Endogenous Path-Dependence and the Sufficiency of Past Returns for Implied Volatility

The authors regress index implied and realized volatility on two features of the past return path — a weighted sum of past returns and the square root of a weighted sum of past squared returns, each under time-shifted power-law kernels — and explain as much as 90% of the variance of implied volatility endogenously. We read this as the empirical complement to roughness: the structure that produces rough-looking paths is, to first order, a deterministic function of observable history.

Volatility is modelled as σ_t = β_0 + β_1 R_{1,t} + β_2 √R_{2,t}, with trend feature R_{1,t} = Σ K_1(t−s) r_s and activity feature R_{2,t} = Σ K_2(t−s) r_s² under power-law (time-shifted) kernels K_1, K_2 capturing both fast and slow decay. A 4-factor Markovian approximation reproduces the joint SPX/VIX smile.

The economically interesting claim is the asymmetric weighting: a signed trend feature and an unsigned activity feature jointly suffice. The model is feature-light yet competitive with rough models on the SPX/VIX joint calibration, which suggests the two literatures are describing one phenomenon from two coordinate systems.

Supports treating the recent realized path as a first-class memo input. Where roughness motivates path-conditioning in principle, this construction gives an explicit, low-dimensional feature set the desk can actually compute.

The regressions are index-level; single-name path-dependence is noisier and the kernel parameters are not stable out-of-sample without regularization. Endogenous explanatory power is not the same as forecasting power.