- Dynamic risk measure
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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.
In a binomial model, the risk at any node only depends on the possible final values branching off from that node. Therefore the conditional risk at a time in the future is the random variable on the nodes at that time. That is, when that future time is reached, the risk analyst would be able[clarification needed] to read off the risk from the previous calculation (and inputting the current state of the market).[clarification needed]
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]
Contents
Conditional risk measure
A conditional risk measure can be written as the worst conditional expected loss over a set of penalized probability measures. A mapping is a conditional convex risk measure if it has the following properties:
- Conditional cash invariance
- Monotonicity
- Conditional convexity
- Normalization
- ρt(0) = 0
If it is a conditional coherent risk measure then it will also have the property:
- Conditional positive homogeneity
Acceptance set
Main article: Acceptance setThe acceptance set at time t associated with a conditional risk measure is
- .
If you are given an acceptance set at time t then the corresponding conditional risk measure is
where essinf is the essential infimum.[2]
Time consistent property
Main article: Time consistencyA dynamic risk measure is time consistent if and only if .[3]
Example: dynamic superhedging price
The dynamic superhedging price has conditional risk measures of the form: . It is a widely shown result that this is also a time consistent risk measure.
References
- ^ Acciaio, Beatrice; Penner, Irina (February 22, 2010) (pdf). Dynamic risk measures. http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf. Retrieved July 22, 2010.
- ^ Penner, Irina (2007) (pdf). Dynamic convex risk measures: time consistency, prudence, and sustainability. http://wws.mathematik.hu-berlin.de/~penner/penner.pdf. Retrieved February 3, 2011.
- ^ Cheridito, Patrick; Stadje, Mitja (October 2008). Time-inconsistency of VaR and time-consistent alternatives.
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