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. Partitioning the state space at observed strikes and at the threshold, with one added variable for mass at the threshold itself, makes this set the projection of a finite linear system. The projection is exact and requires no outcome grid. Support programs recover the joint frontier, give sharp upper and lower values for any portfolio of digital and put payoffs, and produce dual certificates at the quote level. Vertical sections report the conditional depths consistent with a candidate crash probability. In weekly SPX panels from 2013 to 2023, 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, and the joint region covers about 63% of the benchmark formed from separate marginal bounds.