  We propose a mathematical model for the word-of-mouth communications among
stock investors through social networks and explore how the changes of the
investors' social networks influence the stock price dynamics and vice versa.
An investor is modeled as a Gaussian fuzzy set (a fuzzy opinion) with the
center and standard deviation as inputs and the fuzzy set itself as output.
Investors are connected in the following fashion: the center input of an
investor is taken as the average of the neighbors' outputs, where two investors
are neighbors if their fuzzy opinions are close enough to each other, and the
standard deviation (uncertainty) input is taken with local, global or external
reference schemes to model different scenarios of how investors define
uncertainties. The centers and standard deviations of the fuzzy opinions are
the expected prices and their uncertainties, respectively, that are used as
inputs to the price dynamic equation. We prove that with the local reference
scheme the investors converge to different groups in finite time, while with
the global or external reference schemes all investors converge to a consensus
within finite time and the consensus may change with time in the external
reference case. We show how to model trend followers, contrarians and
manipulators within this mathematical framework and prove that the biggest
enemy of a manipulator is the other manipulators. We perform Monte Carlo
simulations to show how the model parameters influence the price dynamics, and
we apply a modified version of the model to the daily closing prices of fifteen
top banking and real estate stocks in Hong Kong for the recent two years from
Dec. 5, 2013 to Dec. 4, 2015 and discover that a sharp increase of the combined
uncertainty is a reliable signal to predict the reversal of the current price
trend.
