Dynamic risk measure

Consider a portfolio's returns at some terminal time ${\displaystyle T}$ as a random variable that is uniformly bounded, i.e., ${\displaystyle X\in L^{\infty }\left({\mathcal {F}}_{T}\right)}$ denotes the payoff of a portfolio. A mapping ${\displaystyle \rho _{t}:L^{\infty }\left({\mathcal {F}}_{T}\right)\rightarrow L_{t}^{\infty }=L^{\infty }\left({\mathcal {F}}_{t}\right)}$ is a conditional risk measure if it has the following properties for random portfolio returns ${\displaystyle X,Y\in L^{\infty }\left({\mathcal {F}}_{T}\right)}$:[3][4]

Conditional cash invariance

${\displaystyle \forall m_{t}\in L_{t}^{\infty }:\;\rho _{t}(X+m_{t})=\rho _{t}(X)-m_{t}}$

Monotonicity

${\displaystyle \mathrm {If} \;X\leq Y\;\mathrm {then} \;\rho _{t}(X)\geq \rho _{t}(Y)}$

Normalization

${\displaystyle \rho _{t}(0)=0}$

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

Conditional convexity

${\displaystyle \forall \lambda \in L_{t}^{\infty },0\leq \lambda \leq 1:\rho _{t}(\lambda X+(1-\lambda )Y)\leq \lambda \rho _{t}(X)+(1-\lambda )\rho _{t}(Y)}$

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

Conditional positive homogeneity

${\displaystyle \forall \lambda \in L_{t}^{\infty },\lambda \geq 0:\rho _{t}(\lambda X)=\lambda \rho _{t}(X)}$

The acceptance set at time ${\displaystyle t}$ associated with a conditional risk measure is

${\displaystyle A_{t}=\{X\in L_{T}^{\infty }:\rho _{t}(X)\leq 0{\text{ a.s.}}\}}$.

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

${\displaystyle \rho _{t}={\text{ess}}\inf\{Y\in L_{t}^{\infty }:X+Y\in A_{t}\}}$

where ${\displaystyle {\text{ess}}\inf }$ is the essential infimum.[5]

A conditional risk measure ${\displaystyle \rho _{t}}$ is said to be regular if for any ${\displaystyle X\in L_{T}^{\infty }}$ and ${\displaystyle A\in {\mathcal {F}}_{t}}$ then ${\displaystyle \rho _{t}(1_{A}X)=1_{A}\rho _{t}(X)}$ where ${\displaystyle 1_{A}}$ is the indicator function on ${\displaystyle A}$. Any normalized conditional convex risk measure is regular.[3]

The financial interpretation of this states that the conditional risk at some future node (i.e. ${\displaystyle \rho _{t}(X)[\omega ]}$) 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.

A dynamic risk measure is time consistent if and only if ${\displaystyle \rho _{t+1}(X)\leq \rho _{t+1}(Y)\Rightarrow \rho _{t}(X)\leq \rho _{t}(Y)\;\forall X,Y\in L^{0}({\mathcal {F}}_{T})}$.[6]

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