An implied volatility surface is the options market's map of risk for a single underlying: one volatility number for every strike and expiry, drawn as a landscape rather than a single line. Its shape encodes how the market prices uncertainty across price and time.
Strike (x) is the option's exercise price; tenor (y) is time to expiry in years; height & colour (z) are annualised implied volatility: cool indigo is calm, warm amber is stressed. The tilt across strikes is the skew / smile; the slope across tenors is the term structure.
For each near-the-money option we invert the bid/ask mid price through a Black-Scholes-Merton model (QuantLib) to recover its implied vol, assuming flat r 3% and q 1%. Those points are fitted into a QuantLib Black-variance surface, then sampled on a fine strike × tenor grid.
Price and hedge options consistently across the whole chain, spot rich / cheap strikes and expiries, gauge how the market prices crash risk via the put skew, and read its expectation of near- versus long-term turbulence from the term structure.
Figures are model-derived from delayed CBOE quotes with flat rate and dividend assumptions, so they are illustrative, not trading-grade marks.