In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future.
A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra.
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] Consider a portfolio's returns at some terminal time
as a random variable that is uniformly bounded, i.e.,
denotes the payoff of a portfolio.
is a conditional risk measure if it has the following properties for random portfolio returns
:[3][4] If it is a conditional convex risk measure then it will also have the property: A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies: The acceptance set at time
associated with a conditional risk measure is If you are given an acceptance set at time
then the corresponding conditional risk measure is where
is the essential infimum.
[5] A conditional risk measure
ρ
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.
ρ
) 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
ρ
[6] The dynamic superhedging price involves conditional risk measures of the form
It is shown that this is a time consistent risk measure.