Insider information and arbitrage profits via enlargements of filtrations

Insider information and arbitrage profits via enlargements of filtrations Claudio Fontana Laboratoire de Probabilités et Modèles Aléatoires Université Paris Diderot XVI Workshop on Quantitative Finance Parma, January 29-30, 2015 This work was supported by a Marie Curie IEF Fellowship within the 7th EC Framework Programme. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 1 / 14

Private information and arbitrage profits Insider trading There exist market participants who possess additional private information which is not shared with the rest of the market; this private information allows for profitable trading opportunities. Let the publicly available information be represented by a filtration F. (In other words, F represents the information of a typical market participant.) Let G = (G t ) t 0 be a filtration such that G t F t, for all t 0. interpretation: G represents the point of view of a better informed agent (insider trader). enlargement of filtrations (seminal papers from the 70-80s: Azéma, Barlow, Brémaud, Jacod, Jeulin, Song, Yoeurp, Yor.) Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 2 / 14

Private information and arbitrage profits Insider trading There exist market participants who possess additional private information which is not shared with the rest of the market; this private information allows for profitable trading opportunities. Let the publicly available information be represented by a filtration F. (In other words, F represents the information of a typical market participant.) Let G = (G t ) t 0 be a filtration such that G t F t, for all t 0. interpretation: G represents the point of view of a better informed agent (insider trader). enlargement of filtrations (seminal papers from the 70-80s: Azéma, Barlow, Brémaud, Jacod, Jeulin, Song, Yoeurp, Yor.) The question: Suppose that no arbitrage profits can be realized by typical market participants on the basis of the information contained in the filtration F. Does the additional knowledge of G allow for arbitrage profits? And, if yes, in what sense? Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 2 / 14

A basic example Let W = (W t ) t 0 be a standard Brownian motion on (Ω, F, F, P), with F = F W ; let S = (S t ) t 0 represent the discounted price of a risky asset, with: ds t = S t σ dw t, S 0 = s (0, ); Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 3 / 14

A basic example Let W = (W t ) t 0 be a standard Brownian motion on (Ω, F, F, P), with F = F W ; let S = (S t ) t 0 represent the discounted price of a risky asset, with: ds t = S t σ dw t, S 0 = s (0, ); define a random time τ : Ω [0, ] (with τ < P-a.s.) as follows: { τ := sup t 0 : S t = sup S u }. u 0 τ is not an F-stopping time! Define the progressively enlarged filtration G = (G t ) t 0 as: G t := s>t G 0 s where G 0 t := F t σ(τ t), for all t 0. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 3 / 14

A basic example Let W = (W t ) t 0 be a standard Brownian motion on (Ω, F, F, P), with F = F W ; let S = (S t ) t 0 represent the discounted price of a risky asset, with: ds t = S t σ dw t, S 0 = s (0, ); define a random time τ : Ω [0, ] (with τ < P-a.s.) as follows: { τ := sup t 0 : S t = sup S u }. u 0 τ is not an F-stopping time! Define the progressively enlarged filtration G = (G t ) t 0 as: G t := s>t G 0 s where G 0 t := F t σ(τ t), for all t 0. An arbitrage strategy in the enlarged filtration G: Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 3 / 14

A basic example Let W = (W t ) t 0 be a standard Brownian motion on (Ω, F, F, P), with F = F W ; let S = (S t ) t 0 represent the discounted price of a risky asset, with: ds t = S t σ dw t, S 0 = s (0, ); define a random time τ : Ω [0, ] (with τ < P-a.s.) as follows: { τ := sup t 0 : S t = sup S u }. u 0 τ is not an F-stopping time! Define the progressively enlarged filtration G = (G t ) t 0 as: G t := s>t G 0 s where G 0 t := F t σ(τ t), for all t 0. An arbitrage strategy in the enlarged filtration G: buy at t = 0 and sell at t = τ. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 3 / 14

A basic example: discussion The key features of the previous example: the filtration F is continuous; the financial market based on F is arbitrage-free (NFLVR) and complete; the random time τ is a honest time: for every t > 0, there exists an F t -measurable random variable ζ t such that τ = ζ t on {τ < t}. Indeed, we can take ζ t := sup { u [0, t] : S u = sup r [0,t] S r } Ft ; for all F-stopping times σ, it holds that P(τ = σ) = 0. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 4 / 14

A basic example: discussion The key features of the previous example: the filtration F is continuous; the financial market based on F is arbitrage-free (NFLVR) and complete; the random time τ is a honest time: for every t > 0, there exists an F t -measurable random variable ζ t such that τ = ζ t on {τ < t}. Indeed, we can take ζ t := sup { u [0, t] : S u = sup r [0,t] S r } Ft ; for all F-stopping times σ, it holds that P(τ = σ) = 0. Under the above assumptions, one always faces the situation of the basic example. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 4 / 14

Arbitrage opportunities arising with honest times Lemma (Nikeghbali & Yor, 2006) Let τ be an honest time such that P(τ = σ) = 0 for all F-stopping times σ. Then, (if the filtration F is continuous), there exists a (continuous) non-negative F-local martingale N = (N t ) t 0 with N 0 = 1 and lim t N t = 0 P-a.s. such that Z t = P(τ > t F t ) = N t /N t, where N t := sup u t N u. Moreover: τ = sup{t 0 : N t = N t } = sup{t 0 : N t = N } Proposition Under the assumptions on the previous slide, the stopped process S τ admits arbitrage opportunities in the filtration G (i.e., NA fails in G). Proof: replicate the local martingale N up to τ. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 5 / 14

Arbitrage opportunities arising with honest times Lemma (Nikeghbali & Yor, 2006) Let τ be an honest time such that P(τ = σ) = 0 for all F-stopping times σ. Then, (if the filtration F is continuous), there exists a (continuous) non-negative F-local martingale N = (N t ) t 0 with N 0 = 1 and lim t N t = 0 P-a.s. such that Z t = P(τ > t F t ) = N t /N t, where N t := sup u t N u. Moreover: τ = sup{t 0 : N t = N t } = sup{t 0 : N t = N } Proposition Under the assumptions on the previous slide, the stopped process S τ admits arbitrage opportunities in the filtration G (i.e., NA fails in G). Proof: replicate the local martingale N up to τ. Related works: Imkeller (2002) and Zwierz (2007) have shown the existence of immediate arbitrage opportunities by trading as soon as τ occurs. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 5 / 14

A different notion of arbitrage Arbitrage of the first kind Definition A random variable ξ L 0 +(G T ), for some T > 0, with P(ξ > 0) > 0 is said to be an arbitrage of the first kind in G if for all v > 0 there exists a strategy θ v such that V (v, θ v ) := v + θ v S 0 and V (v, θ v ) T ξ P-a.s. If such a random variable does not exist we say that NA1(G) holds. Remarks: NA1 is the minimal condition for a meaningful solution of portfolio optimization problems (Karatzas & Kardaras, 2007); NA1 NUPBR (Karatzas & Kardaras, 2007) BK (Kabanov, 1997); NA1 is also a necessary condition for the application of the Benchmark Approach (see Platen & Heath, 2006). Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 6 / 14

No Arbitrage of the First Kind Theorem (Kardaras, 2012; Takaoka & Schweizer, 2014) NA1(G) holds if and only if there exists a local martingale deflator, i.e., a G-local martingale L = (L t ) t 0 with L 0 = 1 and L t > 0 P-a.s. for all t 0 such that LS is a G-local martingale. In our previous setting: does the insider information of G give rise to arbitrages of the first kind on [0, τ]? Proposition The process 1/N τ is a local martingale deflator in G for the stopped process S τ (and, hence, NA1(G) holds for S τ ). However, 1/N τ is not a u.i. G-martingale. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 7 / 14

Arbitrages arising with honest times The continuous case...summing up: Assume that: the filtration F is continuous; the financial market based on F is arbitrage-free (NFLVR) and complete; the random time τ is an honest time such that P(τ = σ) = 0, for every F-stopping time σ. Then, in the progressively enlarged filtration G, it holds that there exist arbitrage opportunities on [0, τ]; there do not exist arbitrages of the first kind on [0, τ]. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 8 / 14

...towards a general semimartingale theory for NA1 What can we say for: a general (P-a.s. finite) random time τ; a general (non-negative) semimartingale model ( Ω, F, F, P; S = (S t ) t 0 ). We shall restrict our attention to what happens in G on the time horizon [0, τ], where G is a right-continuous filtration satisfying the following properties: (i) F t G t, for all t 0; (ii) the random time τ is a G-stopping time; (iii) G t {τ > t} = F t {τ > t}, for all t 0. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 9 / 14

Canonical pair associated to a random time τ If τ is an arbitrary random time, P(τ > t F t ) = N t /Nt may not hold. However... Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 10 / 14

Canonical pair associated to a random time τ If τ is an arbitrary random time, P(τ > t F t ) = N t /Nt may not hold. However... Theorem (Itô-Watanabe, 1965; Kardaras, 2013; Penner & Reveillac, 2014) Let τ be a P-a.s. finite random time. Then it holds that where Z t = P(τ > t F t ) = N t (1 K t ), for all t 0, N = (N t ) t 0 is a non-negative F-local martingale with N 0 = 1, K = (K t ) t 0 is a non-decreasing F-optional process with 0 K 1, such that, for every non-negative F-optional process V = (V t ) t 0, [ ] E[V τ ] = E V t N t dk t. R + Moreover, it holds that N > 0 on [[0, τ]] and {K < 1} {N = 0}. (N, K) is called the canonical pair associated to the random time τ; if τ is a honest time that avoids all F-stopping times, then 1 K = 1/N. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 10 / 14

Behaviour of F-local martingales in G on [0, τ] Classical result: Jeulin & Yor (1978) decomposition formula. Let us define the following F-stopping times: ζ := inf{t 0 : Z t = 0 or Z t = 0}; η := ζ 1 Λ + 1 Λ c, where Λ := {ζ <, K ζ < 1, N ζ > 0, K ζ = 0}. Proposition Let X be a non-negative F-local martingale such that [[η, [[ {X = 0}. Then, the process X τ /N τ is a G-local martingale. The following are equivalent: (i) for every non-negative F-local martingale X, the process X τ /N τ is a G-local martingale; (ii) the process 1/N τ is a G-local martingale; (iii) P(η < ) = 0. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 11 / 14

Behaviour of F-local martingales in G on [0, τ] Classical result: Jeulin & Yor (1978) decomposition formula. Let us define the following F-stopping times: ζ := inf{t 0 : Z t = 0 or Z t = 0}; η := ζ 1 Λ + 1 Λ c, where Λ := {ζ <, K ζ < 1, N ζ > 0, K ζ = 0}. Proposition Let X be a non-negative F-local martingale such that [[η, [[ {X = 0}. Then, the process X τ /N τ is a G-local martingale. The following are equivalent: (i) for every non-negative F-local martingale X, the process X τ /N τ is a G-local martingale; (ii) the process 1/N τ is a G-local martingale; (iii) P(η < ) = 0. In particular, 1/N τ is always a strictly positive G-supermartingale; the behaviour of 1/N τ fully determines the behaviour of F-local martingales in G on the time horizon [0, τ]. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 11 / 14

No Arbitrage of the First Kind in G on [0, τ] Theorem Let S be an F-semimartingale satisfying NA1 in F. If P(η <, S η 0) = 0, then the stopped process S τ satisfies NA1 in G. In particular, NA1 is preserved in all continuous semimartingale models. Theorem The following are equivalent: (i) for every F-semimartingale S satisfying NA1 in F, the stopped process S τ satisfies NA1 in G; (ii) P(η < ) = 0. Remarks: idea of the proof of (ii) (i): if Y is a LMD for S in F then Y τ /N τ is a LMD for S τ in G; alternative proofs of analogous results: Aksamit et al. (2014), Song (2014). Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 12 / 14

Initially enlarged filtrations NA1 stability under Jacod s density hypothesis For some E-valued random variable J, consider the initially enlarged filtration G = (G t ) t 0 defined as the right-continuous augmentation of G 0 t := F t σ(j). Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 13 / 14

Initially enlarged filtrations NA1 stability under Jacod s density hypothesis For some E-valued random variable J, consider the initially enlarged filtration G = (G t ) t 0 defined as the right-continuous augmentation of G 0 t := F t σ(j). Assumption (Jacod, 1985) Let γ be the unconditional law of J and γ t the F t -conditional law of J, for t 0. γ t γ P-a.s. for all t 0. Under the above assumption, there exists a nicely measurable density p x such that γ t (dx) = p x t γ(dx) P-a.s. for all 0. Define η x := inf{t 0 : p x t > 0 and p x t = 0}. Theorem 1 Let S be an F-semimartingale satisfying NA1 in F. If P(η x <, S η x 0) = 0 holds for γ-a.e. x E, then S satisfies NA1 in G. 2 The following are equivalent: for every E-indexed family (S x ) x E of F-semimartingales S satisfying NA1 in F, the process S J satisfies NA1 in G; P(η x < ) = 0 for γ-a.e. x E. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, 2015 13 / 14

Thank you for your attention! For more information (http://sites.google.com/site/fontanaclaud/) F., Jeanblanc & Song (2014): On arbitrages arising with honest times. Acciaio, F. & Kardaras (2014): Arbitrage of the first kind and filtration enlargements in semimartingale financial models.