Joint Modelling and Calibration of SPX and VIX by Optimal Transport

32 Pages Posted: 3 Sep 2021

See all articles by Ivan Guo

Ivan Guo

Monash University - School of Mathematical Sciences

Gregoire Loeper

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

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

Ivan Guo

Monash University - School of Mathematical Sciences ( email )

Clayton Campus
Victoria, 3800
Australia

Gregoire Loeper

BNP Paribas ( email )

Paris
France

Monash University - School of Mathematical Sciences ( email )

Clayton Campus
Victoria, 3800
Australia

Monash University - Monash Centre for Quantitative Finance and Investment Strategies ( email )

9 Rainforest Walk
Clayton Campus
Monash University, Victoria 3800
Australia

HOME PAGE: http://https://www.monash.edu/science/quantitative-finance

Ecole Polytechnique, Palaiseau - CMAP CNRS-UMR 7641 and Ecole Polytechnique ( email )

Route de Saclay
Palaiseau, 91128
France

Jan K. Obloj

University of Oxford - Mathematical Institute ( email )

AWB, ROQ, Woodstock Rd
Oxford, OX2 6GG
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

Eagle House
Walton Well Road
Oxford, Oxfordshire OX2 6ED
United Kingdom

University of Oxford - Saint John's College ( email )

St Giles
Oxford, Oxon OX1 3JP
United Kingdom

Shiyi Wang (Contact Author)

Monash University - School of Mathematics ( email )

Clayton Campus
Victoria, 3800
Australia

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