This page is for the course entitled **Models for Financial
Economics**. The course is offered in the spring, in the months of May
and June.

Class Notices:

- My office is Leacock 507.

The class outline is here as a PDF file. The information it contains is also given below.

The course is directed to students who wish to learn the mathematical techniques used in modern finance theory. The course will also include the basic theory of asset pricing, in particular, the pricing of derivative assets, such as options. If time permits, more elaborate models will also be discussed. The introductory material starts with measure theory, a topic not always treated in courses of mathematics for economists. Measure theory is however a necessary prerequisite for the sort of probability theory needed for financial applications. In particular, we will treat stochastic processes in continuous time, of which the simplest example is Brownian motion.

A brief list of the foundational topics we will treat is as follows.

- Measure theory and the Lebesgue integral;
- Probability based on Sigma-algebras and filtrations;
- Conditional expectations;
- Theory of martingales and arbitrage-free pricing;
- Markov processes and stopping times;
- Generalised probability density and the Radon-Nikodym theorem;
- Brownian motion and Ito's stochastic calculus;
- Stochastic differential equations;
- Kolmogorov's backward and forward equations;
- Girsanov's theorem.

On the more applied side, we will consider

- Hedging a portfolio;
- European and American options;
- Arbitrage-free pricing;
- Specific models, such as Black-Scholes, Cox-Ingersoll-Ross.

We will follow the two-volume set entitled *Stochastic Calculus for
Finance*, by Steven Shreve, in the Springer Finance series. The first
volume contains no sophisticated mathematics, but allows readers to
develop valuable intuition by a detailed treatment of the so-called
binomial model, the simplest of all models of derivative pricing. We will
make use of many of the examples in that volume. The second volume is
where most of the material for the course is to be found. It combines
mathematical developments with some quite sophisticated financial models.

I have from time to time drawn attention to misprints and errors in Volume 2 of Shreve's book. I have located Shreve's own list of errata, which is in fact a lot more comprehensive than my own observations would have led me to think. Here is the PDF file containing these errata .

Assignments:

Solutions to assignments:

Ancillary readings:

I found this review of Shreve's texts, written by Darrell Duffie of Stanford. In it, you will read how good a set of two texts these books are!

By chance I came across an article (in French) written by a Parisian probabilist on the "History of Martingales". It gives a fairly complete account of the numerous senses of the word "martingale", and explains the best modern theories as to why the word means what it does in Probability theory. The article is well written and amusing, as well as being scholarly. It can be found here as a PDF file.

The article found by following this link, by Jarrow and Protter, gives a history of the development of stochastic calculus and its application to mathematical finance. It includes the sad tale of Doeblin, and explains why a Frenchman had a German name.

In order to encourage the use of the Linux operating system, here is a link to an article by James MacKinnon, in which he gives valuable information about what software is appropriate for the various tasks econometricians wish to undertake.

To send me email, click **here** or write directly to
**Russell.Davidson@mcgill.ca**.

URL: http://russell-davidson.arts.mcgill.ca/e765