IMPLICIT METHODS FOR SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH APPLICATION TO BLACK- SCHOLES MERTON OPTION PRICE MODEL

Abstract

This research examined the solutions of linear stochastic differential equations using im- plicit methods. The implicit methods developed include; Drift-Implicit Euler-Maruyama Method (DIEMM), Drift-Implicit Milstein Method (DIMLSTM) and Semi-Implicit Mil- stein Method (SIMLSTM). These methods were developed using Itô Lemma. Two linear stochastic differential equations in the form of Black-Scholes Merton Option Price Model (BSMOPM) used in financial setting were considered. The effect of varying stepsize of the methods were also examined. The implementation of the methods was carried out us- ing the developed methods to determine the numerical solution of Black-Scholes Merton Option Price Model. Absolute errors were obtained and the performance of the methods were compared using mean absolute error criterion. The order of convergence of each method was examined and the accuracy of the methods were compared with the existing explicit methods. The results of this research showed that the DIEMM performed bet- ter than the existing Explicit Euler Maruyama Method (EEMM) in literature for some selected linear stochastic differential equations in form of BSMOPM. The results also showed that the DIMLSTM and SIMLSTM compares favorably with the existing Ex- plicit Milstein Method (EMLSTM) for some selected problems. In reality, the financial market can be highly volatile. This volatility can present significant investment risks. The results obtained in this work also showed that higher volatility corresponds to a higher possibility of a declining market which results to instability within the market. Graphical solutions of the model were considered using the developed methods.