Unified Approach for the Affine and Non-affine Models: An Empirical Analysis on the S&P 500 Volatility Dynamics

Being able to generate a volatility smile and adequately explain how it moves up and down in response to changes in risk, stochastic volatility models have replaced BS model. A single-factor volatility model can generate steep smiles or flat smiles at a given volatility level, but it cannot generate both for given parameters. In order to match the market implied volatility surface precisely, Grasselli introduced a 4/2 stochastic volatility model that includes the Heston model and the 3/2 model, performing as affine and non-affine model respectively. The present paper is intended to further investigate the 4/2 model, which falls into four parts. First, we apply Lewis’s fundamental transform approach instead of Grasselli’s method to deduce PDEs, which is intuitional and simple; Then, we use a result derived by Craddock and Lennox using Lie Symmetries theory for PDEs, and the results are more objective and reasonable; Finally, through adopting the data on S&P 500, we estimate the parameters of the 4/2 model; Furthermore, we investigate the 4/2 model along with the Heston model and the 3/2 model and compare their different performances. Our results illustrate that the 4/2 model outperforms the Heston and the 3/2 model for the fitting problem.