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 is a conditional risk measure if it has the following properties:

Conditional cash invariance
Monotonicity
Normalization

If it is a conditional convex risk measure then it will also have the property:

Conditional convexity

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

Conditional positive homogeneity

Acceptance set

Main article: Acceptance set

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

.

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

where is the essential infimum.[3]

Regular property

A conditional risk measure is said to be regular if for any and then where is the indicator function on . 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. ) 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 article: Time consistency

A dynamic risk measure is time consistent if and only if .[5]

Example: dynamic superhedging price

The dynamic superhedging price involves conditional risk measures of the form . 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".
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