Financial derivative securities are securities whose payoffs are well defined functions of other securities or variables. The keyword here is "well defined," that is, there exists an exact relation that expresses the cash produced by the derivative security based on the values of other variables. Numerous examples include equity-based derivatives, such as stock options and futures; interest-rate derivatives, such as bonds, swaps, interest-rate options, caps, floors, swaptions, etc.; credit-based derivatives, such as credit swaps; commodity derivatives, e.g., corn and oil futures; and other instruments, e.g., weather- and sports-based instruments.

The primary purpose of these instruments is, of course, to provide a convenient vehicle for hedging the various risks associated with the financial system, commodity prices, weather, etc. However, these instruments have been also used for speculation, which is a risk-taking behavior, opposite of hedging. In either case, it is important to properly model the underlying processes that drive the values of these derivative instruments.

For some of these underlying processes, the geometric Brownian motion has been applied (e.g., the famous Black-Scholes formula), while for others a mean-reverting process is necessary (e.g., many interest-rate and volatility-based derivatives). Many applications require a non-negativity constraint as well (e.g., stock prices, interest rates, volatility, etc.).

The choice of the proper underlying process primarily depends on the client's needs. Some choices are easy to implement using the well-known Black-Scholes formula. More often, however, the particular applications we face are considerably more complex and impose underlying processes that require numerical solutions to include:

• multinomial trees (lattices);
• Monte-Carlo simulations;
• numerical solutions to partial differential equations.