Joint Modelling and Calibration of SPX and VIX by Optimal Transport

Guo, Ivan; Loeper, Gregoire; Obłój, Jan; Wang, Shiyi

doi:10.2139/ssrn.3568998

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Joint Modelling and Calibration of SPX and VIX by Optimal Transport

32 Pages Posted: 3 Sep 2021 Last revised: 11 Sep 2020

See all articles by Ivan Guo

Ivan Guo

Monash University - School of Mathematical Sciences

BNP Paribas; Monash University - School of Mathematical Sciences; Monash University - Monash Centre for Quantitative Finance and Investment Strategies; Ecole Polytechnique, Palaiseau - CMAP CNRS-UMR 7641 and Ecole Polytechnique

Jan Obłój

University of Oxford - Mathematical Institute; University of Oxford - Oxford-Man Institute of Quantitative Finance; University of Oxford - Saint John's College

Shiyi Wang

Monash University - School of Mathematics

Date Written: April 3, 2020

Abstract

This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton–Jacobi–Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.

Keywords: Optimal Transport, Volatility calibration, S&P 500, VIX, Joint Calibration

JEL Classification: C60, C61, C63, G13

Suggested Citation: Suggested Citation

Guo, Ivan and Loeper, Gregoire and Obloj, Jan K. and Wang, Shiyi, Joint Modelling and Calibration of SPX and VIX by Optimal Transport (April 3, 2020). Available at SSRN: https://ssrn.com/abstract=3568998 or http://dx.doi.org/10.2139/ssrn.3568998