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On dynamic measures of risk

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Author Info
Ioannis Karatzas () (Departments of Mathematics and Statistics, Columbia University, New York, NY 10027, USA Manuscript)
Jaksa Cvitanic (Department of Statistics, Columbia University, New York, NY 10027, USA)

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Abstract

In the context of complete financial markets, we study dynamic measures of the form \[ \rho(x;C):=\sup_{\nu\in\D} \inf_{\pi(\cdot)\in\A(x)}{\bf E}_\nu\left(\frac{C-X^{x, \pi}(T)}{S_0(T)}\right)^+, \] for the risk associated with hedging a given liability C at time t = T. Here x is the initial capital available at time t = 0, ${\cal A}(x)$ the class of admissible portfolio strategies, $S_0(\cdot)$ the price of the risk-free instrument in the market, ${\cal P}=\{{\bf P}_\nu\}_{\nu\in{\cal D}}$ a suitable family of probability measures, and [0,T] the temporal horizon during which all economic activity takes place. The classes ${\cal A}(x)$ and ${\cal D}$ are general enough to incorporate capital requirements, and uncertainty about the actual values of stock-appreciation rates, respectively. For this latter purpose we discuss, in addition to the above "max-min" approach, a related measure of risk in a "Bayesian" framework. Risk-measures of this type were introduced by Artzner, Delbaen, Eber and Heath in a static setting, and were shown to possess certain desirable "coherence" properties.

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Publisher Info
Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 3 (1999)
Issue (Month): 4 ()
Pages: 451-482
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Handle: RePEc:spr:finsto:v:3:y:1999:i:4:p:451-482

Note: received: February 1998; final version received: February 1999
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Related research
Keywords: Dynamic measures of risk; Bayesian risk; hedging; capital requirements; value-at-risk;

Find related papers by JEL classification:
G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

Cited by:
(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. Michael Monoyios & Alberto Montagnoli, 2002. "Efficient Option Pricing with Transaction Costs," Public Policy Discussion Papers 02-22, Economics and Finance Section, School of Social Sciences, Brunel University. [Downloadable!]
  2. Tak Kuen Siu, Hailiang Yang, 2000. "A PDE approach to risk measures of derivatives," Applied Mathematical Finance, Taylor and Francis Journals, vol. 7(3), pages 211-228, September. [Downloadable!] (restricted)
  3. Alexander Cherny, 2007. "Pricing and hedging European options with discrete-time coherent risk," Finance and Stochastics, Springer, vol. 11(4), pages 537-569, October. [Downloadable!] (restricted)
  4. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January. [Downloadable!] (restricted)
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  5. Tak Siu & Howell Tong & Hailiang Yang, 2004. "On Bayesian Value at Risk: From Linear to Non-Linear Portfolios," Asia-Pacific Financial Markets, Springer, vol. 11(2), pages 161-184, June. [Downloadable!] (restricted)
  6. Michael Monoyios & Alberto Montagnoli, 2002. "Efficient Option Pricing with Transaction Costs," Economics and Finance Discussion Papers 02-22, Economics and Finance Section, School of Social Sciences, Brunel University. [Downloadable!]
  7. H. Föllmer, . "Probabilistic Aspects of Financial Risk," Sonderforschungsbereich 373 2000-103, Humboldt Universitaet Berlin.
  8. Leonel Pérez-Hernández, . "On the Existence of Efficient Hedge for an American Contingent Claim: Discrete Time Market," School of Economics Working Papers EC200505, Universidad de Guanajuato. [Downloadable!]
  9. Alexander Melnikov & Yuliya Romanyuk, 2006. "Efficient Hedging and Pricing of Equity-Linked Life Insurance Contracts on Several Risky Assets," Working Papers 06-43, Bank of Canada. [Downloadable!]
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  10. Stephen Lawrence, 2000. "Value At Risk Incorporating Dynamic Portfolio Management," Computing in Economics and Finance 2000 147, Society for Computational Economics. [Downloadable!]
  11. Frank Riedel, 2003. "Dynamic Coherent Risk Measures," Working Papers 03004, Stanford University, Department of Economics. [Downloadable!]
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