×

On dynamic measure of risk. (English) Zbl 0982.91030

The paper deals with the situation when in a complete continuous-time financial market an agent starts with initial capital \(x\) less than the amount \(C(0)=E[C/S_0(T)]\) required for perfect hedging the liability (without risk) at terminal time \(t=T\). The authors present a solution to the problem of minimizing the expected discounted loss as a solution to the relevant stochastic control problem. Also the supremum of the minimal expected loss, i.e. \[ \rho(x;C)=\sup_{\nu\in D} \inf_{\pi(\cdot)\in A(x)} E_{\nu}\left({{C-X^{x,\pi}(T)}\over{S_0(T)}}\right)^+, \] is proposed as a measure of the risk associated with hedging a given liability \(C\) at time \(t=T\). Here \(A(x)\) is the class of admissible portfolio strategies, \(S_0\) is a price of the risk-free instrument in the market; \({\mathcal P}=\{P_{nu}\), \(\nu\in D\}\) is a suitable family of probability measures (“scenarios”), \([0,T]\) is the temporal horizon during which economic activity take place. In addition to this “max-min” approach a related measure of risk in the “Bayesian” framework is discussed. Examples are worked out under various “capital requirement” and possible interpretations are analysed. Certain open problems are pointed out.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B70 Stochastic models in economics
93E20 Optimal stochastic control
60H30 Applications of stochastic analysis (to PDEs, etc.)
PDFBibTeX XMLCite
Full Text: DOI