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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.

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.

A different approach to dynamic risk measurement has been suggested by Novak.

Conditional risk measure

Consider a portfolio's returns at some terminal time T {\displaystyle T} as a random variable that is uniformly bounded, i.e., X ∈ L ∞ ( F T ) {\displaystyle X\in L^{\infty }\left({\mathcal {F}}_{T}\right)} denotes the payoff of a portfolio. A mapping ρ t : L ∞ ( F T ) → L t ∞ = L ∞ ( F t ) {\displaystyle \rho _{t}:L^{\infty }\left({\mathcal {F}}_{T}\right)\rightarrow L_{t}^{\infty }=L^{\infty }\left({\mathcal {F}}_{t}\right)} is a conditional risk measure if it has the following properties for random portfolio returns X , Y ∈ L ∞ ( F T ) {\displaystyle X,Y\in L^{\infty }\left({\mathcal {F}}_{T}\right)} :

Conditional cash invariance
∀ m t ∈ L t ∞ : ρ t ( X + m t ) = ρ t ( X ) − m t {\displaystyle \forall m_{t}\in L_{t}^{\infty }:\;\rho _{t}(X+m_{t})=\rho _{t}(X)-m_{t}}
Monotonicity
I f X ≤ Y t h e n ρ t ( X ) ≥ ρ t ( Y ) {\displaystyle \mathrm {If} \;X\leq Y\;\mathrm {then} \;\rho _{t}(X)\geq \rho _{t}(Y)}
Normalization
ρ t ( 0 ) = 0 {\displaystyle \rho _{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 ) {\displaystyle \forall \lambda \in L_{t}^{\infty },0\leq \lambda \leq 1:\rho _{t}(\lambda X+(1-\lambda )Y)\leq \lambda \rho _{t}(X)+(1-\lambda )\rho _{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 ) {\displaystyle \forall \lambda \in L_{t}^{\infty },\lambda \geq 0:\rho _{t}(\lambda X)=\lambda \rho _{t}(X)}

Acceptance set

Main article: Acceptance set

The acceptance set at time t {\displaystyle t} associated with a conditional risk measure is

A t = { X ∈ L T ∞ : ρ t ( X ) ≤ 0  a.s. } {\displaystyle A_{t}=\{X\in L_{T}^{\infty }:\rho _{t}(X)\leq 0{\text{ a.s.}}\}} .

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

ρ t = ess inf { Y ∈ L t ∞ : X + Y ∈ A t } {\displaystyle \rho _{t}={\text{ess}}\inf\{Y\in L_{t}^{\infty }:X+Y\in A_{t}\}}

where ess inf {\displaystyle {\text{ess}}\inf } is the essential infimum.

Regular property

A conditional risk measure ρ t {\displaystyle \rho _{t}} is said to be regular if for any X ∈ L T ∞ {\displaystyle X\in L_{T}^{\infty }} and A ∈ F t {\displaystyle A\in {\mathcal {F}}_{t}} then ρ t ( 1 A X ) = 1 A ρ t ( X ) {\displaystyle \rho _{t}(1_{A}X)=1_{A}\rho _{t}(X)} where 1 A {\displaystyle 1_{A}} is the indicator function on A {\displaystyle 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 ] {\displaystyle \rho _{t}(X)} ) 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 ρ t + 1 ( X ) ≤ ρ t + 1 ( Y ) ⇒ ρ t ( X ) ≤ ρ t ( Y ) ∀ X , Y ∈ L 0 ( F T ) {\displaystyle \rho _{t+1}(X)\leq \rho _{t+1}(Y)\Rightarrow \rho _{t}(X)\leq \rho _{t}(Y)\;\forall X,Y\in L^{0}({\mathcal {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 ] {\displaystyle \rho _{t}(-X)=\operatorname {*} {ess\sup }_{Q\in EMM}\mathbb {E} ^{Q}} . It is shown that this is a time consistent risk measure.

References

  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. ^ 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.
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