Local volatility of volatility for the VIX market (Q385648)

This article deals with the volatility index (VIX) market. More precisely, the authors propose and investigate a new approach, for directly modelling the VIX index. In fact, the traditional stochastic volatility models cannot fit an array of different VIX instruments. Therefore, researchers usually model an instantaneous variance or forward variances. However, in this work, the authors suggest modelling directly the VIX index itself in a mean-reverting local-volatility-of-volatility model. The authors obtain the key properties of their model and show theoretically how their model can provide a global fit to the VIX market. Following the existing literature for local volatility without mean reversion, they first set-up a relation between the local volatility-of-volatility surface and the surface of VIX options prices. Then, they construct the local volatility-of-volatility surface by adapting the results of [\textit{P. Carr}, ``Local variance gamma'', Working paper (2008)] and [\textit{J. Andreasen} and \textit{B. N. Huge}, ``Volatility interpolation'', Preprint (2010; \url{doi:10.2139/ssrn.1694972})] to processes with mean reversion. Finally, they show how to calibrate the local volatility-of-volatility model to the real market data.