Chapter 1: Background and Preliminaries

The financial industry is enormously important to state, national and world economies. This industry relies extensively on mathematical modeling of underlying instruments. Computational techniques become helpful in designing related algorithms that enable us to understand how markets function and also lend themselves to highly relevant research problems.

To understand the application of a specific computational technique to the particular financial instrument, it becomes indispensable to have a clear perception of the underlying theory. However, because of vastness of the theoretical literature, some selection becomes necessary.

We choose the highly attractive field of option pricing, a core task of computational finance and risk analysis. An option is a contract that gives the holder the right, but not an obligation, to buy or sell an asset at a pre-agreed price, the strike price, on or before the date of expiry, the maturity date. The broad field of option pricing is quite ambitious and diverse enough to call for a wide range of computational tools. Confining mostly to option pricing enables us to have a more coherent and comprehensive textbook, to a large extent, and avoids being distracted away from computational issues. An option is a standard example of a derivative, a financial instrument whose value relies on one or more assets that are usually referred to as underlying assets. Generally, it takes the form of a contract to buy or sell an asset or item like commodity, property, etc. at the strike price, on or before the expiration date. Other examples of a derivative include bonds, futures contracts, forward contracts and swaps. (For detailed discussions see, among others, Jarrow and Turnbull, 2000.) The financial market in which derivatives are traded may be designated as the derivatives market. We assume a perfect market in the sense that there are no costs of transactions, no restrictions on short sales and existence of a common borrowing and lending rate. Short selling is a business tactic that involves borrowing an asset and selling it immediately, repurchasing the asset (hopefully at a lower price) and returning it to the lender to close the process. (Chapters 11 and 15 of this monograph provide further discussions along this line.)