Throughout
these lessons we present an introduction to the main concepts and
results of Mathematical Finance, a discipline of Applied Mathematics
that has a well-defined theoretical structure worthy of explanation.

Louis
Jean-Baptiste Alphonse Bachelier: Founder of Mathematical Finance

Mathematical
Finance
is the branch of mathematics that
studies
financial markets, which are mechanisms to trade financial securities
such as for example stocks or bonds. Stock markets date back to at
least 1531 when one was started in Antwerp, Belgium. Mathematical
modeling of financial markets by using probability however only came
hundreds of years later with the publication in 1900 of

Louis
Bachelier's thesis titled

The
Theory of
Speculation.
Bachelier's work
was controversial and generated a hostile response from the Paris
Mathematical Community, including his instructor Henri Poincaré, who
considered the thesis' topic remote from the ones used to be treated by
candidates at the time. Now we realize that Bachelier basically
introduced Brownian motion five years earlier than Albert Einstein.

It was only
until
the 1950's that statistician Savage
tipped off
economist Paul Samuelson about Bachelier's work. In his 1965 article
Samuelson made further developments to Bachelier's initial theory by
modeling stock prices according to geometric Brownian motions. After
Samuelson's work, developments in Mathematical Finance are closely
related to advances in the theory of Stochastic Integration. For
instance, the celebrated Robert Merton's

Black-Scholes
formula for
pricing European options is derived by using

Itô's stochastic calculus.
Also, the results of Paul Meyer and the Strasbourg School on general
semimartingale theory proved themselves of great value to model and
understand more complicated financial markets. In the subsequent years
the topic has attracted the attention of a large number of researchers
from different areas such as Economics, Mathematics and Statistics. For
a more detailed history of the topic we refer the reader to a recent
article by Professors Jarrow and Protter from Cornell University titled
"

A short history of
stochastic and mathematical finance: The early
years".

Module 1

In order to
avoid
the use of some technical tools of Probability and
Stochastic Calculus needed to understand the general theory, we will
introduce all the concepts by using The Binomial Model,
a simple model
that reveals clearly the beauty of the subject.

Module 2

The binomial
model
or CRR model considered in module 1 allowed us to
introduce some of the main concepts in mathematical finance. Namely,
arbitrage opportunities, risk neutral measures, contingent claims,
financial derivatives, hedging strategies and complete markets.
However, this model is very simplistic and it is not a good model for
real financial markets. There are many aspects that have to be
considered in order to achieve a more realistic modeling of financial
markets. Among them are, dividend payments, enlargement of the market
by considering more risky assets, time dependent riskless bond interest
rates and rates of return of the risky asset, constraints over the set
of strategies and more general rates of return over the risky assets.
In this module we will briefly explain thorough some examples how some
of the characteristics of the market change after considering these
factors.