This paper proposes a new method for describing the probabilities that financial markets use to price options. Instead of working with the usual probability density or cumulative distribution, the author directly models the quantile function—the inverse of the cumulative distribution, which maps a probability level to a price return. That choice gives a simple mathematical space to work in and makes it easier to build flexible but still small models that control the shape of the implied volatility curve. The implied volatility curve is the standard way traders summarize option prices across strike prices; it can show complex local features such as central concavity or multiple peaks.

The core idea is to parameterize the risk‑neutral quantile function with a few interpretable parameters that each control behavior in a specific range of moneyness (moneyness is a way to compare the option strike to the underlying price). The paper explains how option prices come from that quantile function by a change of variables and gives formulae that link the quantile, its derivative, and the usual implied volatility. The author also shows how to construct valid risk‑neutral quantiles from simple building blocks. Examples in the text include a family built by conic combinations of exponential quantiles that yields an asymmetric logistic quantile, and operations like monotone transformations and segmental construction that preserve validity.

The model is deliberately parsimonious and interpretable. It is designed to capture a wide variety of implied volatility shapes, including U‑shaped, W‑shaped, inverse‑U, and S‑shaped curves with local concavity that have become common in short‑dated option markets, sometimes around scheduled events. The paper reports empirical calibration to about a quarter of a million implied‑volatility curves taken from two years of Standard & Poor’s 500 index option data. The fitted parameters were reported as accurate and stable across different maturities (tenors) and strikes, and the author says those stable patterns allow safe interpolation across maturities and building a dynamic model of parameters without introducing static arbitrage (an internal inconsistency in prices).