Dynamic risk measure

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

; Conditional cash invariance

: $\backslash forall\; m\_t\; \backslash in\; L^\{\backslash infty\}\_t:\; \backslash ;\; \backslash rho\_t(X\; +\; m\_t)\; =\; \backslash rho\_t(X)\; -\; m\_t${{clarify|reason=symbols not defined|date=June 2017}}

; Monotonicity

: $\backslash mathrm\{If\}\; \backslash ;\; X\; \backslash leq\; Y\; \backslash ;\; \backslash mathrm\{then\}\; \backslash ;\; \backslash rho\_t(X)\; \backslash geq\; \backslash rho\_t(Y)${{clarify|reason=symbols not defined|date=June 2017}}

; Normalization

: $\backslash rho\_t(0)\; =\; 0${{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

: $\backslash forall\; \backslash lambda\; \backslash in\; L^\{\backslash infty\}\_t,\; 0\; \backslash leq\; \backslash lambda\; \backslash leq\; 1:\; \backslash rho\_t(\backslash lambda\; X\; +\; (1-\backslash lambda)\; Y)\; \backslash leq\; \backslash lambda\; \backslash rho\_t(X)\; +\; (1-\backslash lambda)\; \backslash rho\_t(Y)${{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

: $\backslash forall\; \backslash lambda\; \backslash in\; L^\{\backslash infty\}\_t,\; \backslash lambda\; \backslash geq\; 0:\; \backslash rho\_t(\backslash lambda\; X)\; =\; \backslash lambda\; \backslash rho\_t(X)${{clarify|reason=symbols not defined|date=June 2017}}

{{main|Acceptance set}}

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

: $A\_t\; =\; \backslash \{X\; \backslash in\; L^\{\backslash infty\}\_T:\; \backslash rho\_t(X)\; \backslash leq\; 0\; \backslash text\{\; a.s.\}\backslash \}$.

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

: $\backslash rho\_t\; =\; \backslash text\{ess\}\backslash inf\backslash \{Y\; \backslash in\; L^\{\backslash infty\}\_t:\; X\; +\; Y\; \backslash in\; A\_t\backslash \}$

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

A conditional risk measure $\backslash rho\_t$ is said to be regular if for any $X\; \backslash in\; L^\{\backslash infty\}\_T$ and $A\; \backslash in\; \backslash mathcal\{F\}\_t$ then $\backslash rho\_t(1\_A\; X)\; =\; 1\_A\; \backslash rho\_t(X)$ where $1\_A$ is the indicator function on $A$. Any normalized conditional convex risk measure is regular.{{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 }}

The financial interpretation of this states that the conditional risk at some future node (i.e. $\backslash rho\_t(X)[\backslash omega]$) 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.

{{main|Time consistency}}

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

The dynamic superhedging price involves conditional risk measures of the form

$\backslash rho\_t(-X)\; =\; \backslash operatorname*\{ess\backslash sup\}\_\{Q\; \backslash in\; EMM\}\; \backslash mathbb\{E\}^Q[X\; |\; \backslash mathcal\{F\}\_t]$.

It is shown that this is a time consistent risk measure.

{{Reflist}}

Category:Financial risk modeling