A general continuous auction system in presence of insiders

Transcription

1 A general coninuous aucion sysem in presence of insiders José M. Corcuera (based on join work wih G. DiNunno, G. Farkas and B. Oksendal) Faculy of Mahemaics Universiy of Barcelona BCAM, Basque Cener for Applied Mahemaics, Ocober 211

2 Ouline The model

3 Ouline The model The equilibrium

4 Ouline The model The equilibrium Enlargemen of filraions

5 The model We consider a financial marke wih a risky asse S and a bank accoun wih ineres rae r =. V, indicaes he fundamenal value of S and P,, is marke price.

6 The model We consider a financial marke wih a risky asse S and a bank accoun wih ineres rae r =. V, indicaes he fundamenal value of S and P,, is marke price. The rading period is [, ).

7 The model We consider a financial marke wih a risky asse S and a bank accoun wih ineres rae r =. V, indicaes he fundamenal value of S and P,, is marke price. The rading period is [, ). A ime (random) an announcemen reveals he fundamenal value of S, V, and he game is finished.

8 The model There are hree kinds of raders:

9 The model There are hree kinds of raders: An insider who has informaion abou V all he ime.

10 The model There are hree kinds of raders: An insider who has informaion abou V all he ime. Noise raders who rade randomly.

11 The model There are hree kinds of raders: An insider who has informaion abou V all he ime. Noise raders who rade randomly. Marke makers, who se prices and clear he marke.

12 Elemenary facs Definiion Le X be a random variable (r.v.) wih (finie) expecaion, and F a σ-field of evens, E(X F )= xp X F (dx). Some properies: E(E(X F ))) = E(X) If X is independen of F, E(X F )=E(X) If X is F -measurable, E(X F )=X For any Y F -measurable and bounded, E(XY F )) = YE(X F ) If E(XY )= for all Y F -measurable and bounded hen E(X F )= If F s F hen E(E(X F ) F s )=E(X F s )

13 Elemenary facs Definiion Le (M ) be a family of r.v. adaped o a filraion F =(F ) (M is F -measurable), and wih expecaion, hen i is an F-maringale if E(M F s )=M s, for all s. Or equivalenly E(M M s F s )=.

14 The model A ime, he informaion of he insider is given by H, where H = σ(p s, V s, s, s ). We assume ha V is a coninuous H-maringale, where H =(H ). The insider ries o maximize his final wealh. His demand process is X and we assume ha i is an H-adaped process of he form dx = θ d. (1)

15 The model A ime, he informaion of he insider is given by H, where H = σ(p s, V s, s, s ). We assume ha V is a coninuous H-maringale, where H =(H ). The insider ries o maximize his final wealh. His demand process is X and we assume ha i is an H-adaped process of he form dx = θ d. (1) Z is he demand process of he noise raders. We assume ha Z is an H-maringale, independen of V and Z =.

16 The model A ime, he informaion of he insider is given by H, where H = σ(p s, V s, s, s ). We assume ha V is a coninuous H-maringale, where H =(H ). The insider ries o maximize his final wealh. His demand process is X and we assume ha i is an H-adaped process of he form dx = θ d. (1) Z is he demand process of he noise raders. We assume ha Z is an H-maringale, independen of V and Z =. Finally, he marke makers observe Y = X + Z, he oal demand, and clear he marke (so heir demand process is Y ), and fix a compeiive (raional) price, given by P = E(V Y s, s ), <.

17 The model Marke makers fix prices hrough a pricing rule, in erms of formulas, P = H(,ξ ), < wih ξ := λ(s)dy s = λ(s)(θ(s)ds + dz s ) where λ is a posiive deerminisic funcion and H(, ) is sricly increasing for every. We indicae a pricing rule by he pair (H,λ).

18 This model generalizes many cases: If V V, 1 and Z is a Brownian moion hen we have Kyle-Back s model (1992). If V V, 1 and Z is a Brownian moion wih variance depending on ime we have Aase, Bjuland and Øksendal s (27). If V V 1 and dz = a(y )db where B is a Brownian moion, hen we have he Campi, Danilova and Cein s (21).

19 If V = 1 { >1}, is an H-sopping ime, = 1 and is known by he insider, his is he Campi and Cein s (27). If V and Z are independen Brownian moions and an independen sopping ime wih exponenial disribuion, hen we have Caldeney and Sachei s (21)...

20 The wealh process Consider firs a discree case where rades are made a imes i = 1, 2,...N, and where N is random. If a ime i 1, here is an order of buying X i X i 1 shares, is cos will be P i (X i X i 1 ), so, here is a change in he bank accoun given by P i (X i X i 1 ). Then he oal change is N P i (X i X i 1 ), i=1 and jus afer ime N, here is he exra income: X N V N (we assume ha X = ).

21 So, he oal wealh generaed is W N = = N P i (X i X i 1 )+X N V N i=1 N P i 1 (X i X i 1 ) i=1 i=1 N (P i P i 1 )(X i X i 1 )+X N V N

22 Analogously, in he coninuous-ime model, W = X V = = X dv + P dx [P, X] (V P ) dx + V dx +[V, X] P dx [P, X] X dv +[V, X] [P, X]

23 Elemenary facs [X, Y ] = P lim s X s Y s. If X is an F-adaped process and M a coninuous F-maringale wih T E(X 2 s )d[m, M] s < hen and is an F-maringale. X s dm s = L 2 lim s X s dm s, T X s M s.

24 The equilibrium Definiion A demand process X is said o be opimal given a pricing rule (H,λ) if i maximizes E(W ). Definiion The riple (H,λ,X) is an equilibrium, if he price process P = H(,ξ ) is compeiive, given X, and he sraegy X is opimal, given (H,λ).

25 The wealh a ime is given by W := (V P )θ d + X dv, where P = H(,ξ ), wih ξ := λ(s)(θ(s)ds + dz s), and where Z is an H-maringale.

26 Opimizaion Then we wan o maximize J(θ) :=E (W )=E (V H(,ξ ))θ d, hen, if θ is opimal, for all β, such ha θ + εβ is anoher insider s sraegy, wih ε> small enough, we will have

27 = d J(θ + εβ) dε = d dε E (V H(, λ s ((θ s + εβ s )ds + dz s ))) (θ + εβ ) d = E (V H(,ξ ))β d + E 2 H(,ξ )θ λ s β s ds d = E (V H(,ξ )) λ 2 H(s,ξ s )θ s ds β d = E 1 [,] ()(V H(,ξ )) λ 1 [,] (s) 2 H(s,ξ s )θ s ds β d

28 = d J(θ + εβ) dε = d dε E (V H(, λ s ((θ s + εβ s )ds + dz s ))) (θ + εβ ) d = E (V H(,ξ ))β d + E 2 H(,ξ )θ λ s β s ds d = E (V H(,ξ )) λ 2 H(s,ξ s )θ s ds β d = E 1 [,] ()(V H(,ξ )) λ 1 [,] (s) 2 H(s,ξ s )θ s ds β d Then, since we can ake β = 1 [u,u+h] () α u, wih α u H u -measurable,

29 u+h E α u 1[,] ()(V H(,ξ )) u λ 1 [,] (s) 2 H(s,ξ s )θ s ds d =, herefore u E(1[,] ()V H ) E(1 [,] ()H(,ξ ) H ) λ E 1 [,] (s) 2 H(s,ξ s )θ s H ds d, u is an H-maringale, and consequenly

30 E(1 [,] ()V H ) E(1 [,] ()H(,ξ ) H ) λ E 1 [,] (s) 2 H(s,ξ s )θ s H ds =,. Now, since is an H-sopping ime, and F P,V H, we can wrie, in he se { <}, V H(,ξ ) λ E 2 H(s,ξ s )θ s ds H =,

31 Moreover if is known a, V H(,ξ ) λ E ( 2 H(s,ξ s )θ s H ) ds =,. and, in paricular, lim H(,ξ )=V,

32 Moreover if is known a, V H(,ξ ) λ E ( 2 H(s,ξ s )θ s H ) ds =,. and, in paricular, lim H(,ξ )=V, This is he case in Campi and Çein (27), where hey ake V = 1 { >1}, is an H-sopping ime and = 1 and is known by he insider, ha is H. Then hey obain 1 { >1} H( 1,ξ 1 )=, Also hey assume ha is he firs passage ime of a sandard Brownian moion ha is independen of Z.

33 In Caldeney and Sacchei (21) auhors assume ha V is an arihmeic Brownian moion and follows an exponenial disribuion independen of V and P hen, on he se < V H(,ξ ) λ e µ(s ) E ( 2 H(s,ξ s )θ s H ) ds =, a.e.

34 Price pressure ( known) From he equaion V H(,ξ ) λ E ( 2 H(s,ξ s )θ s H ) ds =,. and aking ino accoun ha Z is an H-maringale, E 2 H(s,ξ s )θ s ds H = E 1 λ s 2 H(s,ξ s )dξ s H and we can conclude ha λ() =λ, and ha H(, y) saisfies he equaion (assuming, for simpliciy, Z is coninuous) 1 H(s, y) H(s, y)λ 2 sσ 2 s =,, and where σ 2 s := d[z,z ]s ds.

35 Iô s formula For f (, y) C 1,2 we have f (, M ) = f (, M ) f (s, M s )ds + 2 f (s, M s )dm s 22 f (s, M s )d[m, M] s.

36 Compeiive prices Since 1 H(s, y) H(s, y)λ 2 σ2 s = we have ha dp = dh(s,λ Y )= 2 H(s,λ Y )λ dy, in such a way ha prices are compeiive (F Y -maringale) if and only if Y is an F Y -maringale.

37 Enlargemen of filraions The aggregae demand process Y is given by Y = Z + θ s ds, where Z is an H-maringale and θ is H-adaped. So, he r.h.s is he Doob-Meyer decomposiion of he F Y -maringale w.r. o he filraion H F Y.

38 Example (Back 92) Assume ha Z is a Brownian moion wih variance σ 2 and V V 1. If he sraegy is opimal V 1 = H(1, Y 1 ), and if V 1 has coninuous cdf we can assume ha Y 1 N(,σ 2 ) by choosing H(1, ) convenienly. We assume ha V 1 (and consequenly Y 1 ) is independen of Z. By he resuls abou iniial enlargemen of filraions we know ha Y 1 Y s Y = Z + 1 s ds, 1 is a Brownian moion wih variance σ 2. So, here is an equilibrium wih he sraegy θ = Y 1 Y. 1

39 Example (Aase e. al (27)) Z = σ s dw s where σ is deerminisic and V Y 1 is a N(, 1 σ2 sds) independen of Z. Then Y = Z + g (s)(y s Y 1 ) ds, g(s) wih g() = 1 σ 2 sds, has he same law as Z. We have a similar resul if σ is random.

40 Example (Campi, Cein, Danilova 29) In fac if dz = σ(y )dw and V V 1, he final value of V = σ(v s)db s, and independen of Z, hen dy = σ(y )dw + σ 2 (Y ) yg(1, Y, V 1 ) G(1, Y, V 1 ) d where G(, y, z) is he ransiion densiy of V, is a maringale.

41 Example (Campi and Cein (27)) If we wan he aggregae process Y o be a Brownian moion ha reaches he value 1 for he firs ime a ime, and Z is also a Brownian moion hen 1 Y = Z Y s Y s s [, ] (s)ds, so, in his case he sopping ime is known and V = 1 >1.

42 REFERENCES [1] Aase, Knu K., Bjuland, Terje and Oksendal, Bern. Sraegic Insider Trading Equilibrium: A Forward Inegraion Approach. Finance and Sochasics, NHH Dep. of Finance & Managemen Science Discussion Paper No. 27/24. [2] Back, K. Insider rading in coninuous ime. Rev. Financial Sud. 5(3), (1992). [3] Caldeney, R. and Sacchei, E. (21). Insider rading wih a random deadline. Economerica, Vol. 78, No. 1, [4] Campi, L., Cein, U. Insider rading in an equilibrium model wih defaul: a passage from reduced-form o srucural modelling. Finance and Sochasics, 4, (27).

43 REFERENCES [5] Corcuera, J.M., Farkas, G., Di Nunno, G., Oksendal, B. (21). Kyle-Back s model wih Lévy noise. Preprin. [6] Corcuera, J.M., Farkas, G., Di Nunno, G., Oksendal, B. (211). A general coninuous aucion model wih insiders. Preprin. [7] Corcuera, J.M., Imkeller, P., Kohasu-Higa, A. Nualar, D. Addiional uiliy of insiders wih imperfec dynamical informaion. Finance and Sochasics, 8, (24). [8] Cho, K.H. Coninuous aucions and insider rading: uniqueness and risk aversion. Finance and Sochasics, 7,47-71, (23). [9] Jeulin, Thierry. Semi-maringales e Grossissemen d une Filraion. LNM 833, Springer, 198.