The term Black–Scholes refers to three closely related concepts:
The Black–Scholes model is a mathematical model of the market for an equity, in which the equity's price is a stochastic process.
The Black–Scholes PDE is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity.
The Black–Scholes formula is the result obtained by solving the Black-Scholes PDE for a European call option.
Fischer Black and Myron Scholes first articulated the Black-Scholes formula in their 1973 paper, "The Pricing of Options and Corporate Liabilities." The foundation for their research relied on work developed by scholars such as Jack L. Treynor, Paul Samuelson, A. James Boness, Sheen T. Kassouf, and Edward O. Thorp. The fundamental insight of Black-Scholes is that the option is implicitly priced if the stock is traded.
Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model and coined the term "Black-Scholes" options pricing model.
Merton and Scholes received the 1997 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for this and related work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy.
Black–Scholes in practice
The normality assumption of the Black–Scholes model does not capture extreme movements such as stock market crashes.The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely used as a useful approximation, but proper use requires understanding its limitations – blindly following the model exposes the user to unexpected risk.
Among the most significant limitations are:
the underestimation of extreme moves, yielding tail risk, which can be hedged with out-of-the-money options;
the assumption of instant, cost-less trading, yielding liquidity risk, which is difficult to hedge;