The second-order variance estimate earns its extra strike. Bourgey and Gatheral's tests give a desk less reason to pay for the higher-order stencils: orders 4 and 5 sometimes price variance and gamma worse than order 2. I would build a trading signal on the low orders.
What the strikes estimate
The variance contract, M, is expected quadratic variation, equal to minus twice the expected log return. Weight that quantity by S_T/S_0 and it becomes the gamma contract, G. Fukasawa expresses each contract as a Gaussian average of total implied variance, using a normalized strike coordinate. The representation is exact up to interpolation and extrapolation. It also requires strike inversion and integration across the entire extrapolated smile. The usual one-point shortcut takes total variance where d− = 0 for variance, or where d+ = 0 for gamma. Taking the square root of the first gives the Rolloos and Arslan zero-vanna volatility-swap estimate.
Bourgey and Gatheral use the forest expansion of Alòs, Gatheral and Radoičić alongside the Bergomi-Guyon smile expansion. At order n, the total-variance correction is a polynomial in log-strike with degree exactly n. They replace its Taylor coefficients with finite differences at k0 + jΔk, where j belongs to {0, ±1, ±2}. These are the "magic strikes". For variance, k0 = -M/2 and Δk = √M; for gamma, k0 = G/2 and Δk = √G. Because each set of strikes depends on the contract value being estimated, the formulas are fixed-point equations. Orders 1 through 5 read 1, 2, 3, 5 and 5 strikes. A general theorem gives 2⌈n/2⌉+1 as a sufficient strike count at order n. Flip the sign of every odd finite difference in the variance formula and it becomes the gamma formula. Power payoffs (S_T/S_0)^p, with p in [0,1], run between the contracts.
The authors test two Heston and two rough Bergomi parameter sets, each across 15 maturities from 0.05 to 1.0 years. Their rough Bergomi gamma benchmark uses Monte Carlo with 300 time steps and 5 batches of 3×10^5 paths. For the market test, they use four SPX snapshots from 2025 and a Fukasawa integral evaluated with 10 quadrature nodes. They add Vola Dynamics marks for 2 July. We found the accuracy results in plots only, with no error table.
The zero-vanna volatility-swap estimate is exact to first order, and to second order when ρ = 0. The authors also shift trees in a way that preserves the maturity-T smile while changing the volatility contract by -ε²δ/(2M^{3/2}) to leading order. Strikes from one expiry cannot recover that correction. The volatility swap therefore cannot be obtained from that smile alone.
The two-strike calculation
Order 2 is easy to work through. Start with a guess for M, then read total variance at -M/2 - √M and -M/2 + √M. Average those readings and add their squared difference divided by 8M. Repeat until M stops changing.
Since Δk² = M, the centre strike disappears from the calculation. With a flat smile, order-2 weights put 0 on the centre and 1/2 on either side. Keep the same spacing at order 5 and the paper's weight formula gives 1/2, 1/6 and 1/12; the outer readings now reach ±2Δk. The authors show that weights remain nonnegative through order 5 when the spacing multiplier lies in [1, √3]. They choose 1.
Does order five earn its three extra strikes?
Only sometimes, on the paper's evidence.
In Heston Set II (η = 1.99, ρ = -0.68), orders 2 and 3 outperform orders 4 and 5 across much of the maturity range. The higher orders overestimate both contracts. As the authors put it, "Increasing the order does not always improve the estimate, however." They suggest that widening the stencil to 2Δk may leave a low-degree polynomial fitting the smile poorly.
Rough Bergomi Set II (H = 0.11, η = 2.39, ρ = -0.86) gives a divided result. Order 4 follows the variance benchmark closely. For gamma, orders 2 and 3 are closer; orders 4 and 5 overestimate at longer maturities.
The SPX plots divide the same way. On 10 February and 2 July, orders 4 and 5 are closest to the Fukasawa variance value yet sit above its gamma value. Orders 2 through 5 cluster on the two April dates. A signal whose sign changes with the contract being priced gives me little to trade. I would reconsider if a time series of tabulated errors showed orders 4 and 5 consistently inside the bid-ask band.
The SPX comparison has two limits. Its Fukasawa benchmark comes from the same interpolated, flat-extrapolated smile, so agreement measures proximity to that smile's full integral. Only the Vola Dynamics marks on 2 July provide an outside reference. Speed is also a narrower advantage than the paper's description suggests: its runtimes give the Fukasawa integral 0.022 s, against 0.037 s to 0.19 s for the magic-strike orders. The strikes instead avoid extrapolation by staying within two standard deviations of k0.
Moving the spread to SPY options
We cannot trade SPX index options or OTC variance and gamma contracts. Our strategy therefore uses SPY listed options, and it does not replicate the paper's SPX results. SPX accuracy does not carry over to SPY; the pricing mechanism needs only a smile.
For each expiry, we take the end-of-day SPY mid smile and calculate M2, M3 and the Fukasawa integral on that same interpolated smile. The resulting spread combines the value of the wing beyond the magic strikes with truncation error from the expansion itself. Heston Set II shows why that distinction matters: truncation error alone can push orders 4 and 5 above the exact value. When the spread is positive, we sell a listed out-of-the-money wing strip, buy vega-matched options at the magic strikes and delta-hedge daily in SPY.
We require every magic strike to fall inside the quoted strike range; beyond it, the method loses its independence from extrapolation. Skew and vol-of-vol push those strikes outward. Each must have a two-sided quote. As the paper does for its shaded Fukasawa band, we recalculate on bid and ask smiles and trade only when the spread clears the band. We discard a slice if M2 and M3 differ by more than that band.
Pricing disagreement is not P&L
The paper's numerical results compare prices. Its benchmarks are Heston closed forms, rough Bergomi Monte Carlo for gamma, a Fukasawa integral and Vola Dynamics marks on one date. The market evidence spans four SPX snapshots, without a time series. It cannot show whether the spread persists or covers the spreads on a multi-leg SPY strip. Our backtest must establish that out of sample and net of costs; it is still running. Until those results arrive, I regard the paper as a pricing tool. Its order-2 estimate beats the one-strike rule in every test shown, while the trading edge remains untested.