Time-inhomogeneous Hawkes processes and its financial applications

We consider time-inhomogeneous Hawkes processes with an exponential kernel, and we analyze some properties of the model. Time-inhomogeneity for the Hawkes process is indispensable for short rate models or for other calibration purposes, while financial applications for the time-homogeneous case already well known. Distributional properties for such a model generate computational tractability for a financial application. In this paper, moments and the Laplace transform of time-inhomogeneous Hawkes processes are obtained from the distributional properties of the underlying processes. As an applications to finance, we investigate the pricing formula for zero-coupon bonds when short-term interest rates are governed by the time-inhomogeneous Hawkes process. Numerical illustrations are also provided. As an illustrative example, we apply the derived moments and Laplace transform of time-inhomogeneous Hawkes processes to the pricing of zero-coupon bonds within a financial context. By considering the short-term interest rate as driven by inhomogeneous Hawkes processes, we develop explicit formulae for valuing zero-coupon bonds. This application is particularly relevant for modeling interest rate dynamics in real-world scenarios, allowing for a more nuanced understanding of pricing dynamics. Through numerical illustrations, we demonstrate the computational tractability of our approach, showcasing its practical utility for financial practitioners and providing insights into the intricate interplay between time-inhomogeneous Hawkes processes and bond pricing in dynamic markets.