Overview

Every quant who prices a European option uses Black-Scholes. Every quant who looks at the market knows Black-Scholes is wrong: the single number $\sigma$ it demands cannot simultaneously price options of different strikes and maturities. The market's verdict is the implied volatility surface — the function $\Sigma(K, T)$ that maps strike and maturity to the volatility you must plug into Black-Scholes to recover each quoted price. This surface is not flat. It has a downward slope across strikes (the skew) that steepened sharply after the 1987 crash, and a term structure that varies from short-dated panic premium to long-dated calm. Every model in this unit — local volatility, Heston, SABR, jump-diffusion — is an attempt to explain or reproduce it.

This topic is the entry point. It establishes what the surface is, what it reveals about the inadequacy of constant-volatility GBM, and what constraints any no-arbitrage surface must satisfy. The derivation sub-note pins down the mathematical result that ties the whole unit together: local variance is the risk-neutral conditional expectation of instantaneous variance, which is why local vol models can be calibrated to any surface and why that calibration tells us something real.