Financial Derivatives
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 equitybased derivatives, such as stock options and futures; interestrate derivatives, such as bonds, swaps, interestrate options, caps, floors, swaptions, etc.; creditbased derivatives, such as credit swaps; commodity derivatives, e.g., corn and oil futures; and other instruments, e.g., weather and sportsbased 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 risktaking 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 BlackScholes formula), while for others a meanreverting process is necessary (e.g., many interestrate and volatilitybased derivatives). Many applications require a nonnegativity 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 wellknown BlackScholes formula. More often, however, the particular applications we face are considerably more complex and impose underlying processes that require numerical solutions to include:
