Pricing rules under asymmetric information
ESAIM: Probability and Statistics, Volume 11 (2007), pp. 80-88.

We consider an extension of the Kyle and Back’s model [Back, Rev. Finance Stud. 5 (1992) 387-409; Kyle, Econometrica 35 (1985) 1315-1335], meaning a model for the market with a continuous time risky asset and asymmetrical information. There are three financial agents: the market maker, an insider trader (who knows a random variable V which will be revealed at final time) and a non informed agent. Here we assume that the non informed agent is strategic, namely he/she uses a utility function to optimize his/her strategy. Optimal control theory is applied to obtain a pricing rule and to prove the existence of an equilibrium price when the insider trader and the non informed agent are risk-neutral. We will show that if such an equilibrium exists, then the non informed agent’s optimal strategy is to do nothing, in other words to be non strategic.

DOI: 10.1051/ps:2007007
Classification: 49N30,  60H10,  93E20
Keywords: equilibrium, optimal control, asymmetric information
@article{PS_2007__11__80_0,
     author = {Ogawa, Shigeyoshi and Pontier, Monique},
     title = {Pricing rules under asymmetric information},
     journal = {ESAIM: Probability and Statistics},
     pages = {80--88},
     publisher = {EDP-Sciences},
     volume = {11},
     year = {2007},
     doi = {10.1051/ps:2007007},
     mrnumber = {2299648},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2007007/}
}
TY  - JOUR
AU  - Ogawa, Shigeyoshi
AU  - Pontier, Monique
TI  - Pricing rules under asymmetric information
JO  - ESAIM: Probability and Statistics
PY  - 2007
DA  - 2007///
SP  - 80
EP  - 88
VL  - 11
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps:2007007/
UR  - https://www.ams.org/mathscinet-getitem?mr=2299648
UR  - https://doi.org/10.1051/ps:2007007
DO  - 10.1051/ps:2007007
LA  - en
ID  - PS_2007__11__80_0
ER  - 
%0 Journal Article
%A Ogawa, Shigeyoshi
%A Pontier, Monique
%T Pricing rules under asymmetric information
%J ESAIM: Probability and Statistics
%D 2007
%P 80-88
%V 11
%I EDP-Sciences
%U https://doi.org/10.1051/ps:2007007
%R 10.1051/ps:2007007
%G en
%F PS_2007__11__80_0
Ogawa, Shigeyoshi; Pontier, Monique. Pricing rules under asymmetric information. ESAIM: Probability and Statistics, Volume 11 (2007), pp. 80-88. doi : 10.1051/ps:2007007. http://archive.numdam.org/articles/10.1051/ps:2007007/

[1] J. Amendinger, Martingale representation theorems for initially enlarged filtrations. Stoch. Proc. Appl. 89 (2000) 101-116. | Zbl

[2] J. Amendinger, P. Imkeller and M. Schweizer, Additional logarithmic utility of an insider. Stoch. Proc. Appl. 75 (1998) 263-286. | Zbl

[3] K. Back, Insider trading in continuous time. Rev. Financial Stud. 5 (1992) 387-409.

[4] B. Biais, T. Mariotti, G. Plantin and J.C. Rochet, Dynamic security design. Rev. Economic Stud. to appear. | MR

[5] K.H. Cho and N. EL Karoui, Insider trading and nonlinear equilibria:uniqueness: single auction case. Annales d'économie et de statistique 60 (2000) 21-41.

[6] K.H. Cho, Continuous auctions and insider trading: uniqueness and risk aversion. Finance and Stochastics 7 (2003) 47-71. | Zbl

[7] M. Chaleyat-Maurel and T. Jeulin, Grossissement gaussien de la filtration brownienne, in Séminaire de Calcul Stochastique 1982-83, Paris, Lect. Notes Math. 1118 (1985) 59-109. | Zbl

[8] N. El Karoui, Les aspects probabilistes du contrôle stochastique, in Ecole d'été de Saint Flour 1979, Lect. Notes Math. 872 (1981) 73-238. | Zbl

[9] H. Föllmer and P. Imkeller, Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann. Inst. Henri Poincaré 29 (1993) 569-586. | Numdam | Zbl

[10] W.H. Fleming and R.W. Rishel, Deterministic and Stochastic Optimal Control. Springer, Berlin (1975). | MR | Zbl

[11] A. Grorud and M. Pontier, Comment détecter le délit d'initié ? CRAS, Sér. 1 324 (1997) 1137-1142. | Zbl

[12] A. Grorud and M. Pontier, Insider trading in a continuous time market model. IJTAF. 1 (1998) 331-347. | Zbl

[13] A. Grorud and M. Pontier, Probabilité neutre au risque et asymétrie d'information. CRAS, Sér. 1 329 (1999) 1009-1014. | Zbl

[14] A. Grorud and M. Pontier, Asymmetrical information and incomplete markets. IJTAF. 4 (2001) 285-302.

[15] C. Hillairet, Existence of an equilibrium with discontinuous prices, asymmetric information and non trivial initial σ-fields. Math. Finance 15 (2005) 99-117. | Zbl

[16] J. Jacod, Grossissement initial, Hypothèse H' et Théorème de Girsanov, in Séminaire de Calcul Stochastique 1982-83, Paris, Lect. Notes Math. 1118 (1985) 15-35. | Zbl

[17] T. Jeulin, Semi-martingales et grossissement de filtration. Springer-Verlag (1980). | MR | Zbl

[18] A.S. Kyle, Continuous auctions and insider trading. Econometrica 53 (1985) 1315-1335. | Zbl

[19] I. Karatzas and I. Pikovsky, Anticipative portfolio optimization. Adv. Appl. Probab. 28 (1996) 1095-1122. | Zbl

[20] G. Lasserre, Quelques modèles d'équilibre avec asymétrie d'information. Thèse soutenue à l'université de Paris VII, le 16 décembre 2003.

[21] G. Lasserre, Asymmetric information and imperfect competition in a continuous time multivariate security model, Finance and Stochastics 8 (2004) 285-309. | Zbl

[22] P. Protter, Stochastic Integration and Differential Equations. Springer-Verlag (1990). | MR | Zbl

[23] M. Schweizer, On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stochastic Anal. Appl. 13 (1995) 573-599. | Zbl

[24] M. Yor, Grossissement de filtrations et absolue continuité de noyaux, in Séminaire de Calcul Stochastique 1982-83, Paris, Lect Notes Math. 1118 (1985) 6-14. | Zbl

Cited by Sources: