models for portfolio optimization," Annals of Operations Research, Springer, vol. 152(1), pages 227-256, July.
    10. Hanqing Jin & Zuo Quan Xu & Xun Yu Zhou, 2008. "A Convex Stochastic Optimization Problem Arising From Portfolio Selection," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 171-183.
    11. Ogryczak, Wlodzimierz & Ruszczynski, Andrzej, 1999. "From stochastic dominance to mean-risk models: Semideviations as risk measures," European Journal of Operational Research, Elsevier, vol. 116(1), pages 33-50, July.
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    1. repec:gam:jrisks:v:5:y:2017:i:3:p:36-:d:105112 is not listed on IDEAS
    2. Balbás, Alejandro & Balbás, Beatriz & Heras, Antonio, 2011. "Stable solutions for optimal reinsurance problems involving risk measures," European Journal of Operational Research, Elsevier, vol. 214(3), pages 796-804, November.
    3. Heras, Antonio & Balbás, Alejandro & Balbás, Beatriz, 2010. "Stability of the optimal reinsurance with respect to the risk measure," DEE - Working Papers. Business Economics. WB wb100201, Universidad Carlos III de Madrid. Departamento de Economía de la Empresa.
    4. Balbás, Alejandro & Garrido, José & Okhrati, Ramin, 2016. "Good deal measurement in asset pricing: Actuarial and financial implications," INDEM - Working Paper Business Economic Series 23546, Instituto para el Desarrollo Empresarial (INDEM).
    5. Balbás, Alejandro & Blanco, Iván & Navarro, Eliseo, 2013. "Equity, commodity and interest rate volatility derivatives," INDEM - Working Paper Business Economic Series id-13-02, Instituto para el Desarrollo Empresarial (INDEM).
    6. Balbás, Alejandro & Balbás, Beatriz & Balbás, Raquel, 2010. "CAPM and APT-like models with risk measures," Journal of Banking & Finance, Elsevier, vol. 34(6), pages 1166-1174, June.
    7. Alejandro Balbás & Iván Blanco & José Garrido, 2014. "Measuring Risk When Expected Losses Are Unbounded," Risks, MDPI, Open Access Journal, vol. 2(4), pages 1-14, September.


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