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.

In a binomial model, the risk at any node only depends on the possible final values branching off from that node. Therefore the conditional risk at a time in the future is the random variable on the nodes at that time. That is, when that future time is reached, the risk analyst would be able^{[clarification needed]} to read off the risk from the previous calculation (and inputting the current state of the market).^{[clarification needed]}

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]

Conditional risk measure

A conditional risk measure can be written as the worst conditional expected loss over a set of penalized probability measures. 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 convex 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

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

Conditional convexity

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

Normalization

ρ_t(0) = 0

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

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

$A_t = \{X \in L^{\infty}_T: \rho(X) \leq 0 \text{ a.s.}\}$.

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

$\rho_t = \text{ess}\inf\{Y \in L^{\infty}_t: X + Y \in A_t\}$

where essinf  is the essential infimum.^[2]

Time consistent property

Main article: Time consistency

A dynamic risk measure is time consistent if and only if $\rho_{t+1}(X) \leq \rho_{t+1}(Y) \Rightarrow \rho_t(X) \leq \rho_t(Y) \; \forall X,Y \in L^{0}(\mathcal{F}_T)$.^[3]

Example: dynamic superhedging price

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

References

1. ^ Acciaio, Beatrice; Penner, Irina (February 22, 2010) (pdf). Dynamic risk measures. http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf. Retrieved July 22, 2010. 
2. ^ Penner, Irina (2007) (pdf). Dynamic convex risk measures: time consistency, prudence, and sustainability. http://wws.mathematik.hu-berlin.de/~penner/penner.pdf. Retrieved February 3, 2011. 
3. ^ Cheridito, Patrick; Stadje, Mitja (October 2008). Time-inconsistency of VaR and time-consistent alternatives.