ON MILSTEIN SCHEME FOR SOLVING GEOMETRIC BROWNIAN MOTION (GBM) EQUATION
Keywords:Stochastic Differential Equations (SDEs), strong and weak convergence, Milstein method
This study showed the application of explicit Euler and Milstein for solving a Geometric Brownian Motion (GBM). Using the Euler explicit scheme, it was observed that when the price of an asset at the initial time is positive, then the volatility of the asset is always positive; while if the price of the asset at the initial time is negative, the volatility of the asset is also negative. The GBM, discretizing in time T= 2, using the explicit Euler Scheme with constant volatility and drift shows the effect of random walk in stock prices. This shows that the degree of random walk is not entirely centered, and as such with timely variation of the drift and parameters can savage the stock price situation also the GBM through explicit Milstein scheme produced a chaotic process whose random walk is clustered with constant drift and volatility parameters. This suggests that the stock price situation will unlikely be savaged if the stock price market is sabotaged.