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.