The Physical Crash Frontier: What Finite Option Quotes Can and Cannot Reveal

Jirong Zhuang

Papers from arXiv.org

Abstract: Physical crash probabilities recovered from option prices depend on a pricing kernel and on a risk-neutral distribution that finitely many bid and ask quotes do not identify. For a power utility investor, we characterize the pairs of physical crash probability and expected loss below the crash threshold that the quotes admit; the boundary of this set is the physical crash frontier. Both coordinates are ratios of moments, yet when the index is bounded above the closed set is convex, and under an interior regularity condition second-order cone programs compute it exactly at the benchmark risk aversion of two. In a decade of weekly S&P 500 cross sections, the quotes beyond the two puts nearest a 10 percent decline shrink the range of admissible crash probabilities by about 80 percent, yet its upper end remains two to three times its lower end. Under strict quote slack and the interior-mass condition stated below, removing the cap without further tail control drives the lower bounds on crash probability and unconditional shortfall to zero for risk aversion above one. A vanishing probability far in the right tail inflates the normalizing moment while every quote remains inside its spread. In this setting, a positive floor requires additional tail information, supplied here by the support cap.

Date: 2026-08, Revised 2026-10
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