Mathematics and Financial Economics

, Volume 12, Issue 4, pp 589–614 | Cite as

Arbitrage and utility maximization in market models with an insider

  • Huy N. Chau
  • Wolfgang J. Runggaldier
  • Peter Tankov


We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses an additional information in the form of an \(\mathscr {F}_T\)-measurable discrete random variable G, we give criteria for the no unbounded profits with bounded risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided. For the case when G is a continuous random variable, we consider the notion of no asymptotic arbitrage of the first kind (NAA1) and give an explicit construction for unbounded profits if NAA1 fails.


Initial enlargement of filtration Optimal arbitrage No unbounded profits with bounded risk Incomplete markets Hedging Utility maximization 

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Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Acciaio, B., Fontana, C., Kardaras, C.: Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Stochast. Process. Appl. 126, 1761–1784 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aksamit, A., Choulli, T., Jeanblanc, M.: On an optional semimartingale decomposition and the existence of a deflator in an enlarged filtration, in In Memoriam Marc Yor-Séminaire de Probabilités XLVII, pp. 187–218. Springer (2015)Google Scholar
  3. 3.
    Amendinger, J., Imkeller, P., Schweizer, M.: Additional logarithmic utility of an insider. Stochast. Process. Appl. 75, 263–286 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Amendinger, J., Becherer, D., Schweizer, M.: A monetary value for initial information in portfolio optimization. Finance Stochast. 7, 29–46 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ankirchner, S., Zwierz, J.: Initial enlargement of filtrations and entropy of Poisson compensators. J. Theor. Probab. 24, 93–117 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ankirchner, S., Dereich, S., Imkeller, P.: The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34, 743–778 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baldeaux, J., Platen, E.: Liability driven investments under a benchmark based approach. Preprint (2013)Google Scholar
  8. 8.
    Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Berlin (1981)CrossRefMATHGoogle Scholar
  9. 9.
    Chau, N.H.: A Study of Arbitrage Opportunities in Financial Markets without Martingale Measures. Ph.D. thesis, Università degli Studi di Padova / Université Paris Diderot, Paris 7 (2016)Google Scholar
  10. 10.
    Chau, H.N., Tankov, P.: Market models with optimal arbitrage. SIAM J. Financ. Math. 6, 66–85 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Choulli, T., Deng, J., Ma, J.: How non-arbitrage, viability and numéraire portfolio are related. Finance Stochast. 19(4), 719–741 (2015)CrossRefMATHGoogle Scholar
  12. 12.
    Danilova, A., Monoyios, M., Ng, A.: Optimal investment with inside information and parameter uncertainty. Math. Financ. Econ. 3, 13–38 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stochast. Stochast. Rep. 53(3–4), 213–226 (1995)CrossRefMATHGoogle Scholar
  15. 15.
    Fernholz, D., Karatzas, I.: On optimal arbitrage. Ann. Appl. Probab. 20, 1179–1204 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fontana, C.: No-arbitrage conditions and absolutely continuous changes of measure. In: Hillairet, C., Jeanblanc, M., Jiao, Y. (eds.) Arbitrage, Credit and Informational Risks: Peking University Series in Mathematics, vol. 5, pp. 3–18. World Scientific, Singapore (2014)CrossRefGoogle Scholar
  17. 17.
    Fontana, C.: Weak and strong no-arbitrage conditions for continuous financial markets. Int. J. Theor. Appl. Finance 18, 1550005 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Grorud, A., Pontier, M.: Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1, 331–347 (1998)CrossRefMATHGoogle Scholar
  19. 19.
    Imkeller, P., Perkowski, N.: The existence of dominating local martingale measures. Finance Stochast. 19(4), 685–717 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Imkeller, P., Pontier, M., Weisz, F.: Free lunch and arbitrage possibilities in a financial market model with an insider. Stochast. Process. Appl. 92, 103–130 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jacod, J.: Grossissement initial, hypothèse (H\(^{\prime }\)) et théorème de Girsanov, in Grossissements de filtrations: exemples et applications. Springer, pp. 15–35 (1985)Google Scholar
  22. 22.
    Jacod, J.: Intégrales stochastiques par rapport à une semimartingale vectorielle et changements de filtration, Séminaire de Probabilités XIV 1978/79, pp. 161–172. Springer, Berlin (1980)Google Scholar
  23. 23.
    Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  24. 24.
    Jeulin, T.: Semi-Martingales et Grossissement d’une Filtration, vol. 833 of Lecture Notes in Mathematics. Springer (1980)Google Scholar
  25. 25.
    Kabanov, Y., Kardaras, C., Song, S.: No arbitrage and local martingale deflators. ArXiv preprint arXiv:1501.04363 (2015)
  26. 26.
    Kabanov, Y.M.: On the FTAP of Kreps–Delbaen–Schachermayer, in Statistics and Control of Random Processes, pp. 191–203. World Scientific, The Liptser Festschrift (1997)MATHGoogle Scholar
  27. 27.
    Kabanov, Y.M., Kramkov, D.O.: Large financial markets: asymptotic arbitrage and contiguity. Theory Probab. Appl. 39, 182–187 (1994)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stochast. 11, 447–493 (2007)CrossRefMATHGoogle Scholar
  29. 29.
    Kardaras, C.: Market viability via absence of arbitrage of the first kind. Finance Stochast. 16, 651–667 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kardaras, C., Kreher, D., Nikeghbali, A.: Strict local martingales and bubbles. Ann. Appl. Probab. 25, 1827–1867 (2015)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kohatsu-Higa, A., Yamazato, M.: Insider models with finite utility in markets with jumps. Appl. Math. Optim. 64, 217–255 (2011)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Levental, S., Skorohod, A.V.: A necessary and sufficient condition for absence of arbitrage with tame portfolios. Ann. Appl. Probab. 5, 906–925 (1995)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Meyer, P.-A., Jacod, J.: Sur un théoreme de. Séminaire de probabilités de Strasbourg 12, 57–60 (1978)Google Scholar
  35. 35.
    Mostovyi, O.: Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption. Finance Stochast. 19, 135–159 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Pikovsky, I., Karatzas, I.: Anticipative portfolio optimization. Adv. Appl. Probab. 28, 1095–1122 (1996)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2003)Google Scholar
  38. 38.
    Ruf, J., Runggaldier, W.: A systematic approach to constructing market models with arbitrage. In: Hillairet, C., Jeanblanc, M., Jiao, Y. (eds.) Arbitrage, Credit and Informational Risks: Peking University Series in Mathematics, vol. 5, pp. 19–28. World Scientific, Singapore (2014)CrossRefGoogle Scholar
  39. 39.
    Song, S.: An alternative proof of a result of Takaoka. arXiv:1306.1062 (2013)
  40. 40.
    Takaoka, K., Schweizer, M.: A note on the condition of no unbounded profit with bounded risk. Finance Stochast. 18, 393–405 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Huy N. Chau
    • 1
  • Wolfgang J. Runggaldier
    • 2
  • Peter Tankov
    • 3
  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  2. 2.Department of MathematicsUniversity of PaduaPaduaItaly
  3. 3.CREST-ENSAE Paris TechPalaiseauFrance

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