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. <ref>{{cite journal|last1=Acciaio |first1=Beatrice |last2=Penner |first2=Irina |year=2011 |journal=Advanced Mathematical Methods for Finance |pages=1–34 |title=Dynamic risk measures |url=http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf |accessdate=July 22, 2010 |url-status=dead |archiveurl=https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf |archivedate=September 2, 2011 }}</ref>
A different approach to dynamic risk measurement has been suggested by Novak.<ref>{{cite book|last1=Novak|first1=S.Y.|title=On measures of financial risk|journal=In: Current Topics on Risk Analysis: ICRA6 and RISK 2015 Conference, M. Guillén et al. (Eds)|pages=541–549|year=2015|isbn=978-849844-4964}}</ref>
==Conditional risk measure== Consider a portfolio's returns at some terminal time <math>T</math> as a random variable that is uniformly bounded, i.e., <math>X \in L^{\infty}\left(\mathcal{F}_T\right)</math> denotes the payoff of a portfolio. A mapping <math>\rho_t: L^{\infty}\left(\mathcal{F}_T\right) \rightarrow L^{\infty}_t = L^{\infty}\left(\mathcal{F}_t\right)</math> is a conditional risk measure if it has the following properties for random portfolio returns <math>X,Y \in L^{\infty}\left(\mathcal{F}_T\right)</math>:<ref name="DS05"/><ref>{{cite journal|last1=Föllmer |first1=Hans |last2=Penner |first2=Irina |year=2006 |title=Convex risk measures and the dynamics of their penalty functions |journal=Statistics & Decisions |volume=24 |issue=1 |pages=61–96|doi=10.1524/stnd.2006.24.1.61 |citeseerx=10.1.1.604.2774 |s2cid=54734936 }}</ref>
; Conditional cash invariance : <math>\forall m_t \in L^{\infty}_t: \; \rho_t(X + m_t) = \rho_t(X) - m_t</math>{{clarify|reason=symbols not defined|date=June 2017}}
; Monotonicity : <math>\mathrm{If} \; X \leq Y \; \mathrm{then} \; \rho_t(X) \geq \rho_t(Y)</math>{{clarify|reason=symbols not defined|date=June 2017}}
; Normalization : <math>\rho_t(0) = 0</math>{{clarify|reason=symbols not defined|date=June 2017}}
If it is a conditional convex risk measure then it will also have the property:
; Conditional convexity : <math>\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)</math>{{clarify|reason=symbols not defined|date=June 2017}}
A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies:
; Conditional positive homogeneity : <math>\forall \lambda \in L^{\infty}_t, \lambda \geq 0: \rho_t(\lambda X) = \lambda \rho_t(X)</math>{{clarify|reason=symbols not defined|date=June 2017}}
==Acceptance set== {{main|Acceptance set}} The acceptance set at time <math>t</math> associated with a conditional risk measure is : <math>A_t = \{X \in L^{\infty}_T: \rho_t(X) \leq 0 \text{ a.s.}\}</math>.
If you are given an acceptance set at time <math>t</math> then the corresponding conditional risk measure is : <math>\rho_t = \text{ess}\inf\{Y \in L^{\infty}_t: X + Y \in A_t\}</math> where <math>\text{ess}\inf</math> is the essential infimum.<ref>{{cite journal|last=Penner |first=Irina |year=2007 |title=Dynamic convex risk measures: time consistency, prudence, and sustainability |url=http://wws.mathematik.hu-berlin.de/~penner/penner.pdf |accessdate=February 3, 2011 |url-status=dead |archiveurl=https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf |archivedate=July 19, 2011 }}</ref>
==Regular property== A conditional risk measure <math>\rho_t</math> is said to be ''regular'' if for any <math>X \in L^{\infty}_T</math> and <math>A \in \mathcal{F}_t</math> then <math>\rho_t(1_A X) = 1_A \rho_t(X)</math> where <math>1_A</math> is the indicator function on <math>A</math>. Any normalized conditional convex risk measure is regular.<ref name="DS05">{{cite journal|last1=Detlefsen|first1=K.|last2=Scandolo|first2=G.|title=Conditional and dynamic convex risk measures|journal=Finance and Stochastics|volume=9|issue=4|pages=539–561|year=2005|doi=10.1007/s00780-005-0159-6|citeseerx=10.1.1.453.4944|s2cid=10579202 }}</ref>
The financial interpretation of this states that the conditional risk at some future node (i.e. <math>\rho_t(X)[\omega]</math>) 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|Time consistency}} A dynamic risk measure is time consistent if and only if <math>\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)</math>.<ref>{{cite journal|last1=Cheridito|first1=Patrick|last2=Stadje|first2=Mitja|title=Time-inconsistency of VaR and time-consistent alternatives|journal=Finance Research Letters|volume=6|issue=1|pages=40–46|year=2009|doi=10.1016/j.frl.2008.10.002}}</ref>
==Example: dynamic superhedging price== The dynamic superhedging price involves conditional risk measures of the form <math>\rho_t(-X) = \operatorname*{ess\sup}_{Q \in EMM} \mathbb{E}^Q[X | \mathcal{F}_t]</math>. It is shown that this is a time consistent risk measure.
==References== {{Reflist}}
Category:Financial risk modeling