Black-Scholes Model

The model's assumptions are known to be false — constant volatility, no jumps, lognormal returns, continuous trading, no transaction costs — and its practical importance survives that because it provides a common language for quoting prices as volatilities. The volatility smile, in which out-of-the-money strikes imply higher volatility than at-the-money ones, is direct empirical evidence that the model's distributional assumption is wrong. In India, single-stock options are American-style and can be exercised early, so the model is an approximation for them. It is used in reverse far more often than forwards, to extract implied volatility from an observed price.