Volatility surface driving

Summary
Detail
See also

Summary

The driver/derived idea is reused for volatility surfaces: one liquid underlying drives an illiquid one, the derived surface a proxy kept up to date as the driver moves. Crucial distinction: the spot level of a cross is deterministic (just the product/ratio through the pivot, no correlation); but the volatility of a cross depends on how its legs co-move, so correlation enters. Driving operates on the parameters of the surface (ATM / RR / STR), not raw vols, and only composes if driver and derived share the same parameterisation and conventions.

Detail

Parameterisation matters

An FX smile at each tenor is parameterised by three pillars: ATM (level), RR (risk reversal — skew), STR (strangle/butterfly — curvature).
Relationships are defined per pillar — they can mix mechanisms (correlation for ATM; basis for RR/STR) — which only composes if every pillar is a well-defined quantity in a shared parameter space.
Interpolation is done in parameter space (interpolate ATM/RR/STR across tenor); correlations are held as discrete points (see correlation management).

Delta, ATM and premium conventions

Each pillar is anchored to a strike via a delta; the convention is part of the parameterisation: spot vs forward delta (tenor-dependent), the ATM convention (delta-neutral straddle short-dated, ATMF long-dated), and premium-adjusted delta (when premium is paid in the foreign currency).
The premium-adjusted-vs-unadjusted convention can differ between driver and derived; if not reconciled, RR/STR refer to different strikes on the two surfaces and the derivation silently produces an inconsistent cross smile.

Correlation vs basis proxying

Correlation proxying: cross vol built from two driver legs sharing a correlation currency (the pivot) plus a per-tenor correlation factor. Variance \(\sigma_{cross}^2 = \sigma_1^2 + \sigma_2^2 + 2 s \rho \sigma_1 \sigma_2\), where \(s = +1\) when the shared currency cancels as a product (EUR/JPY from the product of EUR/USD and USD/JPY) and \(s = -1\) when it cancels as a ratio (EUR/GBP from EUR/USD over GBP/USD). Drives ATM and (correlation-weighted) the skew.
Basis proxying: derived \(=\) driver \(+\) an offset (a deterministic spread, no correlation). E.g. USD/BRL basis-derived from EUR/USD plus offsets.
Common practice: correlation for ATM, basis for RR/STR. Basis options: a Link ties the derived value to the driver's change; a Flip reverses the offset sign (mainly for RR).

ATM vol spread (basis) methods

Percentage spread (term structure of multiplicative offsets, bounded); absolute spread (additive vol points, bounded, never negative); historic (realised-vol difference over windows); fixed historic basis (ignore the driver, use own history + fixed basis).

Multiple drivers, marks, tenors, rebasing

A derived surface may have multiple drivers. Marks (Skew/Call/Put/Cuts) track parts of the driver's shape (require sharing the driver's tenor set).
Tenor propagation is deterministic: adding a tenor to a correlation driver adds it to the derived pair (other driver interpolated); ATM vol entered on a correlation-derived pair is ignored and re-implied; deslaving/slaving detaches/attaches pairs; drivers and derived need not share tenor structure; decay should only decay driver vols.
Same graph discipline as the spot CRM: chain freely but detect cycles and block both-ends edits (topology). The derived curve must be rebased as spot moves; large driver/derived spot divergence has P&L impact (a known sticky-strike limitation).