Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Beatrice Acciaio

Arbitrage of the first kind and filtration enlargements in semimartingale financial models Beatrice Acciaio the London School of Economics and Political Science (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 / 31

Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 3 / 31

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 / 31

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 / 31

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 ( S t = exp σw t 1 ) 2 σ2 t, 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 Ft W -measurable s.t. τ = ξ t on {τ t} (e.g., ξ t := sup{u t : S u = sup r t S r }). See: Fontana,Jeanblanc,Song(2013), Imkeller(2002), Zwierz(2007) 5 / 31

Different notions of arbitrage Admissible wealth processes X (F, S) : class of all non-negative processes of the type X x,h := x + 0 H tds t. We recall the notions of: arbitrage: X 1,H X (F, S) s.t. P [ X 1,H 1 ] = 1, P [ X 1,H > 1 ] > 0. If such strategies do not exist we say that NA(F, S) holds. free lunch with vanishing risk: ɛ > 0, 0 δ n 1, X 1,Hn X (F, S) s.t. P [ ] [ X 1,Hn > δ n = 1, P X 1,H n > 1 + ɛ ] ɛ. If such strategies do not exist we say that NFLVR(F, S) holds. arbitrage of the first kind: ξ 0 with P [ξ > 0] > 0 s.t. for all x > 0, X X (F, S) with X 0 = x s.t. P [X ξ] = 1. If such strategies do not exist we say that NA1(F, S) holds. Remark. NA1 (Kardaras, 2010) BK (Kabanov, 1997) NUPBR (Karatzas,Kardaras 2007) 6 / 31

Martingale measures and deflators NFLVR NA + NA1 7 / 31

Martingale measures and deflators NFLVR NA + NA1 NFLVR equivalent local martingale measure for S 7 / 31

Martingale measures and deflators NFLVR NA + NA1 NFLVR equivalent local martingale measure for S NA1 supermartingale deflator (Karatzas,Kardaras 2007): Y > 0, Y 0 = 1 s.t. YX is a supermartingale X X local martingale deflator (Takaoka 2013, Song 2013): Y > 0, Y 0 = 1 s.t. YX is a local martingale X X treadable local martingale deflator (A.F.K. 2014): Y local martingale deflator s.t. 1/Y X (up to Q P) 7 / 31

Why NA1? - Let me try to convince you As seen in the basic example, NA and NFLVR easily fail under additional information 8 / 31

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 semimartingale characteristics of S NA1 is the minimal condition in order to proceed with utility maximization NA1 is stable under change of numéraire NA1 is also equivalent to the existence of a numéraire portfolio X (= growth optimal portfolio = log optimal portfolio), in which case 1/X is a supermartingale deflator. 8 / 31

Some related work (NA1 preservation) On progressive enlargement: Aksamit, Choulli, Deng, Jeanblanc 2013: S quasi-left-continuous local martingale, using optional stochastic integral 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 On initial enlargement: nothing in the literature that we are aware of. Some work in progress by Jeanblanc et al. 9 / 31

Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 10 / 31

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 / 31

Our main tools Consider the two F-supermartingales associated to τ: Z t := P [τ > t F t ], Zt := P [τ t F t ] (Z = Azéma supermartingale associated to τ) 12 / 31

Our main tools Consider the two F-supermartingales associated to τ: Z t := P [τ > t F t ], Zt := P [τ t F t ] (Z = Azéma supermartingale associated to τ) A := F-dual optional projection of I [τ, [, M t := E[A F t ], Z t = M t A t, Zt = M t A t, A t = Z t Z t, A σ = P [τ = σ F σ ] for all F-stopping times σ 12 / 31

Our main tools Recall that Z t := P [τ > t F t ] and ζ := inf {t R + Z t = 0}. Define Λ := {ζ <, Z ζ > 0, A ζ = 0} F ζ = set where Z jumps to zero after τ 13 / 31

Our main tools Recall that Z t := P [τ > t F t ] and ζ := inf {t R + Z t = 0}. Define Λ := {ζ <, Z ζ > 0, A ζ = 0} F ζ = set where Z jumps to zero after τ and define η := ζi Λ + I Ω\Λ Note that τ < η; η = time when Z jumps to zero after τ. 13 / 31

Representation pair associated with τ Theorem (Itô,Watanabe 1965, Kardaras 2014). The following multiplicative optional decomposition holds for the Azéma supermartingale Z: where: Z t = P [τ > t F t ] = L t (1 K t ), t 0, L is a nonnegative F-local martingale with L 0 = 1, K is a nondecreasing F-adapted process with 0 K 1, The local martingale L coming from this decomposition will play a main role in our results. 14 / 31

Back to the basic example Asset price process: S t = exp ( σw t 1 ) 2 σ2 t Random time: τ := sup{t : S t = S } In this case Z t = P [τ > t F t ] = S t S t (that is, L = S in the previous multiplicative decomposition) and Y := 1/L τ = 1/S τ is a local martingale deflator for S τ in G. Therefore: NA1 holds while NA and NFLVR fail. Remarks. 1) An analogous situation occurs if we consider the random time τ := sup{t : S t = a}, for some 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) 15 / 31

Local martingales in the progressively enlarged filtration Remember: η is the time when Z jumps to zero after τ. Proposition. Let X be a nonnegative F-local martingale such that X = 0 on [[η, [[. Then X τ /L τ is a G-local martingale. Main tool in the proof: For any nonnegative optional processes V on (Ω, F), [ ] E[V τ ] = E V t L t dk t, R + where L, K come from the multiplicative decomposition of Z. As an immediate consequence we have the following Key-Proposition. Suppose there exists a local martingale deflator M for S in F such that M = 0 on [[η, [[. Then M τ /L τ is a local martingale deflator for S τ in G. 16 / 31

An important lemma = To have preservation of the NA1 property, given a deflator for S in F, we want to kill it from η on. We will do it with the help of the following lemma. Lemma. Let D be the F-predictable compensator of I [η, [. Then: D < 1 P-a.s. ( E( D) > 0 and nonincreasing); E( D) 1 I [0,η [ is a local martingale on (Ω, F, P). Main idea: for any predictable time σ on (Ω, F), D σ = P [η = σ F σ ] < 1 on {σ < }. 17 / 31

NA1 under progressive enlargement: one fixed S Theorem (one fixed S). Suppose that P [η <, S η 0] = 0. If NA1(F, S)holds, then NA1(G, S τ )holds. That is: S does not jump when Z jumps to zero. Corollary. If NA1(F, S η ) holds, then NA1(G, S τ )holds. Remark. Condition P [η <, S η 0] = 0 is equivalent to evanescence of the set {Z > 0, Z = 0, S 0}. (See Aksamit et al. (2013), where S is a quasi-left-continuous local martingale.) 18 / 31

Proof of the theorem Recall: D is the F-predictable compensator of I [η, [. NA1(F, S) X X (F, S) s.t. Y := (1/ X ) is a local martingale deflator for S in F ( Y = 0 when S = 0). In order to apply the Key-Proposition, we need a deflator for S in F that vanishes on the set [[η, [[. Let M := Y E( D) 1 I [0,η [ ( {M > 0} = [[0, η[[). By the Lemma, MS [ E( D) 1 I [0,η [, YS ] F-local martingale. We want M to be a deflator for S in F, so we need to show that the quadratic covariation part is an F-local martingale. S η = 0 (YS) η = 0 [..,..] = [ E( D) 1, YS ], which is indeed an F-local martingale. 19 / 31

NA1 under progressive enlargement: any S Theorem (general stability). TFAE: 1) for any S s.t. NA1(F, S) holds, NA1(G, S τ )holds; 2) η = P-a.s.; 3) For every nonnegative local martingale X on (Ω, F, P), the process X τ /L τ is a local martingale on (Ω, G, P); 4) The process 1/L τ is a local martingale on (Ω, G, P). Remark. Condition 2) is equivalent to evanescence of the set {Z > 0, Z = 0} = {Z > 0, Z = 0, A = 0}. (See Aksamit et al. (2013).) 20 / 31

Proof of the theorem 2) 1): from previous Theorem. 1) 2): suppose P [η < ] > 0. Define S := E( D) 1 I [0,η [. Then S is a F-local martingale, and S τ is nondecreasing with P [S τ > 1] > 0. Hence NA1(F, S)holds, but NA1(G, S τ )fails. 2) 3): from the Proposition. 3) 4): trivial. 4) 2): uses properties of the processes L and K appearing in the multiplicative decomposition of Z. 21 / 31

On the H -hypothesis Proposition. Let X be a nonnegative F-supermartingale. Then, the process X τ /L τ is a G-supermartingale. Remark. This can be used to establish that for any semimartingale X on (Ω, F, P), the process X τ is a semimartingale on (Ω, G, P). Indeed: By the Proposition, X nonnegative bounded F-local martingale X τ /L τ and 1/L τ are G-semimartingales X τ is a G-semimartingale. From the semimartingale decomposition + localisation, same result for any F-semimartingale X. 22 / 31

A partial converse A common assumption is that τ avoids all F-stopping times: P [τ = σ < ] = 0 for all stopping times σ on (Ω, F). Theorem. Suppose that τ avoids all stopping times on (Ω, F, P). If there exists a local martingale deflator for S τ on G, then there is a local martingale deflator for S on F that vanishes on [[η, [[. Proof. Let M be a local martingale deflator for S τ on G. Let C be the G-predictable compensator of I [τ, [. Then also U := ME( C) 1 I [0,τ [ is a local martingale deflator for S τ on G. Let Y be the optional projection of U on (Ω, F, P). Then Y is a local martingale deflator for S on F, with Y = 0 on [[η, [[. 23 / 31

Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 24 / 31

Initial enlargement of filtrations Let J be an F-measurable random variable taking values in a Lusin space (E, B E ), where B E denotes the Borel σ-field of E. Let G = (G t ) t R+ be the right-continuous augmentation of the filtration G 0 = (Gt 0 ) t R+ defined by G 0 t := F t σ(j), t R +. Let γ : B E [0, 1] be the law of J (γ [B] = P [J B], B B E ). For all t R +, let γ t : Ω B E [0, 1] be a regular version of the F t -conditional law of J. 25 / 31

Our main tools O(F) (resp. P(F)) is the F-optional (resp. pred.) σ-field on Ω R + Lemma. There exists a B E O(F)-measurable function E Ω R + (x, ω, t) p x t (ω) [0, ), càdlàg in t R + s.t.: - t R +, γ t (dx) = p x t γ(dx) holds P-a.s; - x E, p x = (pt x ) t R+ is a martingale on (Ω, F, P). For every x E define ζ x := inf{t R + p x t = 0}. Let Λ x := {ζ x <, p x ζ x > 0} F ζ x and define η x := ζ x I Λ x + I Ω\Λ x, x E Note that η x (= time at which p x jumps to zero) is a stopping time on (Ω, F). 26 / 31

NA1 under initial enlargement Similar results ( see ) for the martingale deflators lead to: Theorem (one fixed S). Let P [η x <, S η x If NA1(F, S)holds, then NA1(G, S) holds. 0] = 0 γ-a.e. Theorem (general stability). TFAE: 1) η x = P-a.s. for γ-a.e x E. 2) for all X 0 B E O(F)-meas. s.t. X x F-loc.martingale vanishing on [[η x, [[ γ-a.e., X J /p J is a G-loc.martingale 3) The process 1/p J is a G-loc.martingale And 1) For any S s.t. NA1(F, S)holds, NA1(G, S) also holds. Some care for the converse ( see ); we can derive H -hyp. ( see ). 27 / 31

Problem formulation and motivation Progressive enlargement of filtration Initial enlargement of filtration Examples 28 / 31

Example 1: progressively enlarged filtration Consider ζ : Ω R + such that P [ζ > t] = exp( t), t R +. Let F = (F t ) t R+ be the smallest filtration that satisfies the usual hypotheses and makes ζ a stopping time. Define τ := ζ/2. Note that Z t := P [τ > t F t ] = exp( t)i {t<ζ} for all t R +. Note that ζ = inf {t 0 Z t = 0} =: η < P-a.s. The F-pred. comp. of I [η, [ is D := (η t) t R+. 29 / 31

Example 2: initially enlarged filtration Consider a Poisson(λ) process N stopped at time T (0, ). Let F be the right-cont. filtration generated by N and J := N T. Then (Grorud,Pontier 2001) p x T = e λt x!/(λt ) x I {NT =x} and p x t = e λt ( λ(t t) ) x Nt (λt ) x x! (x N t )! I {N t x}, t [0, T ). S t := exp ( N t λt(e 1) ), for all t [0, T ]. S is a strictly positive F-martingale NA1(F, S) holds. Define the G-stopping time σ := inf {t [0, T ] N t = N T }. For all t [0, T ], we get ( I ]σ,t ] S) t = I {t>σ} exp ( N σ λσ(e 1) )( 1 exp ( λ(t σ)(e 1) )). I ]σ,t ] S is nondecreasing, P [σ < T ] = 1 NA1(G, S) fails. 30 / 31

Thank you for your attention! 31 / 31

An example with η accessible Let ζ : Ω N s.t. p k := P [ζ = k] (0, 1) k N, k p k = 1. Set F = (F t ) t R+ to be the smallest filtration that satisfies the usual hypotheses and makes ζ a stopping time. Since ζ is N-valued, it is an accessible time on (Ω, F, P). Define τ := ζ 1. Z t = 0 holds on {ζ t}. Moreover, with q k := n=k+1 p n k {0, 1,...}, and denoting the integer part, on {t < ζ} Z t = P [τ > t F t ] = P [ζ > t + 1 F t ] = P [ζ > t + 1 F t ] = q t+1 q t. ζ = inf {t R + Z t = 0 or Z t = 0}. Z ζ = q ζ /q ζ 1 > 0. η = ζ; in particular, η is accessible on (Ω, F, P). 32 / 31

Another example where NA fails and NA1 holds ( S t = exp σw t 1 ) 2 σ2 t. For a given constant a (0, 1), define τ := sup{t : S t = a}. Then ( ) St Z t := P [τ > t F t ] = 1 = N t a Nt, t 0 N = E ( 1 1 a Z 1 {S<a}dS Note: τ := sup{t : N t = N }. ). Since S is continuous, NA1(G, S τ ) holds. On the other hand, the following strategy realizes a classical arbitrage in the enlarged filtration at time τ (see Aksamit et al.): ψ = 1 a 1 {S<a}. 33 / 31

Émery-Jeanblanc example Let S = E(σW ) and τ := sup{t 1 : S 1 2S t = 0}, that is, the last time before 1 when S equals half of its value at time 1. Here both NA(G, S τ ) and NA1(G, S τ ) hold NFLVR(G, S τ ). Indeed, { {τ t} = inf 2S s S 1 t s 1 S t S t }. Therefore, [ ] P [τ t F t ] = P inf 2S s t S 1 t = F (1 t), t s 1 where F (u) = P [inf s u 2S s S u ]. Then Z t deterministic, decreasing τ pseudo-stopping time and S τ is a G-martingale. On the other hand: after τ there are arbitrages and arbitrages of the first kind: at τ we know the value of S 1, and S t > S τ t (τ, 1]. 34 / 31

Local martingales in the initially enlarged filtration For x E, D x denotes the F-predictable compensator of I [[η x, [[. Lemma. D can be chosen B E P(F)-measurable and: D x < 1 P-a.s. ( E( D x ) > 0 and nonincreasing); E( D x ) 1 I [0,η x [ is a F-local martingale. Proposition. Let X 0 be B E O(F)-measurable, such that X x F-local martingale vanishing on [[η x, [[ γ-a.e. Then X J /p J is a G-local martingale. As an immediate consequence we have the following Key-Proposition. Suppose there is M 0, B E O(F)-measurable s.t. M x 0 = 1, Mx and M x S are F-local martingales vanishing on [[η x, [[ γ-a.e. Then, M J /p J is a G-local martingale deflator. Back to NA1.in 35 / 31

Some converse implication Recall that D x denotes the F-predictable compensator of I [η x, [ and define S x := E( D x ) 1 I [0,η x [, x E. Theorem. Let E P [ηx < ] γ(dx) > 0. Then NA1(F, S x ) holds for every x E, but NA1(G, S J ) fails. Indeed, S x are F-local martingales, S J = E( D J ) 1 is nondecreasing and P [ S J t = S J 0, t R +] < 1. An insider with knowledge of J takes at time zero a position on a single unit of the stock with index J, and keeps it indefinitely. (The insider identifies from the beginning a single asset in the family (S x ) x E which will not default and can therefore arbitrage.) Some particular cases depending on the law of J, see here. Back to NA1.in 36 / 31

Remarks If k N P [J = x k] = 1 holds for some family (x k ) k N E, E P [ηx < ] γ(dx) > 0 κ : P [η xκ < ] > 0; since P [ ζ J < ] = 0, then P [J = x κ, η xκ < ] = 0; the buy-and-hold strategy I {J=xκ} in the single asset S xκ results in the arbitrage I {J=xκ} S xκ. (NA1(F, S xκ ) holds while NA1(G, S xκ ) fails) If the law γ has a diffuse component, one can still obtain an arbitrage of the first kind, under the stronger hypothesis: B B E with γ [B] > 0 s.t. P [ η B < ] > 0, where η B is the time when the martingale (γ t [B]) t R+ jumps to zero. Indeed, denoting D B the F-predictable compensator of I [η B, [, S := E( D B ) 1 I [0,η B [ is a F-local martingale, I {J B} S is nondecreasing, and P [S t = S 0, t R + ] < 1. (NA1(F, S) holds while NA1(G, S) fails). Back to conv.in or NA1.in 37 / 31

On the H -hypothesis Proposition. Let X 0 be B E O(F)-measurable, such that X x F-supermartingale γ-a.e. Then X J /p J is a G-supermartingale. (cf. concept of universal supermartingale density in Imkeller, Perkowski 2013) Remark. This can be used to establish that any semimartingale X on (Ω, F, P) remains a semimartingale on (Ω, G, P). Back to NA1.in 38 / 31