Mathematics 16:643:627 High-Frequency Finance and Stochastic Control


The course is normally offered during the Fall semester.
  • Class meeting dates: Please visit the University's academic calendar.
  • Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
  • Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.

Course Abstract

This course is an introduction to the mathematical models useful in understanding and developing automated trading systems (often known as high frequency trading).  The fundamental theories and models of market microstructure such as the Glosten-Milgrom model, Roll model and Kyle models are covered.   These are then related to practical concerns that arise in the implementation of automated trading strategies such as adverse selection models, detection of informed trading, and the tradeoff between passive and aggressive trading.  The nature of high frequency data in various markets is discussed, and mathematical and statistical techniques commonly used in modeling such data (such as ARIMA models, logit regression, Kalman filter and cointegration) are covered.  Notions used in trading large orders such as implementation shortfall, slippage and VWAP are covered as well, alongside optimal trading strategies and trade scheduling algorithms.  The emphasis is on practical mathematical and statistical work, with many homework assignments which involve working with actual financial data.

Additional topics include an introduction to stochastic control and estimation with emphasis on applications to problems from mathematical finance. Kalman filtering and linear quadratic control, the classical PDE approach to dynamic programming and the Hamilton-Jacobi-Bellman equation, with applications to Merton portfolio allocation, investment-consumption problems, and the martingale approach with applications to portfolio optimization.


Math 16:643:621 (Mathematical Finance I), Math 16:643:573 (Numerical Analysis I), ECE 16:332:503 (Programming Methodology (C++) for Finance), Stat 16:960:563 (Regression Analysis), or equivalent courses or background.


Co-requisites: Econ 16:220:508 (pdf) (Econometrics II) or Stat 16:960:565 (Applied Time Series Analysis), or equivalent courses or background.

Primary Textbooks

  1. Hasbrouck, Empirical Market Microstructure (primary text),
  2. Kissell and Glantz, Optimal trading strategies (supplementary text).


All course content – lecture notes, homework assignments and solutions, exam solutions, supplementary articles, and computer programs – are posted on Sakai and available to registered students.


The recommended grading policy for elective courses is as follows: class attendance 5%, homework 15%, two midterm exams at 20% each, and final project 40%. Individual instructors may select an alternatve grading policy after consultation with the Mathematical Finance Program Director.

Class Policies

Please see the MSMF common class policies.

Library Reserves

All textbooks referenced on this page should be on reserve in the Hill Center Mathematical Sciences Library (1st floor). Please contact the instructor if reserve copies are insufficient or unavailable.

Weekly Lecturing Agenda and Readings

Please consult Sakai for the most recent lecture agenda and readings.

Introduction. What is high frequency? Who are the players? Market size? Recent press coverage. Structure of various markets. US equity market, European equity market, FX market, futures markets. Compare and contrast. Structure of a typical high frequency trading system. Basic adverse selection model.  
Financial models 1 (Hasbrouck). Intro to trade: what the field is about. Walrasian auction versus price discovery. Other mechanisms. Roll model for security prices. Sequential trade models. Kyle model.  
Stochastic control theory and Hamilton-Jacobi-Bellman equation.  
Financial models 2. Limit order fulfillment (survival analysis, Lo/MacKinlay/Zhang). Probability of informed trading. Other topics in microstructure.  
Alpha modeling techniques. Returns versus prices. Linear regression. Panel regression/cross sectional regression.  
Cointegration, unit root tests, spreads.  
Tracking error and minimization. Index arbitrage.
Final Project Assignment
Midterm Exam
8 Stochastic control and Kalman filtering.  
9 Algorithmic trading (Kissell and Glantz). TWAP/VWAP, and their limitations. Implementation shortfall.  
10 Stochastic control applied to market making and algorithmic trading.  
11 Working with real world data. Level 1 data, trade data. Lee-Ready algorithm. MBO data types, ITCH.  
12 Some empirical studies. Price impact research, for example, Heston on intraday price patterns, Moellemi on latency.  
13 Managing a high frequency system. Research methodologies: simulation versus experimentation
How to simulate? Numbers to look at.
14 More empirical studies.