Dynamic risk measure

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

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_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

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 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.

References

References

1. (2011). "Dynamic risk measures". Advanced Mathematical Methods for Finance.
2. (2015). "On measures of financial risk". In: Current Topics on Risk Analysis: ICRA6 and RISK 2015 Conference, M. Guillén et al. (Eds).
3. (2006). "Convex risk measures and the dynamics of their penalty functions". Statistics & Decisions.
4. Penner, Irina. (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability".
5. (2005). "Conditional and dynamic convex risk measures". Finance and Stochastics.
6. (2009). "Time-inconsistency of VaR and time-consistent alternatives". Finance Research Letters.