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Kyle equilibrium under random price pressure

  • José Manuel Corcuera
  • Giulia Di Nunno
  • José FajardoEmail author
Article
  • 16 Downloads

Abstract

We study the equilibrium in the model proposed by Kyle (Econometrica 53(6):1315–1335, 1985) and extended to the continuous-time setting by Back (Rev Financ Stud 5(3):387–409, 1992). The novelty of this paper is that we consider a framework where the price pressure can be random. We also allow for a random release time of the fundamental value of the asset. This framework includes all the particular Kyle models proposed in the literature. The results enlighten the equilibrium properties shared by all these models and guide the way of finding equilibria in this context.

Keywords

Kyle model Market microstructure Equilibrium Insider trading Stochastic control Enlargement of filtrations 

Mathematics Subject Classification

60G35 62M20 93E10 94Axx 

JEL Classification

C61 D43 D44 D53 G11 G12 G14 

Notes

References

  1. Applebaum, D.: Lévy processes and stochastic calculus. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  2. Aase, K.K., Bjuland, T., Øksendal, B.: Strategic insider trading equilibrium: a filter theory approach. Afr. Mat. 23(2), 145–162 (2012a)CrossRefGoogle Scholar
  3. Aase, K.K., Bjuland, T., Øksendal, B.: Partially informed noise traders. Math. Financ. Econ. 6, 93–104 (2012b)CrossRefGoogle Scholar
  4. Amendinger, J., Imkeller, P., Schweizer, M.: Additional logarithmic utility of an insider. Stoch. Process Appl. 75, 263–286 (1998)CrossRefGoogle Scholar
  5. Back, K.: Insider trading in continuous time. Rev. Financ. Stud. 5(3), 387–409 (1992)CrossRefGoogle Scholar
  6. Back, K.: Asymmetric information and options. Rev. Financ. Stud. 6(3), 435–472 (1993)CrossRefGoogle Scholar
  7. Back, K., Baruch, S.: Information in securities markets: Kyle meets Glosten and Milgrom. Econometrica 72(2), 433–465 (2004)CrossRefGoogle Scholar
  8. Back, K., Pedersen, H.: Long-lived information and intraday patterns. J. Financ. Mark. 1, 385–402 (1998)CrossRefGoogle Scholar
  9. Bank, P., Baum, D.: Hedging and portfolio optimization in financial markets with a large trader. Math. Financ. 14(1), 1–18 (2004)CrossRefGoogle Scholar
  10. Biagini, F., Øskendal, B.: A general stochastic calculus approach to insider trading. App. Math. Optim. 52, 167–181 (2005)CrossRefGoogle Scholar
  11. Biagini, F., Øskendal, B.: Minimal variance hedging for insider trading. Int. J. Theor. Appl. Financ. 9, 1351–1375 (2006)CrossRefGoogle Scholar
  12. Biagini, F., Hu, Y., Meyer-Brandis, T., Øksendal, B.: Insider trading equilibrium in a market with memory. Math. Financ. Econ. 6, 229–247 (2012)CrossRefGoogle Scholar
  13. Caldentey, R., Stacchetti, E.: Insider trading with a random deadline. Econometrica 78(1), 245–283 (2010)CrossRefGoogle Scholar
  14. Campi, L., Çetin, U.: Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling. Financ. Stoch. 4, 591–602 (2007)CrossRefGoogle Scholar
  15. Campi, L., Çetin, U., Danilova, A.: Equilibrium model with default and dynamic insider information. Financ. Stoch. 17(3), 565–585 (2013)CrossRefGoogle Scholar
  16. Cho, K.: Continuous auctions and insider trading: uniqueness and risk aversion. Financ. Stoch. 7, 47–71 (2003)CrossRefGoogle Scholar
  17. Collin-Dufresne, P., Fos, V.: Insider trading, stochastic liquidity, and equilibrium prices. Econometrica 84(4), 1441–1475 (2016)CrossRefGoogle Scholar
  18. Corcuera, J.M., Imkeller, P., Kohatsu-Higa, A., Nualart, D.: Additional utility of insiders with imperfect dynamical information. Financ. Stoch. 8, 437–450 (2004)CrossRefGoogle Scholar
  19. Corcuera, J.M., Farkas, G., Di Nunno, G., Øksendal, B.: Kyle–Back’s model with Lévy noise. Preprint series. Pure Math. http://urn.nb. no/URN: NBN: no-8076 (2010)
  20. Corcuera, J.M., Di Nunno, J.: Kyle-Back’s model with a random horizon. J. Theor. Appl. Financ. 21, 1850016 (2018).  https://doi.org/10.1142/S0219024918500164 CrossRefGoogle Scholar
  21. Cuoco, D., Cvitanić, J.: Optimal consumption choices for a ‘large’investor. J. Econ. Dyn. Control 22(3), 401–436 (1998)CrossRefGoogle Scholar
  22. Danilova, A.: Stock market insider trading in continuous time with imperfect dynamic information. Stoch. Int. J. Probab. Stoch. Process. 82(1), 111–131 (2010)CrossRefGoogle Scholar
  23. Di Nunno, G., Pamen, O.M., Øksendal, B., Proske, F.: A general maximum principle for anticipative stochastic control and application to insider trading. In: Di Nunno, G., Øksendal, B. (eds.) Advanced Mathematical Methods for Finance, pp. 181–221. Springer, Berlin (2011)CrossRefGoogle Scholar
  24. Di Nunno, G., Kohatsu-Higa, A., Meyer-Brandis, T., Øksendal, B., Proske, F., Sulem, A.: Anticipative stochastic control for Lévy processes with application to insider trading. In: Bensoussan, A., Zhang, G. (eds.) Mathematical Modelling and Numerical Methods in Finance. Handbook of Numerical Analysis, North Holland (2008)Google Scholar
  25. Di Nunno, G., Meyer-Brandis, T., Øksendal, B., Proske, F.: Optimal portfolio for an insider in a market driven by Lévy processes. Quant. Financ. 6(1), 83–94 (2006)CrossRefGoogle Scholar
  26. Draouil, O., Øksendal, B.: Optimal insider control and semimartingale decomposition under enlargement of filtration. Stoch. Anal. Appl. 34, 1045–1056 (2016)CrossRefGoogle Scholar
  27. Ernst, P.A., Rogers, L.C.G., Zhou, Q.: The value of foresight. Stoch. Process. Appl. 127, 3913–3927 (2017)CrossRefGoogle Scholar
  28. Grorud, A., Pontier, M.: Insider trading in a continuous time market model. Int. J. Theor. Appl. Financ. 1(03), 331–347 (1998)CrossRefGoogle Scholar
  29. Grorud, A., Pontier, M.: Asymmetrical information and incomplete markets. Int. J. Theor. Appl. Financ. 4(02), 285–302 (2001)CrossRefGoogle Scholar
  30. Grorud, A., Pontier, M.: Financial market model with influential informed investors. Int. J. Theor. Appl. Financ. 8(06), 693–716 (2005)CrossRefGoogle Scholar
  31. Imkeller, P., Pontier, M., Weisz, F.: Free lunch an arbitrage possibilities in a financial market with an insider. Stoch. Process. Appl. 92, 103–130 (2001)CrossRefGoogle Scholar
  32. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (1987)CrossRefGoogle Scholar
  33. Karatzas, I., Pikovski, I.: Anticipative portfolio optimization. Adv. Appl. Probab. 28, 1095–1122 (1996)CrossRefGoogle Scholar
  34. Kohatsu-Higa, A.: (2007) Models for insider trading with finite utility. In: Paris–Princeton Lectures on Mathematical Finance Series: Lect. Notes in Maths, pp. 103–172. Springer, Berlin (1919)Google Scholar
  35. Kyle, A.S.: Continuous auctions and insider trading. Econometrica 53(6), 1315–1335 (1985)CrossRefGoogle Scholar
  36. Lasserre, G.: Asymmetric information and imperfect competition in a continuous time multivariate security model. Financ. Stoch. 8(2), 285–309 (2004)CrossRefGoogle Scholar
  37. Liptser, R.S.; Shiryaev, A.N.: Statistics of random processes II. Applications (translated from the 1974 Russian original by A. B. Aries). Second, revised and expanded edn. Applications of Mathematics (New York), vol. 6. Stochastic Modelling and Applied Probability. Springer, Berlin (2001)Google Scholar
  38. Malkiel, B.G.: A Random Walk Down Wall Street: The Time-Tested Strategy for successful Investing. WW Norton & Company, New York (2007)Google Scholar
  39. Revuz, D., Yor, M.: Continuous Martingales and Brownian motion, 3rd edn. Springer, New York (1999)CrossRefGoogle Scholar

Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2019

Authors and Affiliations

  1. 1.Universitat de BarcelonaBarcelonaSpain
  2. 2.Department of MathematicsUniversity of OsloOsloNorway
  3. 3.NHH, School of EconomicsBergenNorway
  4. 4.Brazilian School of Public and Business AdministrationGetulio Vargas FoundationRio de JaneiroBrazil

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