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

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 $\rho _{t}:L^{\infty }\left({\mathcal {F}}_{T}\right)\rightarrow L_{t}^{\infty }=L^{\infty }\left({\mathcal {F}}_{t}\right)$ is a conditional risk measure if it has the following properties for random portfolio returns $X,Y\in L^{\infty }\left({\mathcal {F}}_{T}\right)$ :^[3]^[4]

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

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

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

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

Acceptance set

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

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

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

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

Regular property

A conditional risk measure $\rho_t$ is said to be regular if for any $X\in L_{T}^{\infty }$ 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.^[3]

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

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

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

↑ Acciaio, Beatrice; Penner, Irina (2011). "Dynamic risk measures" (PDF). Advanced Mathematical Methods for Finance: 1–34. Archived from the original (pdf) on September 2, 2011. Retrieved July 22, 2010.
↑ 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.
1 2 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.
↑ Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions". Statistics & Decisions. 24 (1): 61–96.
↑ Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (PDF). Archived from the original (pdf) on July 19, 2011. Retrieved February 3, 2011.
↑ Cheridito, Patrick; Stadje, Mitja (2009). "Time-inconsistency of VaR and time-consistent alternatives". Finance Research Letters. 6 (1): 40–46.

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[1] Acciaio, Beatrice; Penner, Irina (2011). "Dynamic risk measures" (PDF). Advanced Mathematical Methods for Finance: 1–34. Archived from the original (pdf) on September 2, 2011. 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.

[DS05-3] 1 2 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.

[4] Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions". Statistics & Decisions. 24 (1): 61–96.

[5] Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (PDF). Archived from the original (pdf) on July 19, 2011. Retrieved February 3, 2011.

[6] Cheridito, Patrick; Stadje, Mitja (2009). "Time-inconsistency of VaR and time-consistent alternatives". Finance Research Letters. 6 (1): 40–46.