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.
