Mathematical finance

Mathematical finance, also known as quantitative finance, is a field of mathematics that deals with investing and the financial markets. The financial markets encompass the marketplace where people trade equities, bonds, and currencies, collectively known as securities. The New York Stock Exchange (NYSE) and NASDAQ Stock Market are examples of financial markets. Trillions of dollars in securities are traded each day on the financial markets. The main goal of investing is to make money. Mathematical finance helps investors, professional traders, and financial analysts focus on numbers instead of other intangibles. They use mathematical methods based on probability and statistics to quantify financial decisions to maximize profits while minimizing risk.

Background

Mathematical finance is a highly developed, technical, and abstract branch of math in the financial world that helps people make decisions in uncertain situations. The math part of the field seeks to use theorems, probabilities, and practical numerical output to explain "random" practices in financial risk and decision-making. Mathematical finance uses formulas to account for the probability of various financial outcomes. Most people well-versed in mathematical finance are also educated in other fields, such as engineering or other math disciplines. Unlike computational finance, mathematical finance does not revolve around algorithms and algorithmic decision-making.

Decision-making has interested mathematicians for centuries. In the sixteenth century, Italian mathematician and gambling enthusiast Girolamo Cardano explored gambling and introduced the concept of probability in his work Liber de Ludo Aleae (The Book on Games of Chance). Later mathematicians, such as Blaise Pascal, Pierre de Fermat, and Daniel Bernoulli, expanded upon the development of the probability theory, mostly driven by their gambling interests.

Bernoulli's uncle, Jacob Bernoulli, collected theories on probability in his eighteenth-century work Ars Conjectandi (1713; The Art of Conjecturing). In it, he introduced the law of large numbers, which proposed that the average of the results from repeating the same experiment is close to the expected value the more times the experiment is performed. For example, if a person rolls a die repeatedly, the average of the values (sample mean) is typically close to 3.5. This precision increases the more times the die is rolled.

Mathematicians continued to explore probability theory in the centuries that followed. It was not fully developed, however, until the twentieth century by Russian mathematician Andrey Kolmogorov. He postulated that probability was a measure of a collection of events not based on the frequency of events. Probability and measure theory applications were applied to modern financial theory and eventually the branch of mathematical finance.

American economist Harry Markowitz furthered the interest in mathematical finance with his paper "Portfolio Selection," which was published in the Journal of Finance in 1952. He used math theorems to quantify diversification in investing and applied mathematical models to investing. Fellow American economist Robert Merton researched math theories and applications for pricing derivatives. In 1997, he won the Nobel Prize in economics for his work on the Black-Scholes option pricing formula. The work of these two men and the others before them helped lay the groundwork for the quantitative approach to investing.

Overview

In finance, investors rely on two approaches when deciding where to invest money: quantitative and qualitative methods. Some investors use a mix of quantitative and qualitative approaches to increase returns and mitigate risk. The quantitative method relies on the approaches of mathematical finance for investment decisions. Quantitative investment analysts only care about numbers; they do not involve themselves in the qualitative aspects of the companies in which they invest money. Quantitative strategies are the focus of high-frequency trading (HFT), which uses math to determine investments. On the other hand, instead of basing their decisions on just numbers, qualitative investment analysts spend time researching companies, their products, sales prospects, employees, and more to help them gain knowledge about a company. They feel this knowledge gives them a competitive edge in investment decisions.

Advances in computer technology furthered the popularity of the quantitative approach in the late twentieth and early twenty-first centuries. Computers are used to calculate complex algorithms and large volumes of data quickly. Investors use this data to see which securities establish patterns, make models of these patterns, and use these models to predict the probability of the securities making or losing money. They can use this data to set up automatic buys and sells of particular securities based on these patterns. For example, a stock can be set up to automatically sell or buy when it hits or drops to a certain price. Since computers do much of the work in the quantitative approach, these types of firms generally do not need to employ large teams of analysts and portfolio managers; they also do not need to send employees to inspect and research companies to assess potential investments.

The quantitative approach is not only used to earn money but also to mitigate risk. Since investors base their decisions on numbers, quantitative analysts typically choose investments that carry the least amount of mathematical risk. They generally keep their emotions out of the trading process and rely solely on numbers and patterns. However, qualitative analysts, who are usually well-versed in factors other than just numbers, may recommend a more risky investment. This is because they have invested research into a company that may have allowed emotions about a particular security guide them in the investment process.

While the quantitative method relies on numbers, it is not an exact science. No guarantees exist that investments chosen in this way will perform well and deliver gains. Quantitative analysts rely on data and patterns, and sometimes this method is not always foolproof when it comes to investments. For example, the Great Recession of 2007 to 2009 greatly affected the markets and changed once-consistent patterns, leading to the loss of financial investments. Data also does not tell investors the whole story of a particular company and its securities. Qualitative analysts who have spent much time researching certain companies might be able to spot scandals or issues within a company that could potentially lead to great loss.

Interest has grown in the field of mathematical finance into the twenty-first century. Many universities offer degree programs in the discipline to prepare students for careers that balance both math and finances in a variety of settings, such as banks, investment firms, and stock exchanges. Careers in mathematical finance include risk managers, quantitative portfolio managers, quantitative financial analysts, and traders.

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