Dynamic Risk Measure

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.

A different approach to dynamic risk measurement has been suggested by Novak.

Conditional risk measure

Consider a portfolio's returns at some terminal time $T$ as a random variable that is uniformly bounded, i.e., $X \in L^\left(\mathcal_T\right)$ denotes the payoff of a portfolio. A mapping $\rho_t: L^\left(\mathcal_T\right) \rightarrow L^_t = L^\left(\mathcal_t\right)$ is a conditional risk measure if it has the following properties for random portfolio returns $X,Y \in L^\left(\mathcal_T\right)$:

; Conditional cash invariance
: $\forall m_t \in L^_t: \; \rho_t(X + m_t) = \rho_t(X) - m_t$

; Monotonicity
: $\mathrm \; X \leq Y \; \mathrm \; \rho_t(X) \geq \rho_t(Y)$

; Normalization
: $\rho_t(0) = 0$

If it is a conditional convex risk measure then it will also have the property:

; Conditional convexity
: $\forall \lambda \in L^_t, 0 \leq \lambda \leq 1: \rho_t(\lambda X + (1-\lambda) Y) \leq \lambda \rho_t(X) + (1-\lambda) \rho_t(Y)$

A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies:

; Conditional positive homogeneity
: $\forall \lambda \in L^_t, \lambda \geq 0: \rho_t(\lambda X) = \lambda \rho_t(X)$

Acceptance set

The acceptance set at time $t$ associated with a conditional risk measure is
: $A_t = \$.

If you are given an acceptance set at time $t$ then the corresponding conditional risk measure is
: $\rho_t = \text\inf\$
where $\text\inf$ is the essential infimum.

Regular property

A conditional risk measure $\rho_t$ is said to be ''regular'' if for any $X \in L^_T$ and $A \in \mathcal_t$ then $\rho_t(1_A X) = 1_A \rho_t(X)$ where $1_A$ is the indicator function on $A$. Any normalized conditional convex risk measure is regular.

The financial interpretation of this states that the conditional risk at some future node (i.e. $\rho_t(X)$omega/math>) 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 $\rho_(X) \leq \rho_(Y) \Rightarrow \rho_t(X) \leq \rho_t(Y) \; \forall X,Y \in L^(\mathcal_T)$.

Example: dynamic superhedging price

The dynamic superhedging price involves conditional risk measures of the form
\rho_t(-X) = \operatorname*_ \mathbb^Q \mathcal_t/math>.
It is shown that this is a time consistent risk measure.

References

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Financial risk modeling