Economics 765
Spring 2024
Russell Davidson
Course Outline
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.
Textbook
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.
Academic Honesty
You'll have seen the following in all of your course outlines, because the
McGill Senate requires that it should appear in all of them. I used to think of
it as a pure formality, but a disturbing number of cases of plagiarism have
been detected in recent years, not especially at McGill, but in other North
American universities. So, please take seriously all the admonitions in the
following text.
-
Right to submit in English or French written work that is to be graded
[approved by Senate on 21 January 2009]:
In accord with McGill University's Charter of Students' Rights, students
in this course have the right to submit in English or in French any
written work that is to be graded.
This right applies to all written work that is to be graded, from
one-word answers to dissertations.
-
According to Senate regulations, instructors are not permitted to make
special arrangements for final exams. Please consult the calendar,
section 4.7.2.1, General University Information and Regulations, at
http://www.mcgill.ca.
-
Academic Integrity statement [approved by Senate on 29 January 2003]:
McGill University values academic integrity. Therefore all students must
understand the meaning and consequences of cheating, plagiarism and other
academic offences under the Code of Student Conduct and Disciplinary
Procedures.
(see
http://www.mcgill.ca/students/srr/honest/
for more information).
Et en français:
L'université McGill attache une haute importance à l'honnêteté
académique. Il incombe par conséquent à tous les étudiants de comprendre
ce que l'on entend par tricherie, plagiat et autres infractions
académiques, ainsi que les conséquences que peuvent avoir de telles
actions, selon le Code de conduite de l'étudiant et des procédures
disciplinaires (pour de plus amples renseignements, veuillez consulter le
site
http://www.mcgill.ca/students/srr/honest/)
