**Title:** Dispersion-Constrained Martingale Schrödinger Problems and the Joint S&P 500/VIX Smile Calibration Puzzle

**Abstract:**

__LaTeX-free version__:

The very high liquidity of S&P 500 (SPX) and VIX derivatives requires that financial institutions price, hedge, and risk-manage their SPX and VIX options portfolios using models that perfectly fit market prices of both SPX and VIX futures and options, jointly. This is known to be a very difficult problem. Since VIX options started trading in 2006, many practitioners and researchers have tried to build such a model. So far the best attempts, which used parametric continuous-time jump-diffusion models on the SPX, could only produce approximate fits. In this talk we solve this longstanding puzzle for the first time using a completely different approach: a nonparametric discrete-time model. Given a VIX future maturity T1, we build a joint probability measure on the SPX at T1, the VIX at T1, and the SPX at T2 = T1 + 30 days which is perfectly calibrated to the SPX smiles at T1 and T2, and the VIX future and VIX smile at T1. Our model satisfies the martingality constraint on the SPX as well as the requirement that the VIX at T1 is the implied volatility of the 30-day log-contract on the SPX.

The model is cast as the unique solution of what we call a Dispersion-Constrained Martingale Schrödinger Problem which is solved by duality using an extension of the Sinkhorn algorithm, in the spirit of (De March and Henry-Labordère, Building arbitrage-free implied volatility: Sinkhorn’s algorithm and variants, 2019). We prove that the existence of such a model means that the SPX and VIX markets are jointly arbitrage-free. The algorithm identifies joint SPX/VIX arbitrages should they arise. Our numerical experiments show that the algorithm performs very well in both low and high volatility environments. Finally, we discuss how to extend our results to continuous-time models, (i) by building a martingale interpolation of the discrete-time model, and (ii) by extending our Schrödinger approach to continuous-time stochastic volatility models, via what we dub VIX-Constrained Martingale Schrödinger Bridges, inspired by the classical Schrödinger bridge of statistical mechanics.

__LaTeX version__:

The very high liquidity of S\&P 500 (SPX) and VIX derivatives requires that financial institutions price, hedge, and risk-manage their SPX and VIX options portfolios using models that perfectly fit market prices of both SPX and VIX futures and options, jointly. This is known to be a very difficult problem. Since VIX options started trading in 2006, many practitioners and researchers have tried to build such a model. So far the best attempts, which used parametric continuous-time jump-diffusion models on the SPX, could only produce approximate fits. In this talk we solve this longstanding puzzle for the first time using a completely different approach: a nonparametric discrete-time model. Given a VIX future maturity $T_1$, we build a joint probability measure on the SPX at $T_1$, the VIX at $T_1$, and the SPX at $T_2 = T_1$ + 30 days which is perfectly calibrated to the SPX smiles at $T_1$ and $T_2$, and the VIX future and VIX smile at $T_1$. Our model satisfies the martingality constraint on the SPX as well as the requirement that the VIX at $T_1$ is the implied volatility of the 30-day log-contract on the SPX.

The model is cast as the unique solution of what we call a \emph{Dispersion-Constrained Martingale Schr\”odinger Problem} which is solved by duality using an extension of the Sinkhorn algorithm, in the spirit of (De March and Henry-Labord\`ere, Building arbitrage-free implied volatility: Sinkhorn’s algorithm and variants, 2019). We prove that the existence of such a model means that the SPX and VIX markets are jointly arbitrage-free. The algorithm identifies joint SPX/VIX arbitrages should they arise. Our numerical experiments show that the algorithm performs very well in both low and high volatility environments. Finally, we discuss how to extend our results to continuous-time models, (i) by building a martingale interpolation of the discrete-time model, and (ii) by extending our Schr\”odinger approach to continuous-time stochastic volatility models, via what we dub \emph{VIX-Constrained Martingale Schr\”odinger Bridges}, inspired by the classical Schr\”odinger bridge of statistical mechanics.

** Short bio**: Julien Guyon is a senior quantitative analyst in the Quantitative Research group at Bloomberg L.P., New York. He is also an adjunct professor in the Department of Mathematics at Columbia University and at the Courant Institute of Mathematical Sciences, NYU, and a Louis Bachelier Fellow. Julien serves as an Associate Editor of SIAM Journal on Financial Mathematics, Finance & Stochastics, and Journal of Dynamics and Games, and as a Managing Editor of Quantitative Finance. Before joining Bloomberg, Julien worked in the Global Markets Quantitative Research team at Societe Generale in Paris for six years, and was an adjunct professor at Universite Paris Diderot and Ecole des Ponts ParisTech.

Julien co-authored the book Nonlinear Option Pricing (Chapman & Hall, 2014) with Pierre Henry-Labordere. He has published more than 20 articles in peer-reviewed journals (including Finance & Stochastics, SIAM Journal on Financial Mathematics, Quantitative Finance, Risk, Journal of Computational Finance, Annals of Applied Probability, Stochastic Processes and their Applications) and is a regular speaker at international conferences, both academic and professional. His main research interests include nonlinear option pricing, volatility and correlation modeling, (nonlinear) optimal transport, and numerical probabilistic methods.

A big soccer fan, Julien has also developed a strong interest in sports analytics, and has published several articles on the FIFA World Cup, the UEFA Champions League, and the UEFA Euro both in academic journals and in top-tier newspapers such as The New York Times, The Times, Le Monde, and El Pais, including a new, fairer draw method for the FIFA World Cup. Some of his suggestions for draws and competition formats have already been adopted by FIFA and UEFA. His paper “Risk of collusion: Will groups of 3 ruin the FIFA World Cup?” won the 2nd prize at the 2021 MIT Sloan Sports Analytics Conference, the biggest sports analytics event in the world.

Here is the zoom link is: https://wse.zoom.us/j/95448608570