Numerical Methods for Finance

Math 799 - Winter 2003

Course Description

This course is an introduction to the numerical techniques used by both academics and practioners in Financial Mathematics. It is intended for graduate students in mathematics, statistics, business, economics or physcal sciences in general, who have an interest in numerical methods for financial engineering. To motivate the methods, we will review some key financial problems such as the time evolution of asset prices, hedging and pricing financial instruments and analysing real data sets, so in principle no extensive previous knowledge of finance is required. These problems will then be addressed and solved by the use of discrete models (binomial and trinomial trees), Monte Carlo methods, numerical solutions of PDE's and SDE's and maximum likelihood estimators. Although all the examples treated will be drawn from finance, the course can be taken by students solely interested in the mathematical side of the techniques above. A strong component of the course will be the implementation of the computer routines in MatLab and occasional use of Maple for symbolic calculations.


For the finance part of the course, no formal prerequisites are necessary. A general knowledge of financial terms such as provided by a cursory reading of the first eight chapters of the book ''Options, Futures and Other Derivatives'', by J. Hull, should be more than enough. As for mathematical prerequisites, students should have taken introductory level courses on partial differential equations, probabilty theory (discrete state space) and statistics. Previous contact with discrete time stochastic processes is welcome but not strictly necessary. More advanced topics like Brownian motion and stochastic differential equation will be introduced in the course in a pragmatic fashion. Concerning computer skills, we will provide tutorials and general help with MatLab, but students are expected to have some previous exposure to writing programms in a high level computer language such as C, Fortran or MatLab itself.


Here is a list of topics covered in the course. The information in square brackets refer to book sections where the material can be found at more or less the same level as we are going to be treating them. We use the following abbreviations for the book authors: B=Brandimarte, Bj=Bjork, B/R=Baxter/Rennie, D=Duffie, H=Heath, O=Oksendal, WHD= Wilmott et al, S=Shaw.

Part I - Lattice Methods

Part II - Review of Continuous Time Finance

Part III - Monte Carlo Methods

Part IV - Numerical Solutions of Parabolic PDE's


We are not going to be following the integrity of any particular book. If the students wish to purchase a book, the following is the closest to what we are going to be doing, both in spirit and content: A highly useful compilation and (most importantly) critical view of all sorts of numerical procedures encountered in Finance, emphasizing the role of Mathematica for symbolic calculation, is The Numerical Analysis developed in the course will be at the level of the following two books: For the Financial Mathematics background needed for the course, we are going to be drawing material mostly from The following are other introductory level books in Financial Maths which might also help to understand the background: These last two books are mathematically more demanding but contain material which will be essential for the students seeking for a deeper view on subject:


The final mark will be calculated based on the solutions for the assignment questions posted during the course (40%) as well as a final project. The assignments should be solved individually, whereas the final project can be done in groups of up three students. It will consist of the implementation of the methods studied in the course to the solution of an specific practical problem of financial engineering. Marks will be given to the fully operating computer code (20%) solving the problem, a short written essay explaining the techniques used, accuracy, estimate of computing time, etc (20%) and an industrial style oral presentation conveying it to the rest of the class (20%). A timetable for returning the projects at the end of the course, as well as for the presentations, will be decided in class. I will be suggesting project ideas along the lectures, and students are encouraged to contact me to discuss their preferences.



These were the final projects presented by students taking the course for credit. Click on the titles to see the pdf form of the submitted written versions when available.

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