# On Deep-Fake Stock Prices and Why Investor Behavior Might Not Matter
## 1 Introduction  
A significant number of researchers attempt to model the behavior of stock prices. The Efficient Market Hypothesis, predicated on rational expectations, has inspired a large number of stochastic models that have revolutionized Finance and Economics. As evidence of investor irrationality and cognitive distortions begins to mount, researchers increasingly use agent-based models to replicate the observed anomalies in the market data. Most of these models are quite elaborate and endow their agents with biases and complex behavior in an attempt to improve prediction.  
The contribution of our research consists of providing one of the most parsimonious agent-based models to date. The model is a cellular automata using 32 agents and is derived from [PERSON]'s Rule 110. The agents engage in buying, selling, and/or holding. Each agent is endowed with a starting balance sheet marked-to-market in each iteration. The simulation allows for margin calls for both buying and selling. During each iteration, the number of buy, hold, and sell positions is aggregated into a market price with the help of a simple, linear formula. The formula generates a price that is dependent on the number of buy and sell positions. By varying the initial conditions and the interaction rules among agents, we are able to generate a very large number of data series.  
Here, we are not concerned with stock price prediction, nor are we concerned with the \"realism\" of agent behavior, nor do we seek to reproduce the true functional relationships of a real market. The resemblance that our model bears to real life represents a psychologicalanchor rather than a functional consideration. We select a small sub-sample of 600 data series, and we notice that 42 of the generated data series bear close resemblance to real market data. Using a battery of commonly used statistical tests, we find it is almost impossible to tell the difference between our agent-based model generated data and real stock prices. It appears that the visual pattern of prices, the pattern of volatility clustering, the characteristics of first order differences, and the return distributions are virtually indistinguishable. We have been able indeed to generate a deep-fake financial asset price.  
To the extent to which our cellular automata rules are the computational equivalent of investor behavior, it appears the debate around the impact of investor behavior on stock prices is overrated. All it takes to generate the observed market data complexity is a simple local interaction rule with feedback. That investors behave irrationally and display all sorts of cognitive distortions instead of rational expectations might be a mere incidental aspect rather than a necessary condition. The most unsettling finding is that we don't even need moral agents to generate an outcome of equivalent complexity with that of the market.  
The findings presented in this paper have implication for the regulation of financial markets as well. To the extent of which agent interaction leads to a critical process poised at the edge of chaos, the behavior of the system becomes undecidable and computationally irreducible. Some policies and regulations would have no discernible impact on the overall market behavior in the long run, whereas others would be impossible to predict. Policy makers and regulators appear forever destined to grapple with the unexpected and unintended consequences of their decisions.  
In the next section we provide a brief theoretical background in which we discuss the trials and tribulations of stock price modeling in finance. In Section 3 we introduce our model and describe its functioning. In Section 4 we present our main results. A detailed discussion of the results and their theoretical and practical implications is provided in Section 5. Section 6 concludes.