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

Conditional cash invariance

$\mathrm{\forall }{m}_t\in L_t^{\mathrm{\infty }}:{\rho }_t(X+m_t)={\rho }_t(X)-m_t$

Monotonicity

$\mathrm{I}\mathrm{f}X\le Y\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}{\rho }_t(X)\ge {\rho }_t(Y)$

Normalization

${\rho }_t(0)=0$

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

Conditional convexity

$\mathrm{\forall }\lambda \in L_t^{\mathrm{\infty }},0\le \lambda \le 1:{\rho }_t(\lambda X+(1-\lambda )Y)\le \lambda {\rho }_t(X)+(1-\lambda ){\rho }_t(Y)$

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

Conditional positive homogeneity

$\mathrm{\forall }\lambda \in L_t^{\mathrm{\infty }},\lambda \ge 0:{\rho }_t(\lambda X)=\lambda {\rho }_t(X)$

Acceptance set

Main article: Acceptance set

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

$A_t=\{X\in L_T^{\mathrm{\infty }}:{\rho }_t(X)\le 0\text{ a.s.}\}$.

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

${\rho }_t=\text{ess}\inf \{Y\in L_t^{\mathrm{\infty }}:X+Y\in A_t\}$

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

Regular property

A conditional risk measure ${\rho }_t$ is said to be regular if for any $X\in L_T^{\mathrm{\infty }}$ and $A\in {{ℱ}}_t$ 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 ${\rho }_{t+1}(X)\le {\rho }_{t+1}(Y)\Rightarrow {\rho }_t(X)\le {\rho }_t(Y)\mathrm{\forall }X,Y\in L^0({{ℱ}}_T)$.

Example: dynamic superhedging price

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