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. ^[1]

A different approach to dynamic risk measurement has been suggested by Novak.^[2]

Conditional risk measure

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:

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

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

Acceptance set

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

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

If you are given an acceptance set at time $t$ 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.^[3]

Regular property

A conditional risk measure $\rho_t$ 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 $1_{A}$ is the indicator function on $A$. Any normalized conditional convex risk measure is regular.^[4]

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.

Time consistent property

A dynamic risk measure is time consistent if and only if ${\displaystyle \rho _{t+1}(X)\leq \rho _{t+1}(Y)\Rightarrow \rho _{t}(X)\leq \rho _{t}(Y)\;\forall X,Y\in L^{0}({\mathcal {F}}_{T})}$.^[5]

Example: dynamic superhedging price

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.

References

1. Acciaio, Beatrice; Penner, Irina (February 22, 2010). "Dynamic risk measures" (pdf). Retrieved July 22, 2010.
2. Novak, S.Y. (2015). "On measures of financial risk". In: Current topics on risk analysis: ICRA6 and RISK 2015 Conference, M. Guillén et al. (eds): 541–549. ISBN 978 849844 4964.
3. Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (pdf). Retrieved February 3, 2011.
4. Detlefsen, K.; Scandolo, G. (2005). "Conditional and dynamic convex risk measures". Finance and Stochastics. 9 (4): 539–561. doi:10.1007/s00780-005-0159-6.
5. Cheridito, Patrick; Stadje, Mitja (October 2008). "Time-inconsistency of VaR and time-consistent alternatives".