A single week's SPX quotes do not pin down one crash probability. Under a power-utility kernel with gamma = 2 and an assumed cap of five times the forward, every retained out-of-the-money quote allows the physical probability of a 10 percent decline at 37 days to lie anywhere in [1.97%, 4.53%]. The figures are medians of the endpoints across 324 cross sections. At all three horizons, the upper endpoint is two to three times the lower one.
The object behind those numbers is unusual. Index puts struck below a crash threshold price insurance, which pushes risk-neutral crash probabilities above physical probabilities. Zhuang removes that premium through a power-utility kernel: dP/dQ proportional to X^gamma, where X is the terminal index level normalized by the forward. He uses gamma = 2, the value Martin and Shi (2026) calibrate on market returns following Martin's (2017) log-investor benchmark.
Instead of fitting one implied-volatility surface and extracting a density from its curvature, Zhuang gathers every probability measure on [0, B] with unit forward mean that prices each retained quote within its bid-ask interval. Every admissible measure produces a physical crash probability q and an expected loss below the threshold. Together, the attainable (q, loss) pairs form a compact, convex set. Zhuang calls its boundary the physical crash frontier.
Finite second-order cone programs deliver the support values. A Charnes-Cooper rescaling removes the moment ratio. The support is then partitioned at the strikes into cells, each subject to truncated moment conditions. With gamma = 2, every localizing matrix is 2x2 and contributes one conic constraint.
The data are OptionMetrics end-of-day SPX bid and ask quotes from January 2013 through August 2023. Wednesdays are sampled at exact horizons of 23, 30 and 37 days, producing 1,115 horizon-specific cross sections across 537 distinct calendar weeks. Median retained OTM quote counts are 105.0, 144.0 and 95.5.
The traded strikes buy a large reduction in the identified range. Moving from the two puts nearest the threshold to all retained OTM quotes shrinks the probability interval by a median of 79.9% [78.4, 81.7] at 23 days, 84.4% [82.9, 85.3] at 30 days and 81.8% [80.3, 83.0] at 37 days. Median widths drop from 7.05, 12.14 and 13.80 percentage points to 1.34, 1.82 and 2.36.
We could not run this.
The method requires each contract's quoted bid and ask. Our option history contains end-of-day prices, implied volatilities and Greeks rather than two-sided quotes. Using a single price would turn quote-consistent partial identification into point calibration, which is a different problem. The discussion below therefore comes from reading the specification.
How much do extra strikes buy?
Most identification comes from strikes near the threshold. The paper measures the gain from each layer against the preceding quote set, rather than against the total reduction. Adding six nearby puts to the sparse pair cuts the put-pair width by a median of 69.6% / 72.2% / 72.5%. The remaining put wing then reduces the width still left by 28.0% / 35.2% / 26.9%. Adding all OTM calls cuts the residual by 2.6% [2.2, 3.3] / 4.2% [3.3, 5.2] / 5.0% [4.1, 6.4]. Because every layer starts from a new base, the layers do not sum to 100%.
The call contribution is small in absolute terms. At 37 days, the full put wing leaves a median width of 2.50 pp. Every OTM call reduces it to 2.36 pp. A reverse exercise across 180 cross sections makes the dependence on the base set clearer: deleting those same six nearby puts from the full OTM set leaves the median width unchanged because neighboring wing strikes replace them.
Spread width binds much harder. Across 24 cross sections, doubling every bid-ask spread around its midpoint moves a probability endpoint by a median 0.79 pp and a depth endpoint by 1.84 pp. Keeping only every other strike moves the corresponding endpoints by 0.073 and 0.122 pp. Mid prices cannot recover this object. The contrast between 0.79 pp and 0.073 pp shows why: identification comes from the width of each band, with the number of bands playing a much smaller role.
Implementation choices remain
Much of the machinery is fixed. The cell moment conditions and scaled quote bands are fully specified. An atom at the threshold has its own variable because continuous payoffs cannot distinguish mass at K from mass just above it. The cell beginning at K carries moment conditions after removing that atom.
The direction grid expands from 32 to at most 128 unit directions. Expansion stops once the gap between the plotted outer polygon and the inner hull of support points is below 1% of the put-pair scale. Across the plotted cross sections, the realized maximum was 0.99%. Inference uses a circular block bootstrap on calendar weeks, with 8-week blocks and 2,000 replications, stratified by horizon. The stated filters require positive strikes, nonnegative bids, positive asks, uncrossed intervals and positive volume.
Three choices remain unclear. The text we read does not specify the boundary between OTM puts and OTM calls. Since everything is normalized by the forward, k = 1 is the natural dividing line, and with roughly a hundred retained quotes in each cross section the choice affects a handful of contracts. The eight-strike set also adds "the six nearest additional put strikes" without explaining how those strikes are divided around K.
Solver treatment is the third issue. Of 1,115 all-OTM programs, 1,098 are reported as solved to verified optimality; the others are dropped rather than approximated. Statistics comparing quote sets use the 976 cross sections solved under all four sets. We did not find the solver's name or the tolerance used to define verification. The depth programs carry the exposure: because the depth scaling scalar has no upper bound, those endpoints are suprema and need not be attained.
B = 5 creates the floor
Without the support cap, the probability lower bound is exactly zero in all 180 sensitivity cross sections, as Proposition 3.6 predicts. Place a vanishing mass of order H^(-3/2) at a state H far above the last strike. Fixed-strike option values shift by O(H^(-1/2)) and remain within their spreads, while the denominator's second moment grows like H^(1/2). Both crash probability and unconditional shortfall move toward the origin.
Conditional depth barely changes. Against B = 5, the largest depth-width ratio in the unbounded program reaches 1.0002. This channel is available for every gamma above one. For gamma at or below one, Jensen and the forward restriction force the denominator to at most one.
Zhuang states the dependency directly in the abstract: "A positive floor requires a tail restriction the quotes cannot supply." The next sentence presents the frontier itself as what the quotes supply. The conclusion is stronger: "A positive lower bound on crash probability is therefore a joint statement about traded prices and the maintained support condition."
I accept that answer with one qualification. Section 4.2 describes B = 5 as "a loose economic bound, and its only bite is to rule out states whose arbitrarily large market payoff dominates the kernel normalization." This defends the selected value while leaving the existence of a cap exposed. Moving from B = 2 to B = 10 reduces a cross section's probability lower bound by a median of 0.0008 pp (0.0053 at the 95th percentile). The cap's exact location changes little. Its presence determines whether the floor exists at all, since removing it sends the lower bound to zero.
A reported floor of 1.97 percent at 37 days is therefore a joint claim about the quotes and an unverifiable cap. Calls cannot restore it. Relative to the full put wing, they reduce the remaining width by only 5.0% [4.1, 6.4] at 37 days, while the diluting mass lies beyond every traded strike.
A frontier for screening scenarios
The joint set has practical value because it removes probability-shortfall pairs that no single admissible distribution can produce. Under the paper's interior regularity condition, verified in 1,111 of 1,115 cross sections, a candidate scenario can be checked with one conic feasibility problem.
Its force is asymmetric. Across the 976 shared cross sections, the median normalized gap between the all-OTM joint set and its marginal rectangle ranges from 2.7% to 4.0% of the put-pair scale in the two directions that weight shortfall. The pooled four-direction median is 0.52%. Shortfall itself is tiny in level, with a median interval of [0.127%, 0.152%] at 37 days. The frontier removes corners when scenarios emphasize loss depth and removes little elsewhere.
Validation remains absent from a trader's perspective. The kernel comes from work whose crash bounds forecast realized crashes, yet we did not find a test of these identified sets against realized outcomes or any P&L. End-of-day quote snapshots also leave uncertainty over whether the recorded bid and ask levels were executable.
Conditional depth remains wide even with every quote included: a median [2.94%, 7.44%] of further decline below the 10 percent threshold at 37 days, with a wider interval at K = 0.85. Depth escapes the tail problem and is also the coordinate the quotes identify least tightly.
The frontier is useful for rejecting scenarios. If a stress narrative combines a crash probability and loss that the March 2020 frontier could not have generated, the conflict matters. Both endpoints rising together during that episode gives the honest representation of stress. Any positive lower bound still belongs to the assumptions written down by the user.