Welcome to MATH 565
Fall 2026
Monte Carlo and Quasi-Monte Carlo Methods
This is the course website for MATH 565 — Fall 2026 at Illinois Institute of Technology.
Course Description
This course teaches Monte Carlo simulation techniques, focusing on applications in financial risk management, uncertainty quantification, and Bayesian inference. These sampling methods are used to compute expected values, quantiles, and densities. Advanced techniques, including quasi-Monte Carlo and Markov chain Monte Carlo methods, are covered. Students will gain experience using popular programming languages for Monte Carlo computations.
Instructor: Fred J. Hickernell
- Office: Galvin Tower 18C3-1
- Office hours: By appointment (book here)
- Phone: 312-567-8983
- Email: hickernell@illinoistech.edu
- Website LinkedIn Google Scholar Mathematics Genealogy Profile
Brief bio: Fred J. Hickernell is professor of applied mathematics. His research focuses on improving the efficiency of computer simulations and developing justifiable stopping criteria. He has spent most of his career at Hong Kong Baptist University and Illinois Institute of Technology, often in academic administrative roles. He has served on the editorial boards of the Journal of Complexity, Mathematics of Computation, and the SIAM Journal on Numerical Analysis. He is a Fellow of the Institute of Mathematical Statistics and the recipient of the 2016 Joseph F. Traub Prize for Achievement in Information-Based Complexity.
Hickernell speaks Cantonese and enjoys Chinese food. He is married with adult children. He enjoys discussing matters of substance. His most important identity is a disciple of Jesus.
Math Tutoring Center: Fall 2026 tutoring is available in RE 129 and online. Enrolled students can find the current tutoring schedule and Zoom link on the course Canvas website.
Recommended resources
Prerequisites/Requirements
- Calculus
- Matrix algebra
- A calculus-based probability course, such as MATH 474 or MATH 475; you should understand
- Discrete and continuous random variables
- Probability mass and density functions, cumulative distribution functions
- Mean, median, standard deviation, quantile, covariance, (in)dependence
- Population versus sample quantities
- Central Limit Theorem
- Facility in numerical programming, meaning
- Programming in Python, or some other language such as MATLAB, or R
- Using an integrated development environment (IDE), such as VS Code
- You are highly encouraged to become familiar with GitHub
- Facility with LaTeX or some other technical document preparation system; Markdown is helpful
Objectives
By the end of this course, students will be able to:
- Understand the basics of Monte Carlo and Quasi-Monte Carlo Methods.
- Understand the basics of Markov chain Monte Carlo (MCMC).
- Understand how these methods are used for computations.
- Assess the performance of Monte Carlo methods and improve their effectiveness.
- Understand basic implementation issues in performing Monte Carlo calculations.
Where to Find It
| This course website | Canvas website |
|---|---|
| Syllabus | Grades |
| Schedule and Lecture Notes | Announcements |
| Notebooks | Discussions |
| Assignments | Assignments and submissions |
| Tests and Exams | |
| Project | |
| Course Policies | |
| Course Repository |
Course Outline
Hours are approximate 50-minute classroom hours; each 75-minute class meeting counts as 1.5 hours. The outline allocates about 40 hours, including projected coverage for the remaining units.
Introduction — 5 hours
- Monte Carlo simulation for uncertainty
- Probability review
- Sampling schemes, estimators, and estimates
- Bias, variance, and mean squared error
- Central limit theorem and confidence intervals
- Conditional Monte Carlo
Generating Samples — 5 hours
- Pseudo-random numbers and quantile transforms
- Discrete, continuous, and mixture distributions
- Multivariate normal sampling: Cholesky and PCA
- Brownian motion, asset paths, and option payoffs
- Low discrepancy sampling
- Transport maps and acceptance–rejection
Markov Chain Monte Carlo and Discrepancy — 8 hours
Markov Chain Monte Carlo slides
- Markov chains and Metropolis–Hastings
- Tuning, diagnostics, and parallel tempering
- Kernel discrepancy and sample quality
- Discrepancy and integration error
- Likelihood and Bayesian inference
- Queueing and event-driven simulation
Improving Efficiency — 16 hours
- Transformations and importance sampling
- Control variates
- Conditional Monte Carlo and density estimation
- Antithetic sampling
- Randomized low discrepancy sampling
- Latin hypercube sampling (optional)
- Asian and lookback option comparisons
- Error assessment, stopping criteria, and computational cost
Selected Topics — 6 hours
- Parallel computing
- Gradient and stochastic-gradient descent
- Two-level and multilevel Monte Carlo
Assessment
Assignments: 15%
Project: 25%
Midterm Tests: 20%
Final Exam (Take-Home (35%) + In-Class (65%)): 40%
- In-Class on the date assigned by the registrar
Extra Credit: From August 24 through December 6, you may earn extra credit by being the first student to report a previously unreported error in
- my lecture slides, Jupyter notebooks, or assignments at least 24 hours after the material was presented or posted; or
- the textbook.
Report each error by creating an issue in the course repository. Each confirmed error earns 0.5 or 1 percentage point, depending on the severity of the error, up to 10 percentage points total.
Extra credit is added to your final weighted course total. Canvas may not display the weighted total correctly until grades have been recorded in every assessment category.