page_title: Black–Scholes model
section_title: Black–Scholes equation
hierarchical_section_title: Black–Scholes model / Black–Scholes equation
caption_attribution_description: English: This image plots stock price simulations based on decile 1, 5, 10 monthly stock returns
context_page_description: The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with some adjustments, by options market participants.
  Based on works previously developed by market researchers and practitioners, such as Louis Bachelier, Sheen Kassouf and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in 1968 that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument.
context_section_description: As above, the Black–Scholes equation is a partial differential equation, which describes the price of the option over time.  The equation is:
  The key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset and the bank account asset (cash) in just the right way and consequently "eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the Black–Scholes formula (see the next section).
caption: Simulated geometric Brownian motions with parameters from market data
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