Computational Methods in Finance, Fall 1999.

A graduate course in the Mathematics in Finance program at the Courarant Institute of Mathematical Sciences of New York University.

Instructor: Professor Jonathan Goodman,

Meeting: Warren Weaver Hall , room 1302, Wednesday evenings 5 to 7 pm.

First Class: September 8, 1999

Prerequisites: Scientific Computing. This courses covers basic numerical analysis and computational methods, including finite differences, error expansions, conditioning, and basic Monte Carlo. Corequisites: Mathematics of Finance II, and Partial Differential Equations in Finance. These courses cover multifactor models, diffusions and their relation to diffusion equations, and dynamic programming (not from a computational point of view). If you have not taken the courses but think you have the background for the course, contact the instructor.

Grading: The grade will be based on weekly assignments, mostly computational.

Course description: Computational techniques for solving mathematical problems arising in finance. Numerical solution of parabolic partial differential equations, basic schemes, general theory, relation to binomial and trinomial trees, boundary conditions for American options, computation of sensitivities, application to one factor and multi factor models. Stochastic simulation and Monte Carlo. Pseudo random number generators, generating random variables with specified distributions, statistical analysis of simulation data and error bars. Numerical solution of stochastic differential equations. Application to pricing, hedging, and portfolio management. Path dependent options. Model calibration and hypothesis testing. Value at risk.

Course bboard: Please check regularly, as there may be course announcements, such as corrections to the homrwork.

Note on downloading:

There is lots of stuff to download here. Anyone may download and print a personal copy. Please do not use them for any other purpose without telling me.

Each document will be posted in three formats: the original LaTeX source, the Postscript file, and a translation of the postscript file into PDF format. Postscript is a language created by Adobe for high end printers. If you have a Postscript printer, you can simply print the Postscript format file. This is the best method. Software for viewing and printing Postscript files on non Postscript printers, mostly under the name "ghostscript" is free for the downloading. LaTeX is a typesetting system used by most mathematicians and physicists. It too is shareware. The LaTeX files are the source files for the documents. The Acrobat reader, which allows you to view and print PDF format files, is an Adobe product and can be downloaded free.


Week 1: Dynamic programming, basic probability and Markov chains, forward and backward inductive computation of probabilities, expected values, and optimal decisions under uncertainty. The notes, in LaTeX format, Postscript format, or PDF format, as of September 5. I will try to make the HTML a little better looking soon.

Week 2: Discussion of diffusion equations and their relation to diffusions. Qualitative properties of solutions of diffusion equations. Duality and the relation between the forward and backward Kolmogorov equations. Introduction to finite difference methods for diffusion equations. The notes, in LaTeX format, Postscript format, or PDF format, as of September 24. It has come to my attention that the pdf files do not work for everyone. Since they work for me, it is hard for me to diagnose the problem, but I'm working on it. Try this PDF file instead. I thank Dan Powers for a long list of corrections to the earlier version.

Homework assignments

Assignment 1, in LaTeX format, postscript format, or PDF format.

Assignment 2, in LaTeX format, postscript format, or PDF format, or another PDF format (if one doesn't work, try the other).

Assignment 3, in LaTeX format, postscript format, or PDF format.

Assignment 4, in LaTeX format, postscript format, or PDF format.

Assignment 5, in LaTeX format, postscript format, or PDF format.

Assignment 6, in LaTeX format, postscript format, or PDF format.

Make sure to check the homework comments and hints page often while doing assignments.

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