Seminar on Probability and Statistics

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Organizer(s) Nakahiro Yoshida, Hiroki Masuda, Teppei Ogihara, Yuta Koike

2024/06/18

13:00-14:10   Room #128 (Graduate School of Math. Sci. Bldg.)
Lorenzo Mercuri (University of Milan)
A compound CARMA(p,q)-Hawkes process for pricing financial derivatives (English)
[ Abstract ]
Recently, a new self-exciting point process with a continuous-time autoregressive moving average intensity process, named CARMA(p,q)-Hawkes model, has been introduced. The model generalizes the well-known Hawkes process by substituting the Ornstein-Uhlenbeck intensity with a CARMA(p,q) model where the associated state process is driven by the counting process itself. The new model maintains the same level of tractability of the Hawkes (e.g., Infinitesimal generator, backward and forward Kolmogorov equation, joint characteristic function and so on). However, it is able to reproduce more complex time-dependency structure observed in several market data.
Starting from this model, we introduce a Compound CARMA(p,q)-Hawkes with a random jump size independent of the counting and the intensity processes. This can be used as the main block for a new option pricing model, due to log-affine structure of the characteristic function of the underlying log-price driven by a pure jump compound CARMA(p,q)-Hawkes.
Further, we extend this model by scaling it with a measurable function of the time and the left-limit of the price itself. Exploiting the Markov structure of the new model, we derive the forward Kolmogorov equation that leads us to a Dupire-like formula. Some numerical results will also be presented.
[ Reference URL ]
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZ0rcOmvpjwuGNHx8ht0rMs1rD3HcEajoJv6