Bond and option pricing for interest rate model with clustering effects

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.