Arbitrage of the first kind and filtration enlargements Beatrice Acciaio LSE (based on a joint work with C. Fontana and C. Kardaras)
Outline of the talk Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 2 / 29
Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 3 / 29
The problem The problem: ⊲ Consider a market without arbitrage profits. ⊲ Suppose some agents have additional information. ⊲ Can they use this information to realize arbitrage profits? 4 / 29
The problem The problem: ⊲ Consider a market without arbitrage profits. ⊲ Suppose some agents have additional information. ⊲ Can they use this information to realize arbitrage profits? Mathematically: ⊲ market : (Ω , F , F , P , S ), with F satisfying the usual conditions, S = ( S i ) i =1 ,..., d non-negative semimartingale, S 0 ≡ 1. ⊲ additional information: - progressive enlargement of filtration (with any random time) - initial enlargement of filtration ⊲ arbitrage profits: ...(some motivation first)... 4 / 29
The basic example ⊲ Let W be a standard Brownian motion on (Ω , F , F W , P ). ⊲ Let S represent the discounted price of an asset and be given by � � σ W t − 1 2 σ 2 t S t = exp , σ > 0 given . ⊲ Let S ∗ t := sup { S u , u ≤ t } and define the random time τ := sup { t : S t = S ∗ ∞ } = sup { t : S t = S ∗ t } ⊲ An agent with information τ can follow the arbitrage strategy “buy at t = 0 and sell at t = τ ” 5 / 29
The basic example ⊲ Let W be a standard Brownian motion on (Ω , F , F W , P ). ⊲ Let S represent the discounted price of an asset and be given by � � σ W t − 1 2 σ 2 t S t = exp , σ > 0 given . ⊲ Let S ∗ t := sup { S u , u ≤ t } and define the random time τ := sup { t : S t = S ∗ ∞ } = sup { t : S t = S ∗ t } ⊲ An agent with information τ can follow the arbitrage strategy “buy at t = 0 and sell at t = τ ” Remark. Here τ is an honest time : ∀ t ≥ 0 ∃ ξ t F W t -measurable s.t. τ = ξ t on { τ ≤ t } (e.g., ξ t := sup { u ≤ t : S u = sup r ≤ t S r } ). 5 / 29
Different notions of arbitrage � · X ( F , S ) admissible wealth processes: X x , H := x + 0 H t d S t ≥ 0 We use the notation: � � X 1 , H NA( F , S ): there is no X 1 , H ∈X ( F , S ) s.t. P ≥ 1 = 1, ∞ � � X 1 , H > 1 > 0. P ∞ NFLVR( F , S ): there are no ǫ > 0, 0 ≤ δ n ↑ 1, X 1 , H n ∈ X ( F , S ) � � � � X 1 , H n X 1 , H n s.t. P > δ n = 1, P > 1 + ǫ ≥ ǫ . ∞ ∞ NA1( F , S ): there is no ξ ≥ 0 with P [ ξ > 0] > 0 s.t. for all x > 0, ∃ X ∈ X ( F , S ) with X 0 = x and P [ X ∞ ≥ ξ ] = 1. Remark. NA1 (Kardaras, 2010) ⇐ ⇒ BK (Kabanov, 1997) ⇐ ⇒ NUPBR (Karatzas,Kardaras 2007) 6 / 29
Martingale measures and deflators ◮ NFLVR ⇐ ⇒ NA + NA1 7 / 29
Martingale measures and deflators ◮ NFLVR ⇐ ⇒ NA + NA1 ◮ NFLVR ⇐ ⇒ ∃ equivalent local martingale measure for S 7 / 29
Martingale measures and deflators ◮ NFLVR ⇐ ⇒ NA + NA1 ◮ NFLVR ⇐ ⇒ ∃ equivalent local martingale measure for S ◮ NA1 ⇐ ⇒ ∃ supermartingale deflator (Karatzas,Kardaras’07): Y > 0 , Y 0 = 1 s.t. YX is a supermartingale ∀ X ∈ X ⇐ ⇒ ∃ loc. martingale deflator (Takaoka,Schweizer’13, Song’13): Y > 0 , Y 0 = 1 s.t. YX is a local martingale ∀ X ∈ X ⇐ ⇒ ∃ treadable loc. martingale deflator (A.F.K.’14): Y local martingale deflator s.t. 1 / Y ∈ X (up to Q ∼ P ) 7 / 29
Why NA1? - Let me try to convince you ⊲ As seen in the basic example, NA and NFLVR easily fail under additional information . 8 / 29
Why NA1? - Let me try to convince you ⊲ As seen in the basic example, NA and NFLVR easily fail under additional information . ⊲ Whereas when an arbitrage exists we are in general not able to spot it, when an arbitrage of the first kind exists we are able to construct (and hence exploit) it (NA1 is completely characterized in terms of the characteristic triplet of S ). 8 / 29
Why NA1? - Let me try to convince you ⊲ As seen in the basic example, NA and NFLVR easily fail under additional information . ⊲ Whereas when an arbitrage exists we are in general not able to spot it, when an arbitrage of the first kind exists we are able to construct (and hence exploit) it (NA1 is completely characterized in terms of the characteristic triplet of S ). ⊲ NA1 is the minimal condition in order to proceed with utility maximization. ⊲ NA1 is stable under change of num´ eraire. eraire portfolio X ∗ ⊲ NA1 is equivalent to the existence of a num´ (= growth optimal portfolio = log optimal portfolio), in which case 1 / X ∗ is a supermartingale deflator. 8 / 29
Some related work (NA1 preservation) On progressive enlargement: Fontana, Jeanblanc, Song 2013: S continuous, PRP, τ honest and avoids all F -stopping times, NFLVR in the original market. Then in the enlarged market: ⊲ on [0 , ∞ ): NA1, NA and NFLVR all fail; ⊲ on [0 , τ ]: NA and NFLVR fail, but NA1 holds. Kreher 2014: all F -martingales are continuous, τ avoids all F -stopping times, NFLVR in the original market. Aksamit, Choulli, Deng, Jeanblanc 2013: using optional stochastic integral, ( S quasi-left-continuous). On initial enlargement: nothing in the literature that we are aware of. Some work in progress by Jeanblanc et al. 9 / 29
Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 10 / 29
Progressive enlargement of filtrations ◮ Let τ be a random time (= positive, finite, F -measurable r.v.). ◮ Consider the progressively enlarged filtration G = ( G t ) t ∈ R + , G t := { B ∈ F | B ∩ { τ > t } = B t ∩ { τ > t } for some B t ∈ F t } . 11 / 29
Progressive enlargement of filtrations ◮ Let τ be a random time (= positive, finite, F -measurable r.v.). ◮ Consider the progressively enlarged filtration G = ( G t ) t ∈ R + , G t := { B ∈ F | B ∩ { τ > t } = B t ∩ { τ > t } for some B t ∈ F t } . ◮ Jeulin-Yor theorem ensures that H ′ -hypothesis holds up to τ : every F -semimartingale remains a G -semimartingale up to time τ (in particular S τ is a G -semimartingale). 11 / 29
Our main tools ⊲ The Az´ ema supermartingale associated to τ : Z t := P [ τ > t | F t ] 12 / 29
Our main tools ⊲ The Az´ ema supermartingale associated to τ : Z t := P [ τ > t | F t ] ⊲ Let A be the F -dual optional projection of I [ [ , so that [ τ, ∞ [ ∆ A σ = P [ τ = σ |F σ ] for all F -stopping times σ . 12 / 29
Our main tools ⊲ The Az´ ema supermartingale associated to τ : Z t := P [ τ > t | F t ] ⊲ Let A be the F -dual optional projection of I [ [ , so that [ τ, ∞ [ ∆ A σ = P [ τ = σ |F σ ] for all F -stopping times σ . ⊲ Define the stopping time ζ := inf { t ∈ R + | Z t = 0 } ≥ τ . 12 / 29
Our main tools ⊲ The Az´ ema supermartingale associated to τ : Z t := P [ τ > t | F t ] ⊲ Let A be the F -dual optional projection of I [ [ , so that [ τ, ∞ [ ∆ A σ = P [ τ = σ |F σ ] for all F -stopping times σ . ⊲ Define the stopping time ζ := inf { t ∈ R + | Z t = 0 } ≥ τ . ⊲ Define Λ := { ζ < ∞ , Z ζ − > 0 , ∆ A ζ = 0 } ∈ F ζ = set where Z jumps to zero after τ 12 / 29
Our main tools ⊲ The Az´ ema supermartingale associated to τ : Z t := P [ τ > t | F t ] ⊲ Let A be the F -dual optional projection of I [ [ , so that [ τ, ∞ [ ∆ A σ = P [ τ = σ |F σ ] for all F -stopping times σ . ⊲ Define the stopping time ζ := inf { t ∈ R + | Z t = 0 } ≥ τ . ⊲ Define Λ := { ζ < ∞ , Z ζ − > 0 , ∆ A ζ = 0 } ∈ F ζ = set where Z jumps to zero after τ ⊲ and define η := ζ I Λ + ∞ I Ω \ Λ Note that τ < η ; η = time when Z jumps to zero after τ . 12 / 29
Representation pair associated with τ Theorem (Itˆ o,Watanabe 1965, Kardaras 2014). The Az´ ema supermartingale Z admits the following multiplicative decomposition: Z = L (1 − K ), where: L is a nonnegative F -local martingale with L 0 = 1, K is a nondecreasing F -adapted process with 0 ≤ K ≤ 1, for any nonnegative optional processes V on (Ω , F ), �� � E [ V τ ] = E V t L t d K t . R + ⊲⊲ Together with the stopping time η , the local martingale L will play a main role in our results. 13 / 29
Back to the basic example � � σ W t − 1 2 σ 2 t Asset price process: S t = exp Random time: τ := sup { t : S t = S ∗ ∞ } In this case Z t = P [ τ > t | F t ] = S t S ∗ t Therefore: ⊲ η = ∞ and L = S ⊲ Y := 1 / L τ = 1 / S τ is a local martingale deflator for S τ in G . ⇒ NA1 holds while NA and NFLVR fail. 14 / 29
Back to the basic example � � σ W t − 1 2 σ 2 t Asset price process: S t = exp Random time: τ := sup { t : S t = S ∗ ∞ } In this case Z t = P [ τ > t | F t ] = S t S ∗ t Therefore: ⊲ η = ∞ and L = S ⊲ Y := 1 / L τ = 1 / S τ is a local martingale deflator for S τ in G . ⇒ NA1 holds while NA and NFLVR fail. Remarks. 1) Analogous situation for τ ′ := sup { t : S t = a } , 0 < a < 1. 2) The decomposition Z t = L t / L ∗ t holds for a wide class of honest times (see Nikeghbali,Yor 2006, Kardaras 2013, A.,Penner 2014) 14 / 29
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