We propose new closed-form pricing formulas for European options such as SPX, VIX and realized variance options. Our approach is based on the methodology of approximating a density function with an orthogonal expansion of polynomials weighted by an auxiliary density, which typically is analytically available. In this work, we expand the class of potential auxiliary densities to include those with an explicit and tractable characteristic function, yet retaining fast and accurate option prices approximations. We apply our method to price efficiently SPX and VIX option in a selection of non-standard models from the SVJJ setup of Duffie et al. (2000) for which traditional Fourier methods are computationally demanding. We conduct a series of joint calibration experiments to find that our option expansions yield essentially the same results as exact Fourier pricing but using only a small fraction of the computational time.