Conditional and Dynamic Preferences

Transcription

1 Conditional and Dynamic Preferences How can we Understand Risk in a Dynamic Setting? Samuel Drapeau Joint work with Hans Föllmer Humboldt University Berlin Jena - March 17th 2009 Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 1/22

2 Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 2/22

3 Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 3/22

4 Dynamic Risk Measures: Disappointment Definition Let (Ω, F, P) be a probability space and F t, t = 0, 1,..., T a filtration. Definition (Conditional Convex Risk Measure) A functional ρ t : L L t is a conditional convex risk measure if it is: Monotone: ρ t (X ) ρ t (Y ) P-a.s. for any X Y Conditionally translation invariant: ρ t (X + m t) = ρ t (X ) m t for any X and m t L t. Conditionally convex: For any X, Y and 0 λ t 1 F t-measurable: P-a.s. ρ t (λ tx + (1 λ t) Y ) λ tρ t (X ) + (1 λ t) ρ t (Y ) P-a.s. Normalized: ρ t (0) = 0 P-a.s. Definition (Time Consistency) (ρ t) t [0,T ] is time consistent if for any X, Y and times t s ρ s (X ) ρ s (Y ) P-a.s. = ρ t (X ) ρ t (Y ) P-a.s. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 4/22

5 Dynamic Risk Measures: Disappointment Dual representation An important result concerning convex risk measures is the dual representation (Static case: Föllmer and Schied. Conditional case: Detlefsen and Scandolo). Theorem If a conditional convex risk measure is continuous from below (i.e. X n X implies ρ t (X n) ρ t (X )) the following representation holds: h i o ρ t (X ) = ess sup ne Q X F t α t (Q) Q<<P Q=P on F t where α t : M 1 (Ω, F, P) L + (Ω, F t, P) is a penalty function. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 5/22

6 Dynamic Risk Measures: Disappointment Disappointment Why are we so disappointed? Time consistency and cash invariance impose some strong conditions when the risk is updated infinitely many times. The only good candidates are entropic- like risk measures, i.e. ρ t (X ) = 1/γ ln E he γx Ft i. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 6/22

7 Dynamic Risk Measures: Disappointment Disappointment Why are we so disappointed? Time consistency and cash invariance impose some strong conditions when the risk is updated infinitely many times. The only good candidates are entropic- like risk measures, i.e. ρ t (X ) = 1/γ ln E he γx Ft i. σ n is a subdivision h of [0, T i ] and the penalty function at time t n is Z α tn (Q) = E ϕ Ft Z tn for some ϕ positive convex. If the filtration is generated by a Brownian motion and the discrete family of risk measures ρ σn t i is time consistent we have: Theorem where γ = 2/ϕ (1) ρ σn t (X ) dp dt 1 i σ n 0 γ ln E he Ftn γx (2.1) A similar result has been shown for any time consistent risk measure which are law invariant by Kupper and Schachermayer. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 6/22

8 Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 7/22

9 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 Joint work with Hans Föllmer Conditional and Dynamic Preferences 8/22

10 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 Weak Preference Order: is reflexive, transitive and complete. Independance: µ ν implies: αµ + (1 α) λ αν + (1 α) λ for any λ M and α ]0, 1]. Continuity: The restriction of to M (B (0, r)) is continuous w.r.t. the weak topology for any r > 0. Numerical Representation There exist a continuous function u : R R such that: where: µ ν U (µ) U (ν) Z U (µ) = u (x) µ (dx) Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 8/22

11 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 Weak Preference Order: is reflexive, transitive and complete. Independance: µ ν implies: αµ + (1 α) λ αν + (1 α) λ for any λ M and α ]0, 1]. Continuity: The restriction of to M (B (0, r)) is continuous w.r.t. the weak topology for any r > 0. Numerical Representation There exist a continuous function u : R R such that: where: µ ν U (µ) U (ν) Z U (µ) = u (x) µ (dx) Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 8/22

12 Preference Orders Savage L. (1954) Instead of measures he considered a preference relation on the set of bounded measurable functions X defined on a measurable space (Ω, F ). Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 9/22

13 Preference Orders Savage L. (1954) Instead of measures he considered a preference relation on the set of bounded measurable functions X defined on a measurable space (Ω, F ). Preference Axioms Weak Preference Order: is reflexive, transitive and complete. Independance: X Y implies: αx + (1 α) Z αy + (1 α) Z for any Z X and α ]0, 1]. + several other technical axioms (archimedian, monotonicity,... ) Numerical Representation There exist a continuous function u : R R and a probability measure Q on (Ω, F ) such that: where: X Y U (X ) U (Y ) U (X ) = E Q [u (X )] Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 9/22

14 Preference Orders Savage L. (1954) Instead of measures he considered a preference relation on the set of bounded measurable functions X defined on a measurable space (Ω, F ). Preference Axioms Weak Preference Order: is reflexive, transitive and complete. Independance: X Y implies: αx + (1 α) Z αy + (1 α) Z for any Z X and α ]0, 1]. + several other technical axioms (archimedian, monotonicity,... ) Numerical Representation There exist a continuous function u : R R and a probability measure Q on (Ω, F ) such that: where: X Y U (X ) U (Y ) U (X ) = E Q [u (X )] Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 9/22

15 Preference Orders Robust version: Gilboa & Schmeidler (89)[3], Maccheroni... (04)[5], Föllmer... (07)[1][2] To overcome Elsberg s paradox, the independence axiom will be weakened. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 10/22

16 Preference Orders Robust version: Gilboa & Schmeidler (89)[3], Maccheroni... (04)[5], Föllmer... (07)[1][2] To overcome Elsberg s paradox, the independence axiom will be weakened. The preference order are now defined on the space X of uniformly bounded stochastic kernels on the real line X (ω, dx) in which X and M b (R) are embedded. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 10/22

17 Preference Orders Robust version: Gilboa & Schmeidler (89)[3], Maccheroni... (04)[5], Föllmer... (07)[1][2] Preference Axioms Weak Preference Order: is reflexive, transitive and complete. Weak Uncertainty Independence: α X + (1 α) µ αỹ + (1 α) µ α X + (1 α) ν αỹ + (1 α) ν for any ν M b (R). Uncertainty Aversion: X Ỹ implies α X + (1 α) Ỹ X for any α [0, 1]. + technical axioms (archimedian, monotonicity, continuity from above) Numerical Representation There exist a numerical representation U of and it is such that: U (X ) = ρ conv (u (X )) for some monotone continuous function u : R R and an unconditional convex risk measure ρ conv. In particular, U (X ) = inf {E Q [u (X )] + α (Q)} Q M 1 (Ω,F) where α : M 1 (Ω, F ) R is penalty function. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 10/22

18 Preference Orders Robust version: Gilboa & Schmeidler (89)[3], Maccheroni... (04)[5], Föllmer... (07)[1][2] Preference Axioms Weak Preference Order: is reflexive, transitive and complete. Weak Uncertainty Independence: α X + (1 α) µ αỹ + (1 α) µ α X + (1 α) ν αỹ + (1 α) ν for any ν M b (R). Uncertainty Aversion: X Ỹ implies α X + (1 α) Ỹ X for any α [0, 1]. + technical axioms (archimedian, monotonicity, continuity from above) Numerical Representation There exist a numerical representation U of and it is such that: U (X ) = ρ conv (u (X )) for some monotone continuous function u : R R and an unconditional convex risk measure ρ conv. In particular, U (X ) = inf {E Q [u (X )] + α (Q)} Q M 1 (Ω,F) where α : M 1 (Ω, F ) R is penalty function. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 10/22

19 Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 11/22

20 Conditional Preference Orders What is a Conditional Preference Order. The question of a conditional preference order has already emerged in the literature (Kreps & Porteus [4], Skiadas [6], Macheroni & al.) but their axiomatic is highly disputable, and is strongly related to their basic setting (Trees). Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 12/22

21 Conditional Preference Orders What is a Conditional Preference Order. The question of a conditional preference order has already emerged in the literature (Kreps & Porteus [4], Skiadas [6], Macheroni & al.) but their axiomatic is highly disputable, and is strongly related to their basic setting (Trees). The key question to address is the completeness. There are, beyond mathematical interest, many reasons for doubting of this assumption: Indeed, Incompleteness does not reflects an unexceptional trait as pointed out by Aumann R.J.: Of all the axiom of utility theory, the completeness axiom is perhaps the most questionable. Like others of the axioms, it is inaccurate as a description of real life, but unlike them we find it hard to accept even from a normative viewpoint. [... ] For example, certain decisions that an individual is asked to make might involve highly hypothetical situations, which he will never face in real life. He might feel that he cannot reach an honest decision in such cases. Other decision problems might be extremely complex, too complex for intuitive insight, and our individual might prefer to make no decision at all in these problems. Is it rational to force decision in such cases? Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 12/22

22 Conditional Preference Orders Axiomatic (Ω, F ) measurable space, G F and P probability measure on (Ω, G ). stochastic kernels. Axiomatic Pre-Order: is a reflexive and transitive relation. G -consistency: For any X, Ỹ hold: Negligibility: For any A, B G with P [A] = P [B] holds: X X Ỹ X1 A Ỹ 1 B Local completeness: There exists a set A G of strict positive probability such that: X1 A Ỹ 1 A or X1 A Ỹ 1 A Union consistency: For any countable family (A n) n N in G, n N, X1 An Ỹ 1 An = X1 { Sn N An} Ỹ 1 { S n N An} Reducibility: For A, B G with A B, X1 A Ỹ 1 A = X1 B Ỹ 1 B Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 13/22

23 Conditional Preference Orders Conditional Completeness Even if we loose completeness, we can manage to deal with in a good way: Lemma Suppose given a conditional preference relation then for each X, Ỹ there exists a partition A, B, C G of Ω such that: 8 >< X G A Ỹ X G B >: Ỹ X G C Ỹ This partition can be chosen maximally. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 14/22

24 Conditional Preference Orders Axiomatic: Rest Axiomatic Uncertainty Aversion: X Ỹ implies α X + (1 α) Ỹ G X for any 0 α 1 G -measurable function. Monotonicity: If Ỹ (ω) X (ω) for any ω, then Ỹ X. Moreover, for reals, x < y iff δ x δ y Weak Certainty Independence: For X, Ỹ X, µ, ν M b (R, G ) and a 0 < α < 1 G -measurable, we have: α X + (1 α) µ αỹ + (1 α) µ α X + (1 α) ν αỹ + (1 α) ν Continuity: For Z Ỹ X, there exists α, β with 0 < α, β < 1 G -measurable such that: α Z + (1 α) X Ỹ β Z + (1 β) X Moreover, the restriction of to M 1 ([ c, c], G ) is continuous with respect to the P-a.s. weak topology. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 15/22

25 Conditional Preference Orders Conditional von Neumann & Morgenstern Considering the restriction of to M b (R, G ) we get a conditional version of the theorem of von Neumann J. & Morgenstern O.: Theorem If is a monotone, independent, and continuous conditional preference relation, it admits then a conditional von Neumann and Morgenstern representation: Z U (µ) = u (, µ) µ (, dx) (4.1) where u (, ) is continuous for P-almost surely in its second coordinate and G -measurable in its first for any x R. In particular, U is G -measurable. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 16/22

26 Conditional Preference Orders Conditional Robust Representation Theorem If is a conditional preference order fulfilling all the axioms aforementioned, there exists then a numerical representation Ũ on X such that: Z «Ũ (X ) = ρ conv u (, x) X (, dx) (4.2) where u is as previously and ρ conv is a conditional convex risk measure. In particular:»z ff Ũ X = ess inf je Q u (, x) X (, dx) G + αmin G (Q) Q M 1,f (P,G ) (4.3) If moreover the induced preference order on X satisfies the following additional continuity property: X n X P-a.s. = For any ε > 0, X n G X ε for all large n (4.4) then the penalty function αmin G is concentrated on M 1 (P, G ) Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 17/22

27 Conditional Preference Orders Risk vs. Model Uncertainty Theorem (Savage Representation Theorem) The following two assertion equivalent: G fulfills the more restrictive uncertainty independence axiom: X G Ỹ α X + (1 α) Ỹ G X Ũ is a conditional version of the Savage representation, i.e. there exists Q M 1,f (P) such that:»z Ũ X = E Q u (, x) X (, ω) G (4.5) On X it reduces to: U (X ) = E Q»u X G (4.6) Moreover, if continuity from above is assumed, the finitely additive set function Q is in fact a probability measure. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 18/22

28 Conditional Preference Orders Risk vs. Model Uncertainty Theorem (Pure Measures of Uncertainty) The following two assertion equivalent: The preference relation fulfills translation axiom: X G Ỹ T m X G T m Ỹ (4.7) where T m is the translation operator of a distribution. The numerical representation is one of the following two types: (1) U (X ) = ρ conv (X ) (2) U (X ) = ρ coh `e ax 1 a If we impose furthermore to the preferences of being strictly convex we get only the first type for a strictly convex risk measure. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 19/22

29 Outline 1 Dynamic Risk Measures: Disappointment 2 Preference Orders 3 Conditional Preference Orders 4 Dynamic of Preferences Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 20/22

30 Dynamic of Preferences On Going Work We consider here some processes (X t) t=0,1...t. At time t, the numerical representation of the preference relation is given by: U t (X ) = ρ t (u t (X 0,..., X T )) where u t is monotone continuous in each coordinate. Temporal Consistency: If X t+1 Y and X = Y up to time t, then X t Y. Under this condition: Theorem U t (X ) = ρ t (f t (X 0,..., X t, U t+1 (X ))) (5.1) where f t is an agregator functional increasing in each of its terms. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 21/22

31 Dynamic of Preferences On Going Work Preference for information Suppose you have under a probability measure Q two processes (0, 0, X 2) and (0, 0, Y 2) such that the distribution under Q of X 2 is equal to Y 2, but X 2 is F 1 measurable whereas Y 2 is F 2-measurable. If you prefer X to Y, then the agregator functional f 1 (x 0, x 1, y) should be convex in y. Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 21/22

32 Bibliography Hans Föllmer and Alexander Schied. Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematik. Walter de Gruyter, Berlin, New York, 2 edition, Hans Föllmer, Alexander Schied, and Stefan Weber. Robust Preferences and Robust Portfolio Choice. Preprint, I. Gilboa and D. Schmeidler. Maximin Expected Utility with a Non-Unique Prior. Journal of Mathematical Economics, 18: , David M Kreps and Evan L Porteus. Temporal Resolution of Uncertainty and Dynamic Choice Theory. Econometrica, 46(1): , January available at Fabio Maccheroni, Massimo Marinacci, and Aldo Rustichini. Ambiguity Aversion, Robustness, and the Variational Representation of Preferences. Econometrica, 74(6): , November Costis Skiadas. Conditioning and aggregation of preferences. Econometrica, 65(2): , March John von Neumann and Oskar Morgenstern. Theory of Games and Economics Behavior. Princeton University Press, 2nd edition, Samuel Drapeau Joint work with Hans Föllmer Conditional and Dynamic Preferences 22/22