FINANCIAL MATHEMATICS AT THE UNIVERSITY OF CHICAGO

THE MATHEMATICS IN FINANCE PROGRAM AT THE COURANT INSTITUTE

USING MATHEMATICS IN FINANCE AT THE UNIVERSITY OF TORONTO

SCIENTISTS AS MARKET ANALYSTS: FEATURE INDEX

**S**cientists on Wall Street? By now you've probably heard about ex-scientists and mathematicians making careers on the trading floors of investment banks. But the idea of ex-propeller-heads wearing red suspenders and yelling buy and sell orders into the telephone seems a little incongruous. What could a scientist offer the world of mammon? Plenty, it turns out.

Finance has always been a quantitative discipline. During the 1950s, Harry Markowitz developed a technique for maximizing the expected return on a portfolio of risky instruments by combining statistics and optimization theory. At first, the technique was mostly used by those few finance professionals who had a scientific background and access to a mainframe computer. Today, of course, these techniques are taught to every aspiring financial professional. When you sit down with your investment adviser to determine which combination of stocks and bonds should allow you to move to Florida at the age of 55, the spirit (if not the math) of Markowitz is clearly evident.

It is always the case that new quantitative techniques spawn new fields of study. Initially, though, few are trained to deal with the new concepts, and it takes a while before universities can rise to the challenge. Today there are two related fields of quantitative finance that are still in the formative stage: derivatives and risk management. Most of the scientific types who work in finance work in one of these two fields. I wouldn't be surprised if in 10 years there are degree programs in risk management. In the meantime, Wall Street has to put up with former particle physicists and number theorists.

A derivative instrument is a contract between two parties to exchange a quantity of money or other asset at some future date. The quantity to be exchanged is usually a function of the future value of some observable financial variable, or in other words is derived from some real (measurable) financial indicator. One example of a derivative is a fund which agrees to pay the holder the return on some stock index, but also agrees to preserve the investor's capital in the event of a downturn in the market.

The fund normally charges a large management fee to cover the cost of this type of insurance. How does the fund figure out what this management fee should be? Well, at first you might think that because this is insurance, you should bring in an actuary. You should price the product so that if you sell enough policies, the average payout will match the sum of the collected premiums, and you'll come out even. Problem is, there ain't no averaging in the market--it affects everyone the same way! If the market rises, the best thing in hindsight would have been for the fund to have bought the stock index and pay out its return. If the market tanks, then in hindsight, the fund should have left the investors' money in the bank. Ideally, the fund needs to come up with a strategy to protect itself no matter what the market does. That's where the math comes in.

It turns out that for well-behaved markets (markets that follow geometric Brownian motion), any derivative instrument can be hedged to a large extent by adopting an appropriate trading strategy. For the example in the previous paragraph, the fund manager should own more and more of the index as its price rises and start selling off the index if it goes down. The management fee needs to be adequate to cover all losses which will naturally occur as the market fluctuates. It is actually possible to calculate how much of the index the fund manager should own for each level of the index at each time during the tenure of the contract.

I spent some time working on a trading desk doing exactly this sort of hedging. I was always getting funny looks from other (nonderivative) traders because the computer would tell me to buy when the market was high and sell after it had gone down. Buy high, sell low! But it worked ... most of the time. Occasionally, the market would refuse to conform to the assumptions that went into the derivation of the models, and losses would result. The typical profile of a derivative trader is someone who goes in to work everyday and makes good money no matter what the market does (and looks like a genius) but then one day gets vaporized.

When a bank decides to get into the derivatives business, it needs to have risk managers to keep tabs on what the trading desks are doing. After spending 2 years on the trading floor at CIBC Wood Gundy, I decided to join their risk management team (no, it wasn't the result of getting vaporized). I have found risk management to be a very rewarding career. I work with computer models (there's a lot of programming involved) and work on improving the ability of CIBC to anticipate potential risk "hot spots."

Risk management exists for two reasons: First, every deposit-taking institution in Canada must have an adequate risk-management system as dictated by the Office of Superintendents of Financial Institutions (OSFI); second, no board member or shareholder of CIBC wants to wake up one morning to find that their firm has blown up. There is also a financial incentive to work on improving risk-management practices. Every bank is required by OSFI to put aside adequate capital to cover potential losses due to market events. If one can figure out a way to decrease the bank's risk profile without affecting its profitability, then the return on capital will increase.

If you are interested in learning about the mathematics of derivatives, I would recommend a book called *Financial Calculus* by Martin Baxter and Andrew Rennie. Another good book is John Hull's *Options, Futures and Other Derivative Securities*. A general introduction to financial topics is contained in the book *Investments* by William Sharpe. If you want to learn about what is happening in risk management, check out the magazine called *Risk*. If you want to get into these fields, you should have some programming skills (C++, Visual Basic, spreadsheets). And if you want to get into risk management, it helps to have some knowledge of databases, as there are an awful lot of data to be crunched!

*Mark Staley has a Ph.D. in physics and worked on the trading floor from 1995 to 1997 at the Canadian Imperial Bank of Commerce (CIBC). He is now a general manager at Global Analytics, Market Risk Management, CIBC.*