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Risk Measure Pricing and Hedging in Incomplete Markets

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Author Info
Mingxin Xu (University of North Carolina at Charlotte)

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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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Paper provided by EconWPA in its series Finance with number 0406004.

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Date of creation: 08 Jun 2004
Date of revision: 06 Apr 2005
Handle: RePEc:wpa:wuwpfi:0406004

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Keywords: 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;

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G - Financial Economics

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Hans FÃllmer & Peter Leukert, 2000. "Efficient hedging: Cost versus shortfall risk," Finance and Stochastics, Springer, vol. 4(2), pages 117-146. [Downloadable!] (restricted)
  2. Ernst Eberlein & Jean Jacod, 1997. "On the range of options prices (*)," Finance and Stochastics, Springer, vol. 1(2), pages 131-140. [Downloadable!] (restricted)
  3. Julien Hugonnier & Dmitry Kramkov & Walter Schachermayer, 2005. "On Utility-Based Pricing Of Contingent Claims In Incomplete Markets," Mathematical Finance, Blackwell Publishing, vol. 15(2), pages 203-212. [Downloadable!] (restricted)
  4. Ioannis Karatzas & Jaksa Cvitanic, 1999. "On dynamic measures of risk," Finance and Stochastics, Springer, vol. 3(4), pages 451-482. [Downloadable!] (restricted)
  5. Kramkov, D.O., 1994. "Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets," Discussion Paper Serie B 294, University of Bonn, Germany. [Downloadable!]
  6. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring. [Downloadable!] (restricted)
  7. Pauline Barrieu & Nicole El Karoui, 2005. "Inf-convolution of risk measures and optimal risk transfer," Finance and Stochastics, Springer, vol. 9(2), pages 269-298, 04. [Downloadable!] (restricted)
  8. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July. [Downloadable!] (restricted)
  9. Jun Sekine, 2004. "Dynamic Minimization of Worst Conditional Expectation of Shortfall," Mathematical Finance, Blackwell Publishing, vol. 14(4), pages 605-618. [Downloadable!] (restricted)
  10. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581. [Downloadable!] (restricted)
  11. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July. [Downloadable!] (restricted)
  12. Kasper Larsen & Traian Pirvu & Steven Shreve & Reha Tütüncü, 2005. "Satisfying convex risk limits by trading," Finance and Stochastics, Springer, vol. 9(2), pages 177-195, 04. [Downloadable!] (restricted)
  13. N. Bellamy & M. Jeanblanc, 2000. "Incompleteness of markets driven by a mixed diffusion," Finance and Stochastics, Springer, vol. 4(2), pages 209-222. [Downloadable!] (restricted)
  14. 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. [Downloadable!] (restricted)
  15. Carr, Peter & Geman, Helyette & Madan, Dilip B., 2001. "Pricing and hedging in incomplete markets," Journal of Financial Economics, Elsevier, vol. 62(1), pages 131-167, October. [Downloadable!] (restricted)
  16. Marek Musiela & Thaleia Zariphopoulou, 2004. "An example of indifference prices under exponential preferences," Finance and Stochastics, Springer, vol. 8(2), pages 229-239, 05. [Downloadable!] (restricted)
  17. Marek Musiela & Thaleia Zariphopoulou, 2004. "A valuation algorithm for indifference prices in incomplete markets," Finance and Stochastics, Springer, vol. 8(3), pages 399-414, 08. [Downloadable!] (restricted)
  18. Foldes, Lucien, 2000. "Valuation and martingale properties of shadow prices: An exposition," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1641-1701, October. [Downloadable!] (restricted)
  19. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447. [Downloadable!] (restricted)
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Michail Anthropelos & Nikolaos E. Frangos & Stylianos Z. Xanthopoulos & Athanasios N. Yannacopoulos, 2008. "On contingent claims pricing in incomplete markets: A risk sharing approach," Quantitative Finance Papers 0809.4781, arXiv.org. [Downloadable!]
  2. Xavier De Scheemaekere, 2008. "Dynamic risk indifference pricing in incomplete markets," Working Papers CEB 08-027.RS, Université Libre de Bruxelles, Solvay Brussels School of Economics and Management, Centre Emile Bernheim (CEB). [Downloadable!]
  3. Michail Anthropelos & Gordan Zitkovic, 2009. "Partial Equilibria with Convex Capital Requirements: Existence, Uniqueness and Stability," Quantitative Finance Papers 0901.3318, arXiv.org. [Downloadable!]
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