Moving basket weights lift short-dated index skew

A constituent-volatility formula understates short-dated index skew when it leaves out moving basket weights. Fukasawa, Maeda and Ogiwara show why the omission matters at H = 1/2, the setting for local vol and classical stochastic vol. The missing contribution has a fixed sign.

Where the extra skew comes from

The basket holds fixed shares: S = ΣS^i, with weights Π^i = S^i/S. Its instantaneous variance is V = ΣΠ^iΠ^j √(V^iV^j) ρ_ij. Constituent variances V^i move at a rate r(t), regularly varying with index H and equal to θ^H in the pure-power case. The weights Π^i move as returns change. Because returns drive that movement directly, the paper puts its contribution at order √t regardless of the volatility model.

The general theorem starts with a joint Gaussian limit for normalized returns and variance fluctuations. It also requires uniform integrability of the normalized variance fluctuations and true-martingale prices. Under those assumptions, the near-the-money expansion is σ(√θ z, θ) = √v̄(θ) + z(ψA/(2H+3)·r(θ) + ψB/4·√θ) + o(r(θ)+√θ). In the Gaussian factor setting, given a positive-definite residual correlation matrix, the authors prove an ATM skew of ψA/(2H+3)·r(θ)/√θ + ψB/4 plus a smaller error. Here ψA measures leverage between basket returns and constituent vol shocks; ψB captures changing weights.

The sign of ψB is settled by its formula. If d_i is the covariance rate between name i's return and the basket, then ψB = 2/v(0)^{3/2}·ΣΠ_0^i(d_i − v(0))^2. This weighted variance gives ψB ≥ 0. It reaches zero only when every name has identical covariance with the index. In a selloff, high-d names fall hardest and lose weight, leaving less of the basket in its most variance-heavy pieces. Thus ψB ≥ 0 pushes skew upward, against the usual negative equity skew.

These are asymptotic claims for log-strike k = z√θ as θ → 0. The paper has no market data or sample. For its specialization, the authors take V^i = f^i(t, Z_t), with Gaussian Volterra factors whose kernels behave like κ r(u)/√u. Multi-factor Bergomi, rough Bergomi and quintic models fall within that class. A Gaussian delta method yields ψ_ki = 2/(2H+1)·γ_ki; γ_ki measures the short-time exposure of name i's vol to name k's return shock.

Does the weight term survive at H = 1/2?

When H < 1/2, r(θ)/√θ explodes. The constituent term grows as θ^{H−1/2} in the pure-power case, whereas ψB/4 stays finite. Provided ψA ≠ 0, a rough-vol user can leave the weight term out at the very short end.

At H = 1/2, r(θ)/√θ is slowly varying, so neither contribution can generally be discarded. In the pure-power case r(θ) = √θ, they have the same order. Local vol uses r(t) = √t, making the coefficient of z√θ equal to (ψA+ψB)/4. A single name has ψB = 0, reducing that coefficient to S_0∂_sσ(0,S_0)/2. The authors' warning is direct: "Omitting the portfolio-weight contribution would in general give an incorrect finite short-maturity skew." They report agreement with Pirjol's large-deviation formula for local vol and say the H = 1/2 result for general stochastic vol "appears to be new."

The roughest constituent sets the skew's explosion rate. Remark 2.6 indicates that heterogeneous H_i give a basket rate of θ^{H−1/2}, with H = min H_i, provided the aggregate leverage coefficient of the roughest names does not vanish.

An SSR limit with a condition

The skew stickiness ratio (SSR) is Bergomi's statistic. It divides the regression coefficient of ATM implied vol moves on log-spot moves, measured by their covariation over the variation of log spot, by the ATM skew. Through a Malliavin representation, the authors obtain the exact identity SSR = A/Y. Its short-maturity numerator is A = φ(0)/2·(ψA r(θ) + ψB√θ); its denominator is Y = φ(0)·(ψA/(2H+3)·r(θ) + ψB/4·√θ).

For H < 1/2, ψA dominates both expressions and the ratio becomes (1/2)/(1/(2H+3)) = H + 3/2. At H = 1/2, 1/(2H+3) is 1/4. The weight and leverage contributions then give the same ratio, with a limit of 2. That coincidence lets the basket inherit Fukasawa's single-asset limit despite having two sources of skew.

The qualification matters. The authors describe a natural nondegeneracy condition in the theorem, while the abstract states the limit without it. At H = 1/2, the required condition is liminf |ψA r(θ)+ψB√θ|/(r(θ)+√θ) > 0. In the Gaussian factor model where they prove the theorem, ψA is built from γ_ki = Σ β^i_a ρ^k_l κ_al. Negative loading of vol factors on return shocks makes ψA negative, as in the equity leverage pattern. Yet ψB ≥ 0. Their opposition in the standard equity case makes a nearly flat short-dated basket skew the place where the denominator can collapse.

Further assumptions apply to the density and SSR results. The residual correlation matrix C⊥ must be positive-definite: the portions of the n return shocks orthogonal to vol factors must be linearly independent across names. Log f^i must be C² near the origin. The SSR limit is proved only at t = 0 as θ → 0, and the Volterra specialization excludes H = 0.

What would make this tradeable

We did not find a Monte Carlo check, a calibration or a stated maturity at which the expansion becomes accurate. Some technical steps, including uniform integrability after localization, receive brief arguments. We could not backtest the result: our data has no listed options on bespoke baskets. Daily closes also give only a noisy estimate of the instantaneous quadratic covariation used by the SSR.

ψB remains the usable result. Initial weights, vols and correlations suffice to calculate its sign-definite correction for a stitched constituent model at H = 1/2. The H + 3/2 limit also constrains how a basket-smile model moves with spot. We would want a classical (H = 1/2) Bergomi basket simulation to show that ψB/4 still changes the one-month skew before wiring the correction into a basket hedge.