Insider information and arbitrage profits via enlargements of filtrations

Transcription

1 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, / 14

2 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. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

3 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, / 14

4 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, / 14

5 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, / 14

6 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, / 14

7 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 Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

8 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! Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

9 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, / 14

10 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, / 14

11 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, / 14

12 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, / 14

13 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, / 14

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 } Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

15 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 } Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

16 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, / 14

17 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, / 14

18 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, / 14

19 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. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

20 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, / 14

21 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, / 14

22 ...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, / 14

23 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, / 14

24 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, / 14

25 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, / 14

26 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, / 14

27 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, / 14

28 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, / 14

29 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, / 14

30 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, / 14

31 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, / 14

32 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, / 14

33 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}. Claudio Fontana (Université Paris Diderot - LPMA) XVI WQF, Parma, January 30, / 14

34 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, / 14

35 Thank you for your attention! For more information ( 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.