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

Option quotes with bid-ask spreads do not point-identify the risk-neutral probability of a crash below a given threshold, nor the expected depth of the crash once the threshold is breached. Bounds computed separately for the two quantities can mislead, because their endpoints may be attained by different risk-neutral distributions. We characterize the closure of the set of jointly attainable probability and loss pairs and call its boundary the crash frontier. Partitioning the state space at observed strikes and at the threshold, with one added coordinate for mass at the threshold itself, makes this set the projection of a finite linear system. The projection is exact, and no discretization of the state space is required. Support programs trace the frontier and give sharp upper and lower values for any portfolio of digital and put payoffs. Each bound comes with a dual certificate, a static portfolio of cash, forward, and retained quotes that super-replicates the target payoff. For a candidate crash probability, the set reports the range of conditional depths consistent with it. In SPX option cross sections sampled weekly from 2013 to 2023, the joint region covers a median 63% of the benchmark formed from separate marginal bounds. About a third of the scenarios allowed by the marginal bounds are therefore jointly infeasible. Quotes far from the threshold contract the joint region by 5.4-18.2% at the median relative to a local set of eight strikes.