ESTIMATION OF A JUMP DIFFUSION PROCESS USING A CLOSED-FORM APPROXIMATION LIKELIHOOD UNDER A STRONG DISCRETIZATION SCHEME OF ORDER ONE

Abstract

The jump diffusion process plays a major role in financial modeling. The process was introduced for the first time by Merton in option pricing, to overcome the shortcoming of the Black Scholes formula. This study provides a closed-form expansion of the likelihood function to estimate the model parameters of the jump-diffusion process. We use the so called Jump adapted discretization to approximate the solution of the SDE under consideration. The approximation converges strongly with order one to the exact solution which is available in an explicit form for a few cases of model. That discretization presents some tractability that we used to derive the characteristic function of the process, and the method of saddlepoint is used thereon to get the transition probability. The process being a Markov process, the joint density is deduced as a product of the transition probabilities . Therefore, using a Monte-Carlo simulation, we carried out a maximization program on the closed likelihood function obtained for parameters estimation followed by a bootstrap to check the efficiency of the estimators.