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February 10, 2010

10:23

Chapter

14

The Black-Scholes Model

Easily the best known model of option pricing, the Black-Scholes model is also one of the most widely used models in practice. It forms the benchmark model for pricing options on a variety of underlying assets including equities, equity indices, currencies, and futures. While not designed as a model of interest rates, a variant of the Black-Scholes model, the Black model, is nonetheless commonly used in practice to price certain interest-rate options like caps and ﬂoors. Technically, the Black-Scholes model is more complex than the binomial or other discrete models because it is set in continuous time, i.e., prices in the model may change continuously rather than only at discrete points in time. Modeling continuous-time uncertainty requires the use of much more sophisticated mathematics than we have employed so far. A ﬁrst question we should ask ourselves is: why bother? The binomial model is a ﬂexible one and is transparent and easy to work with. What do we gain from the additional fancy mathematical footwork? It turns out that there is a point. The Black-Scholes model provides something almost unique at the output level: option prices in the model can be expressed in closed-form, i.e., as particular explicit functions of the parameters. There are many advantages to having closed-forms. Most importantly, closed-forms simplify computation of option prices and option sensitivities and facilitate developing and verifying intuition about option pricing and hedging behavior. In the initial segment of this chapter, we focus on options on equities, the context in which the Black-Scholes model was ﬁrst developed. In later sections, we examine how the model may be modiﬁed to accommodate options on indices, currencies, and futures.

14.1

Introduction

The Main Assumption: Geometric Brownian Motion

The main assumption of the Black-Scholes model concerns the evolution of...