Seminar Lecture by Dr. Carlo Sgarra [CANCELLED]
The seminar lecture is cancelled due to epidemic situation in Norway.
We propose an extension of the Gamma-OU Barndorff-Nielsen and Shephard model taking into account jumps 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 introduce a measure change of Esscher type in order to describe the relation between the risk-neutral and the historical dynamics. 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 model proposed exhibits a larger flexibility in comparison with the Gamma-OU model, in spite of the same number of parameters required. In particular, we illustrate numerically that the left wing implied volatility could be first fit by using the original Gamma-OU model and then the right wing can be arranged by a trigger of the intensity and variance processes. Moreover, implied volatility of variance swap options is upward-sloped due to the self-exciting property of Hawkes processes.