Microscopic process of the rough non-affine Heston model and its application
摘要
Recent research in financial volatility modeling demonstrates that rough volatility models, driven by fractional Brownian motion (fBm) with a small Hurst parameter, play a crucial role in understanding the implied volatility surface and improving option pricing. However, the affine structure of the standard Heston model only provides analytical expressions for the characteristic function and density function of the variance process under specific forms of volatility dynamics. To overcome this limitation, this paper introduces a non-affine coefficient, enabling the model to more flexibly capture complex characteristics of asset returns such as excess kurtosis and heavy tails. At the macro level, we replace the Brownian motion in the non-affine Heston model with fBm, constructing the rough non-affine Heston model. At the micro level, we establish a theoretical foundation for the model by linking nearly unstable Hawkes processes to fractional volatility models. Finally, empowered by a robust Lifted Semi-Implicit Euler scheme and the CRN technique, we empirically calibrate the models using S&P 500 index options during the March 2020 crash. The results demonstrate that the rough non-affine model outperforms its affine counterpart by capturing extreme left-tail skewness and drastically reducing downside pricing errors.