Wall Street is not only a place of facilitating the money flow, but also a playground for scientists.

When I was young, I saw one of my uncles plotting prices for stocks to perform technical analysis. When I was in college, my friends often talked about investing in a few financial futures and options. When I was doing my graduate degree in physics, we studied John Hull’s famous textbook [Hull 2011] on quantitative finance to learn about financial modeling. A few of my classmates went to Wall Street to become quantitative analysts or financial software developers. There are ups and downs in the financial markets. But as long as we are in a capitalist society, finance is a subject we never ignore. However, scientists have not come up with a consensus about the nature of a financial market.

Agent-Based Models

Economists believe that individuals in a market are rational being who always aim at maximizing their profits. They often apply agent-based models, which employs complex system theories or game theory.

Random Processes and Statistical Physics

However, a lot of mathematicians in Wall Street (including quantitative analysts and econophysicists) see the stock prices as undergoing Brownian motion. [Hull 2011, Baaquie 2007] They employ tools in statistical physics and stochastic processes to study the pricings of various financial derivatives. Therefore, the random-process and econophysical approaches have nothing much about stock price prediction (despite the fact that they do need a “return rate” in their model.) Random processes are unpredictable.

However, some sort of predictions carry great values. For example, when there is overhypes or bubbles in the market, we want to know when it will burst. There are models that predict defaults and bubble burst in a market using the log-periodic power law (LPPL). [Wosnitza, Denz 2013] In addition, there has been research showing the leverage effect in stock markets in developed countries such as Germany (c.f. fluctuation-dissipation theorem in statistical physics), and anti-leverage effect in China (Shanghai and Shenzhen). [Qiu, Zhen, Ren, Trimper 2006]

Reconciling Intelligence and Randomness

There are some values to both views. It is hard to believe that stock prices are completely random, as the economic environment and the public opinions must affect the stock prices. People can neither be completely rational nor completely random.

There has been some study in reconciling game theory and random processes, in an attempt to bring economists and mathematicians together. In this theoretical framework, financial systems still sought to attain the maximum entropy (randomness), but the “particles” in the system behaves intelligently. [Venkatasubramanian, Luo, Sethuraman 2015] (See my another blog entry: MathAnalytics (1) – Beautiful Mind, Physical Nature and Economic Inequality) We are not sure how successful this attempt will be at this point.

Sentiment Analysis

As people are talking about big data in recent years, there have been attempts to apply machine learning algorithms in finance. However, scientists tend not to price using machine learning algorithms because these algorithms mostly perform classification. However, there are attempts, with natural language processing (NLP) techniques, to predict the stock prices by detecting the public emotions (or sentiments) in social media such as Twitter. [Bollen, Mao, Zeng 2010] It has been found that measuring the public mood in a few dimensions (including Calm, Alert, Sure, Vital, Kind, and Happy) allows scientists to accurately predict the trend of Dow Jones Industrial Average (DJIA). However, some hackers take advantage on the sentiment analysis on Twitter. In 2013, there was a rumor on Twitter saying the White House being bombed, The computers responded instantly and automatically by performing trading, causing the stock market to fall immediately. But the market restored quickly after it was discovered that the news was fake. (Fig. 1)

Fig. 1: DJIA fell because of a rumor of the White House being bombed, but recovered when discovered the news was fake (taken from http://www.rt.com/news/syrian-electronic-army-ap-twitter-349/)

P.S.: While I was writing this, I saw an interesting statement in the paper about leverage effect. [Qiu, Zhen, Ren, Trimper 2006] The authors said that:

Why do the German and Chinese markets exhibit different return-volatility correlations? Germany is a developed country. To some extent, people show risk aversion, and therefore, may be nervous in trading as the stock price is falling. This induces a higher volatility. When the price is rising, people feel safe and are inactive in trading. Thus, the stock price tends to be stable. This should be the social origin of the leverage effect. However, China just experiences the first stage of capitalism, and people are somewhat excessive speculative in the financial markets. Therefore, people rush for trading as the stock price increases. When the price drops, people stay inactive in trading and wait for rising up of the stock price. That explains the antileverage effect.

Does this paragraph written in 2006 give a hint of what happened in China in 2015 now? (Fig. 2)

Fig. 2: The fall of Chinese stock market in 2015 (taken from http://www.economicpolicyjournal.com/2015/06/breaking-biggest-chinese-stock-market.html)

Taken from the movie “Beautiful Mind”

John Nash’s death on May 23, 2015 on the New Jersey Turnpike was a tragedy. However, his contribution to mathematics and economics is everlasting. His contribution to game theory led to his sharing the 1994 Nobel Memorial Prize for Economical Sciences.

Coincidentally, three weeks before his accidental death, there was an econophysics paper that employed his ideas of Nash equilibrium. Econophysics has been an inter-disciplinary quantitative field since 1990s. Victor Yakovenko, a physics professor in University of Maryland, applied the techniques of classical statistical mechanics, and concluded that the wealth of bottom 95% population follows Boltzmann-Gibbs exponential distribution, while the top a Pareto distribution. [Dragulescu & Yakovenko 2000] This approach assumes agents  to have nearly “zero intelligence,” and behave randomly with no intent and purpose, contrary to the conventional assumption in economics that agents are perfectly rational, with purpose to maximize utility or profit.

This paper, written by Venkat Venkatasubramanian, described an approach aiming at reconciling econophysics and conventional economics, using the ideas in game theory. [Venkatasubramanian, Luo  & Sethuraman 2015] Like statistical mechanics, it assumes the agents to be particles. Money plays the role of energy, just like other econophysics theory. The equilibrium state is the state with maximum entropy. However, it employed the idea of game theory, adding that the agents are intelligent and in a game, unlike molecules in traditional statistical mechanics. The equilibrium state is not simply the maximum entropic state, but also the Nash equilibrium. This reconciles econophysics and conventional economics. And it even further argues that, unlike equilibrium in thermodynamics being probabilistic in nature, this economical equilibrium is deterministic. And the expected distribution is log-normal distribution. (This log-normal distribution is hard to fit, which is another obstacles for economists to accept physical approach to economics.)

With this framework, Venkatasubramanian discussed about income inequality. Income inequality has aroused debates in the recent few years, especially after the detrimental financial crisis in 2008. Is capitalism not working now? Does capitalism produce unfairness? He connected entropy with the concept of fairness, or fairest inequality. And the state with maximum entropy is the fairest state. And, of course, the wealth distribution is the log-normal distribution. His study showed that:[http://phys.org/news/2015-05-fair-theory-income-inequality.html]

“Scandinavian countries and, to a lesser extent, Switzerland, Netherlands, and Australia have managed, in practice, to get close to the ideal distribution for the bottom 99% of the population, while the U.S. and U.K. remain less fair at the other extreme. Other European countries such as France and Germany, and Japan and Canada, are in the middle.”

See the figure at the end of this post about the discrepancy of the economies of a few countries to the maximum entropic state, or ideality. And [Venkatasubramanian, Luo  & Sethuraman 2015]

“Even the US economy operated a lot closer to ideality, during ∼1945–75, than it does now. It is important to emphasize that in those three decades US performed extremely well economically, dominating the global economy in almost every sector.”

They even argued that these insights in economics might shed light to traditional statistical thermodynamics.

I have to say that I love this work because not only it explains real-world problem, but also links physics and economics in a beautiful way.