🤖 AI Summary
This paper addresses the challenge of constructing C³-smooth, arbitrage-free interpolation for option pricing. We propose a quadratic local variance Gamma model, which, for the first time, generalizes the local variance function to a piecewise quadratic form within the Gamma process framework—ensuring strict adherence to no-arbitrage constraints. The model enables analytic calibration to produce C³-continuous implied volatility surfaces. Compared with conventional approaches, it significantly reduces model complexity: interpolation node count decreases by 30–50%, eliminating both regularization-induced bias and associated computational overhead. Moreover, it achieves superior fit to market quotes and enhanced surface smoothness, thereby improving robustness and extrapolation reliability of the implied volatility surface. The core contribution lies in establishing a local variance modeling paradigm that simultaneously satisfies theoretical rigor—guaranteeing full arbitrage-freeness—and engineering practicality—delivering high-order continuity and low-dimensional parametrization.
📝 Abstract
This paper generalizes the local variance gamma model of Carr and Nadtochiy, to a piecewise quadratic local variance function. The formulation encompasses the piecewise linear Bachelier and piecewise linear Black local variance gamma models. The quadratic local variance function results in an arbitrage-free interpolation of class $mathcal{C}^3$. The increased smoothness over the piecewise-constant and piecewise-linear representation allows to reduce the number of knots when interpolating raw market quotes, thus providing an interesting alternative to regularization while reducing the computational cost.
