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
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
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
- ↑ Acciaio, Beatrice; Penner, Irina (February 22, 2010). "Dynamic risk measures" (pdf). Retrieved July 22, 2010.
- ↑ 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.
- ↑ Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (pdf). Retrieved February 3, 2011.
- ↑ 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.
- ↑ Cheridito, Patrick; Stadje, Mitja (October 2008). "Time-inconsistency of VaR and time-consistent alternatives".