Financial Mathematics:

Finding Order from Chaotic Economy

by Yong Wook Kim ‘05

Rocket Scientists. Just hearing the title, we quickly think of physicists or engineers who build space shuttles and satellites. Of course, the majority of them are playing an active role in future rocket research at institutions like NASA (National Administration of Space and Aeronautics). But what if rocket scientists were also on Wall Street, the center of international finance? What is financial mathematics, and why do we even need it?

Rocket Scientists in the Financial Market?

What are rocket scientists, whose training is in the field of physical science, doing on Wall Street, where various financial goods such as stocks, bonds, and foreign currency are traded? Their task is financial mathematics, applying advanced mathematical methods, used in rocket science research, to areas of finance—in other words, mathematically analyzing how the values of financial goods change due to various factors, such as interest rates, foreign currency exchange rates, and price level. Based on this, financial mathematicians calculate the exact current price, and research which goods to investigate in order to give the most satisfaction to individuals and firms. In laymen’s terms, financial mathematics is the use of advanced mathematics to analyze various financial goods and develop new investing methods.

Finance is the science of the management and prediction of money flow and other assets. In this sense, mathematics has had an intimate relationship to the field of finance since its early history.  Abacuses and calculators, which are early creations of applied mathematics, have smoothened financial trades and accounting in the past. However, in our current economy, it is difficult to precisely grasp the details of financial markets because capital markets have become fragmented, developing in various forms such as stocks and bonds. The difficulty also includes uncertainties such as future economy and interest rates, as well as risks that follow from these uncertainties. Therefore, it is impossible to predict and make a model of current financial markets with such simple arithmetic as addition and subtraction. For this reason, financial mathematics was created as a field of mathematics that can deal with complex finance.

In particular, financial mathematics has been recognized recently for systematic administration and preparation for financial risks from poor domestic economies and economics crises in Asia and South America.

Financial mathematics is a ‘fusion’ science, combining mathematics, economics, and business administration to explicitly explains complex financial situations. With mathematics as a cornerstone, financial mathematics uses the newest mathematical methods such as probability, statistics, differential equations, functional analysis, and game theory. These mathematical tools are often appreciated because of their ability to enhance profits in financial markets and to reduce all kinds of uncertainty and risk. In particular, financial mathematics has been recognized recently for systematic administration and preparation for financial risks from poor domestic economies and economics crises in Asia and South America. As seen from the information above, the demand for financial mathematics and indeed for financial mathematicians is increasing.

A Nobel Prize in Economics Developed from a Thermal Conduction Law

Financial mathematics has merited worldwide attention since the late 1970s. During that time period, derivative financial goods, which were a new kind of financial product, had started to develop. A derivative is a newly designed trading method that takes into account a trader’s individual situation.  It guarantees profits in a way that minimizes or avoids the loss or risk often caused by financial goods that are foundational assets, such as agricultural products, foreign currency, and stocks. Since derivatives involve several variables, it is complicated to analyze and predict them.

A popular example of a derivative is an ‘option,’ which is frequently referred to in newspapers.  An ‘option’ is trading the right to buy and sell such financial goods as stocks and currency at a fixed price for a limited period of time. In a broad sense, there are ‘call options,’ which trade the right to buy, and ‘put options,’ which trade the right to sell.

For example, assume that someday in February 2003, when some electronics company’s stock price is $40 per share, some security firm issues a call option, which allows you to buy the stocks of that electronics company at $45 per share until July 2003. Customers who already bought the call option can still buy the stock at $45 even though the price of the stock may soar to $50 a share. However, it can possibly be an incredible loss to the security corporation that issued the call option. As we can see in this case, it is very difficult to set a reasonable price at the time of issuing. If the call option is too expensive, then no one will buy it, and if it is too cheap, the firm that issues options will have to bear a huge amount of loss. Without a doubt, when options were initially created, the problem of setting a reasonable price was a huge topic of discussion for economists and financial managers.

Although Black and Scholes had researched the method of yielding a reasonable price of an option for a long time, they could not resolve the problem.

The people who solved this problem were mathematician Fischer Black and economist Myron Scholes, at Massachusetts Institute of Technology (MIT). Although Black and Scholes had researched the method of yielding a reasonable price of an option for a long time, they could not resolve the problem. However, Black was eventually reminded that the thermal conduction law of classical thermodynamics is similar to the problem of setting a reasonable price for a call option.

Since physicists already had solved the thermal conduction law in the nineteenth century, Black had an idea that the problem of setting a price of an option would also be solved very easily if they adopted the idea of the conduction law. Black’s idea turned out to be right, and with collaboration from Scholes, they completed the ‘Black-Scholes Option Pricing Models,’ which compute a reasonable price for an option.

Black-Scholes Option Pricing Models are praised as revolutionary in the field of finance, comparable to the work of Newton and Einstein in Physics. Their paper on option pricing models, published in 1973, led to their receiving a Nobel Prize in Economics in 1997. Black and Scholes brought the decision criteria of option price, which relied on experience and intuition until then, to a more objective and scientific level. In addition, it introduced a risk-reducing method for taking risks due to a new option or issuing of different derivatives. Actually, numerous financial corporations were afraid of selling options because the issuer of an option bears an incredible risk due to a possible change in the price of fundamental assets. However, if one issues options following the Black-Scholes Option Pricing Model, it is possible to remove the risk of selling options up to approximately 95% in a normal market situation.  From this mathematical innovation, financial corporations began issuing various types and prices of options.

Mathematics as a Basis, but also a Broad over Philosophy and Psychology

It is only possible to solve the thermal conduction law that Black and Scholes used in their model by dealing with complex partial differential equations. Therefore, the sensation of natural science and mathematics was blowing hard in the field of finance around the 1970s. Just then, there was a tremendous change in the world economy and chaos due to the Oil Crisis in the Middle East. In addition, as the U.S. space projects started shrinking in the late 1970s, many scientists at NASA were afraid of losing their jobs. At this time, the success of Black and Scholes provided an alternative vision to natural scientists, showing them that they could work in other fields that demanded complicated problem-solving techniques. Thus, many rocket scientists at NASA proceeded to Wall Street and began developing new financial methods using advanced mathematics.

The level of financial mathematics is still elementary.

The need for financial mathematics is becoming urgent as we go through the transformation to a global economy. There is an increasing demand for financial mathematics in the field of finance, ranging from the investment strategy of individuals to the establishment of the financial policy of a nation, equilibrium between international trades, and planning of macro-scale investments. Classical finance theories have proven to have serious limits in explaining various and complex economic situations. However, methods evolved from natural science, such as Brownian motion which explains irregular motions of gas, computer modeling, probability, and statistics have played a promising role in interpreting many financial situations filled with chaos.

Despite this multitude of achievements, the level of financial mathematics is still elementary. Although many people agree on the usefulness and need for financial mathematics, not very many people actually work and research in financial mathematics. This paradox rises from the difficulty of the subject matter; financial mathematics demands an understanding of complex mathematics, as well as of finance and economics. Even though it is presently unpopular as a course of study, financial mathematics can also be a very rewarding field to work in. A recommended path of mastering financial mathematics is learning advanced mathematical tools such as probability, partial differential equations, numerical analysis, statistics, and computer science as an undergraduate, and then studying financial mathematics extensively in graduate school.

In addition, it is highly recommended to have a broad range of related knowledge, since financial mathematics is the analysis of invisible future investment environments. Because economic situations cannot be explained by simple arithmetic, financial mathematics also requires a broad range of interests in the social and natural sciences, in such areas as history, philosophy, evolutionary biology, and psychology, which help to explain investors’ avarice and psychological panic.

All in all, financial mathematics can be considered a science that provides aviation methods for a pilot to fly safely regardless of any harsh turbulence in the future. I will be looking forward to the day when a solid aviation method built by our financial mathematicians will lead us to a prosperous economy.