Priority of Applying Stochastic Differential Equations over their Ordinary Counterparts in Accurately Predicting Stock Prices

Abstract

Background and aims. Despite the abundance of deterministic models designed to study applied issues in biology, physics, economics, and other fields, which have undergone continuous development and modification to achieve satisfactory future predictions, it is of utmost importance to note that prediction is crucial in the field of economics. It allows sellers and buyers to determine the optimal time for buying and selling. However, as known, market movements are not stable but are subject to numerous and continuous random fluctuations. Therefore, any mathematical model dealing with financial issues must take this into consideration. The aim of this work is to study the solution behavior of one of the most important stochastic economic models, the Black-Scholes model, to demonstrate its superiority over its deterministic counterpart by comparing their results with actual prices. Methods. Prior to studying the stochastic model, its deterministic counterpart was solved, which is represented by an ordinary differential equation. Since calculus of stochastic functions is not very common and differs completely from the known calculus, a series of definitions were provided to explain how to handle them, leading to one of the most important formulas, known as the Itô formula, which was extensively utilized to obtain an explicit solution for the Black-Scholes equation. Additionally, the forms and nature of the solutions were clarified using the MATLAB program. Results. Based on realistic data of a commodity's prices over a four-year period, the necessary coefficients values were calculated, and then substituted into the solution formula for both the deterministic and stochastic models to predict the expected price in both cases after a quarter of a year. The actual price was then revealed, and the results were recorded. Conclusion. By comparing the results presented in the table, it is evident that the expected price using the stochastic model is much closer to the actual price than that predicted by the deterministic model, indicating its superiority in prediction accuracy.