A slice-by-slice SVI fit assigns exactly zero variance-swap delta when the index falls 2%. Bossu and Gaitan-Escarpeta prove the result as a theorem, in a proof short enough to remember.
The theorem inside the smile
The argument starts with the Carr-Madan representation. Fair variance to maturity is the 1/K-squared weighted integral of out-of-the-money puts below the forward and calls above it. Applying Leibniz introduces a moving boundary at K = F. Put-call parity cancels those two boundary terms. The remainder consists of the strip's option deltas and a vega-weighted term that captures how the smile moves with F.
Proposition 2.1 gives the compact version: total delta is a positive-weighted integral of the smile's partial derivative with respect to F. When the smile depends purely on log-moneyness x = ln(K/F), that derivative vanishes everywhere. So does the delta.
Theorem 2.2 strengthens the claim to an equivalence. Under monotonicity of the smile in F, F-independence of the smile and F-independence of fair variance imply each other. The weight on the integrand keeps the same sign, which means a vanishing integral forces the derivative to vanish.
The numerical check uses SVI calibrated to the SPX chain of August 20, 2025 for the September 19 expiry. The inputs are T = 0.082192, F0 = 6412.45, r = 4.8%, q = 1.6%. Five parameters are fitted by constrained least squares on total variance, using 60 random restarts. Relative volatility RMSE is 5.2%. Strip integration then produces fair variance of 0.026303 throughout F in (0, 10000), without visible noise. The resulting variance swap rate is 16.21%, broadly in line with that day's VIX range of 15.57 to 17.19.
The proposed repair takes one line: multiply the smile by (F/F0)^lambda. Near F0, fair variance scales as (F/F0)^(2lambda), while the variance swap rate acquires locally constant price elasticity lambda. An additional vega term, proportional to lambdasigma minus sigma-prime, enters the option delta. Corollary 3.2 says that with negative lambda at the reference forward F = F0, this term pushes delta below the powerless total delta.
Which delta belongs on the hedge?
Proposition 2.5 carries the cleaner practical message. The Black-Scholes variance swap delta sums the strip's Black-Scholes deltas while holding implied vols fixed. Its gap from total delta is exactly the vega-weighted smile response.
For a pure downward-sloping function of moneyness, Black-Scholes delta is negative while total delta is zero. When the smile instead decreases in F, so volatility rises as the market falls, the sign of the gap reverses and Black-Scholes sits above total delta. The assumed smile dynamics decide which delta is larger.
Eid's 2011 conclusion, quoted by the paper, says an unstruck variance swap "is only a pure variance product and should not be hedged with any offsetting delta on the spot". Read now, the statement describes a coordinate choice rather than the market.
Lambda carries a heavy load
The empirical exercise runs a zero-intercept OLS of daily variance swap rate returns on daily S&P 500 returns, separately by maturity, from 2008 to 2025. In the full sample, lambda is -4.76 at 1 month, -3.09 at 3 months, -1.61 at 12 months and -1.13 at 24 months. Standard errors range from 0.02 to 0.07, with R-squared between 0.46 to 0.57.
Results for the 2022-2025 subperiod put lambda at -5.04 for 1 month and retain the same -1.13 at 24 months. The added 1-day tenor gives -8.43, though R-squared falls to 0.11. Its standard error is 0.83, roughly 10% of the coefficient against 3-5% elsewhere. The authors themselves describe that horizon as noise-dominated.
For 2008-2025, the paper presents F-statistics of 3723 to 5872 as evidence of significance. With seventeen years of daily data and one regressor, significance was never in doubt. The paper concedes the figure that matters to a hedger: daily index returns explain "roughly half of the variation in daily variance-swap-rate returns, with the rest left to other factors". The other half is orthogonal to index returns. Any deterministic mapping from F to the smile misses it.
Stability says more. Across rolling one-year windows, average lambda runs from -6.364 at 1 month to -1.275 at 24 months. Dispersion declines monotonically from 2.081 to 0.243. As the authors observe, every rolling mean is more negative than its corresponding full-sample estimate.
The 1-month elasticity therefore changes materially with the chosen window and weighting: -4.76, -5.04 or -6.364, depending on which of the paper's tables supplies the estimate. The 1/sqrt(T) term structure survives better than any individual lambda. Eta is -1.476 under full-sample least squares, -1.59 for 2022-2025 under minimum absolute error and -1.898 when averaged across rolling windows. Those rolling fits have average R-squared of 0.964.
A repair with limited reach
The authors explicitly describe equation (13) as local, requiring F sufficiently close to F0. They also say the multiplier needs practical bounds "to avoid nonsensical results". We did not find a bounding rule specified. We also did not find a no-arbitrage check for the shifted smile away from F0, where a power multiplier could damage a wing.
The recommended hedge ratio raises a separate concern. To place power delta below Black-Scholes delta, in line with Hull and White's evidence that optimal hedge ratios come in low, the paper sets lambda equal to the rolling mean minus two rolling standard deviations. This produces -2.71 at 12 months and -10.53 at 1 month. The chosen lambda delivers the desired delta ordering. Yet the 1-month value of -10.53 exceeds twice both the 1-month full-sample estimate of -4.76 and the 2022-2025 estimate of -5.04. Kurtosis of the rolling estimates is negative at every maturity (-0.307 at 1 month), placing that value in a tail the paper's own statistics characterize as thin.
We could not run our own version. The underlying object is an OTC variance swap and its strip, while our data contain end-of-day listed option prices, implied vols and Greeks, with no variance swap quotes at all. Replicating with SPY options on a finite strike grid and rehedging at the close would create a different position with different truncation error. Its backtest would not test what the authors wrote.
The theorem deserves full credit. It should change how traders interpret a per-slice SVI fit used for a variance product. The power smile also offers a defensible hedging convention, with one parameter and a clean term structure.
The hedging claim remains untested. The paper says only that the lower power delta "may help produce better hedge ratios", and Figures 8 through 10 show initiation curves. One thing would change my view: a hedge-error comparison among Black-Scholes, powerless total and power deltas on a 1-month SPX strip, with lambda estimated solely from data preceding each hedging window and results net of the strip's spread. If power delta wins there with lambda near -4.8 rather than -10.53, the convention earns its place.