T. R. (Tom) Hurd

## Professor of Mathematics

Financial Mathematics: what's it all about?

Modern financial mathematics has grown enormously in tandem with the importance of derivative securities, ever since the revolutionary work of Black--Scholes--Merton in 1973. Derivatives of the financial variety are essentially contracts whose value becomes determined by the performance of an underlying security or collection of securities. The basic examples are simple put and call options: a put option gives the purchaser the option to sell a security at a specified price at a specified time; likewise, a call option is the option to buy.

While the popular perception of derivatives as dangerous instruments used to speculate wildly on stocks, their pragmatic use in industry is the exact opposite. By purchasing the right combination of derivatives, market participants can reduce or eliminate their natural exposure to risk. For example, the canny hog farmer may choose to buy a number of put options on pork bellies which mature around the date he intends to put his produce on the market. This way he protects himself against a sharp decline in the market value of his livestock. Of course, he'll have to pay a price to offset his risk: puts can be expensive.

The problem with derivatives is that while straightforward to use, they are difficult to price. It was the brilliance of Black, Scholes and Merton which led to the development of arbitrage pricing theory for derivatives. However, the pricing of derivatives is only as good as the stochastic modeling of the underlying securities, which may be any of a multitude of different varieties: stocks, bonds, FX rates, commodities, etc.

This is where math really starts: adequate modeling of any of these problems will require at minimum graduate level mathematics, probability and statistics, plus a very strong capacity for computation. This explains why the financial industry continues to cry out for graduates in mathematics or physics: Ph.D. level preparation is a minimum requirement for many of the most attractive careers in the financial industry.

My research

• Fat tails The standard modeling of equities assumes log normal distributions of prices, whereas all statistical studies of financial time series show much different behaviour. Most notably, large fluctuations in say the daily price changes happen much more frequently than the standard models allow, a phenomenon popularly known as "fat tails". An important problem in my research is to determine manageable models with fat tails which improve on the Black-Scholes formula.

• Multivariate modeling for extreme events When real markets are in turmoil, it is extremely difficult to understand the dependences possible between all types of securities. The short answer is that no straightforward extension exists when standard multivariate normal models break down. Thus it remains an open problem to characterize multivariate distributions which do a reasonably good job under extreme conditions.

• Portfolio risk management deals with the problem of forecasting the profit/loss distribution of large financial portfolios over short medium and long term horizons. Portfolios in the banking industry typically include a large range ofproducts: equities, sovereign and corporate bonds and other fixed income securities (often in foreign currencies), mortgages and loans, commodities, plus increasing varieties of derivative securities. It follows that global portfolio risk is an extremely complicated business, which includes modeling all of the basic classes of financial instruments. Our special interest in the area of risk management is to improve multivariate methods to better estimate the correlations exhibited during extreme market conditions when the greatest losses occur.

• Electricity modeling Recent years have seen the deregulation of electricity markets around the world. In the province of Ontario, we will likely see the opening of such a market late in 2001. These markets present a whole new dimension of risk management issues: the special "use it or lose it" character of electricity invalidates much of arbitrage pricing theory. This difficulty coupled with a scarcity of reliable data, means the theory is in its infancy, and the most basic models have yet to be found. Here at McMaster, we are working on understanding the market mechanisms in order to model spot and forward prices, and the pricing of energy derivatives.

### Recent papers

last updated  2/06/04