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Bachelier Finance Society
Bachelier Finance Society One World Seminar
Julien Guyon (Bloomberg)

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

Since VIX options started trading in 2006, many researchers have tried to build a model that jointly and exactly calibrates to 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.
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