We propose an extension of the \(\Gamma \)-OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the \(\Gamma \)-OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the \(\Gamma \)-OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the \(\Gamma \)-OU Barndorff-Nielsen and Shephard model.

考虑到跳跃聚类现象，我们建议扩展\（\ Gamma \）- OU Barndorff-Nielsen和Shephard模型。我们假设霍克斯驱动器的强度过程与方差过程一致，直到一个常数。通过应用带有移民的连续状态分支过程的理论，我们证明了控制资产价格动态的SDE强解的存在性和唯一性。为了描述风险中性与历史动态之间的关系，我们提出了一种自激式Esscher类型的量度变化，表明\（\ Gamma \）-OU Hawkes框架在此概率变化下保持稳定。通过利用模型的仿射特征，我们为资产对数收益的拉普拉斯变换，其二次方差和方差过程的遍历分布提供了一种显式形式。我们表明，与\（\ Gamma \）相比，所提出的模型具有更大的灵活性-OU模型，尽管需要相同数量的参数。我们通过特征函数反演技术对市场原始期权价格模型进行了校准，研究了价格敏感性并提出了精确的仿真方案。主要的财务成就是，由于Hawkes流程的自激特性，在VIX上书写的期权的隐含波动性向上成形，与\（\ Gamma \）- OU Barndorff-Nielsen和Shephard所表现出的通常的向下倾斜形成对比模型。