From Arbitrage to Arbitrage-Free Implied Volatilities

Journal of Computational Finance 20(3), 1-19, 2016

12 Pages Posted: 24 Nov 2014 Last revised: 5 Jul 2016

See all articles by Lech A. Grzelak

Lech A. Grzelak

Delft University of Technology

Cornelis W. Oosterlee

Center for Mathematics and Computer Science (CWI)

Multiple version iconThere are 2 versions of this paper

Date Written: March 7, 2016

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

Suggested Citation

Grzelak, Lech Aleksander and Oosterlee, Cornelis W., From Arbitrage to Arbitrage-Free Implied Volatilities (March 7, 2016). Journal of Computational Finance 20(3), 1-19, 2016. Available at SSRN: https://ssrn.com/abstract=2529684 or http://dx.doi.org/10.2139/ssrn.2529684

Lech Aleksander Grzelak (Contact Author)

Delft University of Technology ( email )

Netherlands
00310655731315 (Phone)

Cornelis W. Oosterlee

Center for Mathematics and Computer Science (CWI) ( email )

P.O. Box 94079
Amsterdam, NL-1090 GB
Netherlands

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