An Investigation into Variations of Brownian Motion: Towards a Deeper Understanding of Financial Asset Pricing

Modelling the asset returns distribution has been the focal point of modern finance for almost a century. The extensively studied and applied Geometric Brownian Motion (GBM) modelling process provides the returns distribution to asset prices which is normally distributed. However historical asset returns are skewed and possess excess kurtosis, indicating that a returns distribution has a thicker tail when compared with the normal distribution. Numerous alternate distributions have been proposed to model asset returns, however these distributions are imposed on data exogenously with complex equations for parameter estimation. This innovative research modifies the GBM model by embedding an extra factor to capture leptokurtosis of historic data. This extra factor incorporates a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, substantially outperform the basic GBM model in terms of fitting the returns distribution of historic data price indices. Furthermore this research provides an interpretation of the additional stochastic term in relation to irrational behaviour in financial markets. An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, substantially outperform the basic Geometric Brownian Motion model in terms of fitting the returns distribution of historic data price indices. Furthermore we attempt to provide an interpretation of the additional stochastic term in relation to irrational behaviour in financial markets and outline the importance of this novel model.
Following a Geometrical Brownian Motion extension into an Irrational fractional Brownian motion model, we re-examine irrational agent behaviour reacting to time dependent news on the log-returns for modifying a financial market evolution. We specifically discuss the role of financial news or economic information positive or negative feedback of such irrational (or contrarian) agents upon the price evolution. We observe a kink-like effect reminiscent of soliton behaviour, suggesting how analysts' forecasts errors induce stock prices to adjust accordingly, thereby proposing a measure of the irrational force in a market.
This research also reports a new methodology and results on the forecast of the numerical value of the fat tail(s) in asset returns distributions using the Irrational fractional Brownian Motion Model. Optimal model parameter values are obtained from fits to consecutive daily two-year period returns of S&P500 index over [1950-2016], generating 33-time series estimations. Through an econometric model, the kurtosis of returns distributions is modelled as a function of these parameters. Subsequently an auto-regressive analysis on these parameters advances the modelling and forecasting of kurtosis and returns distributions, providing the accurate shape of returns distributions and measurement of Value at Risk.