Interest Rates Term Structure Models Driven by Hawkes Processes

This paper includes a marked Hawkes process in the original Heath–Jarrow–Morton (HJM) setup and investigates the impact of this assumption on the pricing of the popular vanilla fixed-income derivatives. Our model exhibits a smile that can fit the implied volatility of swaptions for a given key rate (tenor). We harness the log-normality of the model, conditionally with respect to jumps, and derive formulae to evaluate both caplets/floorlets and swaptions. Our model exhibits negative jumps on the zero-coupon (hence positive on the rates). Therefore, its behavior is compatible with the situation where globally low interest rates can suddenly show a cluster of positive jumps in case of tensions on the market. One of the main difficulties when dealing with the HJM model is to keep a framework that is Markovian. In this paper we show how to preserve the relevant features of the Hull and White version, especially the reconstruction formula that provides the zero-coupon bonds in terms of the underlying model factors.