# On dynamic measures of risk

## Author

Listed:
• 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.

## Suggested Citation

• Ioannis Karatzas & Jaksa Cvitanic, 1999. "On dynamic measures of risk," Finance and Stochastics, Springer, vol. 3(4), pages 451-482.
• 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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## References listed on IDEAS

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1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
2. Constantinides, George M, 1992. "A Theory of the Nominal Term Structure of Interest Rates," Review of Financial Studies, Society for Financial Studies, vol. 5(4), pages 531-552.
3. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305 World Scientific Publishing Co. Pte. Ltd..
4. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. " Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-430, March.
5. Alan Brace & Dariusz G¸atarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 127-155.
6. J.E. Kennedy & P.J. Hunt, 1998. "Implied interest rate pricing models," Finance and Stochastics, Springer, vol. 2(3), pages 275-293.
7. Peter Ritchken & L. Sankarasubramanian, 1995. "Volatility Structures Of Forward Rates And The Dynamics Of The Term Structure," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 55-72.
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### Keywords

Dynamic measures of risk; Bayesian risk; hedging; capital requirements; value-at-risk;

### 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

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