Dynamic risk measure


In financial mathematics, a conditional risk measure is a random variable of the financial 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.
A different approach to dynamic risk measurement has been suggested by Novak.

Conditional risk measure

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. A mapping is a conditional risk measure if it has the following properties for random portfolio returns :
; Conditional cash invariance
; Monotonicity
; Normalization
If it is a conditional convex risk measure then it will also have the property:
; Conditional convexity
A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies:
; Conditional positive homogeneity

Acceptance set

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.

Regular property

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.
The financial interpretation of this states that the conditional risk at some future node 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.

Time consistent property

A dynamic risk measure is time consistent if and only if.

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.