A simple approach to determining the Gaussian kernel that constitutes the backbone of the multi-factor Heston model is proposed based on a suitable expansion in powers of volatilities of volatilities. This analysis provides Black-Scholes-like formulas for pricing European vanilla options, allowing for accurate approximations of the option prices under the multi-factor Heston model up to volatilities of volatilities on the order of 50%. The analysis also leads to a simple formula for the implied volatility showing that changes in the convexity of the volatility smile are due only to price skewness, and an easy formula to reproduce volatility indices via the realized volatility. Interestingly, the variance of the Gaussian kernel is equal to the variance of the continuously compounded return in the case of the Heston model. The empirical analyses presented assess the potential of our approach to capture market distortions while adequately forecasting the dynamics of the VIX index
