Implied Volatility
Inverting Black–Scholes against market prices recovers the volatility the market implies, quoted per strike and expiry as the volatility surface.
Implied volatility inverts the question: given a market price, which volatility fed into Black–Scholes reproduces it? No closed form exists for the inversion, so numerical root-finders solve for it, but the idea is profound. Since vanilla prices map one-to-one to implied vol, traders quote and think in vol, a cleaner unit than dollars, and the volatility surface, implied vol plotted against strike and expiry, becomes the market's forecast dressed as a model input. Equity smiles skew downward with strike while currencies smile symmetrically. Those shapes are the seed of the volatility-surfaces track and the proof that every model here is a lens, not the truth.