B-spline techniques for volatility modeling

TL;DR

This paper explores advanced B-spline methods for volatility modeling, including calibration of leverage functions, implied volatility surface parameterization, and solving related PDEs, enhancing smoothness and arbitrage-free properties.

Contribution

It introduces novel B-spline techniques with infinite support basis functions for improved volatility surface calibration and PDE solutions in financial modeling.

Findings

B-spline methods produce smooth, arbitrage-free implied volatility surfaces.

Shape-constrained B-splines improve estimation of conditional expectations.

The Galerkin B-spline finite element approach effectively solves related PDEs.

Abstract

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first come back to the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. An application is the Monte Carlo calibration of stochastic local volatility models by Markov projection. Then we present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon-Nikodym derivative of the underlying's risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon-Nikodym derivative.