# 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
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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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