In the past two decades there has been an explosion in the use of derivative securities by investors, corporations, mutual funds, and financial institutions. Exchange traded derivatives have experienced unprecedented growth in volume while "exotic" securities (i.e., securities with nonstandard payoff patterns) have become more common in the over-the-counter market. Using the most widely accepted financial models, there are many types of securities which cannot be priced in closed-form. This void has created a great need for efficient numerical procedures for security pricing.
Closed-form prices are available in a few special cases. One example is a European option (i.e., an option which can only be exercised at the maturity date of the contract) written on a single underlying asset. The European option valuation formula was derived on the seminal papers of Black & Scholes (1973) and Merton (1973). In the case of American options (i.e., options which can be exercised at any time at or before the maturity date) analytical expressions for the price have been derived, but there are no easily computable, explicit formulas currently available. Researchers and practitioners must then resort to numerical approximation techniques to compute the prices of these instruments. Further complications occur when the pay-off of the derivative security depends on multiple assets or multiple sources of uncertainty. Analytical solutions are often not available for options with path-dependent payoffs and other exotic options.
In this chapter we provide a survey of recent numerical methods for pricing derivative securities. Section 2 focuses on standard American options on a single underlying asset. Section 3 briefly treats barrier and lookback options. Options on multiple assets are covered in Section 4. New computational results are also presented.