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 5:10
to 7 pm.

**First Class: **September 6, 2000

**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.

**TA: **Dmitri Krasnov is the
teaching assistant for this class. Please feel free to contact him with
questions about the class. His office hours are Mondays from 3 to 5.
You may contact him for an appointment at another time.

**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.

**Topics**:

- Dynamic programming for decision problems involving discrete time discrete state space Markov chains. The direction of time and duality.
- Review of diffusion equations for options valuation. The forward and backward Kolmogorov equation stochastic PDE and the duality between them. Qualitative properties such as positivity, short time behavior, smoothing, and the maximum principle. The origin of absorbing and reflecting boundary conditions.
- Introduction to finite difference methods for diffusion equations in finance. The forward Euler method.
- . . . .

** Class bulletin board**
for notes on homework and other announcements, as well as questions. Everyone in the
class should check it often, and all are invited to conribute to it. You may post
questions, or post a message looking for a homework partner, or a general comment
on the class.

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.

**Homework assignments**

Assignment 1, in LaTeX format, postscript format, or PDF format. Please read the comments on homework 1 before going on to homework 2.

Assignment 2, in LaTeX format, postscript format, or PDF format.

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.

Send email to Jonathan Goodman

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