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On dynamic measures of risk

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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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Manuscript received: February 1998; final version received: February 1999

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Cvitanić, J., Karatzas, I. On dynamic measures of risk. Finance Stochast 3, 451–482 (1999). https://doi.org/10.1007/s007800050071

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  • DOI: https://doi.org/10.1007/s007800050071

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