Consistent and efficient pricing of SPX and VIX options under multiscale stochastic volatility

We provide consistent and efficient pricing for both Standard & Poor's 500 Index options and the Chicago Board Options Exchange's Volatility Index options under a multiscale stochastic volatility model. To capture the multiscale, our model adds a fast scale factor to Heston's volatility and we derive approximate analytic formulas for the options under the model. The analytic tractability can greatly improve the efficiency of calibration compared to fitting procedures using a numerical scheme. Our experiment using options data for 3 years shows that the model reduces about 20% of the errors for a single-scale model.