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EMBED for wordpress. Want more? Advanced embedding details, examples, and help! This program allows you to perform quantitative financial analysis of equities. The optimal gains of multiple equity investments are computed. The program decides which of all available equities to invest in at any given time.

It calculates the instantaneous Shannon probability of all equities and uses statistical estimation techniques to estimate the accuracy of the calculated Shannon probability. Entropic techniques are used throughout. A tutorial is presented in the man pages. A companion equity market simulation program is included. There are no reviews yet. Be the first one to write a review.

Software Sites Collection. Note that under optimal conditions, all three equations are identical-only the metric methodology is different. Under non-optimal conditions, Equation 1. Unfortunately, any calculation involving the average of the normalized increments of an equity value time series will be very "sluggish," meaning that practical issues may prevail, suggesting a preference for Equation 1.

However, this would imply that the equities are known to be optimal, ie. There is some possibility that optimality can be verified by metrics:. There have been several heuristic approaches suggested, for example, using the absolute value of the normalized increments as an approximation to the root mean square, rms, and calculating the Shannon probability, P by Equation 1. The statistical estimate in such a scheme should use the same methodology as in the root mean square.

Another alternative is to model equity value time series as a fixed increment fractal, ie. The Shannon probability, P, is then calculated by the quotient of the up movements, divided by the total movements. There is an issue with this model, however. Although not common, there can be adjacent time intervals where an equity's value does not change, and it is not clear how the accounting procedure should work.

There are several alternatives. For example, no changes can be counted as up movements, or as down movements, or disregarded entirely, or counted as both. The statistical estimate should be performed as in Equation 1. We now have a "first order prescription" that enables us to analyze fluctuations in equity values, although we have not explained why equity values fluctuate the way they do. For a formal presentation on the subject, see the bibliography in [ Art95 ] which, also, offers non-mathematical insight into the subject.

Consider a very simple equity market, with only two people holding equities. Equity value "arbitration" ie. There is no other communication between the two people. If the other person buys the equity, then that is the value of the equity at that time.

Obviously, the other person will not buy the equity if the price posted is too high-even if ownership of the equity is desired. For example, the other person could simply decide to wait in hopes that a favorable price will be offered in the future. What this means is that the seller must consider not only the behavior of the other person, but what the other person thinks the seller's behavior will be, ie. Such convoluted logical processes are termed "self referential," and the implication is that the market can never operate in a consistent fashion that can be the subject of deductive analysis [ Pen89 , pp.

As pointed out by [ Art95 , Abstract], these types of indeterminacies pervade economics[ 9 ]. What the two players do, in absence of a deductively consistent and complete theory of the market, is to rely on inductive reasoning. They form subjective expectations or hypotheses about how the market operates. These expectations and hypothesis are constantly formulated and changed, in a world that forms from others' subjective expectations.

What this means is that equity values will fluctuate as the expectations and hypothesis concerning the future of equity values change[ 10 ]. This is a rather interesting conclusion, since analyzing the aggregate actions of many "agents," each operating on subjective hypothesis in a market that is deductively indeterminate, can result in a system that can not only be analyzed, but optimized. The only remaining derivation is to show that the optimal wagering strategy is, as cited above:.

Following [ Rez94 , pp. We assume that the side information which he receives has a probability, P, of being true, and of 1 - P, of being false. Let the original capital of gambler be V 0 , and V n his capital after the n'th wager. Since the gambler is not certain that the side information is entirely reliable, he places only a fraction, f, of his capital on each wager.

Thus, subsequent to n many wagers, assuming the independence of successive tips from the future, his capital is:. These numbers are, in general, values taken by two random variables, denoted by W and L. According to the law of large numbers:. The problem with which the gambler is faced is the determination of f leading to the maximum of the average exponential rate of growth of his capital.

That is, he wishes to maximize the value of:. It was mentioned that it would be useful to model equity prices as a fixed increment fractal, ie. As above, consider a gambler, wagering on the iterated outcomes of an unfair tossed coin game. A fraction, f, of the gambler's capital will be wagered on the outcome of each iteration of the unfair tossed coin, and if the coin comes up heads, with a probability, P, then the gambler wins the iteration, and an amount equal to the wager is added to the gambler's capital, and if the coin comes up tails, with a probability of 1 - P, then the gambler looses the iteration, and an amount of the wager is subtracted from the gambler's capital.

If we let the outcome of the first coin toss, ie. If we let V 0 be the initial value of the gambler's capital, V 1 be the value of the gambler's capital after the first iteration of the game, then:. For the normalized increments of the time series of the gambler's capital, it would be convenient to rearrange these formulas.

For the first iteration of the game:. This section addresses the question "is there reasonable evidence to justify investment in an equity based on data set size? The Shannon probability of a time series is the likelihood that the value of the time series will increase in the next time interval. The Shannon probability is measured using the average, avg, and root mean square, rms, of the normalized increments of the time series.

Using the rms to compute the Shannon probability, P:. However, there is an error associated with the measurement of rms do to the size of the data set, N, ie. The confidence level, c, is the likelihood that this error is less than some error level, e. The error, e, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigma, is:.

From this, the confidence level can be calculated from the cumulative sum, ie. For convenience, let F esigma be the function that given esigma, returns c, ie. Letting a decision variable, decision, be the iteration error created by this equation not being balanced:. From Equations 1. As an example of this algorithm, if the Shannon probability, P, is 0. Likewise, if P is 0. Robustness is an issue in algorithms that, potentially, operate real time.

The traditional means of implementation of statistical estimates is to use an integration process inside of a loop that calculates the cumulative of the normal distribution, controlled by, perhaps, a Newton Method approximation using the derivative of cumulative of the normal distribution, ie. Using the avg to compute the Shannon probability, P:. However, there is an error associated with the measurement of avg do to the size of the data set, N, ie.

There are two radicals that have to be protected from numerical floating point exceptions. The other radical:. This would require:. Obviously, the search algorithm must be prohibited from searching for a solution in this space. The solution is to limit the search of the confidence array to values that are equal to or less than:. Using both the avg and the rms to compute the Shannon probability, P:.

However, there is an error associated with both the measurement of avg and rms do to the size of the data set, N, ie. The error, er, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigmar, is:. For convenience, let F esigmar be the function that given esigmar, returns cr, ie. The error, ea, expressed in terms of the standard deviation of the measurement error do to an insufficient data set size, esigmaa, is:.

For convenience, let F esigmaa be the function that given esigmaa, returns ca, ie. As a final discussion to this section, consider the time series for an equity. Suppose that the data set size is finite, and avg and rms have both been measured, and have been found to both be positive. The question that needs to be resolved concerns the confidence, not only in these measurements, but the actual process that produced the time series. For example, suppose, although there was no knowledge of the fact, that the time series was actually produced by a Brownian motion fractal mechanism, with a Shannon probability of exactly 0.

We would expect a "growth" phenomena for extended time intervals [ Sch91 , pp. Note that, inadvertently, such a time series would potentially justify investment. What the methodology outlined in this section does is to preclude such scenarios by effectively lowering the Shannon probability to accommodate such issues. In such scenarios, the lowered Shannon probability will cause data sets with larger sizes to be "favored," unless the avg and rms of a smaller data set size are "strong" enough in relation to the Shannon probabilities of the other equities in the market.

Note that if the data set sizes of all equities in the market are small, none will be favored, since they would all be lowered by the same amount, if they were all statistically similar. An additional accuracy issue, besides data set size, is the time interval over which the data was obtained. There is some possibility that the data set was taken during an extended run length, either negative or positive, and the Shannon probability will have to be compensated to accommodate this measurement error.

The chances that a run length will exceed time, t, is [ Sch91 , pp. Fortunately, since confidence levels are calculated from the normal probability function, the same lookup table used for confidence calculations ie. Let K be the number of equities in the equity portfolio, and assume that the capital is invested equally in each of the equities, ie.

The portfolio value, over time, would be a time series with a root mean square value of the normalized increments, rmsp, and an average value of the normalized increments, avgp. Obviously, it would be advantageous to optimize the portfolio growth. Note that Equation 1. This is probably not the case, since rmsp will always be less than the individual values of rms for the equities. Again, letting K be the number of equities in the equity portfolio, and assuming that the capital is invested equally in each of the equities, ie.

It is not clear if there is a formal optimization for the distribution, and, perhaps, the applications of simulated annealing or genetic algorithms to the distribution problem may be of some benefit. Additionally, note that Equation 1. Interestingly, plots of Equation 1.

There is little advantage in holding more, and a distinct disadvantage in holding less[ 12 ]. The rationale proceeds as follows. Let l be the run length, ie. Naturally, it would be desirable to buy low and sell high. So, if an equity's price is below average, then the probability of an upward movement is given by Equation 1.

If an equity's price is above average, then, then the probability that it will continue the trend is:. Note that equation 1. Note that there is a heuristic involved in this procedure. The original derivation [ Sch91 , pp. However, simulations of Equation 1. Additionally, note that in the case of a fixed increment Brownian motion fractal, the average is known-zero, by definition.

However, in this procedure, the average is measured, and this can introduce errors, since the average itself is fluctuating slightly, do to a finite data set size. Note, also, that mean reverting functionality was implemented on the infrastructure available in the program, ie. There are probably more expeditious implementations, for example, using a single or multi pole filter as described in APPENDIX 1 to measure the average growth of an equity.

The above derivations assume random walk fractal characteristics as a first order approximation to equity prices. The traditional method of determination of persistence is by metrics on the Hurst exponent, [ Pet91 , pp. As an alternative method, the number of consecutive like movements in an equity's price can be tallied, and the coefficient of the resultant distribution determined.

For a simple random walk fractal, it will be the combinatorics, ie. Note that this is an exploitable attribute in "noise trading". For example, if the Hurst exponent is greater than 0. For simulation, the equities are represented, one per time unit. However, in the "real world," an equity can be represented multiple times in the same time unit, or not at all. This issue is addressed by:. If an equity has multiple representations in a single time unit, ie. If an equity was not represented in a time unit, then at the end of that time unit, the equity is processed as if it was represented in the time unit, but with no change in value.

The advantage of this scheme is that, since fractal time series are self-similar, it does not affect the wagering operations of the equities in relation to one another. Note: The prototype to this program implemented statistical estimates with a single pole filter. The documentation for the implementation was moved to this Appendix. Although the approximation is marginal, reasonably good results can be obtained with this technique. Additionally, the time constants for the filters are adjustable, and, at least in principle, provide a means of adaptive computation to control the operational dynamics of the program.

One of the implications of considering equity prices to have fractal characteristics, ie. The Shannon probability of a equity price time series is the likelihood that a equity price will increase in the next time interval. It is typically 0. However, another implication of considering equity prices to have fractal characteristics is that there are statistical optimizations to maximize portfolio performance.

Also, the optimized average of the normalized increments is equal to the square of the rms. Unfortunately, the measurements of avg and rms must be made over a long period of time, to construct a very large data set for analytical purposes do to the necessary accuracy requirements. Statistical estimation techniques are usually employed to quantitatively determine the size of the data set for a given analytical accuracy.

The calculation of the Shannon probability, P, from the average and root mean square of the normalized increments, avg and rms, respectively, will require require specialized filtering, to "weight" the most recent instantaneous Shannon probability more than the least recent, and statistical estimation to determine the accuracy of the measurement of the Shannon probability.

This measurement would be based on the normalized increments, as derived in Equation 1. The term "sufficiently large" must be analyzed quantitatively. What Table I means is that if a step function, from zero to 0. This means that the Shannon probability, 0. Since half the time the error would be greater than 0. For 33 records, we would use an avg of 0.

What Table II means is that if a step function, from zero to 0. For 2 records, we would use an rms of 0. The average of the necessary poles is 0. Tables V and VI represent similar reasoning, but with a Shannon probability of 0. Table V presents real issues, in that metrics for equities with low Shannon probabilities may not be attainable with adequate precision to formulate consistent wagering strategies.

For example, business days is a little over two millenia-the required size of the data set for day trading. There is some possibility that adaptive filter techniques could be implemented by dynamically change the constants in the statistical estimation filters to correspond to the instantaneous measured Shannon probability.

The equations are defined, below. Another alternative is to work only with the root mean square values of the normalized increments, since the pole frequency is not as sensitive to the Shannon probability, and can function on a much smaller data set size for a given accuracy in the statistical estimate. This may be an attractive alternative if all that is desired is to rank equities by growth, ie.

These metrics would require identical statistical estimate filters for both the average and the root mean squared filters, ie. The Shannon probability can be calculated by several methods using Equations 1. Note that in Equation 1. However, with a two order of magnitude difference in the pole frequencies for avg and rms, the response time of the statistical estimate approximation is dominated by the avg pole.

The decision criteria will be based on variations of the Shannon probability, P, and the average and root mean square of the normalized increments, avg and rms, respectively. Note that from Equation 1. P can be calculated from Equations 1. The measurement of the average, avg, and root mean square, rms, of the normalized increments can use different filter parameters than the root mean square of multiplier, ie.

By substitution, Equation 1. These interpretations offer an alternative to the rather sluggish filters shown in Tables I, III, and V, since there can be two sets of filters, one to perform a statistical estimate approximation to the Shannon probability, and the other to perform a statistical estimate on rms, which can be several orders of magnitude faster than the filters used for the Shannon probability, enhancing dynamic operation.

As a review of the methodology used to construct Tables I, II, III, IV, V, and VI, the size of the data set was obtained using the tsstatest 1 program, which can be approximated by a single pole low pass recursive discreet time filter [ Con78 ], with the pole frequency at 0.

The rationale behind this value is that if we consider an equity with a measured Shannon probability of 0. This number comes from the fact that a Shannon probability, P', would be 0. But if such a scenario is set up as an experiment that was performed many times, it would be expected that half the time, the measured value Shannon probability would be greater than 0.

This value is the confidence level in the statistical estimate of the measurement error of the average of the normalized increments, avg, which for a Shannon probability of 0. So, we now have the error level, 0.

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