Risk assessment for uncertain cash flows: Model ambiguity, - PowerPoint PPT Presentation

Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles Beatrice Acciaio University of Perugia and Vienna University (joint work with Hans F¨ ollmer and Irina Penner) AnStAp10, Vienna University, July 12-16, 2010

Dynamical setting We adopt a dynamical setting in order to take into account: • the timing of payments • the information released in time Discrete-time, with finite or infinite time horizon T : • T ∈ N , time axis T = { 0 , 1 , ..., T } • T = ∞ , time axis T = N 0 or T = N 0 ∪ {∞} Multiperiod information structure: (Ω , F , ( F t ) t ∈ T , P ) R ∞ = bounded adapted processes on (Ω , F T , ( F t ) t ∈ T , P ) = cumulated cash flows, value processes R ∞ = cash flows from time t on t

Dynamical setting We adopt a dynamical setting in order to take into account: • the timing of payments • the information released in time Discrete-time, with finite or infinite time horizon T : • T ∈ N , time axis T = { 0 , 1 , ..., T } • T = ∞ , time axis T = N 0 or T = N 0 ∪ {∞} Multiperiod information structure: (Ω , F , ( F t ) t ∈ T , P ) R ∞ = bounded adapted processes on (Ω , F T , ( F t ) t ∈ T , P ) = cumulated cash flows, value processes R ∞ = cash flows from time t on t

Dynamical setting We adopt a dynamical setting in order to take into account: • the timing of payments • the information released in time Discrete-time, with finite or infinite time horizon T : • T ∈ N , time axis T = { 0 , 1 , ..., T } • T = ∞ , time axis T = N 0 or T = N 0 ∪ {∞} Multiperiod information structure: (Ω , F , ( F t ) t ∈ T , P ) R ∞ = bounded adapted processes on (Ω , F T , ( F t ) t ∈ T , P ) = cumulated cash flows, value processes R ∞ = cash flows from time t on t

Conditional convex risk measures Def. ρ t : R ∞ → L ∞ (Ω , F t , P ) is called a conditional convex t risk measure for processes if for all X , Y ∈ R ∞ t : Normalization: ρ t (0) = 0 Monotonicity: X ≤ Y ⇒ ρ t ( X ) ≥ ρ t ( Y ) Conditional convexity: ∀ λ ∈ L ∞ (Ω , F t , P ) , 0 ≤ λ ≤ 1: ρ t ( λ X + (1 − λ ) Y ) ≤ λρ t ( X ) + (1 − λ ) ρ t ( Y ) Conditional cash-invariance: m ∈ L ∞ (Ω , F t , P ) ρ t ( X + m 1 { t , t +1 ,... } ) = ρ t ( X ) − m , → ( ρ t ) t is called dynamic convex risk measure for processes (Cheridito,Delbaen & Kupper 2006)

Acceptance set An important characterization of a conditional convex risk measure is the acceptance set : � � � � ρ t ( X ) ≤ 0 X ∈ R ∞ A t = . t ρ t is uniquely determined through its acceptance set: � � � � X + Y 1 { t , t +1 ,... } ∈ A t Y ∈ L ∞ (Ω , F t , P ) ρ t ( X ) = ess inf . ⇒ ρ t ( X ) is the minimal conditional capital requirement that has to be added to the cash-flow X at time t in order to make it acceptable.

Acceptance set An important characterization of a conditional convex risk measure is the acceptance set : � � � � ρ t ( X ) ≤ 0 X ∈ R ∞ A t = . t ρ t is uniquely determined through its acceptance set: � � � � X + Y 1 { t , t +1 ,... } ∈ A t Y ∈ L ∞ (Ω , F t , P ) ρ t ( X ) = ess inf . ⇒ ρ t ( X ) is the minimal conditional capital requirement that has to be added to the cash-flow X at time t in order to make it acceptable.

Product space and optional filtration Define the product space (¯ Ω , ¯ F , ¯ ¯ P ) as: Ω = Ω × T , � ¯ ¯ � A t ∈ F t , t ∈ T ) , F = σ ( { A t × { t } P = P ⊗ µ, where µ = ( µ t ) t ∈ T is some adapted reference process s.t. µ t > 0 and � P [ X ] := E P [ � t µ t = 1, and E ¯ t X t µ t ] R ∞ = L ∞ (¯ F , ¯ Ω , ¯ = ⇒ P ) Consider the optional filtration ( ¯ F t ) t ∈ T on (¯ Ω , ¯ F ), given by ¯ F t = σ ( { A j × { j } , A t × { t , .. }| A j ∈ F j , j = 0 , .., t − 1 , A t ∈ F t } )

Product space and optional filtration Define the product space (¯ Ω , ¯ F , ¯ ¯ P ) as: Ω = Ω × T , � ¯ ¯ � A t ∈ F t , t ∈ T ) , F = σ ( { A t × { t } P = P ⊗ µ, where µ = ( µ t ) t ∈ T is some adapted reference process s.t. µ t > 0 and � P [ X ] := E P [ � t µ t = 1, and E ¯ t X t µ t ] R ∞ = L ∞ (¯ F , ¯ Ω , ¯ = ⇒ P ) Consider the optional filtration ( ¯ F t ) t ∈ T on (¯ Ω , ¯ F ), given by ¯ F t = σ ( { A j × { j } , A t × { t , .. }| A j ∈ F j , j = 0 , .., t − 1 , A t ∈ F t } )

Product space and optional filtration Define the product space (¯ Ω , ¯ F , ¯ ¯ P ) as: Ω = Ω × T , � ¯ ¯ � A t ∈ F t , t ∈ T ) , F = σ ( { A t × { t } P = P ⊗ µ, where µ = ( µ t ) t ∈ T is some adapted reference process s.t. µ t > 0 and � P [ X ] := E P [ � t µ t = 1, and E ¯ t X t µ t ] R ∞ = L ∞ (¯ F , ¯ Ω , ¯ = ⇒ P ) Consider the optional filtration ( ¯ F t ) t ∈ T on (¯ Ω , ¯ F ), given by ¯ F t = σ ( { A j × { j } , A t × { t , .. }| A j ∈ F j , j = 0 , .., t − 1 , A t ∈ F t } )

Risk measures viewed on the optional filtration Proposition. There is a one-to-one correspondence between conditional convex risk measures for processes ρ t : R ∞ → L ∞ (Ω , F t , P ) t conditional convex risk measures for random variables on the product space ρ t : L ∞ (¯ Ω , ¯ F , ¯ P ) → L ∞ (¯ Ω , ¯ F t , ¯ ¯ P ) The relation is given by ρ t ( X ) = − X 0 1 { 0 } − . . . − X t − 1 1 { t − 1 } + ρ t ( X )1 { t , t +1 ,... } ¯

Representation of risk measures on random variables Theorem. For ρ t : L ∞ (Ω , F , P ) → L ∞ (Ω , F t , P ) TFAE: 1. ρ t is continuous from above: X n ց X ⇒ ρ t ( X n ) ր ρ t ( X ) 2. ρ t has the following robust representation : ρ t ( X ) = ess sup ( E Q [ − X |F t ] − α t ( Q )) Q ∈Q t where � � � � Q = P | F t Q t = Q ≪ P , and the minimal penalty function α t is given by α t ( Q ) = ess sup ( E Q [ − X |F t ] − ρ t ( X )) X ∈ L ∞ ( F ) (Detlefsen and Scandolo (2005))

Optional random measures For any measure Q ≪ loc P we introduce: ◮ the set Γ( Q ) of optional random measures γ on T which are normalized with respect to Q : γ = ( γ t ) t ∈ T nonnegative adapted process s.t. � t ∈ T γ t = 1 Q -a.s. with the additional property � � dQ � � γ ∞ = 0 Q -a.s. on lim F t = ∞ if T = N 0 ∪ {∞} � dP t →∞

Predictable discounting processes ◮ the set D ( Q ) of predictable discounting processes D : D = ( D t ) t ∈ T predict. non-increasing, D 0 = 1, D ∞ = lim t →∞ D t Q -a.s. where D ∞ = 0 Q -a.s. if T = N 0 , � � � dQ � D ∞ = 0 Q -a.s. on lim F t = ∞ if T = N 0 ∪ {∞} � dP t →∞ ◮◮ There is a one-to-one correspondence between random measures in Γ( Q ) and predictable discounting in D ( Q ): γ t = D t − D t +1 , t < ∞ , γ ∞ = D ∞

Predictable discounting processes ◮ the set D ( Q ) of predictable discounting processes D : D = ( D t ) t ∈ T predict. non-increasing, D 0 = 1, D ∞ = lim t →∞ D t Q -a.s. where D ∞ = 0 Q -a.s. if T = N 0 , � � � dQ � D ∞ = 0 Q -a.s. on lim F t = ∞ if T = N 0 ∪ {∞} � dP t →∞ ◮◮ There is a one-to-one correspondence between random measures in Γ( Q ) and predictable discounting in D ( Q ): γ t = D t − D t +1 , t < ∞ , γ ∞ = D ∞

Decomposition of measures on the optional σ -field Theorem. For any probability measure ¯ Q on (¯ Ω , ¯ F ) we have: Q ≪ ¯ ¯ P if and only if there exist a probability measure Q on (Ω , F T ), Q ≪ loc P an optional random measure γ ∈ Γ( Q ) (resp. D ∈ D ( Q )) such that � T �� � � � X ∈ R ∞ E ¯ Q [ X ] = E Q γ t X t = E Q D t ∆ X t , t ∈ T t =0 (combining the Itˆ o-Watanabe factorization with an extension theorem for standard systems) In this case we write: ¯ Q = Q ⊗ γ = Q ⊗ D

Robust representation Theorem. For ρ t : R ∞ → L ∞ (Ω , F t , P ) TFAE: t 1. ρ t continuous from above: X n s ց X s ∀ s ≥ t ⇒ ρ t ( X n ) ր ρ t ( X ) 2. ρ t has the following robust representation : T � � � � � � � F t ρ t ( X ) = ess sup ess sup − D s ∆ X s − α t ( Q ⊗ D ) , E Q Q ∈Q loc D ∈D t ( Q ) s = t t ր տ model discounting ambiguity ambiguity where Q loc t = { Q ≪ loc P : Q = P | F t } , D t ( Q )= { D ∈ D ( Q ) : D s = 1 s ≤ t } and the minimal penalty function α t is given by: � � γ s � � � � � F t α t ( Q ⊗ D ) = Q-ess sup E Q − X s − ρ t ( X ) D t X ∈R ∞ s ≥ t t

Time consistency X ∈ R ∞ → ( ρ t ( X )) t describes the evolution of risk over time. Question: How should risk measurement be updated as more information becomes available? Def. ( ρ t ) t is called (strongly) time consistent if for all t ≥ 0 X t = Y t and ρ t +1 ( X ) ≤ ρ t +1 ( Y ) ⇒ ρ t ( X ) ≤ ρ t ( Y ) An equivalent characterization is recursiveness : ρ t ( X ) = ρ t ( X t 1 { t } − ρ t +1 ( X )1 { t +1 ,... } ) ∀ t ≥ 0 Remark. ( ρ t ) t on R ∞ is time consistent ⇐ ⇒ the corresponding ρ t ) t on L ∞ (¯ Ω , ¯ F , ¯ (¯ P ) is time consistent

Supermartingale properties Proposition. Let ( ρ t ) t on R ∞ be continuous from above and time consistent . Then, ∀ ¯ Q = Q ⊗ D ≪ ¯ P such that α 0 ( Q ⊗ D ) < ∞ , ( D t α t ( ¯ the discounted penalty process Q )) t ∈ T ∩ N 0 the ‘global risk’ process of X ∈ R ∞ t � D t ( ρ t ( X − X t ) + α t ( ¯ Q )) − D s ∆ X s , t ∈ T ∩ N 0 s =0 are Q -supermartingales.