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.

Conditional risk measure

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:

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

Main article: Acceptance set

The acceptance set at time $t$ associated with a conditional risk measure is

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): A_t = \{X \in L^{\infty}_T: \rho(X) \leq 0 \text{ a.s.}\}

.

If you are given an acceptance set at time $t$ then the corresponding conditional risk measure is

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \rho_t = \text{ess}\inf\{Y \in L^{\infty}_t: X + Y \in A_t\}

where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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

Main article: Time consistency

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 has conditional risk measures of the form: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \rho_t(-X) = \operatorname*{ess\sup}_{Q \in EMM} \mathbb{E}^Q[X | \mathcal{F}_t] . It is a widely shown result that this is also a time consistent risk measure.