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MATH 341-01, Introduction to Financial Mathematics, Fall 2010
Hamrick, Jeff
Hamrick, Jeff
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Abstract
The history of mathematical finance and quantitative
models of risky asset prices can be traced to three individuals during the latter part of the 19th
century: T.N. Thiele of Copenhagen, who more or less created Brownian motion while studying
time series in 1880; Louis Bachelier of Paris, who created a model of Brownian motion in 1900
while trying to describe the behavior of stocks trading on the Paris bourse; and A. Einstein, who
in 1905 used Brownian motion to describe the motion of small particles suspended in a liquid.
Einstein’s work was immediately influential, but Thiele and Bachelier’s work was not recognized
for decades. (Bachelier, incidentally, has become known as the father of mathematical finance,
and is the namesake of one of the main professional organizations to which I belong, the Bachelier
Finance Society.)
The appropriate measure-theoretic and probabilistic tools did not exist during Bachelier’s time. It
took three decades of concentrated effort (interrupted by two world wars and an economic depression)
by mathematicians like A.N. Kolmogorov and P. Levy to create an intellectual environment
conducive to several important achievements. These achievements include, but are not limited to:
the stochastic integral (by K. Ito and his students, H. Kunita and S. Watanabe, in postwar Japan),
a formal proof (in a certain mathematical framework) that properly-anticipated prices fluctuate
randomly (by the famous mathematical economist Paul Samuelson in 1965), a partial differential
equation whose solution describes the price of a European-style call option (by F. Black and M.
Scholes), and a solution to the portfolio optimization problem in continuous-time finance (by R.C.
Merton in 1969).
In 1979, Coss, Ross, and Rubinstein proposed the so-called binomial method for pricing options,
and they found that this method was useful for pricing American-style options (which, as we shall
learn, are different from European-style options). Work by J.M. Harrison, D.M. Kreps, and S.R.
Pliska further modernized the field, which was finally put on axiomatically firm and conceptually
unified ground by F. Delbaen and W. Schachermayer in a series of papers in the late 1980s and
early 1990s. Along the way, incredibly useful models (the Heath-Jarrow-Morton model describing
the evolution of the interest rate curve, the Coss-Ingersoll-Ross model of interest rates, fractional
Brownian motion as an arbitrage-free model under transaction costs, etc.) have been developed
and used to create a panoply of derivative securities.
Today, the field of mathematical finance is remarkably dependent on computation and has (in
some sense) broken off into a number of subfields, including financial engineering and econophysics.
The field is far too important to be of interest only to academics—interacting with practitioners
is a crucial aspect of being a financial mathematician. Though mathematical finance is a “hot,”
intellectually challenging, and lucrative field, it has not been free of controversy (e.g., the role of
financial engineers in precipitating the credit crisis of 2008-2009). Nonetheless, I hope you will
enjoy learning a little bit about the field with me this semester.
Description
This syllabus was submitted to the Office of Academic Affairs by the course instructor. Uploaded by Archives RSA Josephine Hill.