MODELING CHAOTIC BEHAVIOUR OF STOCK PRICE INDEX USING ITO STOCHASTIC DIFFERENTIAL EQUATION

Abstract

Stochastic differential equation (SDE) have become an important tool for modeling the dynamics
of many random phenomena such as financial assets. In real applications, parameters of the
equation are unknown and need to be estimated and many times only discretely sampled data of
the process are available. Financial assets such as stock price are very chaotic and dynamic and
are often represented using stock price index, to reflect overall market sentiments and directions
of stock prices. Investing in stocks or equities is a speculative risk that is complex and
complicated to understand due to its chaotic behaviour. In this paper, attempt was made to study
this chaotic bahaviour via Ito SDE, the forward Kolmogorov equation (FKE). The parameters
estimation was done using Euler-Maruyama method. The model’s mean, variance and Akaike
Information Criterion (AIC) were obtained as 0.08, 896.56 and 4764.08 respectively, as against
ARIMA (1,0,0), (3,1,1) and (6,0,0) having AIC values of 5482.92, 5401.00 and 5433.50,
respectively. Hence the Ito SDE was better in describing stock price index and is therefore
recommended for practitioners and policy makers for sound decision making regarding stocks.
Keywords: Chaos; Diffusion process; Kolmogorov equation; nonlinear dynamics; stochastic
differential equation; Stock price index.