Volatility Surfaces
The smile, the skew, and the models that fit them.
An option’s price is one number, but market fear lives in a whole surface. This track covers how smiles, skews, and term structures arose after 1987, how discrete quotes become arbitrage-free surfaces, the local, stochastic, and SABR models that fit them, honest calibration practice, and the instruments — variance swaps, VIX futures, dispersion — that turn surface geometry into P&L.
Reading Vol
The Volatility Smile & Skew
Implied vol varies by strike because the market prices crashes — 1987 created the skew, and fat tails created the smile.
The Volatility Term Structure
Implied vol also varies by expiry, forming contango, backwardation, and event humps that reveal where the market prices risk in time.
Surface Models
Building the Surface
Quote streams arrive discrete; trading needs a continuous, arbitrage-free surface, so desks interpolate with no-arbitrage constraints baked in.
Local Volatility
Dupire’s formula recovers a volatility function of spot and time that reproduces the smile exactly — but with the wrong smile dynamics.
Stochastic Volatility (Heston)
Heston makes variance itself random and mean-reverting, giving realistic smile dynamics at the price of a heavy five-parameter calibration.
The SABR Model
SABR’s four parameters map to an explicit implied-vol formula, making it the default quoting model for rates and FX options.