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# On dynamic measures of risk

## 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)

## 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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## Bibliographic 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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Web page: http://www.springerlink.com/content/101164/

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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

## References

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## Citations

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Cited by:
1. El Karoui, Nicole & Jeanblanc, Monique & Lacoste, Vincent, 2005. "Optimal portfolio management with American capital guarantee," Journal of Economic Dynamics and Control, Elsevier, vol. 29(3), pages 449-468, March.
2. Zhou, Qing & Wu, Weixing & Wang, Zengwu, 2008. "Cooperative hedging with a higher interest rate for borrowing," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 609-616, April.
3. Monoyios, Michael, 2004. "Option pricing with transaction costs using a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 28(5), pages 889-913, February.
4. Alexander Cherny, 2007. "Pricing and hedging European options with discrete-time coherent risk," Finance and Stochastics, Springer, vol. 11(4), pages 537-569, October.
5. Leonel Pérez-Hernández, 2005. "On the Existence of Efficient Hedge for an American Contingent Claim: Discrete Time Market," Department of Economics and Finance Working Papers EC200505, Universidad de Guanajuato, Department of Economics and Finance.
6. 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.
7. Sabrina Mulinacci, 2011. "The efficient hedging problem for American options," Finance and Stochastics, Springer, vol. 15(2), pages 365-397, June.
8. Mazzoleni, Piera, 2004. "Risk measures and return performance: A critical approach," European Journal of Operational Research, Elsevier, vol. 155(2), pages 268-275, June.
9. Siu, Tak Kuen & Yang, Hailiang, 1999. "Subjective risk measures: Bayesian predictive scenarios analysis," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 157-169, November.
10. 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.
11. Giannopoulos, Kostas & Tunaru, Radu, 2005. "Coherent risk measures under filtered historical simulation," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 979-996, April.
12. 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.
13. Stephen Lawrence, 2000. "Value At Risk Incorporating Dynamic Portfolio Management," Computing in Economics and Finance 2000 147, Society for Computational Economics.
14. Mingxin Xu, 2004. "Risk Measure Pricing and Hedging in Incomplete Markets," Finance 0406004, EconWPA, revised 06 Apr 2005.
15. Pascal François & Geneviève Gauthier & Frédéric Godin, 2012. "Optimal Hedging when the Underlying Asset Follows a Regime-switching Markov Process," Cahiers de recherche 1234, CIRPEE.
16. Tomasz R. Bielecki & Igor Cialenco & Zhao Zhang, 2010. "Dynamic Coherent Acceptability Indices and their Applications to Finance," Papers 1010.4339, arXiv.org, revised May 2011.
17. Ilhan, Aytaç & Jonsson, Mattias & Sircar, Ronnie, 2009. "Optimal static-dynamic hedges for exotic options under convex risk measures," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3608-3632, October.
18. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
19. Frank Riedel, 2003. "Dynamic Coherent Risk Measures," Working Papers 03004, Stanford University, Department of Economics.
20. Balder, Sven & Brandl, Michael & Mahayni, Antje, 2009. "Effectiveness of CPPI strategies under discrete-time trading," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 204-220, January.

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