Convergence, non-negativity and stability of a new Lobatto IIIC-Milstein method for a pricing option approach based on stochastic volatility model

Abstract

Recently, stochastic differential equation (SDE) has been used for many applications in option pricing models which satisfy the non-negativity. So, constructing new numerical method preserves non-negativity for solving SDE is very important. This paper investigates the numerical analyses; convergence, non-negativity and stability of the multi-step Milstein method for SDE. We derive the new general s-stage Milstein method; the Lobatto IIIC-Milstein method for nonlinear SDE and show that the numerical solution preserves non-negativity. Moreover, we prove the strong convergence order 1.0 of the numerical method. The unconditional stability results are proven for SDE. In order to get insight into the numerical analysis of the proposed method; the Black–Scholes model is considered to explain that the exact mean square stability region is totally contained in the numerical region (i.e. the numerical method is stochastically A-stable). In addition, the accuracy and computational cost are discussed. Finally, the Lobatto IIIC-Milstein method was compared with existing Milstein type methods, Monte-Carlo and finite difference methods to examine the efficiency of the proposed method to value the price.