math – An Autonomous Agent

The End of Certainty – Ilya Prigogine

anautonomousagent — Sun, 18 Jan 2015 20:05:10 +0000

The End of Certainty by Ilya Prigogine provides insight into the natural processes which give rise to the novelty of life. Despite being published in 1997, there are so many great quotes and concepts which are still applicable today, that I will just say, “Read the book!” It will also help to read Stuart Kauffman’s book, Investigations, either before or after reading Prigogine’s book. Approaching from a different angle, Kauffman explores biological processes of nature which give rise to novelty and creative adaptive structures. Both books talk heavily about the dynamics of equilibrium and entropy. In the words of Prigogine, on page 67, “…matter at equilibrium is ‘blind,’ but far from equilibrium it begins to ‘see.'” Thus, non-equilibrium systems can think and observe the world, whereas systems in equilibrium are ignorant of all outside processes.

The approach of Prigogine lies in understanding the importance of Poincaré Resonances on dynamics and the construction/destruction of correlations at the microscopic level. How these resonances and these correlations behave leads to macroscopic features and the breaking of time symmetry. He deals with solving these Large Poincare Systems outside of the Hilbert Space; this is a concept which is important to biology and human social sciences. Because in these fields, we are always dealing with a system (human beings) which is far from equilibrium and behaves in novel and creative ways.

In other words, life, as a non-equilibrium dissipative structure emerging from the non-living world, needs to be studied under the auspices of “The End of Certainty.” Irreversible processes and long range correlations are critical to understanding the development of self-organization and the novelty of life.

All social sciences deal with a biological organism (humans), which is a product of non-equilibrium processes. Even Prigogine and the book itself are correlated with the mass of knowledge produced by humanity in the 20th century. In other words, his ideas and those of all scientists are subject to the same non-equilibrium dynamics which Prigogine talks about in his book. Resonances and correlations in the social sphere can lead to amazing discoveries or a lack thereof.

One subject that I think could see development from Prigogine’s ideas is economics. Economics should be considered: “The study of non-equilibirum dissipative structures created by the self-organized social species known as homo sapiens, to reproduce and adapt in the biosphere called ‘Earth.'”

What I thought about the most was the concept of correlation creation and destruction. In terms of self-organizing systems and financial markets, perhaps crashes are correlation destruction events, while bubbles are correlations spreading through time. And after a crash occurs, correlations can be created which makes a crisis even worse.

The Misbehavior of Markets: A Fractal View of Financial Turbulence – Benoit Mandelbrot

anautonomousagent — Wed, 27 Aug 2014 23:45:00 +0000

I’ve been wanting to read The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot for a number of years. Mandelbrot helped to change the way people view financial market dynamics. This book is definitely a must read for people working in the financial industry. However, Mandelbrot was not the first. It is a little surprising that Mandelbrot did not talk in detail about the work of R.N. Elliott or Robert Prechter, among others, which I think complement Mandelbrot’s work on financial fractals. Patrick Harris wrote a short paper discussing whether Mandelbrot should have cited Elliott (link to paper).

To me, the idea of infinite memory processes is one of the most important concepts touched on by Mandelbrot in this book. It suggests that economists and traders should be developing models and theories which value the importance of price series and data going back decades. And it makes sense to me that people and prices do not change their fundamental behavior over extended periods of technological evolution. However, I suspect that organisms do change their fundamental behavior if the time horizon is thousands or millions of years. I am very curious to read more about Hurst’s studies of the Nile.

Dangerous Knowledge – David Malone BBC

anautonomousagent — Wed, 16 Jul 2014 03:04:00 +0000

Dangerous Knowledge, by David Malone, summarizes the work and life of some of the greatest thinkers in late 19th and 20th centuries. Includes the work and life of Cantor, Boltzmann, Godel, and Turing.

Watch it here: Dangerous Knowledge

Kondratieff Waves, Warfare and World Security: Volume 5 NATO Security through Science Series: Human and Societal Dynamics – T.C. Devezas

anautonomousagent — Sun, 13 Jul 2014 17:53:00 +0000

Kondratieff Waves, Warfare and World Security: Volume 5 NATO Security through Science Series: Human and Societal Dynamics by T.C. Devezas is what I am currently reading.

Fascinating to say the least.

Also, see: A Spectral Analysis of World GDP Dynamics: Kondratieff Waves, Kuznets Swings, Juglar and Kitchin Cycles in Global Economic Development, and the 2008–2009 Economic Crisis Korotayev, Andrey V and Tsirel, Sergey V.

Black Holes and Time Warps: Einstein’s Outrageous Legacy – Kip S. Thorne

anautonomousagent — Tue, 28 Jan 2014 17:31:00 +0000

It has been a long time since I read a book on astrophysics and I am now interested in Black Holes and Time Warps: Einstein’s Outrageous Legacy, by Kip S. Thorne. I hope this book will be thought provoking and fun to read.

The Evolution of Physics – Albert Einstein and Leopold Infeld

anautonomousagent — Tue, 28 Jan 2014 17:24:00 +0000

One of the next books I will be reading is The Evolution of Physics by Albert Einstein and Leopold Infeld. Even though the book was published in the 1960’s, most people seem to be in agreement that the book is still a must read for anyone interested in physics and the mind of Einstein.

Our Mathematical Universe – Max Tegmark

anautonomousagent — Tue, 28 Jan 2014 17:12:00 +0000

Max Tegmark’s book, Our Mathematical Universe, takes a deep look at the physical world and shows readers the importance of math throughout the universe. An interesting book regardless of your own personal beliefs.

Price Simulation of Trader Matrix Video

anautonomousagent — Sun, 01 Dec 2013 02:02:00 +0000

The following video is the time evolution of a network of 10,000 traders and their effect on the price movement of a stock.

For more information, please read the following paper:

Non-normal Model of Stock Prices

Atomic Awakening: A New Look at the History and Future of Nuclear Power – James Mahaffey

anautonomousagent — Thu, 28 Nov 2013 20:21:00 +0000

Probably one of the best books I have read in a long time, Atomic Awakening: A New Look at the History and Future of Nuclear Power by James Mahaffey takes the reader on a grand journey. Covering in great detail almost every single development leading to the nuclear reactor, any intellectual will find this book to be a great pleasure to read.

An Interesting Model of Asset Price Behavior using a Tri-Valued Matrix of States

anautonomousagent — Thu, 28 Nov 2013 18:39:00 +0000

Stock prices, as seen by many appear to be purely random processes with no patterns and thus no predictability. And more often than not, people unfamiliar with stochastic processes will equate the work of financial engineers as “hocus-pocus,” “rocket science,” or just plan gambling. Are these people correct? Are asset managers playing roulette with the public’s money? If so, then it would seem that the life work of people such as Ito, Merton, Black, Scholes, Hull, Cox, White, Detemple, Sornette, and others amount to no more than speculation or the ramblings of an inmate in an insane asylum. Assuming otherwise, and that there is method to the madness of these academics, prices are predictable. And stochastic calculus provides a framework with which to build predictions. But how well and how reliable are the calculations? How can we use this predictability to properly manage risk?

There are at least two important responsibilities for risk managers:

1. Risk managers must make rational decisions. An empirical and purely quantitative forecast of future returns should be made with a clear understand of model assumptions.

2. With the foresight provided by the model, a suitable strategy must be taken. If the observations are indicating current market bubble conditions, then managers should be selling, or at the very least not buying.

These responsibilities are general and do not depend on the model being implemented to forecast returns. The accuracy of the forecast depends greatly on the assumptions used to create the model. The majority of models assume:

1. That returns are independent in time.

2. That volatility is constant.

3. That markets are efficient.

4. That prices follow the Geometric Brownian Motion

In other words, most models assume that human emotions, psychology, and behavior are not factors in determining the price movements of assets. How can such critical factors be ignored by financial engineers? The number of papers dealing with the effects of behavior on stochastic equations are few. Instead, the academic community has and continues to brush off these factors with the wrongly conceptualized idea that the sum total of all human interactions will create Brownian Motion. The billions of collisions on a grain of pollen in H₂O are what stirred the creation of this type of stochastic process. So, the question is: Can we equate the process which creates Browian Motion as observed by Brown himself, to the structure of asset price movement? One is created through thermal random thermal interactions; the other through human emotion, psychology, and behaviour.

Risk managers should realize these assumptions and they should be cautious.

If risk managers believe in the predicatablity of prices, then we must, by force of principle assume that there is an underlying structure. In other words, consider the prediction of Earthquakes. The fact that we even consider the ability to predict their occurrence suggests that we have some degree of knowledge regarding their underlying structure. By believing in the predictability of price trajectories, we naturally assume some underlying structure. But what is this structure? How do we monitor the processes at work?

Are returns predictable? According to much research, the answer to this question is a firm, yes. But arbitrage theory says that such predicatablity can not exist. So, what does the risk manager believe? Does he believe that prices are Brownian Motions; that markets are efficient machines, not affected by the emotions, psychology, and behavior of humans; or does he disregard these assumptions in search for a better stochastic model? The answer to this question lies in controversy and may not be resolved for a long time. However, the risk manager has 100 billion of assets which he must manage today. Where is he to look for alternative models which disregard the assumptions of his collegues and predecessors? This paper provides the beginning of one such alternative.

This paper presents a stochastic model of the form: change(price) = function(volume). The volume depends on a matrix with a fixed number of agents. The intuition behind this model comes from the concept that a stock price’s movement should reflect not a normal distribution, which has roots in the idea of Brownian motion and particles in heat transfer, but rather a bi-directional “tug-of-war” between agents who are buying and selling with human emotions, psychology, and behavior. Just as in the youthful tug-of-war game, there are times when it appears that the right side has the upper hand, when, all of the sudden the left side makes a determined effort and brings down the right side. These drastic and rapid phase transitions are unheard of in a Brownian idea of price movements. The Brownian application to asset price movements provide a decent first approximation. However, stochastic processes with Brownian Motion will never evolve in a manner to explain the stylized facts observed from the actual distribution of asset returns. Thus, to truly model price behavior, practitioners should study the fundamental structure of the market.

Consider this thought experiment:

Take a snapshot of all market participants in a security XYZ. This is period zero (t = 0). Suppose there are a fixed number, N, total people in the financial system who can own and trade XYZ. In other words, the price movement of this asset should depend only on the actions of these N traders. Now count the number of traders who currently hold a short position in XYZ and a long position in XYZ. Let S₀and L₀ represent the number of traders who are currently short and long, respectively, the asset XYZ. Let S₀ + L₀< N, thus there are a number of traders, H₀, who hold neither a short nor a long position. Thus, H₀ = N – S₀– L₀. The current price of XYZ is P₀. Assume that all transactions only involve 1 share of XYZ. Further posts will seek to address the important consideration of number of shares.

Now consider the next possible moves for all traders in the market (t = 1). There are assumed to be four possibilities:

An XYZ owner with a long position sells. (L₁ = L₀ – 1 and H₁ = H₀+ 1)
A trader short XYZ covers his/her position. (S₁ = S₀ – 1 and H₁ = H₀+ 1)
A neutral trader initiates a long position. (L₁ = L₀ + 1 and H₁ = H₀– 1)
A neutral trader initiates a short position. (S₁ = S₀ + 1 and H₁ = H₀– 1)

Thus, numbers 1 and 2 increase the number of neutral traders, while 3 and 4 decrease that number. Assume that both 2 and 3 cause the price of XYZ to increase by some factor and 1 and 4 decrease the price of XYZ by the same factor. Thus the movement of the price of XYZ should be a bi-directional “tug-of-war” battle: traders going long and shorts covering versus traders shorting and longs selling.

To represent this computationally, a matrix can be created which holds the current state of all traders. To create the initial state matrix, called StateMatrix₀, form an √N x √N square matrix (for simplification, it would be best to choose N as a perfect square). Then fill the matrix randomly with the value -1 for the correct number of traders short XYZ, S₀. Do likewise with L₀and H₀, represented by 1 and 0, respectively. Thus, the initial StateMatrix₀ will be randomly filled with {-1,0,1}, representing the traders who are short, neutral, and long.

Now, proceed one period to t = 1. To calculate the next StateMatix₁, do the following:

Assume that each trader is influenced by U of his peers and that this influence is proportional to a factor representing the general market “mood.” Call the mood factor for the current period, M_t. To find what the trader will do in the transition from t = 0 to t =1, sum the values of his U neighbors, find their average, multiply by the current M_t and round. In addition, include a possibility for trader_[i,j] to form his own independent value {-1,0,1} and ignore his U neighbors with probability, IDIO (an idiosyncratic change of asset position).

For example, suppose that at t = 0, trader_[i,j] of the matrix was short, i.e. the value was -1 and that his 4 neighbors at i = 0 had values {1,0,1,-1}. Suppose the M₁ at this time is 1.2 and IDIO = 0.20. If this trader_[i,j] does not choose an idiosyncratic position, then he mimics his neighbors. The sum of his four neighbors is thus SumTrader_[i,j] = 1 + 0 + 1 – 1 = 1 and the average is ¼. Thus ¼*1.2 = 0.3. Rounding give a value of 0. Thus, the trader_[i,j] should change his value to 0, in other words the trader_[i,j] will cover his position.

Performing this calculation for every trader on the grid will create the StateMatrix₁. And doing this for T periods will create a StateMatrix_{t=[0,T]} sequence of trader states. To calculate the actual price movement based on this model do the following:

Sum over all values of trader_[i,j]of StateMatrix_t
Sum over all values of trader_[i,j]of StateMatrix_t+1
Calculate the difference, StateMatrix_t+1 – StateMatrix_tand divide by N. Multiply by P_t. This value = C, the change in the price from the previous price
Price_t+1 = Price_t+ C

This will form the evolution of the price through the states of the StateMatrix_{t=[0,T]} sequence.

Note: In order to make the M_tfactor suitable, its values must fall between [0,1.5). If not, then values of the traders_[i,j] may be other than {-1,0,1}. The M_t factor can be calculated at each period with the following calculation:

M_t = #L_t-1/#S_t-1

In other words, the Mood factor M_tfor the next period is the ratio of the number of current traders who are long to the number of current traders who are short. There may be other ways to define this factor.

Look at the previous post, here, to see an example. In that post I used:

N = 1,000,000

T = 700

U = 4

S₀ = 0.4*N

L₀ = 0.4*N

IDIO = 0.19