From Stochastic Orders to Volatility Surfaces: Revisiting the One-X Property

The One-X property, introduced by Zetocha in a 2023 paper, provides a novel stochastic order with direct implications for constructing arbitrage-free implied volatility surfaces. The current work revisits its theoretical foundations and explores its connections with classical stochastic orders, thereby offering a deeper understanding of its mathematical structure and practical significance in calendar-arbitrage-free modeling. We first present an explicit counterexample to a conjecture raised in Zetocha’s previous paper, and then provide a natural and valid enhancement of this conjecture. After discussing the inherent relations between the One-X property and properties such as $T{P}_2$, $R{P}_2$, and unimodality of density ratio (introcuded by Glasserman and Pirjol in their 2024 papers), we further explore some sufficient conditions to achieve the One-X property for random variables of certain mixture types that are frequently seen in applications.