Methods of Monte Carlo Simulation

Dr. Tim Brereton

Teaching assistant
Lisa Handl

Time and Place

Friday, 8-10 am (N24-131)

Excercise session
Monday, 1-2 pm (N24-131)


 2 hours lecture and 1 hour excercise

Credit points:4 for Bachelor students(lecture only)
6 for Master students(lecture and reading course)


Basic knowledge of probability calculus and statistics as taught, for example, in "Elementare Wahrscheinlichkeitsrechnung und Statistik".

Intended Audience

Bachelor students in "Mathematik", "Wirtschaftsmathematik" and "Mathematische Biometrie"; Master students in "Finance"

Master's students in "Mathematik", "Wirtschaftsmathematik" and "Mathematische Biometrie" can also take the course. In order to gain credit, they will be required to do some additional reading (1 - 2 hours per week) and take a different exam, which will be based on the reading material as well as the lectures.


This course gives a broad overview of Monte Carlo methods. These methods include many important tools for students interested in applied probability, finance and statistics. The course will cover the following topics:

  • Pseudo-random numbers and quasi-random numbers 
  • Generating random variables 
  • Generating stochastic processes 
  • Markov Chain Monte Carlo 
  • Variance reduction 
  • Gradient estimation
  • Stochastic optimization

Examples will be given of applications in finance, probability, statistics and the physical sciences.

Reading Course

Master's students taking the reading course will be required to be familiar with Dirk Kroese's lecture notes (with the exception of sections 4.6 - 4.9) . These notes contain several chapters and sections that will not be covered in class. The exam will cover this material. In order to understand this material, students should complete the exercises given in the notes.

The material you need to cover (with study suggestions) is given here:

Study suggestions

Here are some example exam questions for the Master's course (they are a bit harder than the ones you will see on the exam):

Example exam questions

Example answers

Requirements and Exam

In order to participate in the final exam, it is necessary to earn 50% of the points on all problem sheets. Students who want to do so are kindly asked to register for the 'Vorleistung' in the LSF-'Hochschulportal'.

Time and place

First exam:
Thursday, February 27 (10 am - noon) in H14

Second exam:
Friday, April 11 (10 am - noon) in H14

Participants will be allowed to bring one handwritten A4 sheet (both sides) containing useful results from the lecture and exercise classes. Copies and printouts will not be allowed.

Here is a practice exam. NOTE!!! These are not the same questions as those on the actual exam!!!

Practice exam

The second exam is corrected!

You can find the number of points you obtained in the SLC as an extra exercise sheet (marked as "Prüfungsleistung"). The associated marks are indicated in the following tabular (it's the same for first and second exam and for both exam types):

1,038 - 44
1,335,5 - 37,5
1,733 - 35
2,030,5 - 32,5
2,328 - 30
2,726 - 27,5
3,023,5 - 25,5
3,321 - 23
3,718,5 - 20,5
4,016 - 18
5,00 - 15,5

The post-exam review will take place on Wednesday, April 16 from 9 to 10 am in Lisa Handl's office (room 1.42 in Helmholtzstr. 18).

Problem Sheets

Problem Sheet 01     Matlab Solution 01

Problem Sheet 02     Matlab Solution 02

Problem Sheet 03     Matlab Solution 03

Problem Sheet 04     Matlab Solution 04

Problem Sheet 05     Matlab Solution 05

Problem Sheet 06     Matlab Solution 06

Problem Sheet 07     Matlab Solution 07

Problem Sheet 08     Matlab Solution 08    Ex. 2 Plot

Problem Sheet 09     Matlab Solution 09

Problem Sheet 10     Matlab Solution 10

Problem Sheet 11     Matlab Solution 11

Problem Sheet 12     Matlab Solution 12

In order to receive points for your problem sheets, a registration at SLC is required.

Lecture Notes

The first part of the lecture notes is now online. They will be updated frequently over the next week or so. Please email me if you find any mistakes.

 Lecture Notes

The lecture are loosely based on the lecture notes of Prof. Dr. Dirk P. Kroese from the University of Queensland, Brisbane, Australia.

Dirk Kroese's Lecture Notes

Information on discrete-time Markov chains can be found here:


 Asmussen, S. and P. Glynn. Stochastic Simulation. Springer, 2007.

 Fishman, G. Monte Carlo: Algorithms and Applications. Springer, 2003.

 Gamerman, D. and H. Lopes. Stochastic Simulation for Bayesian Inference, Second Edition. Chapman & Hall, 2006.

 Glasserman, P. Monte Carlo Methods in Financial Engineering. Springer, 2004.

 Kroese, D. P., T. Taimre and Z. Botev. Handbook of Monte Carlo Methods. Wiley, 2011.

 Robert, C. P. and G. Casella. Monte Carlo Statistical Methods, Second Edition. Springer, 2005.

 Ross, S. M. Simulation, Fifth Edition. Academic Press, 2012.

 Rubinstein, R. and D. P. Kroese. Simulation and the Monte Carlo Method. Wiley, 2007.



Teaching assistant


If you missed the post-exam review and still want to have a look at your exam, you can do so on Wednesday, April 23 from 11 to 11:30 am in Lisa Handl's office. Please understand that we won't offer any extra appointments besides that one.


The second exam is corrected! You can see the points you obtained in your SLC account, a tabular with the corresponding marks is on this page (section "Requirements and Exam").


We now have printed copies of the lecture notes: You can get them (for free) in Dr. Brereton's office (room 1.43, Helmholtzstr. 18).


The lecture notes have been updated again.


A practice exam is now online and so are some study suggestions for the reading course.


The second exam has been shifted to Friday, April 11 (10 am - noon) in H14. It will NOT take place on Good Friday (Karfreitag)!


The exam dates are fixed:

The first exam will take place on Thursday, February 27 (10 am - noon) in H14, the second exam on Friday, April 11 (10 am - noon) in H14.