Markets quote options in implied vol, but quoting conventions vary by asset class, and a surface is only readable if you know the axes. This note establishes the standard machinery for working with the surface: total variance, log-moneyness, the key quantities ATM level / ATM skew / ATM curvature, and the two competing theories of how the smile moves as spot moves. These conventions are used throughout every subsequent topic — Heston calibration, SABR parameterisation, arbitrage constraints, and exotic pricing all express their outputs in these terms.
The note also introduces the concept of forward-starting variance, which links the current surface to the forward smile that exotic options actually depend on.
Learning objectives. By the end of this note you should be able to:
- convert between the raw surface , total variance , and log-moneyness ;
- read off ATM vol, ATM skew (slope), and ATM curvature from a surface parameterisation;
- compute forward-starting variance from two term-structure quotes;
- distinguish sticky-strike from sticky-delta dynamics and state which each predicts will happen to an option's implied vol when spot moves.
In this lesson
- 2.1 The standard coordinates
- 2.2 Term structure and forward variance
- 2.3 Sticky-strike vs sticky-delta
- 2.4 Reading the surface at a glance
Before you start
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