Mathematics and Financial Economics

, Volume 3, Issue 1, pp 13–38 | Cite as

Optimal investment with inside information and parameter uncertainty

Article

Abstract

An optimal investment problem is solved for an insider who has access to noisy information related to a future stock price, but who does not know the stock price drift. The drift is filtered from a combination of price observations and the privileged information, fusing a partial information scenario with enlargement of filtration techniques. We apply a variant of the Kalman–Bucy filter to infer a signal, given a combination of an observation process and some additional information. This converts the combined partial and inside information model to a full information model, and the associated investment problem for HARA utility is explicitly solved via duality methods. We consider the cases in which the agent has information on the terminal value of the Brownian motion driving the stock, and on the terminal stock price itself. Comparisons are drawn with the classical partial information case without insider knowledge. The parameter uncertainty results in stock price inside information being more valuable than Brownian information, and perfect knowledge of the future stock price leads to infinite additional utility. This is in contrast to the conventional case in which the stock drift is assumed known, in which perfect information of any kind leads to unbounded additional utility, since stock price information is then indistinguishable from Brownian information.

Keywords

Insider trading Enlargement of filtration Filtering Optimal investment Partial information 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aase, K.K., Bjuland, T., Øksendal, B.: Strategic insider trading equilibrium: a forward integration approach. Preprint (2007)Google Scholar
  2. 2.
    Amendinger J., Becherer D., Schweizer M.: A monetary value for initial information in portfolio optimization. Financ. Stoch. 7, 29–46 (2003)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Amendinger J., Imkeller P., Schweizer M.: Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263–286 (1998)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ankirchner S., Dereich S., Imkeller P.: The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34, 743–778 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Back K.: Insider trading in continuous time. Rev. Financ. Stud. 5, 387–409 (1992)CrossRefGoogle Scholar
  6. 6.
    Baudoin F., Nguyen-Ngoc L.: The financial value of a weak information on a financial market. Financ. Stoch. 8, 415–435 (2004)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Björk T., Davis M.H.A., Landén C.: Optimal investment under partial information. Math. Methods Oper. Res. 71, 371–399 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Brendle S.: Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Campi L.: Some results on quadratic hedging with insider trading. Stochastics 77, 327–348 (2005)MathSciNetMATHGoogle Scholar
  10. 10.
    Campi L., Çetin U.: Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling. Financ. Stoch. 11, 591–602 (2007)CrossRefGoogle Scholar
  11. 11.
    Cho K.-H.: Continuous auctions and insider trading: uniqueness and risk aversion. Financ. Stoch. 7, 47–71 (2003)MATHCrossRefGoogle Scholar
  12. 12.
    Corcuera J.M., Imkeller P., Kohatsu-Higa A., Nualart D.: Additional utility of insiders with imperfect dynamical information. Financ. Stoch. 8, 437–450 (2004)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Danilova A.: Stock market insider trading in continuous time with imperfect dynamic information. Stochastics 82, 111–131 (2010)MathSciNetMATHGoogle Scholar
  14. 14.
    Hillairet C.: Comparison of insiders’ optimal strategies depending on the type of side-information. Stoch. Process. Appl. 115, 1603–1627 (2005)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Imkeller P.: Random times at which insiders can have free lunches. Stoch. Stoch. Rep. 74, 465–487 (2002)MathSciNetMATHGoogle Scholar
  16. 16.
    Imkeller, P.: Malliavin’s calculus in insider models: additional utility and free lunches. Math. Financ. 13, 153–169 (2003). Conference on Applications of Malliavin Calculus in Finance, Rocquencourt, 2001Google Scholar
  17. 17.
    Jacod J.: Grossissement initial, hypotheèse (H’) et théorème de Girsanov. In: Jeulin, T., Yor, M. (eds) Grossissements de filtrations: exemples etapplications. Lecture Notes in Mathematics, vol. 1118, pp. 15–35. Springer-Verlag, Berlin (1985)CrossRefGoogle Scholar
  18. 18.
    Karatzas I.: Lectures on the Mathematics of Finance. CRM Monograph Series, vol. 8. American Mathematical Society, Providence, RI (1997)Google Scholar
  19. 19.
    Kohatsu-Higa A., Sulem A.: Utility maximization in an insider influenced market. Math. Financ. 16, 153–179 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kramkov D., Schachermayer W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kyle A.: Continuous auctions and insider trading. Econometrics 53, 1315–1336 (1985)MATHCrossRefGoogle Scholar
  22. 22.
    Lakner P.: Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97 (1998)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Liptser R.S., Shiryaev A.N.: Statistics of Random Processes. I: General Theory, 2nd edn. Springer- Verlag, Berlin (2001)Google Scholar
  24. 24.
    Mansuy R., Yor M.: Random Times and Enlargements of Filtrations in a Brownian Setting. Lecture Notes in Mathematics, vol. 1873. Springer-Verlag, Berlin (2006)Google Scholar
  25. 25.
    Monoyios, M.: Utility-based valuation and hedging of basis risk with partial information. Appl. Math. Financ. (2010)Google Scholar
  26. 26.
    Pham H., Quenez M.-C.: Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11, 210–238 (2001)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Pikovsky I., Karatzas I.: Anticipative portfolio optimization. Adv. Appl. Probab. 28, 1095–1122 (1996)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Rogers L.C.G.: The relaxed investor and parameter uncertainty. Financ. Stoch. 5, 131–154 (2001)MATHCrossRefGoogle Scholar
  29. 29.
    Xiong J., Zhou X.Y.: Mean-variance portfolio selection under partial information. SIAM J. Control Optim. 46, 156–175 (2007)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Yor, M.: Some Aspects of Brownian Motion. Part II, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (1997). (Some recent martingale problems)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsLondon School of Economics and Political ScienceLondonUK
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations