Dynamic Risk Measures and Conditional Robust Utility Representation - PowerPoint PPT Presentation

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures and Conditional Robust Utility Representation How can we Understand Risk in a Dynamic Setting? Samuel Drapeau IRTG — Disentis Summer School 2008 Juli 22th 2007 Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 1/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 2/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 3/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Definition - Static case Let (Ω , F , P ) be a probability space. Definition (Convex Risk Measure — Artzner & al , F¨ ollmer & Schied ) A functional ρ : ▲ ∞ : → ❘ is a convex risk measure if it is: Monotone: For X , Y ∈ ▲ ∞ , X ≥ Y then ρ ( X ) ≤ ρ ( Y ) Translation invariant: For X ∈ ▲ ∞ and m ∈ ❘ , ρ ( X + m ) = ρ ( X ) − m Convex: For X , Y ∈ ▲ ∞ and λ ∈ [0 , 1]: ρ ( λ X + (1 − λ ) Y ) ≤ λρ ( X ) + (1 − λ ) ρ ( Y ) Normalized: ρ (0) = 0 Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 4/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Definition - Conditional case Let (Ω , F , P ) be a probability space and F t a sub- σ -algebra of F . Definition (Conditional Convex Risk Measure) A functional ρ t : ▲ ∞ → ▲ ∞ is a conditional convex risk measure if it is: t Monotone: For X , Y ∈ ▲ ∞ , X ≥ Y then ρ t ( X ) ≤ ρ t ( Y ) P -a.s. Conditionally translation invariant: For X ∈ ▲ ∞ and m t ∈ ▲ ∞ t , ρ t ( X + m t ) = ρ t ( X ) − m t P -a.s. Conditionally convexe: For X , Y ∈ ▲ ∞ and 0 ≤ λ t ≤ 1 F t -measurable: ρ t ( λ t X + (1 − λ t ) Y ) ≤ λ t ρ t ( X ) + (1 − λ t ) ρ t ( Y ) P -a.s. Normalized: ρ t (0) = 0 P -a.s. Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 5/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Dual representation An important result concerning convex risk measures is the dual representation (Static case: F¨ ollmer and Schied . Conditional case: Detlefsen and Scandolo ). Theorem If a conditional convex risk measure is continuous from below (i.e. X n ց X implies ρ t ( X n ) ր ρ t ( X ) ) the following representation holds: n h ˛ i o ρ t ( X ) = ess sup E Q − X ˛ F t − α t ( Q ) ˛ Q ∼ P Q = P over F t where α t : M 1 (Ω , F , P ) → ▲ ∞ + (Ω , F t , P ) ∪ ∞ is a penalty function. Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 6/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Time consistency Considering a family of conditional risk measures ( ρ t ) t ∈ [0 , T ] on a filtrated probability space, the property of time consistency is understood as follow: Definition The family of conditional convex risk measures, is said to be time consistent if for all X , Y ∈ ▲ ∞ and times 0 ≤ t ≤ s ≤ T , holds: ρ s ( X ) ≥ ρ s ( Y ) P -a.s. = ⇒ ρ t ( X ) ≥ ρ t ( Y ) P -a.s. This definition is equivalent to the following dynamic programing principle: ρ t ( X ) = ρ t ( − ρ s ( X )) Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 7/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Disappointment Why are we so disappointed? The time consistency together with cash invariance impose some very strong conditions in the continuous case such that infinitely many of them lead to e − γ X ˛ “ h i” some entropic-“like” risk measures, i.e. ρ t ( X ) = 1 /γ ln E ˛ F t . ˛ Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 8/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Disappointment Why are we so disappointed? The time consistency together with cash invariance impose some very strong conditions in the continuous case such that infinitely many of them lead to e − γ X ˛ “ h i” some entropic-“like” risk measures, i.e. ρ t ( X ) = 1 /γ ln E ˛ F t . ˛ Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 8/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Disappointment Why are we so disappointed? The time consistency together with cash invariance impose some very strong conditions in the continuous case such that infinitely many of them lead to e − γ X ˛ “ h i” some entropic-“like” risk measures, i.e. ρ t ( X ) = 1 /γ ln E ˛ F t . ˛ For a subdivision σ n of the interval [0 , T ], take as penalty function h “ ” ˛ i Z α t ( Q ) = E ϕ ˛ F t for a positive convex function ϕ twice differentiable ˛ Z t in a neighborhood of 1 and with inf ϕ ( x ) = ϕ (1) = 0. The filtration is generated by a Brownian motion. If we imposed for the corresponding discrete family of risk measures ρ σ n to be t i time consistent we have: Theorem 1 “ h e − γ X ˛ i” dP ⊗ dt ρ σ n ( X ) − − − − → γ ln E ˛ F t (2.1) ˛ t | σ n |→ 0 where γ = 2 /ϕ ′′ (1) Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 8/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Dynamic Risk Measures: Disappointment Disappointment Moreover, Kupper and Schachermayer proved in the restrictive framework of law invariance a general result: Theorem For an infinite family ρ n of law invariant risk measures on an atom free filtration ( F n ) n ∈ ◆ . If the family is time consistent, there exists then γ ∈ ❘ + ∪ ∞ such that: ρ n ( X ) = 1 “ h e − γ X ˛ i” γ ln E ˛ F n ˛ Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 9/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 10/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Preference Orders von Neumann J. & Morgenstern O. (1944)[7] The preference order is defined by a binary relation � on the set of measures with bounded support M b ( S , S ) ≡ M . Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 11/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Preference Orders von Neumann J. & Morgenstern O. (1944)[7] The preference order is defined by a binary relation � on the set of measures with bounded support M b ( S , S ) ≡ M . Preference Axioms Numerical Representation Weak Preference Order: � is reflexive, transitive and complete. There exist a continuous function u : ❘ �→ ❘ such that: Independance: For any µ ≻ ν holds: µ � ν ⇔ U ( µ ) ≥ U ( ν ) αµ + (1 − α ) λ ≻ αν + (1 − α ) λ where: for any λ ∈ M and α ∈ ]0 , 1]. Z Continuity: The restriction of � to U ( µ ) = u ( x ) µ ( dx ) M ( B (0 , r )) is continuous w.r.t. the weak topology for any r > 0. Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 11/21

Dynamic Risk Measures: Disappointment Preference Orders Conditional Preference Orders Dynamic of Preferences Preference Orders von Neumann J. & Morgenstern O. (1944)[7] The preference order is defined by a binary relation � on the set of measures with bounded support M b ( S , S ) ≡ M . Preference Axioms Numerical Representation Weak Preference Order: � is reflexive, transitive and complete. There exist a continuous function u : ❘ �→ ❘ such that: Independance: For any µ ≻ ν holds: µ � ν ⇔ U ( µ ) ≥ U ( ν ) αµ + (1 − α ) λ ≻ αν + (1 − α ) λ where: for any λ ∈ M and α ∈ ]0 , 1]. Z Continuity: The restriction of � to U ( µ ) = u ( x ) µ ( dx ) M ( B (0 , r )) is continuous w.r.t. the weak topology for any r > 0. Samuel Drapeau — Dynamic Risk Measures and Conditional Robust Utility Representation 11/21