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Black-Scholes model

The Black-Scholes model is a pricing model for European-style options developed in 1973 by Fisher Black and Myron Scholes, and independently by Robert Merton. It set the foundation of modern finance, as it allowed the development of the derivatives market and still constitutes the root of most derivative pricing models. Following its success, in 1997 Scholes and Merton were awarded winners of the Nobel Prize in economics (Black died two years earlier).

The model is based on the Black-Scholes equation, a partial differential equation that describes the price dynamics of a European option (call or put) as a function of underlying’s price, market risk-free rate, time to expiry of the contract and underlying’s volatility. To have an explicit formula for the price of the option, the Black-Scholes formula is obtained by solving the Black-Scholes equation isolating the option price.

The Black-Scholes formula relies on the existence of a replicating portfolio for the option. An option can be replicated by taking a position in the underlying asset and one in the risk-free bond. Those two positions can be set in such way that the option is perfectly hedged, and create a portfolio whose value is not dependent on the price of the underlying. The existence of this hedge confirms there is only one price for the option. Perfect replication ensures there are no arbitrage opportunities.

The original formulation of the Black-Scholes model has a number of assumptions that limit its ability to match market prices with precision. It assumes no dividends, no transaction costs or frictions, perfectly liquid market, normal distribution of the underlying asset, constant risk-free rate and constant volatility. The latter, which is refuted by empirical evidence, has attracted countless studies for over three decades which led to solutions such as the use of stochastic volatility or local volatility.

Moreover, it assumes the option cannot expire before maturity, excluding American-style options from its domain.

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