The rule your desk uses to clean Deribit quotes already changes its gamma. Zhong measures the size of that choice.
One feasible set, five ways to repair prices
Crypto option quotes can be wide and sparse, with prices that violate strike arbitrage. Desks project those quotes into a feasible set, called A in the paper. Membership requires each price to fall within its bid-ask interval intersected with the model-free bounds. Adjacent call spreads must cost between 0 and 1 per unit of normalized strike, and discrete butterflies must be nonnegative. These constraints apply separately to each maturity. Any price vector in A is an "admissible repair".
A is a polytope, and its constraints can leave room for many repairs. Earlier work cited by the paper projects traded prices into feasible sets, assessing feasibility, fit and smoothness. Zhong asks whether repairs that satisfy the same constraints also agree on Greeks, hedges and tail estimates.
The sample uses hourly Deribit BTC and ETH inverse (coin-settled) chains, taking the 12:00 UTC snapshot. There are 375,894 quotes in the main 12:00 UTC sample. The classical design draws 18 calendar months from 2022-11 to 2026-05, with repairs limited to 2 < DTE ≤ 30. It compares five rules:
- weighted L1 and L2 projections of the mid onto A
- L2 with a squared-second-difference penalty, weighted by ρ in {0.01, 0.1, 1, 10}, with Smooth10 at the top
- NN-Proj, a small neural net fitted to log total implied variance, then L2-projected into A
- an SVI-target repair
Each rule gets the same slices, feasible set A and book templates. Only the repair rule changes. L1 against L2 supplies the control.
The headline measure takes the p90 spot-gamma gap and scales it by each slice's median absolute gamma. Its median across 108 evaluations is 0.0039 for L1 against L2. For L2 against Smooth10, it reaches 0.0726, roughly 18.6 times the control. The paper calls this "nearly twenty". The excess is positive in 18 of 18 calendar months, with a two-sided sign test p = 7.63e-6. In 108 of 108 evaluations, gamma gaps are nondecreasing in ρ. Against L2, the SVI target produces 0.0834.
Why can similar prices leave so much curvature unresolved?
Finite differencing supplies the mechanism. Two admissible repairs within price distance ε can have second differences that differ by 4ε/h², where h denotes local spacing in K/F. Theorem 1 shows that two repairs inside A attain this bound whenever a feasible surface has local slack at the strike. The result holds for ε up to a limit determined by that slack. Wide quotes at densely spaced strikes can therefore leave curvature poorly identified, even when both repairs look acceptable in price.
Zhong measures the entire feasible curvature range at selected interior strikes by solving two LPs per strike. Across 51,189 high-score interior strikes, the range has a Spearman correlation of 0.8722 with the observed L2-Smooth10 curvature gap. The log-log R² is 0.6112. Tighter recomputation leaves no observation above the range.
There is a distinction worth keeping in view. Both the theorem and the LP address the second difference of normalized price. The headline gamma is Black gamma, calculated after each repaired price is inverted to implied vol. The paper explicitly assumes no algebraic equality between these quantities. The LP evidence validates the curvature gaps directly; its support for the gamma gaps rests on analogy.
The score used to rank slices, relative spread over h², also explains little at individual strikes. Per-evaluation regressions put the gamma gap against log relative spread and log(1/h²). Both coefficients are positive in 100 of 108 evaluations, yet the median R² is 0.0290. The paper accordingly uses the score only for slice ranking.
An 18.62 bp hedge disagreement, before any loss
The risk-reversal book supplies the economic figure. Across 42 evaluations, L2 and Smooth10 have a median p90 hedge-notional gap of 18.62 bps of gross spot notional. This amounts to USD 1,862 of delta per USD 1 million of gross spot notional. The figure measures disagreement over the hedge position. Any P&L follows from the next spot move, and transaction costs remain zero throughout.
The wings carry the exposure. Across all classical repair pairs, the median p90 scaled hedge-gap impact is 0.0002 for the ATM straddle, against 0.0330 for the risk reversal. For VaR-style estimates comparing Smooth10 with L2, the put butterfly is most sensitive. Its common-scenario p90 relative differences run from 0.13 to 0.17 across selection groups. The straddle records 0.00 in all three.
The paper uses two different VaR calculations. Under the common scenario, repairs change only the hedge: the scaled gap has median 0.0111 and p90 0.0933 across 432 cells. Repair-specific revaluation also allows each repair to determine book values. Those figures are 0.0904 and 0.3882 across 417 cells. The authors flag that the valid cells differ between the calculations. Each cell contains 10 to 50 next-day losses, and they describe these as descriptive comparisons rather than regulatory backtests.
Five neural seeds produce 19 times the control's curvature disagreement
NN-Proj scores well on its fitting measure: median held-out R² in log total variance is 0.941. Before projection, however, a median 55.9% of prices fall outside bid-ask, with a median error of 2.21 spreads. Projection into A leaves a gamma gap of 0.1890 against L2 and 0.1957 against Smooth10. The control is 0.0108. This check covers only 10 evaluations across 4 months.
The larger seed replication is harder to dismiss. Five refits use identical data, architecture and window. Their spread-scaled RMSEs differ by at most 8.17%, with a median difference of 1.10%. Yet their median p90 curvature gap reaches 1.3822, compared with 0.0719 for the L1-L2 control. The excess is positive in 17 of 18 months (p = 1.45e-4).
A desk retraining a learned surface can thus change its curvature through the random seed while moving spread-scaled RMSE by a median 1.10%. This replication measures curvature alone. How much of the disagreement reaches hedges remains untested.
What would make this a model choice?
Smooth10 occupies the top of the ρ grid. As the paper puts it, ρ "has no economic interpretation independent of the strike grid". The authors defend L1 against L2 as the matched projection-rule control: the weighting rules differ, yet their spot-gamma gap stays small. Still, both target the same midpoints and differ only in norm and weights, so a small gap is expected. A also excludes cross-maturity arbitrage. A 12-pair calendar screen leaves dispersion unchanged, with a median ratio of 1.000. That suggests calendar constraints would leave the gap open, though the sample is thin.
The paper calculates next-day hedged losses for each repair. It reports the distance between their 95% quantiles, leaving unanswered which repair hedges better. The evidence establishes disagreement without selecting a rule to run. Its conclusion asks for the same limited response: measure dispersion alongside fit. Evidence that one repair systematically hedges the risk reversal better would turn this governance check into a model choice.
We could not reproduce any of it.
Our data lacks Deribit inverse chains with recorded bid-ask intervals and forwards. Substituting US-listed options would change the market as well as the inverse-option accounting.
For a desk running a learned vol surface on crypto chains, the seed result alone warrants comparing repaired curvature across refits, along with the Greeks derived from it, before trusting them.