Finance: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
Consider a portfolio's returns at some terminal time [math]\displaystyle{ T }[/math] as a random variable that is uniformly bounded, i.e., [math]\displaystyle{ X \in L^{\infty}\left(\mathcal{F}_T\right) }[/math] denotes the payoff of a portfolio. A mapping [math]\displaystyle{ \rho_t: L^{\infty}\left(\mathcal{F}_T\right) \rightarrow L^{\infty}_t = L^{\infty}\left(\mathcal{F}_t\right) }[/math] is a conditional risk measure if it has the following properties for random portfolio returns [math]\displaystyle{ X,Y \in L^{\infty}\left(\mathcal{F}_T\right) }[/math]:[3][4]
- Conditional cash invariance
- [math]\displaystyle{ \forall m_t \in L^{\infty}_t: \; \rho_t(X + m_t) = \rho_t(X) - m_t }[/math][clarification needed]
- Monotonicity
- [math]\displaystyle{ \mathrm{If} \; X \leq Y \; \mathrm{then} \; \rho_t(X) \geq \rho_t(Y) }[/math][clarification needed]
- Normalization
- [math]\displaystyle{ \rho_t(0) = 0 }[/math][clarification needed]
If it is a conditional convex risk measure then it will also have the property:
- Conditional convexity
- [math]\displaystyle{ \forall \lambda \in L^{\infty}_t, 0 \leq \lambda \leq 1: \rho_t(\lambda X + (1-\lambda) Y) \leq \lambda \rho_t(X) + (1-\lambda) \rho_t(Y) }[/math][clarification needed]
A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies:
- Conditional positive homogeneity
- [math]\displaystyle{ \forall \lambda \in L^{\infty}_t, \lambda \geq 0: \rho_t(\lambda X) = \lambda \rho_t(X) }[/math][clarification needed]
Acceptance set
The acceptance set at time [math]\displaystyle{ t }[/math] associated with a conditional risk measure is
- [math]\displaystyle{ A_t = \{X \in L^{\infty}_T: \rho_t(X) \leq 0 \text{ a.s.}\} }[/math].
If you are given an acceptance set at time [math]\displaystyle{ t }[/math] then the corresponding conditional risk measure is
- [math]\displaystyle{ \rho_t = \text{ess}\inf\{Y \in L^{\infty}_t: X + Y \in A_t\} }[/math]
where [math]\displaystyle{ \text{ess}\inf }[/math] is the essential infimum.[5]
Regular property
A conditional risk measure [math]\displaystyle{ \rho_t }[/math] is said to be regular if for any [math]\displaystyle{ X \in L^{\infty}_T }[/math] and [math]\displaystyle{ A \in \mathcal{F}_t }[/math] then [math]\displaystyle{ \rho_t(1_A X) = 1_A \rho_t(X) }[/math] where [math]\displaystyle{ 1_A }[/math] is the indicator function on [math]\displaystyle{ A }[/math]. 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. [math]\displaystyle{ \rho_t(X)[\omega] }[/math]) 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 [math]\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) }[/math].[6]
Example: dynamic superhedging price
The dynamic superhedging price involves conditional risk measures of the form [math]\displaystyle{ \rho_t(-X) = \operatorname*{ess\sup}_{Q \in EMM} \mathbb{E}^Q[X | \mathcal{F}_t] }[/math]. It is shown that this is a time consistent risk measure.
References
- ↑ Acciaio, Beatrice; Penner, Irina (2011). "Dynamic risk measures". Advanced Mathematical Methods for Finance: 1–34. Archived from the original on September 2, 2011. https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf. Retrieved July 22, 2010.
- ↑ Novak, S.Y. (2015). On measures of financial risk. 541–549. ISBN 978-849844-4964.
- ↑ 3.0 3.1 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.
- ↑ Föllmer, Hans; Penner, Irina (2006). "Convex risk measures and the dynamics of their penalty functions". Statistics & Decisions 24 (1): 61–96. doi:10.1524/stnd.2006.24.1.61.
- ↑ Penner, Irina (2007). Dynamic convex risk measures: time consistency, prudence, and sustainability. Archived from the original on July 19, 2011. https://web.archive.org/web/20110719042923/http://wws.mathematik.hu-berlin.de/~penner/penner.pdf. Retrieved February 3, 2011.
- ↑ 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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