Modelling the stock market using a multi-scale approach

Mathematical modelling is one of the fundamental elements in the modern financial industry, playing vital roles in terms of decision making, risk management, financial innovation, and government regulation. The financial market has attracted extensive research interest from academics due to its strategic importance to the global economy. With the increased understanding of the market, the financial industry has developed towards a direction of a structured and refined system, thanks to a large number of financial instruments carefully engineered to meet the demand from the investors. On the other hand, fundamental research such as risk modelling remains challenging. The advancement of financial models does not prevent the market from financial crisis or mitigate the consequence of crash. Bearing in mind that modelling is an abstract of reality, the current research takes a step back to examine one of the corner stones of financial modelling: the efficient market hypothesis, the Gaussian statistics and the Brownian motion, as well as the process of data analyses for the modelling inputs. Using the context of the stock market, a systematic approach is adopted based on a variety of data from different stock markets during different periods. Some interesting statistical findings are presented in a quantitative manner, providing both the confirmation of the non-Gaussian statistics and empirical understandings to the market movements. Meanwhile, two different research methodologies are adopted to model the empirical findings: 1) macroscopic and phenomenological modelling based on analysing statistical data, and 2) microscopic and mechanism-based modelling based on understanding the behaviours of the market players. By taking advantage of both modelling methodologies, a multi-scale modelling approach is proposed in the current research. A step by step method is used to pin down the essential mechanisms that lead to the market inefficiency and non-Gaussian statistics. It is shown that the proposed approach requires a small number of input parameters by maximising the information obtained from market performance data and market microstructure. It is also shown that the multi-scale modelling approach, facilitated by a systematic empirical study, will greatly enhance both our understandings on the micro-foundations of the stock market and the applicability of the classical models widely used by the modern financial industry.