SYLLABUS for : IEOR 4703. Monte Carlo Simulation

Fall 2007, Professor K. Sigman

This course serves as an introduction to Monte Carlo stochastic simulation with its main focus on finance applications. Examples include simulating various random variables and then stochastic processes (random walks, point processes, geometric Brownian motion, other diffusians, binomial lattice model, etc.) for the purpose of numerically estimating quantities of interest (option prices, probabilities, other expected values and integrals, etc.) Methods to make the simulations more efficient (variance reduction methods), and statistical output analysis (confidence intervals) will be explored too. Although the main focus is on financial applications, other examples will sometimes be provided. Computer programming in MATLAB will be used.

Prerequisites

IEOR 4701: Stochastic Models for FE
IEOR 4007: Optimization Models and Methods for FE
Computer programming skills: simulation using MATLAB will be the focus, and the TA will give you a primer in it; but feel free to use a higher level language if you so wish (C++, for eample). HMWKs will often require you to do programming. MATLAB is available on the computers in Columbia's computer labs. If you wish to buy MATLAB, buy the student edition; you will need it with the Statistics toolbox, but not the other fancier stuff. MATLAB ONLINE TUTORIALS:
http://www.eece.maine.edu/mm/matweb.html

Reference Textbooks

There are no required texts; lecture notes will be posted on the course website.

Sheldon Ross (2006). Simulation. 4th Edition, Academic Press, N.Y.
G. S. Fishman (2006). A First Course in Monte Carlo. Duxbury, CA
Peter Jackel (2002). Monte Carlo Methods in Finance. Wiley, N.J.
Paul Glasserman (2004). Monte Carlo Methods in Financial Engineering. Springer-Verlag, N.Y.

Homework and Exams

There will be weekly or sometimes biweekly homework assignments, but they are not to be turned in and do not count towads course credit; solutions will be posted.

Tentative course schedule

Review of probability, overview of Monte Carlo with basic applications in finance and other fields
Generating univariate random variables; application to simulating random walks, point processes, the binomial lattice model, more general Markov chains
Generating univariate random variables continued (acceptance rejection method, for example)
Algorithms for generating normal rvs in one and higher dimensions (multivariate normals), simulating stochastic processes such as Brownian motion, geometric Brownian motion
Other kinds of models and other kinds of simulation (long-run ``steady-state" simulation, discrete-event simulation)
Output analysis (using confidence intervals, etc.)
Variance reduction techniques (common random numbers, antithetic variates, control variates, conditional Monte Carlo, importance sampling, etc.)
Variance reduction techniques continued
Variance reduction techniques continued
Simulating stochastic differential equations, diffusions
Mention of the difficulty in pricing American options