AMAT 601.21

• 2011-10-19 Wed 10:21 I finally figured out why my antithetic error estimate was not showing any improvement. While the estimate itself was being computed perfectly correctly, the antithetic samples need to be averaged before computing the sample std. (duhh!). Here is a working version of the code.
• 2011-10-15 Sat 21:21 The second assignment is now available.
• 2011-10-15 Sat 21:21 Here too is the code for fixing correlation matrices I promised some time ago.
• 2011-10-13 Thu 13:37 I have put up some suggestions for possible project topics below.
• 2011-09-19 Mon 10:54 The first assignment is now available.
• 2011-09-12 Mon 14:40 Class times have been set: lectures take place in MS 522, Monday 11:00-12:15 and Tuesday 15:30-16:45
• 2011-09-12 Mon 10:34 There is/was a scheduling meeting in my office at 11:00 on Monday 12th September.

All references to future events should be taken as provisional.

The links "p:c:s" refer to `plain', `compact' and `spread' versions of the week's lecture notes, respectively.

• Lectures take place in MS 522, Monday 11:00-12:15 and Tuesday 15:30-16:45.
• The course text is Monte Carlo Methods in Financial Engineering by Paul Glasserman. There is also a paperback version available.
• I will hold office hours in MS 586: TBA.

• Course outline

There will be three assignments through the semester, each worth 20% of the final grade. There will also be a final assignment worth 40% which will involve writing a short paper and giving a presentation during the final week of classes.

This is a list of suggestions for topics for the final project/presentation. You will be required to give a 15 minute presentation on your chosen topic. These will be scheduled during the final week of classes. You should also submit a short paper (between four and ten pages, including graphics), together with working source code for any programs you have used, and references for any material you used.

These can be joint projects (up to two people), if desired. Please consult with me before finalising your chosen topic (which need not be from this list).

• Monte Carlo computation of the greeks. Describe in more detail the Smoking Adjoints approach of Giles and Glasserman, and reproduce their LIBOR computations (more details here). What is vibrato Monte Carlo?
• Multi-level Monte Carlo (more details here).
• Low-discrepancy sequences: coding of Sobol sequences.
• Quasi-Monte Carlo methods (here's a recent review that goes into much more depth than we were able to cover in lectures).
• Methods for finding nearby positive-definite correlation matrices. Here's a (very) recent lecture on the topic by Nick Higham, and there's an older one here, and a paper by Nick here. Peter Jackel and Ricardo Rebonato have also written about it, and here's another recent article.
• Using copulas to capture co-movement between random variables. For example, you could take a look at: Bivariate option pricing with copulas.
• Monte Carlo option pricing under Levy processes (see Section 2 of this paper for an overview and elsewhere in the paper for comparisons with other methods; for specific examples see also this paper, this paper, this paper, and this one about pricing barrier options under Levy processes).
• Efficient ways of sampling from the true transition density for the Heston model.
• MCMC.
• The history of Monte Carlo methods. If you want to hear it from the horse's mouth, there are several articles related to early Monte Carlo methods from a 1987 special issue of Los Alamos Science that was dedicated to one of the original developers of the method, Stanislaw Ulam. Paul Wilmott has also spoken on the history of Monte Carlo.

• For background information on the subject of Monte Carlo methods. The Wikipedia page is not a bad place to start. Also, RiskGlossary.com has a very nice account of the history and basic concepts of the Monte Carlo method.
• If you want to hear it from the horse's mouth, there are several articles related to early Monte Carlo methods from a 1987 special issue of Los Alamos Science that was dedicated to one of the original developers of the method, Stanislaw Ulam.
• Paul Wilmott on the history of Monte Carlo.
• If you want to brush up on your understanding of the mathematics of randomness, the Virtual Laboratories in Probability and Statistics from the University of Alabama in Huntsville are a remarkable resource for learning everything you might want to know about the subject.
• Peter Forsyth's An introduction to computational finance without the agonizing pain is a relatively gentle but authoritative introduction to computational derivative pricing and risk management. There is more here than you will need for this course, but chapters 1-4 and 8 contain relevant material.
• RiskGlossary.com site also has a wealth of information on financial options and many other relevant topics.
• Other books:
  • Peter Jackel: Monte Carlo Methods in Finance is a comprehensive text by an accomplished practitioner with a wealth of useful information about implementing Monte Carlo methods.
  • Hammersley and Hanscomb's book on Monte Carlo methods (pdf version) is an old but classic text on Monte Carlo methods.
• Online course materials

Some good online course materials are:

• Oxford Professor Mike Giles works on Monte Carlo methods for finance, in a high-performance computing context, and has a course on Monte Carlo methods.
• Dr. Michael Mascagni's course from ETH Zurich, with lecture notes available.
• Marco A.G. Dias' introduction to quasi-Monte Carlo methods, with some demonstrations.
• Miscellaneous links
• Buffon's needle (mathworld site)
• Sobol sequence generator.
• Random and pseudorandom - a discussion from the BBC's In Our Time programme. They also have a lot more mathematics-related content.