INTRODUCTION

In Chapter 9 we examined the binomial model for the pricing of derivatives where the stock price takes on only two values in a period and trading takes place at discrete time points. In a realistic stock price evolution process, the number of trading intervals is large and the time between trades is small. As the number of trading intervals increases, the stock price is likely to vary over a larger number of values and trading takes place almost continuously. In the limit as the length of the trading period becomes infinitesimally small, the trading process becomes continuous. The analytical formulation of the majority of option theory is based on continuous-time set-up.

In this chapter we study the most well-known continuous-time model, the Black–Scholes–Merton model. This model, developed by Fischer Black and Myron Scholes, with help from Robert C. Merton, is a breakthrough in the theory of option pricing (see Black and Scholes 1973 and Merton 1973). This model describes the formulae for European call and put options on an asset. The next section of the chapter presents the Black–Scholes–Merton partial differential equation. The Black-Scholes pricing formulae for European options are discussed in Section 11.3. Section 11.4 is concerned with comparative static analysis of these formulae. Finally, a discussion on volatility is presented in Section 11.5.