Dear all, Tuesday, May 18 at 4 PM (CEST) there will be the following online seminar. Speaker: Julien Guyon (Bloomberg L.P.); below the short bio. Title: Dispersion-Constrained Martingale Schrodinger Problems and the Joint S&P 500/VIX Smile Calibration Puzzle Abstract: 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 long standing 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 Schrodinger Problem which is solved by duality using an extension of the Sinkhorn algorithm, in the spirit of (De March and Henry-Labordere, 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 our technique extends to continuous-time stochastic volatility models, via what we dub VIX-Constrained Martingale Schrodinger Bridges, inspired by the classical Schrodinger bridge of statistical mechanics. The resulting stochastic volatility model is numerically implemented and is shown to achieve joint calibration with very high accuracy. Time permitting, we will also briefly discuss a few related topics: (i) a remarkable feature of the SPX and VIX markets: the inversion of convex ordering, and how classical stochastic volatility models can reproduce it; (ii) why, due to this inversion of convex ordering, and contrary to what has often been stated, among the continuous stochastic volatility models calibrated to the market smile, the local volatility model does not maximize the price of VIX futures. 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. The link for joining the meeting is: https://teams.microsoft.com/l/meetup-join/19%3ameeting_MzdlZDljNmMtZDU3Zi00Z...