In this paper we propose semiclosed‐form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. We illustrate the advantage of the proposed analytic expressions by comparing them with those obtained from benchmark Monte–Carlo simulations. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.

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Leveraged exchange-traded funds (LETF) are newly introduced ETFs that have become increasingly popular. It closely tracks the value of an underlying index while allowing for additional leverage. In this paper, we consider the valuation of options written on LETF under two popular affine GARCH models, the Heston–Nandi model and the inverse Gaussian GARCH model. We also calibrate the two models using market data, and demonstrate the superior pricing performance.