A Stochastic Model of the Dynamics of Change in Stock Price

Abstract

The solutions of many mathematical models resulting in stochastic differential equations are based on the assumption that the drift and the volatility coefficients were linear functions of the solutions. We formulated a model whose basic parameters could be derived from observations over discretized time intervals rather than the assumption that the drift and the volatility coefficients were linear functions of the solutions. We took into consideration the possibility of an asset gaining, losing or stable in a small interval of time instead of the assumption of the Binomial Asset pricing models that posited that the price could appreciate by a factor p or depreciate by a factor 1-p. A multi-dimensional stochastic differential equation was obtained whose drift is the expectation vector and the volatility the covariance of the stocks with respect to each other. The resulting system of stochastic differential equations was solved numerically using the Euler Maruyama Scheme for multi-dimensional stochastic differential equations through the use of a computer program written in MatLab. We obtained a realization of the evolutions of their prices over a chosen interval of time