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

Battle Count: 78

Highly relevant for quantitative risk management and tail-risk-aware trading. The crash frontier provides model-free, sharp joint bounds on crash probability and conditional depth directly from observable option quotes, enabling: (1) validation of tail-risk scenarios against market prices, (2) pricing digital+put payoff packages without model assumptions, (3) identification of infeasible risk scenarios that marginal bounds would allow, (4) dual certificates that translate bounds into executable static hedges. The 5.4-18.2% area contraction from using the full put wing vs. local strikes has direct implications for how many strikes a desk should monitor. However, the method is primarily a risk-assessment tool rather than a signal-generation or alpha-extraction framework.

Moderate-to-high complexity. Requires: (1) LP solver for support programs (standard, e.g., Gurobi/CPLEX), (2) polyhedral computation for adaptive hull reconstruction (vertex enumeration, edge normal computation), (3) careful handling of the threshold atom coordinate and cell decomposition, (4) dual extraction and certificate construction, (5) analytical transformation to conditional-depth coordinates, (6) numerical tolerance management (10^-10 to 10^-8 thresholds). The core LP is finite-dimensional and tractable for typical SPX strike counts (~18-40 puts), but the adaptive hull algorithm requires iterative LP calls. No existing off-the-shelf package implements this exact pipeline.

The paper provides detailed algorithmic descriptions (adaptive hull, support programs, dual certificates), synthetic validation designs, and explicit numerical tolerances. However, the primary data source (OptionMetrics) is commercial and not freely available. No code repository is mentioned. The mathematical framework is fully specified with proofs, enabling independent implementation. Sampling rules, tie-breaking, and robustness specifications are documented in the appendix.