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., $$X \in L^{\infty}\left(\mathcal{F}_T\right)$$ denotes the payoff of a portfolio. A mapping $$\rho_t: L^{\infty}\left(\mathcal{F}_T\right) \rightarrow L^{\infty}_t = L^{\infty}\left(\mathcal{F}_t\right)$$ is a conditional risk measure if it has the following properties for random portfolio returns $$X,Y \in L^{\infty}\left(\mathcal{F}_T\right)$$:


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


 * Monotonicity
 * $$\mathrm{If} \; X \leq Y \; \mathrm{then} \; \rho_t(X) \geq \rho_t(Y)$$


 * Normalization
 * $$\rho_t(0) = 0$$

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


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

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


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

Acceptance set
The acceptance set at time $$t$$ associated with a conditional risk measure is
 * $$A_t = \{X \in L^{\infty}_T: \rho_t(X) \leq 0 \text{ a.s.}\}$$.

If you are given an acceptance set at time $$t$$ then the corresponding conditional risk measure is
 * $$\rho_t = \text{ess}\inf\{Y \in L^{\infty}_t: X + Y \in A_t\}$$

where $$\text{ess}\inf$$ is the essential infimum.

Regular property
A conditional risk measure $$\rho_t$$ is said to be regular if for any $$X \in L^{\infty}_T$$ and $$A \in \mathcal{F}_t$$ 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. $$\rho_t(X)[\omega]$$) 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 $$\rho_{t+1}(X) \leq \rho_{t+1}(Y) \Rightarrow \rho_t(X) \leq \rho_t(Y) \; \forall X,Y \in L^{0}(\mathcal{F}_T)$$.

Example: dynamic superhedging price
The dynamic superhedging price involves conditional risk measures of the form $$\rho_t(-X) = \operatorname*{ess\sup}_{Q \in EMM} \mathbb{E}^Q[X | \mathcal{F}_t]$$. It is shown that this is a time consistent risk measure.