Given the assumption of risk-neutrality, and the boundary conditions imposed by the option payoff formula, there is only one solution to the s.d.e. There is no corresponding analytical solution in the case of an American put or American call on a dividend paying stock. The price of the underlying stock follows a lognormal distribution. The underlying stock pays no dividends. There are no taxes or transaction costs.
All securities are perfectly divisible.
These mainly involve relaxation of the assumptions of the model, or adaptations to special cases. This was developed by Fischer Black to price European options on commodity futures. In the Black Model, s is the volatility of the future, f is the future's value. The remaining variables are as the BS model. However, this model gives the same price as the BS model on the underlying stock.