Black–Scholes Model and Option Pricing

This chapter focuses on Itô's Lemma to derive the celebrated option pricing formula by Black and Scholes in the early 1970s. This formula has far-reaching consequences and plays a fundamental role in modern option pricing theory. To illustrate the Black–Scholes formula, the chapter discusses some fundamental concepts in a one period binomial model from which a risk-neutral argument is introduced. The Black–Scholes option pricing equation has initiated modern theory of finance. Its development has triggered an enormous amount of research and revolutionized the practice of finance. The equation was developed under the assumption that the price fluctuation of the underlying security can be described by a diffusion process. The logic behind the equation is conceptually identical to the binomial lattice: at each moment two available securities are combined to construct a portfolio that reproduces the local behavior of a contingent claim. Historically, the Black–Scholes theory predates the binomial lattice.