A Self-Exciting Threshold Jump-Diffusion Model for Option Valuation

Abstract: 

In this talk, we shall discuss a self-exciting threshold jump-diffusion model for option valuation. This model incorporates regime switches without introducing an exogenous stochastic factor process. A generalized version of the Esscher transform is adopted to select a pricing kernel. The valuation of both the European and American contingent claims is considered. A piecewise linear partial differential-integral equation governing a price of a standard European contingent claim is obtained. For an American contingent claim, a formula decomposing a price of the American claim into the sum of its European counterpart and the early exercise premium is provided. An approximate solution to the early exercise premium based on the quadratic approximation technique is obtained for a particular case where the jump component is absent. Some numerical results for European and American options are presented for the case without jumps.