What Is Financial Mathematics?

Some people interested in mathematics have curiosity about how mathematics is used in finance, and perhaps even have an eye toward a career in quantitative finance. Others are already pursuing a career in finance and the capital markets, and have an interest in learning more of the underlying mathematics.
All around the world Financial Mathematics and Financial Engineering programs are appearing, filling a growing educational need, and the umbrella organization for these programs is the International Association of Financial Engineers (http://www.iafe.org/home.php).
The proliferation of financial mathematics was the subject of a Wall Street Journal article ("Wall Street Warms To Finance Degree With Focus on Math", 14 November 2006), by Ronald Alsop. It was also the cover story of the 23 January 2007 issue of Business Week. The subject matter was a main focus of a one-quarter program entitled "Quantitative Modeling in Finance and Econometrics" held in Spring 2004 at the Institute for Mathematics and Its Applications, see http://www.ima.umn.edu/complex/#spring .
The basic mathematics that underlies the subject is probability theory, with its strong connections to PDE and numerical analysis. On the finance side, the main topics of importance are the pricing of derivatives, the evaluation of risk, and the management of portfolios. In fact, in today's world, many aspects of capital markets management are becoming more quantitatively and computationally sophisticated, but it all began with derivatives.
A derivative is a financial instrument whose value is derived from some other instrument, called "the underlying." A simple example: Suppose I own a single share of stock that is selling for $1 today. Suppose I offer you a contract, called a "forward", that commits me to sell you this share for $1.03 one year from today; no money changes hands now. Suppose you have access to a bank that offers 6% effective interest annually. Finally, suppose you have a friend who, like me, owns a share of the stock, and who has no plans to sell it in the next year. Suppose your friend is willing to loan it to you for a year, but then wants a share returned at that time, along with a fee of two cents.
Then you can make guaranteed money: You sign the forward with me, borrow your friend's stock, sell it for $1, and put that $1 in the bank. A year later, you have $1.06. Honoring your forward with me only costs you $1.03, and you're left with a stock share and three cents. You give your friend back that share and with two of the three cents. You're left with a penny over which your heirs can squabble.
Here's the point: I mispriced my forward at $1.03. Its correct valuation should have been $1.04, and I gave you an opportunity to earn a penny off my mistake. Finding the correct value of the forward is dependent on knowing other prices, and so it "derives" its value from other market variables, like the price of the underlying stock. It is therefore referred to as a "derivative" (quite different from derivative in the sense of calculus!). This contrasts with the underlying stock, whose value comes from the hard work and toil of the good people running the company that issued it.
While there are difficulties that can come up in the pricing of forwards, they are not nearly as complicated as some other derivatives, and many clever tools exist to light the way in the pricing of these more sophisticated financial products.
In the example above of the mispriced forward, there was guaranteed risk-free money to be had -- assuming all parties honor their commitments. Often things are not as simple as that. Sometimes one is in a situation where one has to accept a certain amount of risk; not to put too fine a point on it, risk even appears in the assumption that individuals and companies will not default on their obligations.
Measuring the risk of individual assets held by a company is a difficult mathematical task, but one also has to be aware that risks are not additive; they sometimes cancel each other, but sometimes don't. Suppose you've invested in 100 risky ventures, each of which has a 10% chance of costing you $1,000, but a 90% chance of earning you $1,000. Your feeling of safety might be undermined if you find out that a rise in the price of oil could cause all of the ventures to go bad, and, for each one, 9% of the bad 10% is driven by oil. That is, the risks are not independent, and you're facing a 9% chance that you'll owe 100 x $1,000. Now suppose you can find 100 risky ventures with the same probabilities (10% and 90%) and same returns (lose $1,000 or earn $1,000), but which are all independent of one another. Then you should trade in your current 100 for this new 100.
The business of managing portfolios via understanding risk and return is another key topic in the area of Financial Mathematics.
The simplicity of the ideas expressed above gives way, in modern finance, to very sophisticated mathematics. For example, a whole new approach was necessary to be able to apply calculus to processes with random elements, such as stock markets and quantum physics. Kiyoshi Ito, the mathematician most responsible for the foundation of stochastic calculus, was just awarded the first ever Gauss medal, a new award that will go every four years "for outstanding mathematical contributions that have found significant applications outside mathematics."
This application of advanced mathematics to finance has had a profound impact on the global economy. Almost any issue that you pick up of a business magazine, such as "Business Week" or the "Economist," will have discussions of new financial products, such as credit derivatives and mortgage-backed securities, that depend on this mathematics. Investment banks and commercial banks now employ thousands of people with advanced degrees (PhDs in mathematics and physics, Masters in Financial Engineering and Financial Mathematics) working on these products and this represents one of the fastest growing segments of this industry. Tens of thousands of computer scientists are employed programming these calculations.
The exciting and rapidly growing new industry of hedge fund management also utilizes many of these ideas and employs many of the same type of graduates. In 1997, Robert Merton and Myron Scholes, two of the pioneers in mathematical finance, received the Nobel Prize in Economics in recognition of the major role this work has had on the world of finance.
With tighter regulation (Sarbanes-Oxley and the Basel accords) and a growing awareness of quantitative risk management, the job prospects of "quants" have been soaring. Those who have the mathematical skills to do this kind of analysis are in greater and greater demand. Moreover in finance, the data environment is the envy of professional statisticians everywhere!
Reprinted with permission of the Mathematical Association of America, 2008. All rights reserved.

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