From Arbitrage to Arbitrage-Free Implied Volatilities

Abstract

We propose a method for determining an arbitrage-free density implied by Hagan’s formula. Our technique is based on the stochastic collocation method. The principle is to determine a few collocation points on the implied survival distribution function (SDF) and project them on a polynomial of an arbitrage-free variable for which we choose the Gaussian variable. In this way we have equality in probability at the collocation points while the generated density is arbitrage-free. Analytic European option prices are available and the implied volatilities stay very close to those initially obtained by Hagan’s formula. The proposed method is very fast and straightforward to implement as it only involves 1D Lagrange interpolation and inversion of a linear system of equations. The technique is generic and may be applied to other variants or other models that generate arbitrage.

Keywords: Arbitrage-free Hagan’s density, Collocation Method, Orthogonal Projection

JEL Classification: C63, G12, G13