  This study provides a consistent and efficient pricing method for both
Standard & Poor's 500 Index (SPX) options and the Chicago Board Options
Exchange's Volatility Index (VIX) options under a multiscale stochastic
volatility model. To capture the multiscale volatility of the financial market,
our model adds a fast scale factor to the well-known Heston volatility and we
derive approximate analytic pricing formulas for the options under the model.
The analytic tractability can greatly improve the efficiency of calibration
compared to fitting procedures with the finite difference method or Monte Carlo
simulation. Our experiment using options data from 2016 to 2018 shows that the
model reduces the errors on the training sets of the SPX and VIX options by
9.9% and 13.2%, respectively, and decreases the errors on the test sets of the
SPX and VIX options by 13.0\% and 16.5\%, respectively, compared to the
single-scale model of Heston. The error reduction is possible because the
additional factor reflects short-term impacts on the market, which is difficult
to achieve with only one factor. It highlights the necessity of modeling
multiscale volatility.
