What problems does it solve?
Over the past two decades, new quantitative techniques have transformed the investment process and the finance industry. Today, banks and other financial institutions gain competitive advantage through technical innovation. Powerful mathematical models are used to measure risk, and value complicated transactions. Computational methods transform these theories into tools that sit at the fingertips of traders, portfolio managers, regulators, and risk managers, bringing greater efficiency and rigor to financial markets. These developments have led to a large and growing demand for talented people trained in the mathematics of finance.
New kinds of financial instruments called "derivative securities" have become major tools of financial planning for corporations, banks, mutual funds, and other large financial institutions. Called derivative securities because their value is derived from values of other commodities in the market, these include (mixtures of) currency repurchase agreements, put and call options, and futures contracts of various kinds. Making mathematical models of these financial instruments has become a rapidly growing part of applied mathematics, and the mathematical models are used to understand the value and the hedging structure of many of the derivatives. Typical models are based upon stochastic (i.e., time-dependent) partial differential equations that are usually solved by numerical techniques. See [PR] for a discussion of this relatively new field of mathematics. Several Mathematics in Finance Master Programs have been established in some of nation's most prestigious institutes, and a list can be found at the weblink below in 3).
What should one study in college?
Mathematical topics that are of particular use in the mathematics of finance are calculus, differential and finite difference equations, probability and statistics, numerical analysis, and modern algebra. Stochastic modeling courses could also be valuable as might courses in mathematics and other departments that study the diffusion, or heat, equation. Other valuable courses outside of mathematics include courses in finance, mathematical economics, and perhaps other social science courses that use game theory to model human behavior.