# On Deep-Fake Stock Prices and Why Investor Behavior Might Not Matter
## 2 Theoretical Background  
There has always been fascination with the pattern of stock price movements. It has been noticed that stock prices could be modeled as stochastic processes [1]. This spawned a revolution of sorts in Finance and Economics. It had two major consequences. On the one hand, it ushered in the concept of market efficiency; on the other hand, it eventually led to a closed-end formula for option-pricing. The Efficient Market Hypothesis (EMH henceforth) rose to preeminence in the 1970s. It represents one of the pillars of modern Finance and is closely linked to rational behavior and rational expectations [2; 3; 4; 5; 6]. The rationalization of the EMH is intricate and relatively arcane [7; 8]. At its very core, it posits that all relevant past information has been already reflected in current market prices. Although the past has been already accounted for, future changes are random. In principle, no one can derive any informational advantage based on public data. In other words, there is no crystal ball. Broad portfolio diversification and passive investment are the more sensible approaches in a world in which stock prices behave more like Brownian particles than the outcome of human craft. At first, econometrics and statistics have idealized price series by reducing them to a random walk (RW henceforth). The conceptual edifice of EMH is in fact a rationalization of this simple process. A random walk is an auto-regressive stochastic process with unit root. It has infinite memory, meaning that at any point in time, the current position of the stock price is the sum of all previous random shocks accumulated over the entire duration of the series [9]. The reasons why economists have for so long clung to the illusion of random assets prices despite evidence to the contrary is not always properly acknowledged. If stock prices were indeed random, we would have to concede a strong from of price predictability in exchange for a weaker and more modest one. It is an attempt to salvage at least some form of prediction. Random prices might not allow money managers to \"beat\" the market, but there would still be some residual form of prediction possible stemming from the underlying probability distributions built into those models. Many of the techniques employed by financial managers, including but not limited to portfolio optimization, dynamic hedging, and Value at Risk, would still be possible. The most important prediction still allowed under the RW hypothesis is that of no black swans. Randomness assigns vanishingly small probabilities to sudden and dramatic increases or crashes in stock prices, the likes of which we have seen in 1929, 1987, and 2010. But where would randomness come from to justify the approach? Stochastic modeling has been predicated on investor decision-making under risk. If events impacting investors' decision arise at random, following certain probability distributions, the randomness of initial conditions would carry over into the overall price-generating process. Even though agent interaction is deterministic, the initial conditions are not. Randomness is viewed as exogenous, yet it eventually drives the entire market behavior. A list of the largest price movements that occurred after World War II was compiled at the end of the 1980s by [PERSON], [PERSON], and [PERSON] [10]. The study attempts to match large price swings in the S&P500 to significant events, as reported in the media. Surprisingly, or perhaps not, these large price changes do not correspond to any momentous developments in finance, economics, or politics. Arguably, the best examples to illustrate this state of affairs are the crash of 19 October 1987 and the Flash Crash of 6 May 2010. The former is the second-largest percentage one day drop in the Dow Jones Industrial Average. The latter is the shortest-lived crash on record, having lasted mere minutes. To this day, scholars are still debating what caused them.