In this paper, we introduce a large system of interacting financial agents in which all agents are faced with the decision of how to allocate their capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model [
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Figure 4. Quantile-quantile plot of logarithmic stock return distribution (left-hand side) and logarithmic return of fundamental prices (right-hand side). The simulation has been performed in the case of long-term investors and a stochastic fundamental price. The risk tolerance has been set to $ \gamma = 0.9 $, the scale to $ \rho = \frac{5}{8} $ and the random seed is chosen to be $\texttt{rng(767)}$. All further parameters are chosen as reported in section A.4 of the Appendix
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Sketch of the modelling process
Example of the value function
Stock price evolution in the long-term investor case with a constant fundamental price
Quantile-quantile plot of logarithmic stock return distribution (left-hand side) and logarithmic return of fundamental prices (right-hand side). The simulation has been performed in the case of long-term investors and a stochastic fundamental price. The risk tolerance has been set to
Stock price distribution in the long-term investor case. The solid lines are analytical solution, whereas the circles are the numerical result
Distribution of the wealth invested in stocks with a Gaussian fit (solid line). Left figure has a linear scale, whereas the right figure shows the distribution in log-log scale
Distribution of the wealth invested in bonds in the special case
Stock price distribution in the high-frequency case (red circles). The fit by the inverse-gamma distribution (solid line) clearly underestimates the tail. This reveals that the full model can create heavier tails than the inverse-gamma distribution
Marginal wealth distributions in the high-frequency investor case. The left hand side illustrates the distribution of investments in stocks and the right-hand side the wealth invested in bonds at
Steady state stock price distribution in the high-frequency investor case (circles) together with the analytically computed steady state of inverse-gamma type (solid line)