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

Andrès, Boumezoued, and Jourdain extend the path-dependent volatility (PDV) programme of Guyon and Lekeufack from the level of instantaneous or realized variance to the entire at-the-money-forward (ATMF) implied volatility term structure. Fitting the SSVI (Surface SVI) parameterization to daily SPX option data and extracting the maturity-dependent ATMF total variance parameter θ(T) for maturities T ranging from one month to two years, the authors regress each θ(T) onto the same two path features that drive the PDV model — the signed trend kernel R_{1,t} = Σ_{s ≤ t} K_1(t−s) r_s and the unsigned activity kernel R_{2,t} = Σ_{s ≤ t} K_2(t−s) r_s² — and find that the maturity-specific path regression explains a substantial fraction of the variance of θ(T) at each maturity, with the explanatory power ranging from approximately 80% at short maturities to 60–70% at two years. The regression therefore extends the two-feature sufficiency claim to the full ATMF term structure: the term structure of implied volatility is, to a first approximation, a deterministic function of the past return path via two scalar features, rather than an independent state variable characterizing the market's forward variance expectations. The Bergomi forward variance curve ξ(T) = E_t[σ²_T] provides the theoretical framing: the PDV features R_1 and R_2 correspond to projections of the Bergomi curve onto the two dominant modes of its variation, so that the path regression is implicitly extracting the two leading principal components of the curve's dynamics directly from the observable return path. The maturity-dependent coefficient functions α_j(T) are estimated independently at each T, revealing that the loading on R_2 (activity) dominates at short maturities — consistent with the near-term variance predictability of realized variance — while the loading on R_1 (trend) grows in relative importance at long maturities — consistent with the contribution of directional drift to longer-horizon expected variance. The residual term-structure covariance, estimated from the regression residuals across maturities, reveals a structured two-factor pattern suggesting that a third path feature — at an intermediate timescale between those of K_1 and K_2 — would substantially close the remaining gap. For the desk, the conclusion is that the ATMF term structure is not an autonomous information source but is largely redundant with the observable return path: a surface read conducted independently of the path-conditioning step is reading largely the same information twice, and any apparent disagreement between the surface-implied term structure and the path-predicted term structure is the anomaly to investigate rather than the surface being a genuine additional signal. The surface's independent information is concentrated in the residual ε(T) — the component of each θ(T) unexplained by the path regression — and in the shape parameters of the SSVI fit (skew, wing asymmetry) that are not constrained by the ATMF regression.

The SSVI total implied variance surface is w(k, T) = (θ(T)/2){1 + ρ_s φ(θ(T)) k + √{(φ(θ(T)) k + ρ_s)² + (1 − ρ_s²)}} where k = log(K/F) is log-moneyness at the forward F(T), ρ_s ∈ (−1,1) and φ(θ) = η θ^{−γ} with (η, γ) calibrated to the market smile. The ATMF total variance is θ(T) = w(0, T). The path regression is θ_t(T) = α_0(T) + α_1(T) R_{1,t} + α_2(T) R_{2,t} + ε_t(T) where R_{j,t} = Σ_{s ≤ t} K_j(t−s) r_s^{p_j} with p_1 = 1, p_2 = 2 and K_j(u) = c_j(1 + u/θ_j)^{−α_j} power-law kernels calibrated by cross-validated regression. The coefficient function vector (α_0(T), α_1(T), α_2(T)) is estimated at each T by ordinary least squares over the daily panel: α̂(T) = (Φ^T Φ)^{-1} Φ^T θ(T) where Φ ∈ R^{n×3} is the design matrix with rows (1, R_{1,t_i}, R_{2,t_i}). The residual covariance Γ(T,T') = E[ε_t(T)ε_t(T')] is estimated from the time series of regression residuals and is a positive semi-definite matrix whose principal eigenvalues quantify the unexplained structured variation of the ATMF term structure.

The maturity-dependent loading structure of the regression coefficients is economically interpretable through the lens of forward variance pricing: at short maturities T ≤ 3 months, the ATMF total variance θ(T) is predominantly driven by the current activity level R_2, because variance swaps expiring soon are highly sensitive to the level of near-term realized variance and hence to the recent activity of squared returns; at longer maturities T ≥ 1 year, the ATMF is more influenced by the directional history R_1 through the leverage-adjusted expected price level, because the term structure at long maturities reflects the expected drift of the process as much as its variance, and R_1 encodes the recent directional trend that predicts near-term price level shifts. The transition between these two regimes — and the specific maturity at which the R_1 loading crosses the R_2 loading — depends on the kernel parameters (θ_j, α_j) and provides a model-implied characterization of the market's effective memory horizon for each type of information. The connection between the path regression and the Bergomi model is made precise by the mode decomposition of the Bergomi forward variance curve: the Bergomi model's forward variance ξ(T) evolves as an infinite sum ξ(T) = Σ_k ξ^k_T where each component ξ^k satisfies dξ^k_T = ω_k ξ^k_T dW^k_t, with the modes indexed by their mean-reversion speed κ_k; the PDV features R_1 and R_2 are essentially the first two moments of the projected Bergomi mode distribution, and the path regression is recovering the dominant Bergomi modes from the observable return path via their impact on realized variance. The residual term-structure covariance Γ(T,T') therefore has rank equal to the number of Bergomi modes not captured by R_1 and R_2, typically two to three in empirical data, providing a quantitative statement about the incompleteness of the two-feature representation. Comparison with rough Bergomi — where the forward variance curve has a singular mode structure driven by the fractional kernel — reveals that the path regression at large lags is approximating the fractional moving average of squared returns implied by RFSV, and the quality of the approximation depends on how well the power-law kernel K_2 matches the fractional kernel at the relevant timescales; for small Hurst index H ≈ 0.1, the fractional kernel decays very rapidly at short lags and slowly at long lags, a combination that the power-law kernel K_2(u) = (1 + u/θ)^{−α} can approximate but not exactly reproduce, so the residual ε_t(T) absorbs some rough-vol dynamics that the two-feature regression cannot capture.

The primary desk implication is that surface reads and path-feature reads should be performed jointly, not sequentially: the correct conditioning procedure is to first compute (R_{1,t}, R_{2,t}) from the return path, use the estimated regression coefficients α̂_j(T) to predict the ATMF term structure θ̂_t(T), and then compare the observed ATMF term structure θ_t(T) from the SSVI fit against the prediction. Agreement within estimation uncertainty indicates that the surface carries no information beyond what the path already contains at those maturities; disagreement — a surface term structure that is above or below its path prediction — indicates genuine new information in the surface that the path features have not captured and that is attributable to flow, positioning, or structural factors not in the return history. The residual ε_t(T) is the desk's primary surface-level signal: it represents the component of implied term-structure level that is not explained by path history, which could arise from large options flows, hedging demand from structured products, or macro factor repricing that has not yet appeared in the realized return path. Monitoring ε_t(T) across maturities provides a leading indicator of when the surface is being moved by forces exogenous to the return path — one of the few genuine surface-minus-path information differentials available in the conditioning protocol.

The linear regression of θ(T) onto (R_1, R_2) is justified by the affine structure of Bergomi-type forward variance models but is only an approximation to any rough volatility model where the ATMF variance is a nonlinear functional of the path kernel; the approximation error grows with the degree of roughness (small H) and the level of correlation (large |ρ_s|), and is most visible at short maturities where the near-singularity of the fractional kernel is most pronounced. Systematic misfits of the linear regression at short maturities — visible as structured autocorrelation in the regression residuals at T ≤ 1 month — indicate that the two-feature linear specification is insufficient and a third feature capturing the high-frequency (rough) component of volatility is needed. The stationarity assumption underlying the OLS regression of θ(T) on (R_1, R_2) is violated over long estimation windows that include fundamental regime changes in the SPX volatility surface — including the sustained low-volatility regime of 2017, the COVID spike of March 2020, and the rate-volatility coupling of 2022 — during which the coefficient functions α_j(T) shift materially. Rolling re-estimation on a 2–3 year window mitigates but does not eliminate this non-stationarity, and the resulting coefficient estimates are a time-weighted average of the true α_j(T) across regimes rather than the current regime's coefficients; during regime transitions the rolling regression is at its least accurate precisely when path-conditioning is most needed.