Before revision:  They are also satisfied empirically by the option data on the S&P 500 over a ten yearsperiod. Our proposed model outperforms the widely used surface stochastic volatility inspired  model on the mean average percentage error in both in-sample and out-of-sample datasets.
After revision:  They are also satisfied empirically by the option data on the S&P 500 index over twenty years. The developed neural network model and its simplified variations outperform the widely used surface stochastic volatility inspired model on the mean average percentage error in both in-sample and out-of-sample datasets.