Financial Math Seminar: Dr. Julien Guyon (Columbia & NYU ) @ Whitehead 304
Title: On the joint calibration of SPX and VIX options
Abstract: Since VIX options started trading in 2006, many researchers have attempted to build a model for the SPX that jointly calibrates to SPX and VIX options. In 2008, Jim Gatheral showed that a diffusive model could approximately, but not exactly, fit both markets. Later, others have argued that jumps in the SPX were needed to jointly calibrate both markets. We revisit this problem, asking the following questions: Does there exist a continuous model on the SPX that jointly calibrates to SPX options, VIX futures, and VIX options? If so, how to build one such model? If not, why? We present a novel approach based on the SPX smile calibration condition. In the limiting case of instantaneous VIX, the answers are clear and involve the timewise convex ordering of two distributions (local variance and instantaneous variance) and a novel application of martingale transport to finance. The real case of a 30-day VIX is more involved, as time-averaging and projection onto a filtration can undo convex ordering. We observe that in usual market conditions the distribution of VIX^2 in the local volatility model and the market-implied distribution of VIX^2 are not in convex order, and we show that fast mean-reverting volatility models and rough volatility models are able to reproduce this surprising behavior.