Textbook(s): There is no unique textbook justifying the words "The Textbook"! I will read rather from my own book [10] in progress as developed as the course advances. Check also better these ones:
Course
Description: This one semester course is a basic
introduction to the analytical theory, numerical methods and applications of
STOCHASTIC CALCULUS APPLIED TO MATHEMATICAL FINANCE. The emphasis is put to
end up in a capacity to understand and to carry out real computations on
computers, and to the study of theoretical backbone as far as the level of
participants makes it possible.
Thus we will hopefully find a student-friendly adapted way of teaching.
The course is directed to graduate students, hence to mathematically
more advanced participants, however newcomers which bring their
unbounded willingness to learn new mathematical techniques are welcomed as well.
We are aiming at developing an almost completely typed manuscript of text [10]
which may be handed to the participants at the end of this course.
MATHEMATICAL FINANCE is a very young and very large field, and we
will certainly not be able to cover all of the important techniques in a
one-semester course, so I intend to let the interests and needs of the
registered students guide the choice of mathematical strength
in specific topics to be studied. A preliminary list includes
a sketch of possible application fields and motivation, properties
of Brownian motion, martingale theory, stochastic integration by Ito-Riemann
sums, Ito's and Dynkin's formula, Burkholder-Davis-Gundy type inequalities,
variation-of-constants inequalities (extensions of Bellman-Gronwall Lemma)
as the key mathematical tools. Then we are able to treat classical
problems of DYNAMIC ASSET AND OPTION PRICING, the theory of
STOCHASTIC INTEREST RATES modelling (SDEs of Vasicek, Cox-Ingersoll-Ross,
Dothan et al), arbitrage free markets and equivalent martingale measures,
selffinancing strategies, the BINOMIAL TREE MODEL of Cox-Rubinstein, the
BLACK-SCHOLES-MERTON MODEL, AMERICAN and EUROPEAN OPTIONS,
PATH-DEPENDENT OPTIONS among many other facts up to simulation issues
of random variables and stochastic processes, numerical techniques
to solve the Ito-Wentzell pricing PDEs by Crank-Nicolson method,
Monte Carlo techniques to estimate functionals of price processes as
seen in DERIVATIVE AND SECURITIES PRICING and practical implementation issues,
and examples, examples, examples, ...
(I am afraid of skipping very important issues of practical simulation tools,
statistical aspects like parametric and nonparametric estimation, time series,
stochastic flows, stochastic stability, stochastic attractors, "stochastic chaos",
and stochastic partial differential equations (SPDEs) due to given time-frame.
However, I will trouble myself to cover as much as possible and in a way
such that the theory and practice of mathematical finance, which is needed
to implement algorithms involving the simulation of continuous time
Stochastic Processes as SDEs on computers.)
Course History: To our knowledge, the course contents have not been taught at SIU before. Thus, it will be somewhat experimental. There are plenty of books related to mathematical finance available in the public market, in fact too much! Thus, stay with me and learn about recent developments.
Prerequisites
and Development of Contents:
This course should be accessible to any student with a strong grasp
of single variable calculus and some prelude in probability.
It is very desirable to have the preknowledge of foundations of
probability theory as taught here at the university (e.g. Math 581
or equivalent, but don't worry too much on that, I will be able to
explain those contents in office hours or so, the main thing is you
are willing to learn with me).
The content of this course itself should very nearly coincide with
my new book [10] in progress, although from time to time I will have
to give some
more details from other standard references on Stochastic Calculus.
It is really expected that the students have had some previous
exposure to aspects of probability, in particular the concept of
random variables, related probabilities and moments, but I guess
that you haven't seen the concepts developed in the way that they
are here. Thus, for your pleasure and convenience, I will give
a concise and relatively selfcontained summary whatever is needed to
understand the related course material, including convergence of random
variables and martingale theory. The plan is for us to cover most of
the rest of the book, as time permits.
For those who want to have more practice, they can study simulations
of stochastic differential equations as met in finance theory as an
project by end
of this course (No need to worry at this stage, I will explain what
to do by means of my books I have already written!). Since it will
be the very first time that I am teaching that topic at this level
here in Minnesota everything will be somewhat experimental, and I
hope the audience
will forgive me that I am not perfect. However, be sure that I
will do my very best to please you and your expectations. My
advantage is that I have already collected experiences in simulations
of Stochastic Processes over the last 17 years, which might not be
the special experience of any other faculty here at the University.
Readings,
Problem Sets, Exams:
Readings and problem sets will be from texts I hand out in the
lecture and from my manuscript, and
it will be assigned in class and additionally published at my
homepage. Exams will cover all material covered in the lectures and/or
the readings. I hope to have put a fair effort into his
presentation from very practically oriented point of view,
and my approach is probably quite different from what you've seen
before (hopefully not). I therefore encourage you to read my
progressing book and related material,
and even better read extra literature for those they want to have
A+ grade at the very end with me.
Exam Dates:
My Grading Strategies - Grade Distribution: Grading relative to the best attained score, no curve fitting
Course Syllabus (Last Update: 06/09/03) - This is in postscript format, you will need ghostview to read it!
Current Course Outline
(Tentative Schedule Updated Each Week) (Last Update: 06/11/03):
Read this
Without Tears:
WHAT IS EXPECTED OF YOU
(From "Teaching at the University Level" by Stephen Zucker, Notices Amer.
Math. Soc. 43 (1996), p. 863))
33 Further
Introductory Readings To Probability and Stochastic Processes For The Ideal Student (optional, for
those they want to have profound readings I recommend to work through this
list, and make your personal preferences, but do not expect to understand
them all immediately, it took me more than 16 years, and we are still working
on it!):
Week 1 : Motivation, Basic Financial Terminologies, Verbal Description
of Financial Markets and Instruments, Basic Securities (Assets, Bonds,
Stocks), Example of Pricing of Bonds, The Yield and Interest Rates,
Derivative Securities (Forward Transactions, Options, Warrants,
Caps, etc.), Basics from Probability and Stochastic Processes
Week 2 : Basics form Stochastic Calculus,
Wiener Process and Stochastic Integration, Ito's Lemma,
Martingale Theory
Week 3 : Arbitrage Free Markets and Equivalent Martingale Measures
Week 4 : The discrete Pricing Model, Binomial Tree Method
Week 5 : Continuous Pricing Model, Black-Scholes-Merton Model
Week 6 : Theory of Stochastic Interest Rates, Path-Dependent Options
Week 7 : Numerical Techniques (Monte Carlo, Crank-Nicolson Scheme)
Week 8 : Simulation of Random Variables and Miscellanea