Mathematical Finance & Financial Data Science Seminar

The Joint S&P 500/VIX Smile Calibration Puzzle Solved

Speaker: Julien Guyon, Bloomberg L.P.

Location: Warren Weaver Hall 1302

Date: Tuesday, April 21, 2020, 5:30 p.m.

Synopsis:

Since VIX options started trading in 2006, many researchers have tried to build a model that jointly and exactly calibrates th the prices of S&P 500 (SPX) options, VIX futures and VIX options.  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 using a completely different approach:  a nonparametric discrete-time model.  The model is cast as a dispersion-constrained martingale transport problem which is solved using the Sinkhorn algorithm. We prove by duality that the existence of such 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 briefly discuss:

(i)  how our technique extends to continuous-time stochastic volatility models;

(ii)  a remarkable feature of the SPX and VIX markets:  the inversion of convex ordering and how classical stochastic volatility models can reproduce it;

(iii)  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 Dupire local volatility model does not maximize the price of VIX futures.

Bio - Julien Guyon

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.  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 Paris Tech.  He co-authored the book Nonlinear Option Pricing (Chapman & Hall, 2014) with Pierre Henry-Labordere.  His main research interests include nonlinear option pricing, volatility and correlation modeling, 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 in top-tier newspapers such as The New York Times, Le Monde, and El Pais, including a new fairer draw method for the FIFA World Cup.