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.

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.

**Lectures:**

**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.

Send email to Jonathan Goodman

Back to Jonathan
Goodman's home page.

Back to the Mathematics Department home
page.

Back to the Courant Institute home page.

Back to the home page or the NYU Graduate
School of Arts and Sciences