SIAM Journal on Financial Mathematics, Volume 11, Issue 4, Page 1063-1097, January 2020.
The rough Heston model has recently attracted the attention of many finance practitioners and researchers as it maintains the basic structure of the classical Heston model while having descriptive capabilities in terms of microstructural foundations of the market. Using the fact that the characteristic function of log-price in this model could be expressed in terms of the solution of a nonlinear parametric fractional Riccati differential equation not admitting a closed-form solution, devising efficient numerical schemes for pricing and calibration under this model has become a crucial need in the computational finance community. Although the fractional Adams method has been used in most of the recent studies on the rough Heston model, this method suffers from some stability and convergence issues in treating the problem. In this paper, we present a numerical method based on Newton--Kantorovich quasi-linearization to solve the nonlinearity issue followed by spectral collocation based on “polyfractonomials" to approximate the fractional derivative in an accurate and efficient manner. We provide sufficient conditions under which our method is convergent and the order of convergence is also obtained. In order to guarantee the specified convergence rate, we first prove some regularity results on the linearized problem and then employ the proposed scheme to solve a practical calibration problem from the SPX options market. The efficiency of the proposed method is illustrated by comparing the obtained results with those of the fractional Adams method as well as a fast hybrid scheme based on fractional power series expansion.

中文翻译：

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用准线性化和谱配点法求解由粗糙Heston模型引起的参数分数阶微分方程

SIAM金融数学杂志，第11卷，第4期，第1063-1097页，2020年1月。
粗略的Heston模型最近吸引了许多金融从业者和研究人员的注意力，因为它保持了经典Heston模型的基本结构，同时在市场的微观结构基础上具有描述能力。利用该模型中对数价格的特征函数可以通过非线性参数分数Riccati微分方程不包含封闭形式的解的表达式来表示的事实，在该模型下设计有效的定价和标定数值方案成为计算金融界的关键需求。尽管分数Adams方法已用于粗糙Heston模型的最新研究中，但该方法在处理该问题时存在一些稳定性和收敛性问题。在本文中，