Over the past few years derivative securities (options, futures, and forward contracts) have become essential tools for corporations and investors alike. Derivatives facilitate the transfer of financial risks. As such, they may be used to hedge risk exposures or to assume risks in the anticipation of profits.

To take a simple yet instructive example, a gold mining firm is exposed to fluctuations in the price of gold. The firm could use a forward contract to fix the price of its future sales. This would protect the firm against a fall in the price of gold, but it would also sacrifice the upside potential from a gold price increase. This could be preserved by using options instead of a forward contract.

Individual investors can also use derivatives as part of their investment strategies. This can be done through direct trading on financial exchanges. In addition, it is quite common for financial products to include some form of embedded derivative. Any insurance contract can be viewed as a *put*; option. Consequently, any investment which provides some kind of protection actually includes an option feature.

Typical examples include deposit insurance guarantees on savings accounts as well as the provision of being able to redeem a savings bond at par at any time. In Ontario, the provincial government has introduced a cap on the price of electricity. If the spot price of electricity goes above the cap price, then the government makes up the difference between the cap price and the spot price. This cap can be regarded as a form of insurance. How much is this insurance worth? Determining the fair market value of these sorts of contracts is a problem in option pricing.

## Stock Prices: A Random Walk

Despite all the efforts to find patterns in stock prices, most studies show that stock prices follow a random walk. In other words, it is futile to try to predict stock prices, gold prices or electricity prices. Rather surprisingly, it is possible to derive an equation which can be used to determine the fair market value of a contract which depends on an underlying asset which follows a random walk. This equation is a Partial Differential Equation, which was developed in 1973 by Fischer Black and Myron Scholes, and is now commonly referred to as the Black-Scholes equation. In addition to determining the value of a derivative contract, the solution of the Black-Scholes equation also contains information on how to hedge portfolios of assets in order to minimize risk.

## Hedging

The hedging strategies determined by the solution to the Black-Scholes equation are dynamic. In other words, assets must be bought and sold at frequent intervals in order to optimally reduce risk. A large financial institution typically has thousands of risky positions, each of which must be dynamically hedged. These hedges are rebalanced at least daily, and sometimes more frequently.

## The Computational Challenge

Except for very simple cases, there is no exact analytical solution to the Black-Scholes equation. Consequently, it is necessary to devise efficient numerical algorithms to solve the Black-Scholes equation. Since new financial products are developed in response to changing market conditions, new algorithms have to be constructed to determine how to price and hedge these products.

As well, financial institutions are required to carry out simulation studies to determine how their hedging strategies would behave under unusual market conditions. The results of these studies, commonly encapsulated in a single number, the Value at Risk, are reported to regulatory bodies to ensure that the financial institution is not exposed to undue risk.

Financial institutions employ teams of quantitative analysts, whose job it is to develop methods for solving the Black-Scholes equation and to test hedging strategies through simulations. A quantitative analyst employed in the financial sector must have an excellent grasp of Computational Mathematics as well as a thorough grounding in modern Finance.