In this paper, we extend the 3/2-model for VIX studied by Goard and Mazur (2013) and introduce generalized 3/2 and 1/2 classes for volatility. Under these models, we study the pricing of European and American VIX options and, for the latter, we obtain an early exercise premium representation using a free-boundary approach and local time-space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type.
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