Equilibrium in the Presence of Many Insider Traders

Reza Habibi

doi:doi:10.11648/j.ml.20251102.11

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Equilibrium in the Presence of Many Insider Traders

Reza Habibi^*

Published in Mathematics Letters (Volume 11, Issue 2)

Received: 17 July 2025 Accepted: 1 August 2025 Published: 19 August 2025

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Abstract

The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximize expected profits while the market maker observes the total order flow and sets prices accordingly. The equilibrium of auction, when there are a noise trader, a insider trader and a price maker is well studied in the literature. However, in practice, there exist more than one insider and noise traders. In this paper, the case of κ insider traders are considered. First, the pure Nash equilibriums are derived and two learning methods namely gradient and partial best response are studied. Then, the effect existence of more than one insider traders in the market on equilibriums and learning methods are considered. Also, mixture equilibriums are derived and corresponding learning method for mixture distributions is derived. Finally, a conclusion is proposed.

Published in	Mathematics Letters (Volume 11, Issue 2)
DOI	10.11648/j.ml.20251102.11
Page(s)	32-40
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Gradient Scheme, Kyle's Equilibrium, Learning Method, Nash Equilibrium, BPR

References

[1]	Aase, K. K., Bjuland, T., and Oksendal, B. (2010). Strategic insider trading equilibrium: a filter theory approach. Technical Report. Centre of Mathematics for Applications (CMA), Dept of Mathematics, University of Oslo, Norway.
[2]	Aksamit, A. and Hou, Z. (2016). Robust framework for quantifying the value of information in pricing and hedging. arXiv:1605.02539.
[3]	Back, K. (1992). Insider trading in continuous time. The Review of Financial Studies 5, 387-409.
[4]	Back, K., Cocquemas, F., Ekren, I. and Lioui, A. (2021). Optimal transport and risk aversion in Kyle’s model of informed Trading. arXiv:2006.09518.
[5]	Biagini, F., and Oksendal, B. (2005). A general stochastic calculus approach to insider trading. Applied Mathematics & Optimization 52, 167-181.
[6]	Borkar, V. S. (2008). Stochastic approximations: a dynamical system viewpoint. Cambridge university press.
[7]	Campi, L. (2005). Some results on quadratic hedging with insider trading. Stochastics 77, 327-348.
[8]	Campi, L. and Cetin, U. (2007), Insider trading in an equilibrium model with default: a passage from reduced-form to structural modeling, Finance and Stochastics 11(4), 591-602.
[9]	Cetin, U. (2018), Financial equilibrium with asymmetric information and random horizon, Finance and Stochastics 22, 97-126.
[10]	Cetin, U. and Danilova, A. (2022), Order routing and market quality: Who benefits from internalisation? arXiv:2212.07827.
[11]	Cetin, U. and Larsen, K. (2023), Is Kyle’s equilibrium model stable? SSRN Electronic Journal.
[12]	Danilova, A. (2010), Stock market insider trading in continuous time with imperfect dynamic information, Stochastics 82(1), 111-131.
[13]	Glosten, L. R., and Milgrom, P. R. (1985). Bid, ask and transaction prices in a specialist market with hetrogeneously informed traders. Journal of Financial Economics 14, 71-100.
[14]	Kramkov, D. and Xu, Y. (2022), An optimal transport problem with backward martingale constraints motivated by insider trading, The Annals of Applied Probability 32(1).
[15]	Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica 53, 1315-1336.
[16]	McLennan, A., Monteiro, P. K. and Tourky, R. (2017). On uniqueness of equilibrium in the Kyle model. Mathematics and Financial Economics 11(2), 161-172.
[17]	Owen, G. (1995). Game theory. Academic press. UK.
[18]	Singh, S., Kearns, M., and Mansour, Y. (1999). Nash convergence of gradient dynamics in general-sum games. Technical Report. AT&T labs. USA.
[19]	Rochet, J.-C. and Vila, J.-L. (2020), Insider trading without normality. The Review of Economic Studies 61(1), 131-152.
[20]	Sgroi, D. (2017). Mixed strategies. Lecture Notes. EC202 Course. University of Warwick.
[21]	Vovk. V. (2012). Continuous-time trading and the emergence of probability. Finance and Stochastic 16, 561-609.

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APA Style

Habibi, R. (2025). Equilibrium in the Presence of Many Insider Traders. Mathematics Letters, 11(2), 32-40. https://doi.org/10.11648/j.ml.20251102.11

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ACS Style

Habibi, R. Equilibrium in the Presence of Many Insider Traders. Math. Lett. 2025, 11(2), 32-40. doi: 10.11648/j.ml.20251102.11

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AMA Style

Habibi R. Equilibrium in the Presence of Many Insider Traders. Math Lett. 2025;11(2):32-40. doi: 10.11648/j.ml.20251102.11

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@article{10.11648/j.ml.20251102.11,
  author = {Reza Habibi},
  title = {Equilibrium in the Presence of Many Insider Traders
},
  journal = {Mathematics Letters},
  volume = {11},
  number = {2},
  pages = {32-40},
  doi = {10.11648/j.ml.20251102.11},
  url = {https://doi.org/10.11648/j.ml.20251102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251102.11},
  abstract = {The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximize expected profits while the market maker observes the total order flow and sets prices accordingly. The equilibrium of auction, when there are a noise trader, a insider trader and a price maker is well studied in the literature. However, in practice, there exist more than one insider and noise traders. In this paper, the case of κ insider traders are considered. First, the pure Nash equilibriums are derived and two learning methods namely gradient and partial best response are studied. Then, the effect existence of more than one insider traders in the market on equilibriums and learning methods are considered. Also, mixture equilibriums are derived and corresponding learning method for mixture distributions is derived. Finally, a conclusion is proposed.},
 year = {2025}
}

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TY  - JOUR
T1  - Equilibrium in the Presence of Many Insider Traders

AU  - Reza Habibi
Y1  - 2025/08/19
PY  - 2025
N1  - https://doi.org/10.11648/j.ml.20251102.11
DO  - 10.11648/j.ml.20251102.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 32
EP  - 40
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20251102.11
AB  - The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximize expected profits while the market maker observes the total order flow and sets prices accordingly. The equilibrium of auction, when there are a noise trader, a insider trader and a price maker is well studied in the literature. However, in practice, there exist more than one insider and noise traders. In this paper, the case of κ insider traders are considered. First, the pure Nash equilibriums are derived and two learning methods namely gradient and partial best response are studied. Then, the effect existence of more than one insider traders in the market on equilibriums and learning methods are considered. Also, mixture equilibriums are derived and corresponding learning method for mixture distributions is derived. Finally, a conclusion is proposed.
VL  - 11
IS  - 2
ER  -

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Author Information

Reza Habibi

Department of Banking, Iran Banking Institute, Central Bank of Iran, Tehran, Iran

Contact Email

http://orcid.org/0000-0001-8268-0326

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APA Style

Habibi, R. (2025). Equilibrium in the Presence of Many Insider Traders. Mathematics Letters, 11(2), 32-40. https://doi.org/10.11648/j.ml.20251102.11

Copy | Download

ACS Style

Habibi, R. Equilibrium in the Presence of Many Insider Traders. Math. Lett. 2025, 11(2), 32-40. doi: 10.11648/j.ml.20251102.11

Copy | Download

AMA Style

Habibi R. Equilibrium in the Presence of Many Insider Traders. Math Lett. 2025;11(2):32-40. doi: 10.11648/j.ml.20251102.11

Copy | Download

@article{10.11648/j.ml.20251102.11,
  author = {Reza Habibi},
  title = {Equilibrium in the Presence of Many Insider Traders
},
  journal = {Mathematics Letters},
  volume = {11},
  number = {2},
  pages = {32-40},
  doi = {10.11648/j.ml.20251102.11},
  url = {https://doi.org/10.11648/j.ml.20251102.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20251102.11},
  abstract = {The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximize expected profits while the market maker observes the total order flow and sets prices accordingly. The equilibrium of auction, when there are a noise trader, a insider trader and a price maker is well studied in the literature. However, in practice, there exist more than one insider and noise traders. In this paper, the case of κ insider traders are considered. First, the pure Nash equilibriums are derived and two learning methods namely gradient and partial best response are studied. Then, the effect existence of more than one insider traders in the market on equilibriums and learning methods are considered. Also, mixture equilibriums are derived and corresponding learning method for mixture distributions is derived. Finally, a conclusion is proposed.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Equilibrium in the Presence of Many Insider Traders

AU  - Reza Habibi
Y1  - 2025/08/19
PY  - 2025
N1  - https://doi.org/10.11648/j.ml.20251102.11
DO  - 10.11648/j.ml.20251102.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 32
EP  - 40
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20251102.11
AB  - The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximize expected profits while the market maker observes the total order flow and sets prices accordingly. The equilibrium of auction, when there are a noise trader, a insider trader and a price maker is well studied in the literature. However, in practice, there exist more than one insider and noise traders. In this paper, the case of κ insider traders are considered. First, the pure Nash equilibriums are derived and two learning methods namely gradient and partial best response are studied. Then, the effect existence of more than one insider traders in the market on equilibriums and learning methods are considered. Also, mixture equilibriums are derived and corresponding learning method for mixture distributions is derived. Finally, a conclusion is proposed.
VL  - 11
IS  - 2
ER  -

Copy | Download