Roscoff, March 19, 2010 NFL2 Yuri Kabanov Laboratoire de Math - PowerPoint PPT Presentation

NA2 (discrete time) NFL2(continuous time Roscoff, March 19, 2010 NFL2 Yuri Kabanov Laboratoire de Math´ ematiques, Universit´ e de Franche-Comt´ e March 19, 2010 Yuri Kabanov NFL2 1 / 15

NA2 (discrete time) NFL2(continuous time Outline NA2 (discrete time) 1 NFL2(continuous time 2 Yuri Kabanov NFL2 2 / 15

NA2 (discrete time) NFL2(continuous time Example Two-asset 1-period model : S 1 0 = S 2 0 = 1, S 1 1 = 1, S 2 1 takes values 1 + ε and 1 − ε > 0 with probabilities 1 / 2. The filtration is generated by S . 1 = R + 1 . Then � K ∗ 0 = cone { (1 , 2) , (2 , 1) } , K ∗ K ∗ 1 = R + S 1 . The process Z with Z 0 = (1 , 1) and Z 1 = S 1 is a strictly consistent price system, so the NA w -property holds. Let v ∈ C where C ∗ = cone { (1 , 1 + ε ) , (1 , 1 − ε ) } ⊆ � K 1 . For ε ∈ ]0 , 1 / 2[ the cone C is strictly larger than � K 0 = K 0 . The investor the initial endowment v ∈ C \ K 0 will solvent at T = 1 though not solvent at the date zero. One can introduce small transaction costs at time T = 1 to get the same conclusion for a model with efficient friction. Yuri Kabanov NFL2 3 / 15

NA2 (discrete time) NFL2(continuous time Arbitrage of the second kind Setting Let G = ( G t ), t = 0 , 1 , ..., T , be an adapted cone-valued process, s := � T A T t = s L 0 ( − G t , F t ). The models admits arbitrage opportunities of the 2nd kind if there exist s ≤ T − 1 and an F s -measurable d -dimensional random variable ξ such that Γ := { ξ / ∈ G s } is not a null-set and ( ξ + A T s ) ∩ L 0 ( G T , F T ) � = ∅ , i.e. ξ = ξ s + ... + ξ T for some ξ t ∈ L 0 ( G t , F t ), s ≤ t ≤ T . If such ξ does exist then, in the financial context where G = � K , an investor having I Γ ξ as the initial endowments at time s , may use the strategy ( I Γ ξ t ) t ≥ s and get rid of all debts at time T . Yuri Kabanov NFL2 4 / 15

NA2 (discrete time) NFL2(continuous time NA2 property Rasonyi theorem (2008) The model has no arbitrage opportunities of the 2nd kind (i.e. has the NA 2-property) if s and ξ ∈ L 0 ( R d , F s ) the intersection ( ξ + A T s ) ∩ L 0 ( G T , F T ) is non-empty only if ξ ∈ L 0 ( G s , F s ). Alternatively, the NA 2-property can be expressed as : L 0 ( R d , F s ) ∩ ( − A T s ) = L 0 ( G s , F s ) ∀ s ≤ T . Theorem Suppose that the efficient friction condition is fulfilled, i.e. G t ∩ ( − G t ) = { 0 } and R d + ⊆ G t for all t. Then the following conditions are equivalent : ( a ) NA 2 ; ( b ) L 0 ( R d , F s ) ∩ L 0 ( G s +1 , F s ) ⊆ L 0 ( G s , F s ) for all s < T ; s +1 ∩ ¯ ( c ) cone int E ( G ∗ O 1 (0) |F s ) ⊇ int G ∗ s (a.s.) for all s < T ; ( d ) for any s < T and η ∈ L 1 ( int G ∗ s , F s ) there is Z ∈ M T s ( int G ∗ ) such that Z s = η ( PCV - ” Prices are consistently extendable” .) Yuri Kabanov NFL2 5 / 15

NA2 (discrete time) NFL2(continuous time Tools Conditional expectations A subset Ξ ∈ L p is called decomposable if with two its elements ξ 1 , ξ 2 it contains also ξ 1 I A + ξ 2 I A c whatever is A ∈ F . Proposition Let Ξ be a closed subset of L p ( R d ) , p ∈ [0 , ∞ [ . Then Ξ = L p (Γ) for some Γ which values are closed sets if and only if Ξ is decomposable, . Proposition Let G be a sub- σ -algebra of F . Let Γ be a measurable mapping which values are non-empty closed convex subsets of ¯ O 1 (0) ⊂ R d . Then there is a G -measurable mapping, E (Γ |G ) , which values are non-empty convex compact subsets of ¯ O 1 (0) and the set of its G -measurable a.s. selectors coincides with the set of G -conditional expectations of a.s. selectors of Γ . Yuri Kabanov NFL2 6 / 15

NA2 (discrete time) NFL2(continuous time Outline NA2 (discrete time) 1 NFL2(continuous time 2 Yuri Kabanov NFL2 7 / 15

NA2 (discrete time) NFL2(continuous time Model We are given set-valued adapted processes G = ( G t ) t ∈ [0 , T ] and G ∗ = ( G ∗ t ) t ∈ [0 , T ] whose values are closed cone in R d , G ∗ t ( ω ) = { y : yx ≥ 0 ∀ x ∈ G t ( ω ) } . “Adapted”means that � � ( ω, x ) ∈ Ω × R d : x ∈ G t ( w ) ∈ F t ⊗ B d . G t are proper ( EF -condition) : G t ∩ ( − G t ) = { 0 } . We assume also that G t dominate R d + , i.e. G ∗ \{ 0 } ⊂ int R d + . In financial context G t = � K t , the solvency cone in physical units. For each s ∈ ]0 , T ] we are given a convex cone Y T s of optional R d -valued processes Y = ( Y t ) t ∈ [ s , T ] with Y s = 0. Assumption : if sets A n ∈ F s form a countable partition of Ω s , then � and Y n ∈ Y T n Y n I A n ∈ Y T s . Yuri Kabanov NFL2 8 / 15

NA2 (discrete time) NFL2(continuous time Notations for d -dimensional processes Y and Y ′ the relation Y ≥ G Y ′ means Y t − Y ′ t ∈ G t a.s. for every t ; 1 = (1 , ..., 1) ∈ R d + ; Y T s , b denotes the subset of Y T s formed by the processes Y dominated from below : Y t + κ 1 ∈ G t for some constant κ ; Y T s , b ( T ) is the set of random variables Y T where Y ∈ Y T s , b ; A T s , b ( T ) = ( Y T s , b ( T ) − L 0 ( G T , F T )) ∩ L ∞ ( R d , F T ) and w is its closure in σ { L ∞ , L 1 } ; A T s , b ( T ) M T s ( G ∗ ) is the set of martingales Z = ( Z t ) t ∈ [ s , T ] evolving in G ∗ , i.e. such that Z t ∈ L 1 ( G ∗ t , F t ). Yuri Kabanov NFL2 9 / 15

NA2 (discrete time) NFL2(continuous time Conditions Standing Hypotheses S 1 E ξ Z T ≤ 0 for all ξ ∈ Y T s , b ( T ), Z ∈ M T s ( G ∗ ), s ∈ [0 , T [. S 2 ∪ t ≥ s L ∞ ( − G t , F t ) ⊆ Y T s , b ( T ) for each s ∈ [0 , T ]. Properties of Interest w ∩ L ∞ ( R d NFL A T + , F T ) = { 0 } for each s ∈ [0 , T [. s , b ( T ) NFL2 For each s ∈ [0 , T [ and ξ ∈ L ∞ ( R d , F s ) w ) ∩ L ∞ ( R d ( ξ + A T s , b ( T ) + , F T ) � = ∅ only if ξ ∈ L ∞ ( G s , F s ). MCPS For any η ∈ L 1 ( int G ∗ s , F s ), there is s ( G ∗ \ { 0 } ) with Z s = η . Z ∈ M T B If ξ is an F s -measurable R d -valued random variable such that Z s ξ ≥ 0 for every Z ∈ M T s ( G ∗ ), then ξ ∈ G s . Yuri Kabanov NFL2 10 / 15

NA2 (discrete time) NFL2(continuous time FTAP Theorem NFL ⇔ M T 0 ( G ∗ \{ 0 } ) � = ∅ . Proof. ( ⇐ ) Let Z ∈ M T 0 ( G ∗ \{ 0 } ). Then the components of Z T are strictly positive and EZ T ξ > 0 for all ξ ∈ L ∞ ( R d + , F T ) except ξ = 0. On the other hand, E ξ Z T ≤ 0 for all ξ ∈ Y T s , b ( T ) and so for w . all ξ ∈ A T s , b ( T ) ( ⇒ ) The Kreps–Yan theorem on separation of closed cones in L ∞ ( R d , F T ) implies the existence of η ∈ L 1 ( int R d + , F T ) such that w , hence, by virtue of the E ξη ≤ 0 for every ξ ∈ A T s , b ( T ) hypothesis S 2 , for all ξ ∈ L ∞ ( − G t , F t ). Let us consider the martingale Z t = E ( η |F t ), t ≥ s , with strictly positive components. Since EZ t ξ = E ξη ≥ 0, t ≥ s , for every ξ ∈ L ∞ ( G t , F t ), it follows that Z t ∈ L 1 ( G t , F t ) and, therefore, Z ∈ M T s ( G ∗ \{ 0 } ). Yuri Kabanov NFL2 11 / 15

NA2 (discrete time) NFL2(continuous time Main Result Theorem The following relations hold : MCPS ⇒ { B , M T 0 ( G ∗ \{ 0 } ) � = ∅} ⇔ { B , NFL } ⇔ B ⇔ NFL2 . If, moreover, the sets Y T s , b ( T ) are Fatou-closed for any s ∈ [0 , T [ . Then all five conditions are equivalent. In the above formulation the Fatou-closedness means that the set Y T s , b ( T ) contains the limit on any a.s. convergent sequence of its elements provided that the latter is bounded from below in the sense of partial ordering induced by G T . Yuri Kabanov NFL2 12 / 15

NA2 (discrete time) NFL2(continuous time Discrete-time model,1 B p If ξ ∈ L 0 ( R d , F s ) and Z s ξ ≥ 0 for any Z ∈ M T s ( G ∗ ) with Z T ∈ L p , then ξ ∈ G s (a.s.), s = 0 , ..., T . L p NAA p A T ∩ L p ( R d 0 , b ( T ) + , F T ) = { 0 } . Lemma The conditions NAA p for p ∈ [1 , ∞ [ are measure-invariant and any of them is equivalent to NAA 0 as well as to the condition NFL (which, in turn, is equivalent, to the existence of a bounded s ( G ∗ \ { 0 } ) . process Z in M T NAA2 p For each s = 0 , 1 , ..., T − 1 and ξ ∈ L ∞ ( R d , F s ) L p ) ∩ L 0 ( R d ( ξ + A T s , b ( T ) + , F T ) � = ∅ only if ξ ∈ L ∞ ( G s , F s ). Yuri Kabanov NFL2 13 / 15

NA2 (discrete time) NFL2(continuous time Discrete-time model,2 Lemma The conditions NAA2 p for p ∈ [1 , ∞ [ are measure-invariant and any of them is equivalent to NAA2 0 as well as to the condition NFL2 (which, in turn, is equivalent to the condition B ). Thus, for the discrete-time model with efficient friction MCPS ⇔ { B , M T 0 ( G ∗ \{ 0 } ) � = ∅} ⇔ { B , NFL } ⇔ B ⇔ NFL2 Formally, all properties above are different from those in the asonyi theorem PCE ⇔ NA2 . Recall that R´ s := � T t = s L 0 ( − G t , F t )) and A T NA2 For each s ∈ [0 , T [ and ξ ∈ L 0 ( R d , F s ) ( ξ + A T s ) ∩ L 0 ( R d + , F T ) � = ∅ only if ξ ∈ L 0 ( G s , F s ). Yuri Kabanov NFL2 14 / 15