Time Consistency and Calculation of Risk Measures in Markets with Transaction Costs Birgit Rudloff ORFE, Princeton University joint work with Zach Feinstein (Princeton University) Probability, Control and Finance A Conference in Honor of Ioannis Karatzas, June 5, 2012
Outline 1 Dynamic set-valued risk measures 2 Time consistency 3 Examples and calculation of risk measures 1 Superhedging under transaction costs 2 AV@R 4 Multi-portfolio time consistency by composition B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs d assets (may include different currencies), discrete time Θ, (Ω , ( F t ) t ∈ Θ , P ) B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs d assets (may include different currencies), discrete time Θ, (Ω , ( F t ) t ∈ Θ , P ) portfolio vectors in physical units (num´ eraire-free): (# of units in d assets) B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs d assets (may include different currencies), discrete time Θ, (Ω , ( F t ) t ∈ Θ , P ) portfolio vectors in physical units (num´ eraire-free): (# of units in d assets) proportional transaction costs at time t : closed convex cone + ⊆ K t ( ω ) ⊆ R d (solvency cone), positions transferrable R d into nonnegative positions B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs d assets (may include different currencies), discrete time Θ, (Ω , ( F t ) t ∈ Θ , P ) portfolio vectors in physical units (num´ eraire-free): (# of units in d assets) proportional transaction costs at time t : closed convex cone + ⊆ K t ( ω ) ⊆ R d (solvency cone), positions transferrable R d into nonnegative positions claim X ∈ L p d ( F T ): payoff (in physical units) at time T B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs d assets (may include different currencies), discrete time Θ, (Ω , ( F t ) t ∈ Θ , P ) portfolio vectors in physical units (num´ eraire-free): (# of units in d assets) proportional transaction costs at time t : closed convex cone + ⊆ K t ( ω ) ⊆ R d (solvency cone), positions transferrable R d into nonnegative positions claim X ∈ L p d ( F T ): payoff (in physical units) at time T a portfolio vector u ∈ M t ( M t ⊆ L p d ( F t ) linear subspace of eligible assets, e.g. Euro & Dollar) compensates the risk of X at time t B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs d assets (may include different currencies), discrete time Θ, (Ω , ( F t ) t ∈ Θ , P ) portfolio vectors in physical units (num´ eraire-free): (# of units in d assets) proportional transaction costs at time t : closed convex cone + ⊆ K t ( ω ) ⊆ R d (solvency cone), positions transferrable R d into nonnegative positions claim X ∈ L p d ( F T ): payoff (in physical units) at time T a portfolio vector u ∈ M t ( M t ⊆ L p d ( F t ) linear subspace of eligible assets, e.g. Euro & Dollar) compensates the risk of X at time t if X + u ∈ A t for some set A t ⊆ L p d ( F T ) of acceptable positions. B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure A set-valued function R t : L p d ( F T ) → P (( M t ) + ) = { D ⊆ M t : D = D + ( M t ) + } is a conditional risk measure if B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure A set-valued function R t : L p d ( F T ) → P (( M t ) + ) = { D ⊆ M t : D = D + ( M t ) + } is a conditional risk measure if 1 Finite at zero: ∅ � = R t (0) � = M t B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure A set-valued function R t : L p d ( F T ) → P (( M t ) + ) = { D ⊆ M t : D = D + ( M t ) + } is a conditional risk measure if 1 Finite at zero: ∅ � = R t (0) � = M t 2 M t translative: R t ( X + m ) = R t ( X ) − m for any m ∈ M t B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure A set-valued function R t : L p d ( F T ) → P (( M t ) + ) = { D ⊆ M t : D = D + ( M t ) + } is a conditional risk measure if 1 Finite at zero: ∅ � = R t (0) � = M t 2 M t translative: R t ( X + m ) = R t ( X ) − m for any m ∈ M t 3 Monotone: if X − Y ∈ L p d ( F T ) + then R t ( X ) ⊇ R t ( Y ) B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure A set-valued function R t : L p d ( F T ) → P (( M t ) + ) = { D ⊆ M t : D = D + ( M t ) + } is a conditional risk measure if 1 Finite at zero: ∅ � = R t (0) � = M t 2 M t translative: R t ( X + m ) = R t ( X ) − m for any m ∈ M t 3 Monotone: if X − Y ∈ L p d ( F T ) + then R t ( X ) ⊇ R t ( Y ) A conditional risk measure is normalized if for any X ∈ L p d ( F T ): R t ( X ) + R t (0) = R t ( X ) B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs M t ⊆ L p ( M t ) + = M t ∩ L p d ( F t ) d ( F t ) + Conditional Set-Valued Risk Measure A set-valued function R t : L p d ( F T ) → P (( M t ) + ) = { D ⊆ M t : D = D + ( M t ) + } is a conditional risk measure if 1 Finite at zero: ∅ � = R t (0) � = M t 2 M t translative: R t ( X + m ) = R t ( X ) − m for any m ∈ M t 3 Monotone: if X − Y ∈ L p d ( F T ) + then R t ( X ) ⊇ R t ( Y ) A conditional risk measure is normalized if for any X ∈ L p d ( F T ): R t ( X ) + R t (0) = R t ( X ) dynamic risk measure : sequence ( R t ) T t =0 of conditional risk measures B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs Primal Representation Risk measures and acceptance sets are one-to-one via R t ( X ) = { u ∈ M t : X + u ∈ A t } A t = { X ∈ L p d ( F T ) : 0 ∈ R t ( X ) } . and B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs Primal Representation Risk measures and acceptance sets are one-to-one via R t ( X ) = { u ∈ M t : X + u ∈ A t } A t = { X ∈ L p d ( F T ) : 0 ∈ R t ( X ) } . and R t A t M t 1 I ∩ A t � = ∅ ∅ � = R t (0) � = M t finite at zero I ∩ ( L p M t 1 d \ A t ) � = ∅ Y − X ∈ L p d ( F T ) + A t + L p monotone d ( F T ) + ⊆ A t ⇒ R t ( Y ) ⊇ R t ( X ) convex convex positively homogeneous cone subadditive A t + A t ⊆ A t closed images directionally closed lsc closed R t ( X ) = R t ( X ) + K M t A t + L p d ( K M t ) ⊆ A t market compatible t t B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs Let G (( M t ) + ) = { D ⊆ M t : D = cl co( D + ( M t ) + ) } . Dual Representation, 1 ≤ p ≤ ∞ A function R t : L p d ( F T ) → G (( M t ) + ) is a closed coherent conditional risk measure if and only if there is a nonempty set W q t,R t ⊆ W q t such that � { E Q R t ( X ) = t [ − X ] + G t ( w ) } ∩ M t . ( Q ,w ) ∈W q t,Rt B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs Let G (( M t ) + ) = { D ⊆ M t : D = cl co( D + ( M t ) + ) } . Dual Representation, 1 ≤ p ≤ ∞ A function R t : L p d ( F T ) → G (( M t ) + ) is a closed coherent conditional risk measure if and only if there is a nonempty set W q t,R t ⊆ W q t such that � { E Q R t ( X ) = t [ − X ] + G t ( w ) } ∩ M t . ( Q ,w ) ∈W q t,Rt Q vector probability measure with components Q i (i=1,...,d), d Q i d Q ∈ L q B. Rudloff Dynamic risk measures in markets with transaction costs
1. Risk measures under transaction costs Let G (( M t ) + ) = { D ⊆ M t : D = cl co( D + ( M t ) + ) } . Dual Representation, 1 ≤ p ≤ ∞ A function R t : L p d ( F T ) → G (( M t ) + ) is a closed coherent conditional risk measure if and only if there is a nonempty set W q t,R t ⊆ W q t such that � { E Q R t ( X ) = t [ − X ] + G t ( w ) } ∩ M t . ( Q ,w ) ∈W q t,Rt Q vector probability measure with components Q i d Q ∈ L q and E Q t [ X d ]) T . t [ X ] = ( E Q 1 t [ X 1 ] , ..., E Q d (i=1,...,d), d Q i B. Rudloff Dynamic risk measures in markets with transaction costs
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