Dynamic risk measure explained

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( l{F} _{T\right)} denotes the payoff of a portfolio. A mapping \rho _{t:} L ^{infty} \left( l{F} _{T\right)} → L ^{inf ty} _{t} = L ^{infty} \left( l{F} _{t\right)} is a conditional risk measure if it has the following properties for random portfolio returns X,Y \in L ^{infty} \left( l{F} _{T\right)} :^[3]

Conditional cash invariance

\forall m _{t} \in L ^{inf ty} _{t:} \rho _{t(X} + m _{t)} = \rho _{t(X)} - m _{t}

Monotonicity

If X \leq Y 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 λ \in L ^{inf ty} _{t,} 0 \leq λ \leq 1 : \rho _{t(λ} X + ( 1-λ ) Y) \leq λ \rho _{t(X)} + ( 1-λ ) \rho _{t(Y)}

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

Conditional positive homogeneity

\forall λ \in L ^{inf ty} _{t,} λ \geq 0 : \rho _{t(λ} X) = λ \rho _{t(X)}

Acceptance set

See main article: Acceptance set. The acceptance set at time

t associated with a conditional risk measure isA _{t} = \{X \in L ^{inf ty} _{T:} \rho _{t(X)} \leq 0 a.s. \} .

If you are given an acceptance set at time

t then the corresponding conditional risk measure is\rho _{t} = ess inf \{Y \in L ^{inf ty} _{t:} X + Y \in A _{t\}} where ess inf is the essential infimum.^[4]

Regular property

A conditional risk measure

\rho _{t} is said to be regular if for any X \in L ^{inf ty} _{T} and A \in l{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.^[5]

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

See main article: Time consistency. A dynamic risk measure is time consistent if and only if

\rho _{t+1} (X) \leq \rho _{t+1} (Y) ⇒ \rho _{t(X)} \leq \rho _{t(Y)} \forall X,Y \in L ^{0} ( l{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} E ^{Q[X} | l{F} _{t]} . It is shown that this is a time consistent risk measure.

Notes and References

1. Acciaio . Beatrice . Penner . Irina . 2011 . Advanced Mathematical Methods for Finance . 1–34 . Dynamic risk measures . July 22, 2010 . dead . https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf . September 2, 2011 .
2. Book: Novak. S.Y.. On measures of financial risk. In: Current Topics on Risk Analysis: ICRA6 and RISK 2015 Conference, M. Guillén et al. (Eds). 541–549. 2015. 978-849844-4964.
3. Föllmer . Hans . Penner . Irina . 2006 . Convex risk measures and the dynamics of their penalty functions . Statistics & Decisions . 24 . 1 . 61–96. 10.1524/stnd.2006.24.1.61 . 10.1.1.604.2774 . 54734936 .
4. Penner . Irina . 2007 . Dynamic convex risk measures: time consistency, prudence, and sustainability . February 3, 2011 . dead . https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf . July 19, 2011 .
5. Detlefsen. K.. Scandolo. G.. Conditional and dynamic convex risk measures. Finance and Stochastics. 9. 4. 539–561. 2005. 10.1007/s00780-005-0159-6. 10.1.1.453.4944. 10579202 .
6. Cheridito. Patrick. Stadje. Mitja. Time-inconsistency of VaR and time-consistent alternatives. Finance Research Letters. 6. 1. 40–46. 2009. 10.1016/j.frl.2008.10.002.