Skip to main content
SpringerLink
Log in

Optimal investment with inside information and parameter uncertainty

  • Published:
Mathematics and Financial Economics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aase, K.K., Bjuland, T., Øksendal, B.: Strategic insider trading equilibrium: a forward integration approach. Preprint (2007)

  2. Amendinger J., Becherer D., Schweizer M.: A monetary value for initial information in portfolio optimization. Financ. Stoch. 7, 29–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amendinger J., Imkeller P., Schweizer M.: Additional logarithmic utility of an insider. Stoch. Process. Appl. 75, 263–286 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Back K.: Insider trading in continuous time. Rev. Financ. Stud. 5, 387–409 (1992)

    Article  Google Scholar 

  6. Baudoin F., Nguyen-Ngoc L.: The financial value of a weak information on a financial market. Financ. Stoch. 8, 415–435 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Björk T., Davis M.H.A., Landén C.: Optimal investment under partial information. Math. Methods Oper. Res. 71, 371–399 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brendle S.: Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Campi L.: Some results on quadratic hedging with insider trading. Stochastics 77, 327–348 (2005)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  11. Cho K.-H.: Continuous auctions and insider trading: uniqueness and risk aversion. Financ. Stoch. 7, 47–71 (2003)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Danilova A.: Stock market insider trading in continuous time with imperfect dynamic information. Stochastics 82, 111–131 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Hillairet C.: Comparison of insiders’ optimal strategies depending on the type of side-information. Stoch. Process. Appl. 115, 1603–1627 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Imkeller P.: Random times at which insiders can have free lunches. Stoch. Stoch. Rep. 74, 465–487 (2002)

    MathSciNet  MATH  Google Scholar 

  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, 2001

    Google Scholar 

  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)

    Chapter  Google Scholar 

  18. Karatzas I.: Lectures on the Mathematics of Finance. CRM Monograph Series, vol. 8. American Mathematical Society, Providence, RI (1997)

    Google Scholar 

  19. Kohatsu-Higa A., Sulem A.: Utility maximization in an insider influenced market. Math. Financ. 16, 153–179 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kramkov D., Schachermayer W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kyle A.: Continuous auctions and insider trading. Econometrics 53, 1315–1336 (1985)

    Article  MATH  Google Scholar 

  22. Lakner P.: Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liptser R.S., Shiryaev A.N.: Statistics of Random Processes. I: General Theory, 2nd edn. Springer- Verlag, Berlin (2001)

    Google Scholar 

  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. Monoyios, M.: Utility-based valuation and hedging of basis risk with partial information. Appl. Math. Financ. (2010)

  26. Pham H., Quenez M.-C.: Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11, 210–238 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pikovsky I., Karatzas I.: Anticipative portfolio optimization. Adv. Appl. Probab. 28, 1095–1122 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rogers L.C.G.: The relaxed investor and parameter uncertainty. Financ. Stoch. 5, 131–154 (2001)

    Article  MATH  Google Scholar 

  29. Xiong J., Zhou X.Y.: Mean-variance portfolio selection under partial information. SIAM J. Control Optim. 46, 156–175 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yor, M.: Some Aspects of Brownian Motion. Part II, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (1997). (Some recent martingale problems)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, London School of Economics and Political Science, London, UK

    Albina Danilova

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Michael Monoyios & Andrew Ng

Authors
  1. Albina Danilova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Monoyios
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew Ng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Monoyios.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Danilova, A., Monoyios, M. & Ng, A. Optimal investment with inside information and parameter uncertainty. Math Finan Econ 3, 13–38 (2010). https://doi.org/10.1007/s11579-010-0025-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11579-010-0025-y

Keywords

Search

Navigation