Financial mathematics is the application of
mathematical
methods to the solution of problems in
finance. (Equivalent names
sometimes used are financial engineering,
mathematical finance, and
computational finance.) It draws on tools from
applied mathematics,
computer science, statistics, and economic
theory. Investment banks,
commercial banks, hedge funds, insurance
companies, corporate
treasuries, and regulatory agencies apply the
methods of financial
mathematics to such problems as derivative
securities valuation,
portfolio structuring, risk management, and
scenario simulation.
Quantitative analysis has brought efficiency
and rigor to financial
markets and to the investment process and is
becoming increasingly
important in regulatory concerns. As the pace
of financial innovation
increases, the need for highly qualified
people with specific training
in financial mathematics intensifies.

Finance as a sub-field of economics concerns
itself with
the valuation of assets and financial
instruments as well as the
allocation of resources. Centuries of history
and experience have
produced fundamental theories about the way
economies function and the
way we value assets. Mathematics comes into
play because it allows
theoreticians to model the relationships
between variables and
represent
randomness in a manner that can lead to useful
analysis. Mathematics,
then, becomes a tool chest from which
researchers can draw to solve
problems, provide insights and make the
intractable model tractable.

Mathematical finance draws from the
disciplines of
probability theory, statistics, scientific
computing and partial
differential equations to provide models and
derive relationships
between fundamental variables such as asset
prices, market movements
and interest rates. These mathematical tools
allow us to draw
conclusions that can be otherwise difficult to
find or not immediately
obvious from intuition. Especially with the
aid of modern computational
techniques, we can store vast quantities of
data and model many
variables simultaneously, leading to the
ability to model quite large
and complicated systems. Thus the techniques
of scientific computing,
such as numerical computations, Monte Carlo
simulation and optimization
are an important part of financial
mathematics.

A large part of any science is the ability to
create
testable hypotheses based on a fundamental
understanding of the objects
of study and prove or contradict the
hypotheses through repeatable
studies. In this light, mathematics is a
language for representing
theories and provides tools for testing their
validity. For example, in
the theory of option pricing due to Black,
Scholes and Merton, a model
for the movement of stock prices is posited,
and in conjunction with
basic theory which states that a riskless
investment will receive the
risk-free rate of return, the researchers
reasoned that a value can be
assigned to an option that is independent of
the expected future value
of the stock.

This theory, for which Scholes and Merton
were awarded
the Nobel prize, is an excellent illustration
of the interaction
between math and financial theory, which
ultimately led to a surprising
insight into the nature of option prices. The
mathematical contribution
was the basic stochastic model (Geometric
Brownian motion) for stock
price movements and the partial differential
equation and its solution
providing the relationship between the
option's value and other market
variables. Their analysis also provided a
completely specified strategy
for managing option investment which permits
practical testing of the
model's consequences. This theory, which would
not have been possible
without the fundamental participation of
mathematics, today plays an
essential role in a trillion dollar industry.