Dynamic risk measure - Alchetron, The Free Social Encyclopedia

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.

Conditional risk measure

A mapping ρ t : L ∞ ( F T ) → L t ∞ = L ∞ ( F t ) is a conditional risk measure if it has the following properties:

Conditional cash invariance

∀ m t ∈ L t ∞ : ρ t ( X + m t ) = ρ t ( X ) − m t

Monotonicity

I f X ≤ Y t h e n ρ t ( X ) ≥ ρ t ( Y )

Normalization

ρ t ( 0 ) = 0

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

Conditional convexity

∀ λ ∈ L t ∞ , 0 ≤ λ ≤ 1 : ρ t ( λ X + ( 1 − λ ) Y ) ≤ λ ρ t ( X ) + ( 1 − λ ) ρ t ( Y )

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

Conditional positive homogeneity

∀ λ ∈ L t ∞ , λ ≥ 0 : ρ t ( λ X ) = λ ρ t ( X )

Acceptance set

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

A t = { X ∈ L T ∞ : ρ ( X ) ≤ 0 a.s. } .

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

ρ t = ess inf { Y ∈ L t ∞ : X + Y ∈ A t }

where ess inf is the essential infimum.

Regular property

A conditional risk measure ρ t is said to be regular if for any X ∈ L T ∞ and A ∈ F t then ρ t ( 1 A X ) = 1 A ρ t ( X ) where 1 A is the indicator function on A . Any normalized conditional convex risk measure is regular.

The financial interpretation of this states that the conditional risk at some future node (i.e. ρ t ( X ) [ ω ] ) 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 ρ t + 1 ( X ) ≤ ρ t + 1 ( Y ) ⇒ ρ t ( X ) ≤ ρ t ( Y ) ∀ X , Y ∈ L 0 ( F T ) .

Example: dynamic superhedging price

The dynamic superhedging price involves conditional risk measures of the form ρ t ( − X ) = * ⁡ e s s sup Q ∈ E M M E Q [ X | F t ] . It is shown that this is a time consistent risk measure.

References

Dynamic risk measure Wikipedia

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