The Risk-Neutral Crash Frontier: Sharp Joint Bounds on Crash Probability and Conditional Depth from Option Bid-Ask Quotes

Jirong Zhuang

Papers from arXiv.org

Abstract: Index put prices are the market's quotes for crash insurance, and a put's value equals the probability of a crash times the expected shortfall given one. The market therefore prices the product of likelihood and depth, not the factors, and finitely many bid and ask quotes leave a range of ways to split it. Fitting one density hides that range, and bounds computed one factor at a time can combine into scenarios that no single risk-neutral distribution could produce. We characterize the closure of the probability and loss pairs generated by distributions that price every quote inside its spread, with depth as their ratio, and call its boundary the risk-neutral crash frontier. Under a strict quote interior condition, a partition at the quoted strikes and the threshold represents the closed joint set as the exact projection of a finite linear system, with no price grid. Linear programs trace the frontier and give sharp bounds for portfolios of the threshold digital and put, with static super- and sub-replicating portfolios of cash, forward, and quoted options certifying the upper and lower bounds. In weekly SPX cross sections from 2013 to 2023, joint restrictions exclude a median 36.6% of the area of a benchmark formed from separate probability and unconditional-loss bounds, adjusted for $0 \leq L \leq Kp$ and measured in probability-depth coordinates. Extending the quote set from the eight strikes nearest the threshold to the complete put wing reduces the joint area by a median 5.4 to 18.2% across the four specifications.

Date: 2026-07, Revised 2026-10
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