Implied Volatility and Smile
Find the volatility that makes Black–Scholes–Merton reproduce an option's market price, then compare implied volatilities across strikes for one expiry.
Learning focus: implied volatility is the market's price of an option restated as a volatility, and because it differs by strike, the model's single constant volatility is only an approximation.
One option
The model price rises steadily with volatility, so each price inside the bounds matches exactly one volatility: where the curve crosses the market price.
The smile across strikes
Sample data, not market data: the default prices are made with Black–Scholes–Merton from an illustrative equity-index-style skew, σ(K) = 20% − 0.30 ln(K ÷ S) + 0.40 [ln(K ÷ S)]², and rounded to cents. Replace them with quotes for one expiry.
If Black–Scholes–Merton held exactly, this line would be flat. Equity-index options usually show a skew: low strikes (out-of-the-money puts) trade at higher implied volatility, reflecting demand for crash protection and the fatter left tail of returns.