Path-Dependence of the At-the-Money-Forward Implied Volatility Term Structure

The paper lifts path-dependent volatility from the spot to the surface: a large fraction of the moves of the at-the-money-forward implied volatility, across maturities out to roughly two years, is explained by the past trajectory of the underlying — its returns and squared returns — and the two SSVI parameters that govern the ATMF level inherit this path-dependence. We read it as evidence that the surface is not an independent state variable but, to first order, a deterministic readout of recent price history.

A parsimonious SSVI surface w(k, θ) is fitted, and its ATMF-controlling parameters are regressed on the path features R_1 = Σ K_1(t − s) r_s and R_2 = Σ K_2(t − s) r_s² — trend and activity — echoing Guyon–Lekeufack at the level of the surface.

The contribution is the lift from one-dimensional realized or implied volatility to the whole ATMF term structure: if implied vol at each maturity is path-readable, then much of what a desk treats as a separate vol view is already contained in the return path it observes.

Strengthens the case for conditioning event-window memos on the recent path rather than on the surface as an independent input; a surface move unaccompanied by a corresponding path move is the anomaly worth flagging.

Index-level and ATMF-focused; skew, wings, and single-name surfaces are not claimed to be as path-readable, and the SSVI parsimony imposes structure that can absorb residual variation.