M2MO: Modélisation Aléatoire, Finance et Data Science

Master en statistique, probabilités et finance - Université Paris 7 - Paris Diderot


Algorithmic trading

Lecturer: O. Guéant
Period: Term 3
Schedule: 3 hours per week (at ENSAE)


The aim of this course is to introduce students to the various modeling issues associated with high frequency trading. In particular, the course will focus on taking into account execution costs and market impact in the construction of optimal execution strategies for large orders (the so-called optimal execution problem). From a mathematical point of view, the course will make extensive use of the notions of optimisation and optimal control (stochastic or not). Practical illustrations will be presented.

Main learning outcomes of the course: at the end of the course, the student will have been familiarized with the problems of high frequency trading. He will know the main block crossing modes as well as the models used. On the mathematical level, he will have deepened his knowledge of deterministic and stochastic optimization, and will have acquired new knowledge in the field of statistics on high-frequency data.


  •  Introduction to stock markets : financial markets operating (order books, different types of orders, competition between platforms, dark pools, ...) optimal execution problematic, utility expectation criterion, CARA functions and mean-variance criteria.

  • Almgren-Chriss model in discrete time : trading curves that invoke only elementary optimization tools.

  • The continuous time Almgren-Chriss model: quadratic execution costs and linear permanent marketimpact ; then the general case with more realistic execution costs (Euler Lagrange equation and Hamiltonian systems in a general framework).

  • Pricing of a block trade by indifference, value of liquidity (Hamilton-Jacobi equations and viscosity solutions).

  • S - POV - Target Close – VWAP : Implementation Shortfall (IS), orders POV, orders Target Close and orders VWAP.

  • Market impact models : permanent, temporary, transient. Dynamic arbitrages and compatible models.

  • Estimation of market impacts, of execution costs and other high-frequency statistics.

  • No-execution risk : stochastic optimization for dark pools and limit orders.


[1] A. Alfonsi, A. Fruth, and A. Schied. Constrained portfolio liquidation in a limit order book model. Banach Center Publ, 83:9-25, 2008.

[2] R. Almgren and N. Chriss. Optimal execution of portfolio transactions. Journal of Risk, 3:5-40, 2001.

[3] R.F. Almgren. Optimal execution with nonlinear impact functions and trading-enhanced risk. Applied Mathematical Finance, 10(1):1{18, 2003.

[4] D. Bertsimas and A. Lo. Optimal control of execution costs. Journal of Financial Markets, 1(1):1{50, 1998.

[5] B. Bouchard, N.M. Dang, and C.A. Lehalle. Optimal control of trading algorithms: a general impulse control approach. 2009.

[6] J. Gatheral, A. Schied, and A. Slynko. Transient linear price impact and Fredholm integral equations. In Preprint available at http://ssrn. com/abstract, volume 1531466, 2010.

[7] O. Gueant, C.A. Lehalle, and Fernandez-Tapia J. Dealing with the Inventory Risk. 2011.

[8] O. Gueant, C.A. Lehalle, and Fernandez-Tapia J. Optimal Portfolio Liquidation with Limit Orders. 2011.

[9] H. He and H. Mamaysky. Dynamic trading policies with price impact. Journal of Economic Dynamics and Control, 29(5):891{930, 2005.

[10] G. Huberman and W. Stanzl. Optimal liquidity trading. Review of Finance, 9(2):165, 2005. 2

[11] C.A. Lehalle. Rigorous strategic trading: Balanced portfolio and mean-reversion. The Journal of Trading, 4(3):40-46, 2009.

[12] J. Lorenz and R. Almgren. Mean-Variance Optimal Adaptive Execution. Preprint, 2010.

[13] Dang N.-M. Optimal trading with transient price impact - An comparison of discrete and continuous approach. Preprint, 2011.

[14] A. Obizhaeva and J. Wang. Optimal trading strategy and supply/demand dynamics, 2005.

[15] A. Schied and T. Schoneborn. Optimal portfolio liquidation for CARA investors. Preprint, TU Berlin.