A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures. [1]
A different approach to dynamic risk measurement has been suggested by Novak.[2]
Conditional risk measure
Consider a
portfolio'sreturns at some terminal time as a
random variable that is
uniformly bounded, i.e., denotes the payoff of a portfolio. A mapping is a conditional risk measure if it has the following properties for random portfolio returns :[3][4]
A conditional risk measure is said to be regular if for any and then where is the
indicator function on . Any normalized conditional convex risk measure is regular.[3]
The financial interpretation of this states that the conditional risk at some future node (i.e. ) only depends on the possible states from that node. In a
binomial model this would be akin to calculating the risk on the subtree branching off from the point in question.
A dynamic risk measure is time consistent if and only if .[6]
Example: dynamic superhedging price
The dynamic
superhedging price involves conditional risk measures of the form
.
It is shown that this is a time consistent risk measure.
References
^Acciaio, Beatrice; Penner, Irina (2011).
"Dynamic risk measures"(PDF). Advanced Mathematical Methods for Finance: 1–34. Archived from
the original(PDF) on September 2, 2011. Retrieved July 22, 2010.
^Cheridito, Patrick; Stadje, Mitja (2009). "Time-inconsistency of VaR and time-consistent alternatives". Finance Research Letters. 6 (1): 40–46.
doi:
10.1016/j.frl.2008.10.002.