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
BlackScholesMerton 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 BlackScholes 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.