Capital Formation

Given a trading stock S in a financial market:

Q1: can a trader become infinitely rich immediately after reaching a point in time?

Theoretically yes, practically is not always simple to achieve.

An intuitive investing strategy is to buy when the stock price is low and sell later when the price is high. For instance in a continuous time set up, when the stock price S(t) follows the geometrical Brownian motion with drift µ and volatility σ, there is always an investing strategy depending on the direction of a price process (“ups” and “downs”) that can achieve a theoretically capital process approximatively equivalent to:

Capital(t) = Exp( µ ²*t /2σ²).

In practice ( not necessary continuous time) it's too hard for the trader to know the optimum move between two successive trading times.
These movements (“ups” and “downs”) are determined only by the market and may act in a certain sense that the theoretical exponential capital formation can not be achieved without risking a potential bankruptcy.

Q2: how a trader has to design an investing strategy with good reasons to prevent big losses?
Q3: how a trader has to select a stock to apply an investing strategy with good reasons to prevent big losses?

The Capital Asset Pricing Model may be used to handle these questions, except that this theory has to make assumptions about the beliefs or preferences of the trader, stock return is determined by a stochastic process and so on. A discussion is in EQUITY RESEARCH-REPORT ESSENTIALS.

Capital Formation via Neural Network Models

Recently, new approaches have been studies without excessive modeling assumptions. In this essay we use the work in Sequential optimizing investing strategy with neural networks that directly consider investing strategies for financial time series based on neural network models and idea to formulate the process of capital formation as the trader is betting against the Market. At each round, n, the trader capital, K(n) = K(n−1)(1 + α(n)x(n)), where α(n) is the ratio of trader’s investment to his previous accumulated capital K(n−1). The investing ratio, α(n), at round n is function of the previous market's moves, x(n-1),x(n-2),x(n-3),x(n-4) ...
The work used neural network models to analytically define the function, α(n), by using the gradient descent method to solve the problem of maximization where the non-linear "activation function" is tanh. The process is an unsupervised neural network and is a sequential optimization of parameter values of the network, these values estimate the function, α(n).

Indeed, it has a drawback, and that probably is regarded as a serious concern, the iterative process needs to start with an initial guess for the function, α(n), values and after sufficient times of iteration the estimated capital converges to a local maximum and not necessary reaching the adequate capital accumulation(formation). The network converges to a poor capital formation. A plausible conjecture, these Stocks that yield poor learning dynamics are not "good" for trading. The stocks that yield good learning dynamics are financially certain for an adequate capital accumulation; it will happen.

The classification "poor/good" is depending on the considered period. It may that the stock is good for a given period and poor for the next ones and vice versa.

A Mathematical reasoning in solving this plausible conjecture is in progress.

Competitive Stocks for Trading: Numerical Testing

In this essay we use the work in Sequential optimizing investing strategy with neural networks to assess historical stock data and to calculate the capital gain that can be fostered by using that unsupervised neural network "investing strategy". Given a number of stocks, we use this capital gain to rank the stocks, potentially good for trading during the considered period.

The constructed network has three-layered neural network models. The input layer has L neurons and they just distribute the previous L moves to every neuron in the hidden layer (M neurons). Each layer in an artificial neural network contains artificial neurons. Each neuron receives as input the outputs of neurons in a previous layer. The inputs are then summed together using weights(need to be calculated) and passed through a non-linear "activation" function(tanh, for instance). As it is difficult to specify (L,M) in advance. We test an ensemble of values assigned to (L,M).

These Ensemble Neural Network Models(ENNM) results are mined to find hidden meaning and insight in huge volumes of data related to a number of stocks. This knowledge offers compelling and differentiated information about the stock data interrelation with the investing strategies provided by the ENNM to identify opportunities about the most competitive stocks for trading during the considered period.

Our numerical testing used the daily stock price data of listed stocks in the Dow Jones Industrial Average plus couple of stocks. The investing period is from January-10-2017 to March-10-2017 for all strategies using L=(5,10,15) and M=(10,15,20,30).

Note that the results depend on the considered period & on the initial guess. It's mainly illustrative to show stocks that yield poor/good learning dynamics using the work in Sequential optimizing investing strategy with neural networks that exploits the price process (“ups” and “downs”) during the considered period.

Data retrieval and the Mathematical formulations are implemented using the languages (Java, C++,Python, for instance) & the insight analysis uses R programming language to process the results ( R is a software environment for statistical computing and graphics supported by the R Foundation for Statistical Computing ).

Ranking Trading Stocks