There are six key assumptions for the Black-Scholes model. $$ $$ European options: the Black-Scholes model is specifically designed to price European options, which can only be exercised at the option expiry. $$ $$ Underlying asset model: the model assumes that the underlying asset obeys the lognormal model. $$ $$ No dividends: the derivation assumes that no dividends of the underlying asset will be paid during the lifetime of the option (extensions of Black-Scholes do exist which incorporate dividends). $$ $$ Constant parameters: the risk-free rate and volatility is assumed to be constant. (Due to the fact that you can sum the variance for uncorrelated random variables, time-dependent volatility can be converted to a constant parameter by taking it's historical root mean square.) $$ $$ No transaction costs: the model assumes frictionless markets, meaning there are no transaction costs or taxes, allowing for continuous risk-free trading. This is necessary for the hedging condition in our derivation - that we can continuously buy or sell the underlying to maintain a delta-hedge. $$ $$ Arbitrage-free markets: markets are assumed to be efficient, with prices reflecting all available information. There is no room for arbitrage opportunities (risk-free profit). This condition enables us to impose that the hedged portfolio must grow at the risk-free rate in our derivation.