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.

## Comments

Mathematics in Finance! is an old subject, not really new. In the 1960s as a new hire at the Boeing Company with a BS in Business and minors in math and physics, I was assigned to their Industrial Engineering department. One of my assignments was to create a predictive cost model for the manhours of design labor required to configure the unique combinations of features purchased by each commercial airline for their partially unique jet airplanes. In the years before I arrived, they had no model or estimating tools for this aspect of the business. The "unpredictable" part was the time needed to design the unique wiring, tubing and ducting for each airline, which was driven by dozens of variables. After many tries, I ended up creating a model with differential equations, which I converted to a set of standard graphs for others to use in predicting those hours based on six numeric variables. It blew the minds of most of the industrial engineers and financial analysts at the time, but I made it very easy to use with the graphs. The DE model worked a lot better than the statistical correlation models I crafted, and rejected. Higher math saved the day, at least it did 40 years ago. And in the 1980s, I designed a huge cost estimating model for an all-new airplane that heavily utilized matrix algebra. Another technique not often seen within the financial community back in the 60s and 80s. Keep 'em coming!!

Mr. Rivers (not verified) | 05/23/08 | 11:48 AM

Professor Adams

Excellent and timely explanation of Financial Mathematics. What may have appeared as a relatively dry topic in May 08 is prophetic in Oct 08 especially your line "This application of advanced mathematics to finance has had a profound impact on the global economy."

Your insight into the current market problems would be welcome to many of us.

Jim Myres

Jim Myres | 10/06/08 | 20:37 PM

I am an independent mathematics student with research ineterest in financial mathematics.

I wish to currently write a research project on the subject``Mathematical Finance`` and willing to hear from all individuals to help me with the subject.

In the mean time, I would like to know how to reduce interest repayment on mortgage refinancing.

I wish to currently write a research project on the subject``Mathematical Finance`` and willing to hear from all individuals to help me with the subject.

In the mean time, I would like to know how to reduce interest repayment on mortgage refinancing.

Bismark Larbi Davis (not verified) | 12/18/08 | 08:06 AM

I studied mathematics and statistics(major) at university and currently works in a treasury department of a bank.I am very interested in financial mathematics to further my career but have no idea on how ,where i can undertake the studies.

Anonymous (not verified) | 01/30/09 | 06:12 AM

i am good at mathematics and would like to read financial mathematics in the university. l am currently reading sociology, philosophy and religion in one of the universities in Ghana and this involves a lot of reading which i am facing problems.I need advices

Asante Eric (not verified) | 02/13/09 | 12:14 PM

The above article reads as if it had been written just before the Crash of 2008, and so is rich in unintended irony. In some ways, we are all in the poo because financial firms became too clever by half with "risk management" and all that. But in other respects, the problem is that the suits kept overruling the quants, who, it seems, issued repeated warnings that their employers were in over their heads.

There are quite a few UK and USA university textbooks on elementary financial mathematics. Most are predictable and not all that good. One I respect is by Simon Benninga. A hard UK bok is by Wilmott. If you have access to hard currency, you can buy any book in English through Amazon Books.

Financial mathematics comes on 3 levels.

1. Algebra, elementary calculus, optimization, and the exponential and natural log functions. This is the time value of money, and all of finance employs these ideas. A good first year uni course in calculus will give you more than what you need.

2. Probability and statistics. Expectation, variance, covariance, linear model, normal distribution, pdf and cdf. Hypothesis tests. Employed in empirical finance and the pricing of shares.

3. Methods for pricing derivative assets. This can get hard, in the form of stochastic partial differential equations.

For more advice, jj5498@gmail.com .

There are quite a few UK and USA university textbooks on elementary financial mathematics. Most are predictable and not all that good. One I respect is by Simon Benninga. A hard UK bok is by Wilmott. If you have access to hard currency, you can buy any book in English through Amazon Books.

Financial mathematics comes on 3 levels.

1. Algebra, elementary calculus, optimization, and the exponential and natural log functions. This is the time value of money, and all of finance employs these ideas. A good first year uni course in calculus will give you more than what you need.

2. Probability and statistics. Expectation, variance, covariance, linear model, normal distribution, pdf and cdf. Hypothesis tests. Employed in empirical finance and the pricing of shares.

3. Methods for pricing derivative assets. This can get hard, in the form of stochastic partial differential equations.

For more advice, jj5498@gmail.com .

cushla (not verified) | 02/13/09 | 17:24 PM

Financial formulas are only helpful to an extent - they can't truly predict the true risk of any financial product. As we have witnessed by the credit-market meltdown, no mathematical formula can predict the human element in a financial transaction.

Advocate (not verified) | 09/11/09 | 00:06 AM

Another thing, mathematical finance is not all that mathematical. At worst, it involves stochastic differential equations. Probably the most valuable skills are probability/statistics, basic linear algebra, and numerical computer programming.

The vast part of the realm of pure mathematics, e.g., analysis and algebra, differential geometry, and

category theory, play no role in mathematical finance. Mathematical finance is a good deal less demanding than physics. Those of your out there with genuine math skills will be disappointed by mathematical finance.

Nothing compels managers to take mathematical finance seriously. Bosses can also direct quant types to come up with math rationales for practices that are very profitable in the short run, but make the firm very vulnerable to a large shock in the longer run. If a quant type balks at doing this, then he will pay a price come the annual performance review. He may find his pay and rank frozen, and he is likely to be made redundant come the next house cleaning. This will be the case no matter how intellectually sound his reasons for delivering what his masters want.

Mathematicians contributing to science and engineering are often respected. I've not heard of such mathematicians being made to feel like whores. I cannot paint as rosy a picture of financial mathematics, where short run profit can easily steamroll the truth.

The vast part of the realm of pure mathematics, e.g., analysis and algebra, differential geometry, and

category theory, play no role in mathematical finance. Mathematical finance is a good deal less demanding than physics. Those of your out there with genuine math skills will be disappointed by mathematical finance.

Nothing compels managers to take mathematical finance seriously. Bosses can also direct quant types to come up with math rationales for practices that are very profitable in the short run, but make the firm very vulnerable to a large shock in the longer run. If a quant type balks at doing this, then he will pay a price come the annual performance review. He may find his pay and rank frozen, and he is likely to be made redundant come the next house cleaning. This will be the case no matter how intellectually sound his reasons for delivering what his masters want.

Mathematicians contributing to science and engineering are often respected. I've not heard of such mathematicians being made to feel like whores. I cannot paint as rosy a picture of financial mathematics, where short run profit can easily steamroll the truth.

cushla (not verified) | 09/11/09 | 17:57 PM

Mike