List the assumptions behind the Black-Scholes formula and name, for each one, a real market that breaks it and the direction in which the model price is then wrong.

List the assumptions behind the Black-Scholes formula and name, for each one, a real market that breaks it and the direction in which the model price is then wrong.

Approach: Go through the derivation and stop at each place a hedge argument is used, since every assumption enters there. Then ask what a violation does to the cost of running the hedge.

Black-Scholes assumes constant volatility, continuous paths, frictionless continuous trading, a constant known rate, a lognormal price and no dividends or borrow frictions, and every one of them fails somewhere in a way the implied volatility smile is already correcting for. Constant volatility fails everywhere, which is why the smile exists at all and why a single volatility number cannot fit two strikes. Continuous paths fail on earnings and on takeovers, so a hedged short option position loses on gaps no matter how often it rebalances, and the model underprices the wings. Frictionless trading fails through the bid ask spread, so the true cost of the hedge exceeds the model price and a market maker must charge for it. A constant rate is harmless for a one-month equity option and material for a ten-year rates option. The practical position is that the formula survives as a quoting convention: it maps a price to an implied volatility monotonically, and traders reason in that coordinate while carrying the model risk in the shape of the surface.

Follow-up: Which single assumption would you relax first for a one-year single stock option, and what would you replace it with?

Key concepts: constant volatility, continuous paths, frictionless trading, model risk, implied volatility smile.