Joint calibration of S&P 500 and VIX options under local stochastic volatility models

It is extremely challenging to design a model calibrating both SPX and VIX option prices. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate these two quantities with a continuous model. So far, most studied continuous time models are affine, so we investigate the conjecture among 14 well-known non-affine local stochastic volatility models in this article. First, we propose a unified efficient willow tree method for S&P500 and VIX option pricing under non-affine models. Second, we compare the joint calibration performance on these 14 models on the S&P500 and VIX option prices data from 2006 to 2019. We find that the VIX option price data can provide extra information of the variance dynamics of the models, and the non-affine structure and the volatility with linear, rather than square-root, diffusion process provide a better fit for the both data sets than the affine counterparts. Among the 14 stochastic models, the SABR model provides the best in- and out-of-sample performance (in terms of mean square error) regardless of the state of the economy. Nevertheless, even for the best-fitted SABR model, the relative error on the VIX option is still around 18%, still quite significant. Therefore, we found the non-affine and local volatility structure improve the joint calibration but are still far from satisfactory.