B-spline techniques for volatility modeling

Sylvain Corlay (1)

1 Bloomberg L.P. Quantitative Finance Research

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.

Mots clés: finite elements; second-order cone programming; calibration; stochastic volatility; local volatility; B-splines; Tikhonov regularization; Radon-Nikodym

Domaines: Probabilités [math.PR]

Sylvain Corlay. B-spline techniques for volatility modeling. 2013. ⟨hal-00830378v4⟩