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

edit

Consider a portfolio's returns at some terminal time   as a random variable that is uniformly bounded, i.e.,   denotes the payoff of a portfolio. A mapping   is a conditional risk measure if it has the following properties for random portfolio returns  :[3][4]

Conditional cash invariance
 [clarification needed]
Monotonicity
 [clarification needed]
Normalization
 [clarification needed]

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

Conditional convexity
 [clarification needed]

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

Conditional positive homogeneity
 [clarification needed]

Acceptance set

edit

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.[5]

Regular property

edit

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.[3]

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

edit

A dynamic risk measure is time consistent if and only if  .[6]

Example: dynamic superhedging price

edit

The dynamic superhedging price involves conditional risk measures of the form  . It is shown that this is a time consistent risk measure.

References

edit
  1. ^ Acciaio, Beatrice; Penner, Irina (2011). "Dynamic risk measures" (PDF). Advanced Mathematical Methods for Finance: 1–34. Archived from the original (PDF) on September 2, 2011. Retrieved July 22, 2010.
  2. ^ Novak, S.Y. (2015). On measures of financial risk. pp. 541–549. ISBN 978-849844-4964. {{cite book}}: |journal= ignored (help)
  3. ^ a b Detlefsen, K.; Scandolo, G. (2005). "Conditional and dynamic convex risk measures". Finance and Stochastics. 9 (4): 539–561. CiteSeerX 10.1.1.453.4944. doi:10.1007/s00780-005-0159-6. S2CID 10579202.
  4. ^ Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions". Statistics & Decisions. 24 (1): 61–96. CiteSeerX 10.1.1.604.2774. doi:10.1524/stnd.2006.24.1.61. S2CID 54734936.
  5. ^ Penner, Irina (2007). "Dynamic convex risk measures: time consistency, prudence, and sustainability" (PDF). Archived from the original (PDF) on July 19, 2011. Retrieved February 3, 2011. {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Cheridito, Patrick; Stadje, Mitja (2009). "Time-inconsistency of VaR and time-consistent alternatives". Finance Research Letters. 6 (1): 40–46. doi:10.1016/j.frl.2008.10.002.