Implied Volatility Smile and Surface
The smile shows how implied volatility varies with strike; the surface extends this across maturities under no-arbitrage constraints.
For each maturity, implied volatility as a function of strike forms a smile or skew, reflecting non-lognormal returns and supply–demand for options.
Across maturities, implied vols form a surface σ_imp(K, T); traders think and quote in terms of this surface rather than a single volatility number.
Arbitrage-free interpolation requires call prices to be decreasing and convex in strike and total variance σ²T to be non-decreasing in maturity.
Parametric forms such as SVI or SABR-style fits are used to build smooth, arbitrage-consistent surfaces that drive pricing, hedging, and risk for vanilla and structured products.
The smile is σ(K) at fixed maturity. The surface extends it across maturities. The diagnostics below check basic no-arbitrage shapes: calls decreasing/convex in strike, and total variance non-decreasing in maturity.
The smile reflects how implied volatility changes with strike, which is shorthand for non-lognormal returns and option supply–demand. The surface extends this across maturities.
The diagnostics are deliberately basic. They are not a full arbitrage-free construction. They act as a guardrail: if you bend the sliders until these fail, you are likely building a surface that cannot correspond to valid call prices.
Practical modelling step: instead of inventing σ(K,T), many desks parameterise total variance w=σ²T and enforce smoothness and calendar structure there.