The **Black–Scholes** /ˌblæk ˈʃoʊlz/[1] or **Black–Scholes–Merton model** is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic 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 given the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes; Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited.

The main principle behind the model is to hedge the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.

The model is widely used, although often with some adjustments, by options market participants.[2]^{: 751 } The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. The insights of the model, as exemplified by the Black–Scholes formula, are frequently used by market participants, as distinguished from the actual prices. These insights include no-arbitrage bounds and risk-neutral pricing (thanks to continuous revision). Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible.

The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models, e.g. for OTC derivatives.

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