What the Black-Scholes assumptions are — constant volatility, no dividends, log-normal prices, no transaction costs

What the Black-Scholes formula prices — European call and put options

Black-Scholes prices European call and put options using volatility, interest rates, and time to expiration

How does the Black-Scholes formula for pricing European call and put options incorporate the concepts of stochastic volatility and risk-neutral valuation? support: The Black-Scholes formula incorporates stochastic volatility by assuming that the volatility of the underlying

is a random process, and risk-neutral valuation through the use of a risk-free interest rate and a discount factor

What volatility smile shows — implied volatility varies with strike price, contradicting Black-Scholes

Volatility smile indicates implied volatility's non-linearity with respect to strike prices

What the three forms of market efficiency are — weak, semi-strong, strong

Weak: Prices reflect all publicly available information; Semi-strong: Prices reflect all public and private information; Strong: Prices reflect all information, including private

What implied volatility tells you — the market's expectation of future price movement

Implied volatility indicates the market's anticipated future price fluctuation of an asset

What the efficient market hypothesis claims — prices reflect all available information

Efficient Market Hypothesis: Prices incorporate all publicly available information