Option Pricing: Valuation Models and Applications

1. Derivatives Markets: Introduction and Definitions Management Science has a long tradition of publishing important research in the finance area, including significant contributions to portfolio optimization; asset-liability management; utility theory and stochastic dominance; and empirical finance and derivative securities. Within the derivatives area, contributions in the journal have advanced our understanding of the pricing, hedging, and risk management of derivative securities in a wide variety of financial markets, including equity, fixed income, commodity, and credit markets. In addition to theoretical advances, several articles have focused on practical applications through the design of efficient numerical procedures for valuing and hedging derivative securities. In this paper we survey the option pricing literature over the last four decades, including many articles that have appeared in the pages of Management Science. We begin with a description of derivative securities and their properties. A derivative security is a financial asset whose payoff depends on the value of some underlying variable. The underlying variable can be a traded asset, such as a stock; an index portfolio; a futures price; a currency; or some measurable state variable, such as the temperature at some location or the volatility of an index. The payoff can involve various patterns of cash flows. Payments can be spread evenly through time, occur at specific dates, or a combination of the two. Derivatives are also referred to as contingent claims. An option is a derivative security that gives the right to buy or sell the underlying asset, at or before some maturity date T , for a prespecified price K, called the strike or exercise price. A call (put) option is a right to buy (sell). Because exercise is a right and not an obligation, the exercise payoff is S−K + ≡max S−K 0 for a call option and K − S + ≡ max K − S 0 for a put option, where S denotes the price of the underlying asset. Options can be European style, which can only be exercised at the maturity date, or American style, where exercise is at the discretion of the holder, at any time before or at the maturity date. Plain vanilla options, such as those described above, were introduced on organized option exchanges such as the Chicago Board of Options Exchange (CBOE) in 1973. Since then, innovation has led to the creation of numerous products designed to fill the needs of various types of investors. Path-dependent options, such as barrier options, Asian options, and lookbacks are examples of contractual forms that have emerged since and are now routinely traded in markets or quoted by financial institutions, or both. Even more exotic types of contracts, whose payoffs depend on multiple underlying assets or on occupation times of predetermined regions, have emerged in recent years and have drawn interest. This paper surveys past contributions in the field and provides an overview of recent trends. We first review the fundamental valuation principles for European-style (§2) and American-style (§3) derivatives. Next, we outline extensions of the basic model (§4) and survey numerical methods (§5). An assessment of the future and comments on open problems are formulated in the concluding section (§6). The appendix treats derivatives in the fixed income