Stochastic volatility models and the pricing of VIX options (Q2847239)

The paper examines and compares empirically the performance of eight continuous-time stochastic volatility models in their ability to describe the behavior of the VIX index. All considered models are nested in a general model \(dV=(c_{1}+\frac{c_{2}}{V}+ c_{3}V\ln V+ c_{4}V+ c_{5}V^{2} )dt+kV^{\gamma}dZ\), where \(Z\) is a Wiener process. The applied estimation technique is the generalized methods of moments from [\textit{L. P. Hansen}, Econometrica 50, 1029--1054 (1982; Zbl 0502.62098)]. The data under scrutiny include the VIX values from the period 1990--2009. The empirical analysis indicates that the best is the \(3/2\) volatility model with a quadratic drift (see, e.g. [\textit{A. L. Lewis}, Option valuation under stochastic volatility. Newport Beach, CA: Finance Press (2000; Zbl 0937.91060)]). The equation of the winning model is \(dV=(aV+bV^{2} )dt+cV^{ \frac{3}{2}}dZ\) and, according to the authors, the diffusion term is crucial for its good fit. The authors provide an analytic solution for the value of a call option on the VIX under the \(3/2\) model, and use it to calibrate at-the-money market option prices.