Joint Modelling and Calibration of SPX and VIX by Optimal Transport
32 Pages Posted: 3 Sep 2021
Date Written: April 3, 2020
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
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