How Likely and How Deep? Sharp Joint Bounds on Risk-Neutral Crash Probability and Conditional Depth from Option Bid-Ask Quotes

Jirong Zhuang

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Abstract: A finite panel of option quotes with bid-ask spreads generally does not point-identify either the risk-neutral probability of breaching a specified threshold or the expected shortfall below that threshold conditional on a breach. Sharp marginal bounds characterize each quantity in isolation but not their jointly attainable combinations: values within the two marginal intervals may require different risk-neutral distributions, so pairing them can produce a tail scenario inconsistent with the option panel. We characterize the closure of the jointly attainable probability and loss pairs under finite bid-ask constraints. Partitioning the state space at observed strikes and at the crash threshold makes the model-implied value of every retained option linear in cell probabilities and first moments, and one additional variable tracks mass at the threshold itself. Under a strict interior condition, the identified set is the projection of a finite linear system, and an adaptive hull algorithm recovers the probability-loss polygon. Vertical sections report the conditional depths consistent with a candidate crash probability, and the support function gives sharp upper and lower values for any portfolio of digital and put payoffs. In 557 weekly SPX panels from January 2013 to August 2023, with 1,966 primary cases, the complete put wing lowers median transformed area by 5.4-18.2% relative to a local set of eight strikes, and the joint set fills a median 63.40% of the benchmark formed from separate marginal bounds and the universal probability-loss restriction.

Date: 2026-07
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