Actuarial Finance

16Introduction to the Black-Scholes-Merton model

The year 1973 shall be remembered as the year of breakthroughs in the history of options and derivatives. First of all, the largest U.S. options market, the Chicago Board Options Exchange (CBOE), was created that year. Moreover, the works from Fischer Black, Myron Scholes and Robert C. Merton, gave birth to modern financial mathematics and thus largely contributed to the shape of today’s derivatives markets.

More precisely, in the paper The Pricing of Options and Corporate Liabilities, published in 1973, Black and Scholes pioneered rational option pricing by dynamically replicating the option payoff until maturity. During the same period, Robert C. Merton published Theory of Rational Option Pricing. He extended Black and Scholes’ approach in several ways, e.g. by pricing options on dividend-paying stocks and down-and-out barrier options. See [] and [].

In his paper, Merton refers to Black and Scholes’ framework as the ”Black-Scholes’ theory of option pricing” and even nowadays the model is widely known as the Black-Scholes’ model. Their work was recognized by the Royal Swedish Academy of Sciences (Nobel Prizes) in 1997.¹ The model is still very popular today on the financial markets.

In this chapter, our main objective is to lay the foundations of the famous Black-Scholes-Merton market model and its pricing formula. We will provide a heuristic approach to this formula by linking as much as possible the derivations to the binomial ...