Black-Scholes Assumptions

What are the assumptions behind the Black-Scholes option pricing formula?

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#The dynamics

The stock price follows geometric Brownian motion,

dS = \mu S\,dt + \sigma S\,dW, \tag{1}

with constant drift $\mu$ and constant volatility $\sigma$ , so prices are lognormal and log returns are normal.

At the center sits geometric Brownian motion, with constant drift and constant volatility, so prices are lognormal. The fluctuations grow with the price because the volatility multiplies S, and the rest of the assumptions make the hedge exact.

#The market

The remaining assumptions idealize the market around that process.

A constant, known risk-free rate $r$ , with unlimited borrowing and lending at it.
No dividends paid over the life of the option.
No transaction costs or taxes, and perfectly divisible assets.
Continuous trading, with short selling allowed.
No arbitrage opportunities.
European exercise, only at maturity.

#What it buys

Together these make the delta hedge exact, so the option price solves the Black-Scholes partial differential equation and has a closed form depending on $S$ , $K$ , $r$ , $\sigma$ , and $T$ . The striking feature is what it leaves out, the drift $\mu$ , since the hedged portfolio earns the risk-free rate no matter which way the stock is expected to move.