From: John Conover <john@email.johncon.com>
Subject: new tsinvest release
Date: Mon, 17 Mar 1997 01:14:57 -0800
There will be a new version of the tsinvest(1) program within the next
couple of days. (It is currently going through regression testing, of
6 market scenarios, each with 4 wagering strategies, at about 4 hours
for each test/strategy. It will be released when the regression tests
finish.) There were no bug fixes, but I had many requests to add
"noise trading" functionality to the program. See the section on "MEAN
REVERTING DYNAMICS" for particulars. (Actually, the program name
should be changed, since this is not an investment strategy, but a
trading strategy. The investment strategies are still included, and
unchanged.)
John
INTRODUCTION
One of the prevailing concepts in financial quantitative analysis,
(eg., "financial engineering,") is that equity prices exhibit "random
walk," (eg., Brownian motion, or fractal,) characteristics. The
presentation by Brian Arthur [Art95] offers a compelling theoretical
framework for the random walk model. William A. Brock and Pedro
J. F. de Lima [BdL95], among others, have published empirical evidence
supporting Arthur's theoretical arguments.
There is a large mathematical infrastructure available for
applications of fractal analysis to equity markets. For example, the
publications authored by Richard M. Crownover [Cro95], Edgar E. Peters
[Pet91], and Manfred Schroeder [Sch91] offer formal methodologies,
while the books by John L. Casti [Cas90], [Cas94] offer a less formal
approach for the popular press.
There are interesting implications that can be exploited if equity
prices exhibit fractal characteristics:
1) It would be expected that equity portfolio volatility would be
equal to the root mean square of the individual equity
volatilities in the portfolio.
2) It would be expected that equity portfolio growth would be
equal to the linear addition of the growths of the individual
equities in the portfolio.
3) It would be expected that an equity's price would fluctuate,
over time, and the range, of these fluctuations (ie., the maximum
price minus the minimum price,) would increase with the square
root of time.
4) It would be expected that the number of equity price
transitions in a time interval, (ie., the number of times an
equity's price reaches a local maximum, then reverse direction and
decreases to a local minimum,) would increase with the square root
of time.
5) It would be expected that the zero-free voids in an equity's
price, (ie., the length of time an equity's price is above
average, or below average,) would have a cumulative distribution
that decreases with the reciprocal of the square root of time.
6) It would be expected that an equity's price, over time, would
be mean reverting, (ie., if an equity's price is below its
average, there would be a propensity for the equity's price to
increase, and vice versa.)
Note that 1) and 2) above can be exploited to formulate an optimal
hedging strategy; 3), and 4) would tend to imply that "market timing"
is not attainable; and 5), and 6) can be exploited to formulate an
optimal buy-sell strategy.
DERIVATION
As a tutorial, the derivation will start with a simple compound
interest equation. This equation will be extended to a first order
random walk model of equity prices. Finally, optimizations will
derived based on the random walk model that are useful in optimizing
equity portfolio performance.
If we consider capital, V, invested in a savings account, and
calculate the growth of the capital over time:
V(t) = V(t - 1)(1 + a(t)) ......................(1.1)
where a(t) is the interest rate at time t, (usually a constant[1].)
In equities, a(t) is not constant, and fluctuates, perhaps being
negative at certain times, (meaning that the value of the equity
decreased.) This fluctuation in an equity's value can be represented
by modifying a(t) in Equation (1.1):
a(t) = f(t) * F(T) ............................(1.2)
where the product f * F is the fluctuation in the equity's value at
time t. An equity's value, over time, is similar to a simple tossed
coin game [Sch91, pp. 128], where f(t) is the fraction of a gambler's
capital wagered on a toss of the coin, at time t, and F(t) is a random
variable[2], signifying whether the game was a win, or a loss, ie.,
whether the gambler's capital increased or decreased, and by how much.
The amount the gambler's capital increased or decreased is f(t) *
F(t).
In general, F(t) is a function of a random variable, with an average,
over time, of avgf, and a root mean square value, rmsf, of unity.
Note that for simple, time invariant, compound interest, F(t) has an
average and root mean square, both being unity, and f(t) is simply the
interest rate, which is assumed to be constant. For a simple, single
coin game, F(t) is a fixed increment, (ie., either +1 or -1,) random
V(t) = V(t - 1)(1 + f(t) * F(t)) ...............(1.3)
and subtracting V(t - 1) from both sides:
V(t) - V(t - 1) = V(t - 1) (1 + f(t) * F(t)) -
V(t - 1) .......................................(1.4)
and dividing both sides by V(t - 1):
V(t) - V(t - 1)
--------------- =
V(t - 1)
V(t - 1) (1 + f(t) * F(t)) - V(t - 1)
------------------------------------- ..........(1.5)
V(t - 1)
and combining:
V(t) - V(t - 1)
--------------- =
V(t - 1)
(1 + f(t) * F(t) ) - 1 = f(t) * F(t) ...........(1.6)
We now have a "prescription," or process, for calculating the
characteristics of the random process that determines an equity's
price, over time. That process is, for each unit of time, subtract
the value of the equity at the previous time from the value of the
equity at the current time, and divide this by the value of the equity
at the previous time. The root mean square[4] of these values are the
root mean square value of the random process. The average of these
values are the average of the random process, avgf. The root mean
square of these values can be calculated by any convenient means, and
will be represented by rms. The average of these values can be found
by any convenient means, and will be represented by avg[5].
Therefore, if f(t) = f, and assuming that it does not vary over time:
rms = f ........................................(1.7)
which, if there are sufficiently many samples, is a metric of the
equity's price "volatility," and:
avg = f * F(t) .................................(1.8)
and if there are sufficiently many samples, the average of F(t) is
simply avgf, or:
avg = f * avgf .................................(1.9)
which are the metrics of the equity's random process. Note that this
is the "effective" compound interest rate from Equation (1.1).
Equations (1.7) and (1.9) are important equations, since they can be
used in portfolio management. For example, Equation (1.7) states that
portfolio volatility is calculated as the root mean square sum of the
individual volatilities of the equities in the portfolio. Equation
(1.9) states that the averages of the normalized increments of the
equity prices add together linearly[6] in the portfolio. Dividing
Equation (1.9) by Equation (1.7) results in the two f's canceling, or:
avg
--- = avgf ....................................(1.10)
rms
There may be analytical advantages to "model" F(t) as a simple tossed
coin game, (either played with a single coin, or multiple coins, ie.,
many coins played at one time, or a single coin played many times[7].)
The number of wins minus the number of losses, in many iterations of a
single coin tossing game would be:
P - (1 - P) = 2P - 1 ..........................(1.11)
where P is the probability of a win for the tossed coin. (This
probability is traditionally termed, the "Shannon probability" of a
win.) Note that from the definition of F(t) above, that P = avgf. For
a fair coin, (ie., one that comes up with a win 50% of the time,) P =
0.5, and there is no advantage, in the long run, to playing the game.
However, if P > 0.5, then the optimal fraction of capital wagered on
each iteration of the single coin tossing game, f, would be 2P - 1.
Note that if multiple coins were used for each iteration of the game,
we would expect that the volatility of the gambler's capital to
increase as the square root of the number of coins used, and the
growth to increase linearly with the number of coins used,
irregardless of whether many coins were tossed at once, or one coin
was tossed many times, (ie., our random generator, F(t) would assume a
binomial distribution and if the number of coins was very large, then
F(t) would assume, essentially, a Gaussian distribution.) Many
equities have a Gaussian distribution for the random process, F(t).
It may be advantageous to determine the Shannon probability to analyze
avg
--- = avgf = 2P - 1 ...........................(1.12)
rms
or:
avg
--- + 1 = 2P ..................................(1.13)
rms
and:
avg
--- + 1
rms
P = ------- ...................................(1.14)
2
where only the average and root mean square of the normalized
increments need to be measured, using the "prescription" or process
outlined above.
Interestingly, what Equation (1.12) states is that the "best" equity
investment is not, necessarily, the equity that has the largest
average growth. A better investment criteria is to choose the equity
that has the largest growth, while simultaneously having the smallest
volatility.
Continuing with this line of reasoning, and rearranging Equation
(1.12):
avg = rms * (2P - 1) ..........................(1.15)
which is an important equation since it states that avg, (and the
parameter that should be maximized,) is equal to rms, which is the
measure of the volatility of the equity's value, multiplied by the
quantity, twice the likelihood that the equity's value will increase
in the next time interval, minus unity.
As derived in the Section, OPTIMIZATION, below, the optimal growth
occurs when f = rms = 2P - 1. Under optimal conditions, Equation
(1.14) becomes:
rms + 1
P = ------- ...................................(1.16)
2
or, sqrt (avg) = rms, (again, under optimal conditions,) and
substituting into Equation (1.14):
sqrt (avg) + 1
P = -------------- ............................(1.17)
2
giving three different computational methods for measuring the
statistics of an equity's value.
Note that, from Equations (1.14) and (1.12), that since avgf = avg /
rms = (2P - 1), choosing the largest value of the Shannon probability,
P, will also choose the largest value of the ratio of avg / rms, rms,
or avg, respectively, in Equations (1.14), (1.16), or (1.17). This
suggests a method for determination of equity selection
criteria. (Note that under optimal conditions, all three equations are
identical-only the metric methodology is different. Under non-optimal
conditions, Equation (1.14) should be used. 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
(1.17).) However, this would imply that the equities are known to be
optimal, ie., rms = 2P + 1, which, although it is nearly true for most
equities, is not true for all equities. There is some possibility that
optimality can be verified by metrics:
2
if avg < rms
then rms = f is too large in Equation (1.12)
2
else if avg > rms
then rms = f is too small in Equation (1.12)
2
else avg = rms
and the equities time series is optimal, ie.,
rms = f = 2P - 1 from Equation (1.36), below
HEURISTIC APPROACHES
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 (1.16), using the absolute value,
abs, instead of the rms. 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., by counting the up movements in an equity's
value. 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 (1.14),
with an rms of unity, and an avg that is the Shannon probability
itself-that is the definition of a fixed increment fractal.
MARKET
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., how equity values are
determined,) is handled by one person posting (to a bulletin board,) a
willingness to sell a given number of equities at a given price, to
the other person. 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., the seller must base the pricing strategy on
the seller's pricing strategy. 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. 101][8]. 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]. The fluctuations created by these indeterminacies in the
equity market are represented by the term f(t) * F(t) in Equation
(1.3), and since there are many such indeterminacies, we would
anticipate F(t) to have a Gaussian distribution. 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.
OPTIMIZATION
The only remaining derivation is to show that the optimal wagering
strategy is, as cited above:
f = rms = 2P - 1 ..............................(1.18)
where f is the fraction of a gambler's capital wagered on each toss of
a coin that has a Shannon probability, P, of winning. Following
[Rez94, pp. 450], consider that the gambler has a private wire into
the future, (ie., an inductive hypothesis,) who places wagers on the
outcomes of a game of chance. 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:
w l
V(n) = (1 + f) (1 - f) V (0) ................(1.19)
where w is the number of times he won, and l = n - w, the number of
times he lost. These numbers are, in general, values taken by two
random variables, denoted by W and L. According to the law of large
numbers:
1
lim - W = P .........................(1.20)
n -> infinity n
1
lim - L = q = 1 - P .................(1.21)
n -> infinity n
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:
1 V(n)
G = lim - ln ---- ...................(1.22)
n -> infinity n V(0)
with respect to f, assuming a fixed original capital and specified P:
W L
G = lim - ln (1 + f) + - ln (1 - f) .(1.23)
n -> infinity n n
or:
G = P ln (1 + f) + q ln (1 - f) ...............(1.24)
which, by taking the derivative with respect to f, and equating to
zero, can be shown to have a maxima when:
dG P - 1 1 - P
-- = P(1 + f) (1 - f) -
df
1 - P - 1
(1 - P)(1 - f) (1 + f)P = 0 ..........(1.25)
combining terms:
P - 1 1 - P
0 = P(1 + f) (1 - f) -
P P
(1 - P)(1 - f) (1 + f ) .....................(1.26)
and splitting:
P - 1 1 - P
P(1 + f) (1 - f) =
P P
(1 - P)(1 - f) (1 + f) ......................(1.27)
then taking the logarithm of both sides:
ln (P) + (P - 1) ln (1 + f) + (1 - P) ln (1 - f) =
ln (1 - P) - P ln (1 - f) + P ln (1 + f) ......(1.28)
and combining terms:
(P - 1) ln (1 + f) - P ln (1 + f) +
(1 - P) ln (1 - f) + P ln (1 - f) =
ln (1 - P) - ln (P) ...........................(1.29)
or:
ln (1 - f) - ln (1 + f) =
ln (1 - P) - ln (P)...........................(1.30)
and performing the logarithmic operations:
1 - f 1 - P
ln ----- = ln ----- ...........................(1.31)
1 + f P
and exponentiating:
1 - f 1 - P
----- = ----- .................................(1.32)
1 + f P
which reduces to:
P(1 - f) = (1 - P)(1 + f) .....................(1.33)
and expanding:
P - Pf = 1 - Pf - P + f .......................(1.34)
or:
P = 1 - P + f .................................(1.35)
and, finally:
f = 2P - 1 ....................................(1.36)
Note that Equation (1.24), which, since rms = f, can be rewritten:
G = P ln (1 + rms) + (1 - P) ln (1 - rms) .....(1.37)
where G is the average exponential rate of growth in an equity's
value, from one time interval to the next, (ie., the exponentiation of
this value minus unity[11] is the "effective interest rate", as
expressed in Equation (1.1),) and, likewise, Equation (1.36) can be
rewritten:
rms = 2P - 1 ..................................(1.38)
and substituting:
G = P ln (1 + 2P - 1) +
(1 - P) ln (1 - (2P - 1)) .................(1.39)
or:
G = P ln (2P) +
(1 - P) ln (2 (1 - P)) ....................(1.40)
using a binary base for the logarithm:
G = P ln (2P) +
2
(1 - P) ln (2 (1 - P)) ....................(1.41)
2
and carrying out the operations:
G = P ln (2) + P ln (P) +
2 2
(1 - P) ln (2) + (1 - P) ln (1 - P)) ......(1.42)
2 2
which is:
G = P ln (2) + P ln (P) +
2 2
ln (2) - P ln (2) + (1 - P) ln (1 - P) ....(1.43)
2 2 2
and canceling:
G = 1 + P ln (P) + (1 - P) ln (1 - P) .........(1.44)
2 2
if the gambler's wagering strategy is optimal, ie., f = rms = 2P - 1,
which is identical to the equation in [Schroder, pp. 151].
FIXED INCREMENT FRACTAL
It was mentioned that it would be useful to model equity prices as a
fixed increment fractal, ie., an unfair tossed coin game.
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., whether it came up
as a win or a loss,) be c(1) and the outcome of the second toss be
c(2), and so on, then the outcome of the n'th toss, c(n), would be:
[win, with a probability of P
c(n) = [
[loose, with a probability of 1 - P
for convenience, let a win to be represented by +1, and a loss by -1:
[+1, with a probability of P
c(n) = [
[-1, with a probability of 1 - P
for the reason that when we multiply the wager, f, by c(n), it will be
a positive number, (ie., the wager will be added to the capital,) and
for a loss, it will be a negative number, (ie., the wager will be
subtracted from the capital.) This is convenient, since the
increment, by with the gambler's capital increased or decreased in the
n'th iteration of the game is f * c(n).
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:
V(1) = V(0) * (1 + c(1) * f(1)) ...............(1.45)
after the first iteration of the game, and:
V(2) = V(0) * ((1 + c(1) * f(1)) *
(1 + c(2) * f(2))) ....................(1.46)
after the second iteration of the game, and, in general, after the
n'th iteration of the game:
V(n) = V(0) * ((1 + c(1) * f(1)) *
(1 + c(2) * f(2)) * ...
* (1 + c(n) * f(n)) *
(1 + c(n + 1) * f(n + 1))) .............(1.47)
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:
V(1) - V(0) = V(0) * (1 + c(1) * f(1)) - V(0) .(1.48)
or
V(1) - V(0) V(0) * (1 + c(1) * f(1)) - V(0)
----------- = ------------------------------- .(1.49)
V(0) V(0)
and after reducing, the first normalized increment of the gambler's
capital time series is:
V(1) - V(0)
----------- = (1 + c(1) * f(1)) - 1
V(0)
= c(1) * f(1) .....................(1.50)
and for the second iteration of the game:
V(2) = V(0) * ((1 + c(1) * f(1)) *
(1 + c(2) * f(2))) .....................(1.51)
but V(0) * ((1 + c(1) * f(1)) is simply V(1):
V(2) = V(1) * (1 + c(2) * f(2)) ...............(1.52)
or:
V(2) - V(1) = V(1) * (1 + c(2) * f(2)) - V(1) .(1.53)
which is:
V(2) - V(1) V(1) * (1 + c(2) * f(2)) - V(1)
----------- = ------------------------------- .(1.54)
V(1) V(1)
and after reducing, the second normalized increment of the gambler's
capital time series is:
V(2) - V(1)
----------- = 1 + c(2) * f(2)) - 1
V(1)
= c(2) * f(2) .....................(1.55)
and it should be obvious that the process can be repeated
indefinitely, so, the n'th normalized increment of the gambler's
capital time series is:
V(n) - V(n - 1)
--------------- = c(n) * f(n) .................(1.56)
V(n)
which is Equation (1.6).
DATA SET SIZE CONSIDERATIONS
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:
rms + 1
P = ------- ...................................(1.57)
2
However, there is an error associated with the measurement of rms do
to the size of the data set, N, (ie., the number of records in the
time series,) used in the calculation of rms. The confidence level, c,
is the likelihood that this error is less than some error level, e.
Over the many time intervals represented in the time series, the error
will be greater than the error level, e, (1 - c) * 100 percent of the
time-requiring that the Shannon probability, P, be reduced by a factor
of c to accommodate the measurement error:
rms - e + 1
Pc = ----------- ..............................(1.58)
2
where the error level, e, and the confidence level, c, are calculated
using statistical estimates, and the product P times c is the
effective Shannon probability that should be used in the calculation
of optimal wagering strategies.
The error, e, expressed in terms of the standard deviation of the
measurement error do to an insufficient data set size, esigma, is:
e
esigma = --- sqrt (2N) ........................(1.59)
rms
c esigma
-------------
50 0.67
68.27 1.00
80 1.28
90 1.64
95 1.96
95.45 2.00
99 2.58
99.73 3.00
Note that the equation:
rms - e + 1
Pc = ----------- ..............................(1.60)
2
will require an iterated solution since the cumulative normal
distribution is transcendental. For convenience, let F(esigma) be the
function that given esigma, returns c, (ie., performs the table
operation, above,) then:
rms - e + 1
P * F(esigma) = -----------
2
rms * esigma
rms - ------------ + 1
sqrt (2N)
= ---------------------- ........(1.61)
2
Then:
rms * esigma
rms - ------------ + 1
rms + 1 sqrt (2N)
------- * F(esigma) = ---------------------- ..(1.62)
2 2
or:
rms * esigma
(rms + 1) * F(esigma) = rms - ------------ + 1 (1.63)
sqrt (2N)
Letting a decision variable, decision, be the iteration error created
by this equation not being balanced:
rms * esigma
decision = rms - ------------ + 1
sqrt (2N)
- (rms + 1) * F(esigma) ...........(1.64)
which can be iterated to find F(esigma), which is the confidence
level, c.
Note that from Equation (1.58):
rms - e + 1
Pc = -----------
2
and solving for rms - e, the effective value of rms compensated for
accuracy of measurement by statistical estimation:
rms - e = (2 * P * c) - 1 .....................(1.65)
and substituting into Equation (1.57):
rms + 1
P = -------
2
rms - e = ((rms + 1) * c) - 1 .................(1.66)
and defining the effective value of rms as rmseff:
rmseff = rms - e ..............................(1.67)
2
avg = rms ....................................(1.68)
or:
2
avgeff = rmseff ..............................(1.69)
As an example of this algorithm, if the Shannon probability, P, is
0.51, corresponding to an rms of 0.02, then the confidence level, c,
would be 0.996298, or the error level, e, would be 0.003776, for a
data set size, N, of 100.
Likewise, if P is 0.6, corresponding to an rms of 0.2 then the
confidence level, c, would be 0.941584, or the error level, e, would
be 0.070100, for a data set size of 10.
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., the formula for the normal distribution:
2
1 - x / 2
f(x) = ------ * e ...................(1.70)
2 * PI
Numerical stability and convergence issues are an issue in such
processes.
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 avg to compute the Shannon probability, P:
sqrt (avg) + 1
P = -------------- ............................(1.71)
2
However, there is an error associated with the measurement of avg do
to the size of the data set, N, (ie., the number of records in the
time series,) used in the calculation of avg. The confidence level, c,
is the likelihood that this error is less than some error level, e.
Over the many time intervals represented in the time series, the error
will be greater than the error level, e, (1 - c) * 100 percent of the
time-requiring that the Shannon probability, P, be reduced by a factor
of c to accommodate the measurement error:
sqrt (avg - e) + 1
Pc = ------------------ .......................(1.72)
2
where the error level, e, and the confidence level, c, are calculated
using statistical estimates, and the product P times c is the
effective Shannon probability that should be used in the calculation
of optimal wagering strategies.
The error, e, expressed in terms of the standard deviation of the
measurement error do to an insufficient data set size, esigma, is:
e
esigma = --- sqrt (N) .........................(1.73)
rms
c esigma
-------------
50 0.67
68.27 1.00
80 1.28
90 1.64
95 1.96
95.45 2.00
99 2.58
99.73 3.00
Note that the equation:
sqrt (avg - e) + 1
Pc = ------------------ .......................(1.74)
2
will require an iterated solution since the cumulative normal
distribution is transcendental. For convenience, let F(esigma) be the
function that given esigma, returns c, (ie., performs the table
operation, above,) then:
sqrt (avg - e) + 1
P * F(esigma) = ------------------
2
rms * esigma
sqrt [avg - ------------] + 1
sqrt (N)
= ----------------------------- .(1.75)
2
Then:
sqrt (avg) + 1
--------------- * F(esigma) =
2
rms * esigma
sqrt [avg - ------------] + 1
sqrt (N)
----------------------------- .............(1.76)
2
or:
(sqrt (avg) + 1) * F(esigma) =
rms * esigma
sqrt [avg - ------------] + 1 .............(1.77)
sqrt (N)
Letting a decision variable, decision, be the iteration error created
by this equation not being balanced:
rms * esigma
decision = sqrt [avg - ------------] + 1
sqrt (N)
- (sqrt (avg) + 1) * F(esigma) .....(1.78)
which can be iterated to find F(esigma), which is the confidence
level, c.
There are two radicals that have to be protected from numerical
floating point exceptions. The sqrt (avg) can be protected by
requiring that avg >= 0, (and returning a confidence level of 0.5, or
possibly zero, in this instance-a negative avg is not an interesting
solution for the case at hand.) The other radical:
rms * esigma
sqrt [avg - ------------] .....................(1.79)
sqrt (N)
and substituting:
e
esigma = --- sqrt (N) .........................(1.80)
rms
which is:
e
rms * --- sqrt (N)
rms
sqrt [avg - ------------------] ...............(1.81)
sqrt (N)
and reducing:
sqrt [avg - e] ................................(1.82)
requiring that:
avg >= e ......................................(1.83)
Note that if e > avg, then Pc < 0.5, which is not an interesting
solution for the case at hand. This would require:
avg
esigma <= --- sqrt (N) ........................(1.84)
rms
Obviously, the search algorithm must be prohibited from searching for
a solution in this space. (ie., testing 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:
avg
--- sqrt (N) ..................................(1.85)
rms
which can be accomplished by setting integer variable, top, usually
set to sigma_limit - 1, to this value.
Note that from Equation (1.72):
sqrt (avg - e) + 1
Pc = ------------------
2
and solving for avg - e, the effective value of avg compensated for
accuracy of measurement by statistical estimation:
2
avg - e = ((2 * P * c) - 1) ..................(1.86)
and substituting into Equation (1.71):
sqrt (avg) + 1
P = --------------
2
2
avg - e = (((sqrt (avg) + 1) * c) - 1) .......(1.87)
and defining the effective value of avg as avgeff:
avgeff = avg - e ..............................(1.88)
2
avg = rms ....................................(1.89)
or:
rmseff = sqrt (avgeff) ........................(1.90)
As an example of this algorithm, if the Shannon probability, P, is
0.52, corresponding to an avg of 0.0016, and an rms of 0.04, then the
confidence level, c, would be 0.987108, or the error level, e, would
be 0.000893, for a data set size, N, of 10000.
Likewise, if P is 0.6, corresponding to an rms of 0.2, and an avg of
0.04, then the confidence level, c, would be 0.922759, or the error
level, e, would be 0.028484, for a data set size of 100.
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
both the avg and the rms to compute the Shannon probability, P:
avg
--- + 1
rms
P = ------- ...................................(1.91)
2
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 number of records
in the time series,) used in the calculation of avg and rms. The
confidence level, c, is the likelihood that this error is less than
some error level, e.
Over the many time intervals represented in the time series, the error
will be greater than the error level, e, (1 - c) * 100 percent of the
time-requiring that the Shannon probability, P, be reduced by a factor
of c to accommodate the measurement error:
avg - ea
-------- + 1
rms + er
P * ca * cr = ------------ ....................(1.92)
2
where the error level, ea, and the confidence level, ca, are
calculated using statistical estimates, for avg, and the error level,
er, and the confidence level, cr, are calculated using statistical
estimates for rms, and the product P * ca * cr is the effective
Shannon probability that should be used in the calculation of optimal
wagering strategies, (which is the product of the Shannon probability,
P, times the superposition of the two confidence levels, ca, and cr,
ie., P * ca * cr = Pc, eg., the assumption is made that the error in
avg and the error in rms are independent.)
The error, er, expressed in terms of the standard deviation of the
measurement error do to an insufficient data set size, esigmar, is:
er
esigmar = --- sqrt (2N) .......................(1.93)
rms
cr esigmar
--------------
50 0.67
68.27 1.00
80 1.28
90 1.64
95 1.96
95.45 2.00
99 2.58
99.73 3.00
Note that the equation:
avg
-------- + 1
rms + er
P * cr = ------------ .........................(1.94)
2
will require an iterated solution since the cumulative normal
distribution is transcendental. For convenience, let F(esigmar) be the
function that given esigmar, returns cr, (ie., performs the table
operation, above,) then:
avg
-------- + 1
rms + er
P * F(esigmar) = ------------ =
2
avg
------------------- + 1
esigmar * rms
rms + -------------
sqrt (2N)
----------------------- ......(1.95)
2
Then:
avg
--- + 1
rms
------- * F(esigmar) =
2
avg
------------------- + 1
esigmar * rms
rms + -------------
sqrt (2N)
----------------------- ................(1.96)
2
or:
avg
(--- + 1) * F(esigmar) =
rms
avg
------------------- + 1 ................(1.97)
esigmar * rms
rms + -------------
sqrt (2N)
Letting a decision variable, decision, be the iteration error created
by this equation not being balanced:
avg
decision = ------------------- + 1
esigmar * rms
rms + -------------
sqrt (2N)
avg
- (--- + 1) * F(esigmar) ..........(1.98)
rms
which can be iterated to find F(esigmar), which is the confidence
level, cr.
The error, ea, expressed in terms of the standard deviation of the
measurement error do to an insufficient data set size, esigmaa, is:
ea
esigmaa = --- sqrt (N) ........................(1.99)
rms
ca esigmaa
--------------
50 0.67
68.27 1.00
80 1.28
90 1.64
95 1.96
95.45 2.00
99 2.58
99.73 3.00
Note that the equation:
avg - ea
-------- + 1
rms
P * ca = ------------ ........................(1.100)
2
will require an iterated solution since the cumulative normal
distribution is transcendental. For convenience, let F(esigmaa) be the
function that given esigmaa, returns ca, (ie., performs the table
operation, above,) then:
avg - ea
-------- + 1
rms
P * F(esigmaa) = ------------ =
2
esigmaa * rms
avg - -------------
sqrt (N)
------------------- + 1
rms
----------------------- .....(1.101)
2
Then:
avg
--- + 1
rms
------- * F(esigmaa) =
2
esigmaa * rms
avg - -------------
sqrt (N)
------------------- + 1
rms
----------------------- ...............(1.102)
2
or:
avg
(--- + 1) * F(esigmaa) =
rms
esigmaa * rms
avg - -------------
sqrt (N)
------------------- + 1 ...............(1.103)
rms
Letting a decision variable, decision, be the iteration error created
by this equation not being balanced:
esigmaa * rms
avg - -------------
sqrt (N)
decision = ------------------- + 1
rms
avg
- (--- + 1) * F(esigmaa) .................(1.104)
rms
which can be iterated to find F(esigmaa), which is the confidence
level, ca.
Note that from Equation (1.94):
avg
-------- + 1
rms + er
P * cr = ------------
2
and solving for rms + er, the effective value of rms compensated for
accuracy of measurement by statistical estimation:
avg
rms + er = ---------------- ..................(1.105)
(2 * P * cr) - 1
and substituting into Equation (1.100):
avg
--- + 1
rms
P = -------
2
avg
rms + er = -------------------- ..............(1.106)
avg
((--- + 1) * cr) - 1
rms
and defining the effective value of avg as rmseff:
rmseff = rms +/- er ..........................(1.107)
Note that from Equation (1.100):
avg - ea
-------- + 1
rms
P * ca = ------------
2
and solving for avg - ea, the effective value of avg compensated for
accuracy of measurement by statistical estimation:
avg - ea = ((2 * P * ca) - 1) * rms ..........(1.108)
and substituting into Equation (1.91):
avg
--- + 1
rms
P = -------
2
avg
avg - ea = (((--- + 1) * ca) - 1) * rms ......(1.109)
rms
and defining the effective value of avg as avgeff:
avgeff = avg - ea ............................(1.110)
As an example of this algorithm, if the Shannon probability, P, is
0.51, corresponding to an rms of 0.02, then the confidence level, c,
would be 0.983847, or the error level in avg, ea, would be 0.000306,
and the error level in rms, er, would be 0.001254, for a data set
size, N, of 20000.
Likewise, if P is 0.6, corresponding to an rms of 0.2 then the
confidence level, c, would be 0.947154, or the error level in avg, ea,
would be 0.010750, and the error level in rms, er, would be 0.010644,
for a data set size of 10.
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 0.5. We would
expect a "growth" phenomena for extended time intervals [Sch91,
pp. 152], in the time series, (in point of fact, we would expect the
cumulative distribution of the length of such intervals to be
proportional to 1 / sqrt (t).) 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.)
To reiterate, in the equation avg = rms * (2P - 1), the Shannon
probability, P, can be compensated by the size of the data set, ie.,
Peff, and used in the equation avgeff = rms * (2Peff - 1), where rms
is the measured value of the root mean square of the normalized
increments, and avgeff is the effective, or compensated value, of
the average of the normalized increments.
PORTFOLIO OPTIMIZATION
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., if
V is the capital value of the portfolio, then the amount invested in
each equity is V / K.) 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
rmsp + 1
Pp = -------- ................................(1.111)
2
where the root mean square value of the normalized increments of the
portfolio value, rmsp, is the root mean square sum of the root mean
square values of the normalized increments of each individual equity:
1 2 1 2
rmsp = sqrt ((-rms ) + (-rms ) + ...
K 1 K 2
1 2
... + (-rms ) ) .......................(1.112)
K K
or:
1 2 2 2
rmsp = - sqrt (rms + rms + ... + rms ) ....(1.113)
K 1 2 K
and Pp is the Shannon probability (ie., the likelyhood,) that the
value of the portfolio time series will increase in the next time
interval.
Note that Equation (1.16) presumes that the portfolio's time series
will be optimal, ie., rmsp = sqrt (avgp). This is probably not the
case, since rmsp will always be less than the individual values of rms
for the equities. Additionally, note that assuming the distribution of
capital, V / K, invested in each equity to be identical may not be
optimal. It is not clear if there is a formal optimization for the
distribution, and, perhaps, the application of simulated annealing,
linear programming, or genetic algorithms to the distribution problem
may be of some benefit.
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., if V is the capital value of the portfolio, then the
amount invested in each equity is V / K.) 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
sqrt (avgp) + 1
Pp = --------------- .........................(1.114)
2
where the average value of the normalized increments of the portfolio
value, avgp, is the sum of the average values of the normalized
increments of each individual equity:
1 1 1
avgp = - avg + - avg + ... + - avg .........(1.115)
K 1 K 2 K K
or:
1
avgp = - (avg + avg + ... + avg ) ..........(1.116)
K 1 2 K
and Pp is the Shannon probability (ie., the likelyhood,) that the
value of the portfolio time series will increase in the next time
interval.
Note that Equation (1.17) presumes that the portfolio's time series
will be optimal, ie., rmsp = sqrt (avgp). This is probably not the
case, since rmsp will always be less than the individual values of rms
for the equities. Additionally, note that assuming the distribution of
capital, V / K, invested in each equity to be identical may not be
optimal. 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.
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., if V is the capital value of the portfolio, then the
amount invested in each equity is V / K.) 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
avgp
---- + 1
rmsp
Pp = -------- ................................(1.117)
2
where the average value of the normalized increments of the portfolio
value, avgp, is the sum of the average values of the normalized
increments of each individual equity, and rmsp is the root mean square
sum of the root mean square values of the normalized increments of
each individual equity:
1 1 1
avgp = - avg + - avg + ... + - avg .........(1.118)
K 1 K 2 K K
and:
1 2 1 2
rmsp = sqrt ((-rms ) + (-rms ) + ...
K 1 K 2
1 2
... + (-rms ) ) .......................(1.119)
K K
or:
1
avgp = - (avg + avg + ... + avg ............(1.120)
K 1 2 K
and:
1 2 2 2
rmsp = - sqrt (rms + rms + ... + rms ) ....(1.121)
K 1 2 K
and dividing:
(avg + avg + ... + avg )
avgp 1 2 K
---- = -------------------------------- ......(1.122)
rmsp 2 2 2
sqrt (rms + rms + ... + rms )
1 2 K
and Pp is the Shannon probability (ie., the likelyhood,) that the
value of the portfolio time series will increase in the next time
interval.
The portfolio's average exponential rate of growth, Gp, would be, from
Equation (1.37):
Gp = Pp ln (1 + rmsp) +
(1 - Pp) ln (1 - rmsp) ..................(1.123)
where the Shannon probability of the portfolio, Pp, is determined by
one of the Equations, (1.111), (1.114), or (1.117).
Note that assuming the distribution of capital, V / K, invested in
each equity to be identical may not be optimal. 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 (1.117) should be used for portfolio management, as opposed
to Equations (1.111) and (1.114), which are not applicable, (Equations
(1.111) and (1.114) are monotonically decreasing on K, the number of
equities held concurrently.)
Interestingly, plots of Equation (1.123) using Equations (1.117) and
(1.123) to calculate the Shannon probability, Pp, of the portfolio,
with the number of equities held, K, as a parameter for various values
of avg and rms, tends to support the prevailing concept that the best
number of equities to hold is approximately 10. There is little
advantage in holding more, and a distinct disadvantage in holding
less[12].
MEAN REVERTING DYNAMICS
It can be shown that the number of expected equity value "high and
low" transitions scales with the square root of time, meaning that the
cumulative distribution of the probability of an equity's "high or
low" exceeding a given time interval is proportional to the reciprocal
of the square root of the time interval, (or, conversely, that the
probability of an equity's "high or low" exceeding a given time
interval is proportional to the reciprocal of the time interval raised
to the power 3/2 [Schroder, pp. 153]. What this means is that a
histogram of the "zero free" run-lengths of an equity's price would
have a 1 / (l^3/2) characteristic, where l is the length of time an
equity's price was above or below "average.") This can be exploited
for a short term trading strategy, which is also called "noise
trading."
The rationale proceeds as follows. Let l be the run length, (ie., the
number of time intervals,) that an equity's value has been above or
below average, then the probability that it will continue to do so in
the next time interval will be:
Pt = 1 / sqrt (l + 1) ........................(1.124)
where Pt is the "transient" probability. 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 (1.124). If an equity's price is above average, then, then
the probability that it will continue the trend is:
Pt = 1 - (1 / sqrt (l + 1)) ..................(1.125)
Equations (1.124) and (1.125) can be used to find the optimal time to
trade one stock for another.
Note that equation (1.37) can be used to find whether an equity's
current price is above, or below average:
G = P ln (1 + rms) + (1 - P) ln (1 - rms)
by exponentiating both sides of the equation, and subtracting the
value from the current price of the equity.
Note that there is a heuristic involved in this procedure. The
original derivation [Schroder, pp. 152], assumed a fixed increment
Brownian motion fractal, (ie., V (n + 1) = V (n) + F (n)), which is
different than Equation (1.3), V (n + 1) = V (n) (1 + F (n)). However,
simulations of Equation (1.3) tend to indicate that a histogram of the
"zero free" run-lengths of an equity's price would have a 1 / (l^3/2)
characteristic, where l is the length of time an equity's price was
above or below "average." Note that in both formulas, with identical
statistical processes, the values would, intuitively, be above, or
below, average in much the same way. 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., the measurement of avg
and rms to determine the average growth of an equity. 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.
OTHER PROVISIONS
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:
1) If an equity has multiple representations in a single time
unit, (ie., multiple instances with the same time stamp,) only the
last is used.
2) 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.
APPENDIX 1
Approximating Statistical Estimates to a Time Series with a
Single Pole Filter
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., random walk or Brownian motion, is that future
prices can not be predicted from past equity price performance. 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 0.51, on a day to day bases, (although, occasionally, it
will be as high as 0.6) What this means, for a typical equity, is that
51% of the time, a equity's price will increase, and 49% of the time
it will decrease-and there is no possibility of determining which will
occur-only the probability.
However, another implication of considering equity prices to have
fractal characteristics is that there are statistical optimizations to
maximize portfolio performance. The Shannon probability, P, is related
to the optimal volatility of a equity's price, (measured as the root
mean square of the normalized increments of the equity's price time
series,) rms, by rms = 2P - 1. 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 (1.6):
V(t) - V(t - 1)
---------------
V(t - 1)
which, when averaged over a "sufficiently large" number of increments,
is the mean of the normalized increments, avg. The term "sufficiently
large" must be analyzed quantitatively. For example, Table I is the
statistical estimate for a Shannon probability, P, of a time series,
vs, the number of records required, based on a mean of the normalized
increments = 0.04, (ie., a Shannon probability of 0.6 that is optimal,
ie., rms = (2P - 1) * avg):
P avg e c n
0.51 0.0004 0.0396 0.7000 27
0.52 0.0016 0.0384 0.7333 33
0.53 0.0036 0.0364 0.7667 42
0.54 0.0064 0.0336 0.8000 57
0.55 0.0100 0.0300 0.8333 84
0.56 0.0144 0.0256 0.8667 135
0.57 0.0196 0.0204 0.9000 255
0.58 0.0256 0.0144 0.9333 635
0.59 0.0324 0.0076 0.9667 3067
0.60 0.0400 0.0000 1.0000 infinity
Table I.
where avg is the average of the normalized increments, e is the error
estimate in avg, c is the confidence level of the error estimate, and
n is the number of records required for that confidence level in that
error estimate. What Table I means is that if a step function, from
zero to 0.04, (corresponding to a Shannon probability of 0.6,) is
applied to the system, then after 27 records, we would be 70%
confident that the error level was not greater than 0.0396, or avg was
not lower than 0.0004, which corresponds to an effective Shannon
probability of 0.51. Note that if many iterations of this example of
27 records were performed, then 30% of the time, the average of the
time series, avg, would be less than 0.0004, and 70% greater than
0.0004. This means that the the Shannon probability, 0.6, would have
to be reduced by a factor of 0.85 to accommodate the error created by
an insufficient data set size to get the effective Shannon probability
of 0.51. Since half the time the error would be greater than 0.0004,
and half less, the confidence level would be 1 - ((1 - 0.85) * 2) =
0.7, meaning that if we measured a Shannon probability of 0.6 on only
27 records, we would have to use an effective Shannon probability of
0.51, corresponding to an avg of 0.0004. For 33 records, we would use
an avg of 0.0016, corresponding to a Shannon probability of 0.52, and
so on.
Following like reasoning, Table II is the statistical estimate for a
Shannon probability, P, of a time series, vs, the number of records
required, based on a root mean square of the normalized increments =
0.2, (ie., a Shannon probability of 0.6 that is optimal, ie., rms =
(2P - 1) * avg):
P rms e c n
0.51 0.02 0.18 0.7000 1
0.52 0.04 0.16 0.7333 1
0.53 0.06 0.14 0.7667 2
0.54 0.08 0.12 0.8000 3
0.55 0.10 0.10 0.8333 4
0.56 0.12 0.08 0.8667 8
0.57 0.14 0.06 0.9000 16
0.58 0.16 0.04 0.9333 42
0.59 0.18 0.02 0.9667 227
0.60 0.20 0.00 1.0000 infinity
Table II.
where rms is the average of the normalized increments, e is the error
estimate in rms, c is the confidence level of the error estimate, and
n is the number of records required for that confidence level in that
error estimate. What Table II means is that if a step function, from
zero to 0.2, (corresponding to a Shannon probability of 0.6,) is
applied to the system, then after 1 records, we would be 70% confident
that the error level was not greater than 0.18, or rms was not lower
than 0.02, which corresponds to an effective Shannon probability of
0.51. Note that if many iterations of this example of 1 records were
performed, then 30% of the time, the root mean square of the time
series, rms, would be less than 0.01, and 70% greater than 0.02. This
means that the the Shannon probability, 0.6, would have to be reduced
by a factor of 0.85 to accommodate the error created by an
insufficient data set size to get the effective Shannon probability of
0.51. Since half the time the error would be greater than 0.02, and
half less, the confidence level would be 1 - ((1 - 0.85) * 2) = 0.7,
meaning that if we measured a Shannon probability of 0.6 on only 1
record, we would have to use an effective Shannon probability of 0.51,
corresponding to an rms of 0.02. For 2 records, we would use an rms of
0.06, corresponding to a Shannon probability of 0.53, and so on.
And curve fitting to Tables I and II:
P avg e c
0.51 0.0004 0.0396 0.7000
0.52 0.0016 0.0384 0.7333
0.53 0.0036 0.0364 0.7667
0.54 0.0064 0.0336 0.8000
0.55 0.0100 0.0300 0.8333
0.56 0.0144 0.0256 0.8667
0.57 0.0196 0.0204 0.9000
0.58 0.0256 0.0144 0.9333
0.59 0.0324 0.0076 0.9667
0.60 0.0400 0.0000 1.0000
P n pole
0.51 27 0.000059243
0.52 33 0.000455135
0.53 42 0.000357381
0.54 57 0.000486828
0.55 84 0.000545072
0.56 135 0.000526139
0.57 255 0.000420259
0.58 635 0.000256064
0.59 3067 0.000086180
0.60 infinity -----------
Table III.
where the pole frequency, fp, is calculated by:
avg
ln (1 - ----)
0.04
fp = - ------------ ..........................(1.126)
2 PI n
which was derived from the exponential formula for a single pole
filter, vo = vi ( 1 - e^(-t / rc)), where the pole is at 1 / (2 PI
rc). The average of the necessary poles is 0.000354700, although
an order of magnitude smaller could be used, as could 50% larger.
P rms e c
0.51 0.02 0.18 0.7000
0.52 0.04 0.16 0.7333
0.53 0.06 0.14 0.7667
0.54 0.08 0.12 0.8000
0.55 0.10 0.10 0.8333
0.56 0.12 0.08 0.8667
0.57 0.14 0.06 0.9000
0.58 0.16 0.04 0.9333
0.59 0.18 0.02 0.9667
0.60 0.20 0.00 1.0000
P n pole
0.51 1 0.016768647
0.52 1 0.035514399
0.53 2 0.028383290
0.54 3 0.027100141
0.55 4 0.027579450
0.56 8 0.018229025
0.57 16 0.011976139
0.58 42 0.006098810
0.59 227 0.001614396
0.60 infinity -----------
Table IV.
where the pole frequency, fp, is calculated by:
rms
ln (1 - ---)
0.2
fp = - ------------ ..........................(1.127)
2 PI n
which was derived from the exponential formula for a single pole
filter, vo = vi ( 1 - e^(-t / rc)), where the pole is at 1 / (2 PI
rc). The average of the necessary poles is 0.019251589, although an
order of magnitude smaller could be used, as could 50% larger.
Tables I, II, III, and IV represent an equity with a Shannon
probability of 0.6, which is about the maximum that will be seen in
the equity markets. Tables V and VI represent similar reasoning, but
with a Shannon probability of 0.51, which is at the low end of the
probability spectrum for equity markets:
P avg e c
0.501 0.000004 0.000396 0.964705882
0.502 0.000016 0.000384 0.968627451
0.503 0.000036 0.000364 0.972549020
0.504 0.000064 0.000336 0.976470588
0.505 0.000100 0.000300 0.980392157
0.506 0.000144 0.000256 0.984313725
0.507 0.000196 0.000204 0.988235294
0.508 0.000256 0.000144 0.992156863
0.509 0.000324 0.000076 0.996078431
0.510 0.000400 0.000000 1.000000000
P n pole
0.501 10285 0.000000156
0.502 11436 0.000000568
0.503 13358 0.000001124
0.504 16537 0.000001678
0.505 22028 0.000002079
0.506 32424 0.000002191
0.507 55506 0.000001931
0.508 124089 0.000001310
0.509 524307 0.000000504
0.510 infinity -----------
Table V.
where the pole frequency, fp, is calculated by:
avg
ln (1 - ------)
0.0004
fp = - --------------- .......................(1.128)
2 PI n
which was derived from the exponential formula for a single pole
filter, vo = vi ( 1 - e^(-t / rc)), where the pole is at 1 / (2 PI
rc). The average of the necessary poles is 0.000001282, although
an order of magnitude smaller could be used, as could 70% larger.
P rms e c
0.501 0.002 0.018 0.964705882
0.502 0.004 0.016 0.968627451
0.503 0.006 0.014 0.972549020
0.504 0.008 0.012 0.976470588
0.505 0.010 0.010 0.980392157
0.506 0.012 0.008 0.984313725
0.507 0.014 0.006 0.988235294
0.508 0.016 0.004 0.992156863
0.509 0.018 0.002 0.996078431
0.510 0.020 0.000 1.000000000
P n pole
0.501 3 0.005589549
0.502 4 0.008878600
0.503 5 0.011353316
0.504 8 0.010162553
0.505 11 0.010028891
0.506 19 0.007675379
0.507 36 0.005322728
0.508 89 0.002878090
0.509 415 0.000883055
0.510 infinity -----------
Table VI.
where the pole frequency, fp, is calculated by:
rms
ln (1 - ----)
0.02
fp = - ------------ ..........................(1.129)
2 PI n
which was derived from the exponential formula for a single pole
filter, vo = vi ( 1 - e^(-t / rc)), where the pole is at 1 / (2 PI
rc). The average of the necessary poles is 0.006974618, although an
order of magnitude smaller could be used, as could 60% larger.
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, 524307
business days is a little over two millenia-the required size of the
data set for day trading.) Another issue is that the pole frequency
changes with magnitude of the Shannon probability, as shown by
comparison of Tables III, V, and IV, VI, respectively. 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., pick the top 10,) since, for a
given data set size, a larger Shannon probability will be chosen over
a smaller. However, this would imply that the equities are known to be
optimal, ie., rms = 2P + 1, which, although it is nearly true for most
equities, is not true for all equities. There is some possibility that
optimality can be verified by metrics:
2
if avg < rms
then rms = f is too large in Equation (1.12)
2
else if avg > rms
then rms = f is too small in Equation (1.12)
2
else avg = rms
and the equities time series is optimal, ie.,
rms = f = 2P - 1 from Equation (1.36)
These metrics would require identical statistical estimate filters for
both the average and the root mean squared filters, ie., the square of
rms would have the same filter pole as avg, which would be at
0.000001282, and would be conservative for Shannon probabilities above
0.51.
The Shannon probability can be calculated by several methods using
Equations (1.6) and (1.14). Equation (1.14):
avg
--- + 1
rms
P = -------
2
has two other useful alternative solutions if it is assumed that the
equity time series is optimal, ie., rms = 2P - 1, and by substitution
into Equation (1.14):
rms + 1
P = ------- ..................................(1.130)
2
and:
sqrt (avg) + 1
P = -------------- ...........................(1.131)
2
Note that in Equation (1.14) the confidence levels listed in Tables I
and II should be multiplied together, and a new table made for the
quotient of the average and the root mean square of the normalized
increments, avg and rms respectively. 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 (1.14),
avg = rms (2P - 1), which can be optimized/maximized. P can be
calculated from Equations (1.14), (1.61), or (1.62). 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., there can be an rms that uses different
filter parameters, than the RMS in the equation, avg = RMS (2P -
1). By substitution, Equation (1.14) will have a decision criteria of
the largest value of RMS * avg / rms, Equation (1.61) will have a
decision criteria of the largest value of RMS * rms, and Equation
(1.62) will have a decision criteria of RMS * sqrt (avg), or avg.
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 0.000053 times the time series sampling frequency, for the average
of the normalized increments of the time series, avg. (The rationale
behind this value is that if we consider an equity with a measured
Shannon probability of 0.51-a typical value-and we wish to include an
uncertainty in the precision of this value based on the size of the
data set, then we must decrease the Shannon probability by a factor of
0.960784314. This number comes from the fact that a Shannon
probability, P', would be (0.5 / 0.51) * P = 0.980392157 * P = 0.51 *
0.980392157 = 0.5, a Shannon probability below which, no wager should
be made, (as an absolute lower limit.) 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 0.51, and half less, than the "real" value of
the Shannon probability. So the Shannon probability must be reduced by
a factor of c = 1 - 2(1 - 0.980392157) = 0.960784314. 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 0.51 is 0.0004, since the root mean square,
rms, of the normalized increments of a time series with a Shannon
probability of 0.51 is 0.02, and, if the time series is optimal, where
avg = (2P - 1) * rms, then avg = 0.0004. So, we now have the error
level, 0.0004, and the required confidence level, 0.960784314, and the
number of required records, ie., the data set size, would be 9773,
The advantage of the discreet time recursive single pole filter
approximation to a statistical estimate is that it requires only 3
lines of code in the implementation-two for initialization, and one in
the calculation construct. A "running average" methodology would offer
far greater accuracy as an approximation to the statistical estimate,
however the memory requirements for the average could be prohibitive
if many equities were being tracked concurrently, (see Table V,) and
computational resource requirements for circular buffer operation
could possible be an issue. The other alternative would be to perform
a true statistical estimate, however the architecture of a recursive
algorithm implementation may be formidable.
The single pole low pass filter is implemented from the following
discrete time equation:
v = I * k2 + v * k1 ....................(1.132)
t + 1 t
where I is the value of the current sample in the time series, v are
the value of the output time series, and k1 and k2 are constants
determined from the following equations:
-2 * p * pi
k1 = e ............................(1.133)
and
k2 = 1 - k1 ..................................(1.134)
where p is a constant that determines the frequency of the pole-a
value of unity places the pole at the sample frequency of the time
series.
APPENDIX 2
Number of Concurrently Held Equities
Note: The prototype to this program was implemented with a user
configurable fixed number of equities in the equity portfolio, as
determined by the reasoning outlined in this Appendix. This
methodology was superceded by dynamically determining the number of
equities held as outlined in the Section, PORTFOLIO OPTIMIZATION.
The remaining issue is the number of equities held
concurrently. Measuring the average and root mean square of the
normalized increments of many equities (600 equities selected from all
three American markets, 1 January, 1993 to 1 May, 1996,) resulted in
an average Shannon probability of 0.52, and an average root mean
square of the normalized increments of 0.03. Only infrequently was a
volatility found that exceed the optimal, ie., where rms = 2P - 1, by
a factor of 3, (approximately a one sigma limit.) However, once,
(approximately a 3 sigma limit,) a factor of slightly in excess of 10
was found for a short interval of time. There is a possibility that
the equities with the maximum Shannon probability and maximum growth
This, also, seems consistent Equation (1.123),
Gp = Pp ln (1 + rmsp) +
(1 - Pp) ln (1 - rmsp)
and substituting Equation (1.117) for Pp:
avgp
---- + 1
rmsp
Gp = -------- ln (1 + rmsp) +
2
avgp
1 - ----
rmsp
-------- ln (1 - rmsp) ..................(1.135)
2
and iterating plots of equities with similar statistical
characteristics as a parameter, (ie., using a P of 0.51, etc., and
plotting the portfolio gain, Gp, with the number of equities held as a
parameter.) There seems to be little advantage in holding more than 10
equities concurrently, which is also consistent with the advice of
many brokers.
FOOTNOTES
[1] For example, if a = 0.06, or 6%, then at the end of the first time
interval the capital would have increased to 1.06 times its initial
value. At the end of the second time interval it would be (1.06 *
1.06), and so on. What Equation (1.1) states is that the way to get
the value, V in the next time interval is to multiply the current
value by 1.06. Equation (1.1) is nothing more than a "prescription,"
or a process to make an exponential, or "compound interest"
mechanism. In general, exponentials can always be constructed by
multiplying the current value of the exponential by a constant, to get
the next value, which in turn, would be multiplied by the same
constant to get the next value, and so on. Equation (1.1) is a
construction of V (t) = exp(kt) where k = ln(1 + a). The advantage of
representing exponentials by the "prescription" defined in Equation
(1.1) is analytical expediency. For example, if you have data that is
an exponential, the parameters, or constants, in Equation (1.1) can be
determined by simply reversing the "prescription," ie., subtracting
the previous value, (at time t - 1,) from the current value, and
dividing by the previous value would give the exponentiating constant,
(1 + at). This process of reversing the "prescription" is termed
calculating the "normalized increments." (Increments are simply the
difference between two values in the exponential, and normalized
increments are this difference divided by the value of the
exponential.) Naturally, since one usually has many data points over a
time interval, the values can be averaged for better precision-there
is a large mathematical infrastructure dedicated to these types of
precision enhancements, for example, least squares approximation to
the normalized increments, and statistical estimation.
[2] "Random variable" means that the process, F(t), is random in
nature, ie., there is no possibility of determining what the next
value will be. However, F can be analyzed using statistical methods
[Fed88, pp. 163], [Sch91, pp. 128]. For example, F typically has a
Gaussian distribution for equity prices [Cro95, pp. 249], in which
case the it is termed a "fractional Brownian motion," or simply a
"fractal" process. In the case of a single tossed coin, it is termed
"fixed increment fractal," "Brownian," or "random walk" process. The
determination of the statistical characteristics of F(t) are the
essence of analysis. Fortunately, there is a large mathematical
infrastructure dedicated to the subject. For example, F could be
verified as having a Gaussian distribution using, perhaps, Chi-Square
techniques. Frequently, it is convenient, from an analytical
standpoint, to "model" F using a mathematically simpler process
[Sch91, pp. 128]. For example, multiple iterations of tossing a coin
can be used to approximate a Gaussian distribution, since the
distribution of many tosses of a coin is binomial-which if the number
of coins tossed is sufficient will represent a Gaussian distribution
to any required precision [Sch91, pp. 144], [Fed88, pp. 154].
[3] Equation (1.3) is interesting in many other respects. For
example, adding a single term, m * V(t - 1), to the equation results
in V(t) = v(t - 1) (1 + f(t) * F(t) + m * V(t - 1)) which is the
"logistic," or 'S' curve equation,(formally termed the "discreet time
quadratic equation,") and has been used successfully in many unrelated
fields such as manufacturing operations, market and economic
forecasting, and analyzing disease epidemics [Mod92, pp. 131]. There
is continuing research into the application of an additional
"non-linear" term in Equation (1.3) to model equity value
non-linearities. Although there have been modest successes, to date,
the successes have not proven to be exploitable in a systematic
fashion [Pet91, pp. 133]. The reason for the interest is that the
logistic equation can exhibit a wide variety of behaviors, among them,
"chaotic." Interestingly, chaotic behavior is mechanistic, but not
"long term" predictable into the future. A good example of such a
system is the weather. It is an important concept that compound
interest, the logistic function, and fractals are all closely related.
[4] In this Section, "root mean square" is used to mean the variance
of the normalized increments. In Brownian motion fractals, this is
computed by sigmatotal^2 = sigma1^2 + sigma2^2 ... However, in many
fractals, the variances are not calculated by adding the squares,
(ie., a power of 2,) of the values-the power may be "fractional," ie.,
3 / 2 instead of 2, for example [Sch91, pp. 130], [Fed88, pp.
178]. However, as a first order approximation, the variances of the
normalized increments of equity prices can be added root mean square
[Cro95, kpp. 250]. The so called "Hurst" coefficient determines the
process to be used. The Hurst coefficient is range of the equity
values over a time interval, divided by the standard deviation of the
values over the interval, and its determination is commonly called "R
/ S" analysis. As pointed out in [Sch91, pp. 157] the errors committed
in such simplified assumptions can be significant-however, for
analysis of equities, squaring the variances seems to be a reasonably
accurate simplification.
[5] For example, many calculators have averaging and root mean square
functionality, as do many spreadsheet programs-additionally, there are
computer source codes available for both. See the programs tsrms(1)
and tsavg(1). The method used is not consequential.
[6] There are significant implications do to the fact that equity
volatilities are calculated root mean square. For example, if capital
is invested in N many equities, concurrently, then the volatility of
the capital will be rms / sqrt (N) of an individual equity's
volatility, rms, provided all the equites have similar statistical
characteristics. But the growth in the capital will be unaffected,
ie., it would be statistically similar to investing all the capital in
only one equity. What this means is that capital, or portfolio,
volatility can be minimized without effecting portfolio growth-ie.,
volatility risk can addressed. There are further applications. For
example, Equation (1.6) could be modified by dividing both the
normalized increments, and the square of the normalized increments by
the daily trading volume. The quotient of the normalized increments
divided by the trading volume is the instantaneous, average, avg, of
the equity, on a per-share basis. Likewise, the square root of the
square of the normalized increments divided by the daily trading
volume is the instantaneous root mean square, rmsf, of the equity on a
per-share basis, ie., its instantaneous volatility of the equity.
(Note that these instantaneous values are the statistical
characteristics of the equity on a per-share bases, similar to a coin
toss, and not on time.) Additionally, it can be shown that the
range-the maximum minus the minimum-of an equity's value over a time
interval will increase with the square root of of the size of the
interval of time [Fed88, pp. 178]. Also, it can be shown that the
number of expected equity value "high and low" transitions scales with
the square root of time, meaning that the cumulative distribution of
the probability of an equity's "high or low" exceeding a given time
interval is proportional to the reciprocal of the square root of the
time interval, (or, conversely, that the probability of an equity's
"high or low" exceeding a given time interval is proportional to the
reciprocal of the time interval raised to the power 3/2 [Schroder,
pp. 153]. What this means is that a histogram of the "zero free"
run-lengths of an equity's price would have a 1 / (l^3/2)
characteristic, where l is the length of time an equity's price was
above or below "average.")
[7] Here the "model" is to consider two black boxes, one with an
equity "ticker" in it, and the other with a casino game of a tossed
coin in it. One could then either invest in the equity, or,
alternatively, invest in the tossed coin game by buying many casino
chips, which constitutes the starting capital for the tossed coin
game. Later, either the equity is sold, or the chips "cashed in." If
the statistics of the equity value over time is similar to the
statistics of the coin game's capital, over time, then there is no way
to determine which box has the equity, or the tossed coin game. The
advantage of this model is that gambling games, such as the tossed
coin, have a large analytical infrastructure, which, if the two black
boxes are statistically the same, can be used in the analysis of
equities. The concept is that if the value of the equity, over time,
is statistically similar to the coin game's capital, over time, then
the analysis of the coin game can be used on equity values. Note that
in the case of the equity, the terms in f(t) * F(t) can not be
separated. In this case, f = rms is the fraction of the equity's
value, at any time, that is "at risk," of being lost, ie., this is the
portion of a equity's value that is to be "risk managed." This is
usually addressed through probabilistic methods, as outlined below in
the discussion of Shannon probabilities, where an optimal wagering
strategy is determined. In the case of the tossed coin game, the
optimal wagering strategy is to bet a fraction of the capital that is
equal to f = rms = 2P - 1 [Sch91, pp. 128, 151], where P is the
Shannon probability. In the case of the equity, since f = rms is not
subject to manipulation, the strategy is to select equities that
closely approximate this optimization, and the equity's value, over
time, on the average, would increase in a similar fashion to the coin
game. As another alternative, various equities can be invested in
concurrently to exercise control over portfolio volatility. The growth
of either investment would be equal to avg = rms^2, on average, for
each iteration of the coin game, or time unit of equity/portfolio
investment. This is an interesting concept from risk management since
it maximizes the gain in the capital, while, simultaneously,
minimizing risk exposure to the capital.
[8] Penrose, referencing Russell's paradox, presents a very good
example of logical contradiction in a self-referential system.
Consider a library of books. The librarian notes that some books in
the library contain their titles, and some do not, and wants to add
two index books to the library, labeled "A" and "B," respectively; the
"A" book will contain the list of all of the titles of books in the
library that contain their titles; and the "B" book will contain the
list of all of the titles of the books in the library that do not
contain their titles. Now, clearly, all book titles will go into
either the "A" book, or the "B" book, respectively, depending on
whether it contains its title, or not. Now, consider in which book,
the "A" book or the "B" book, the title of the "B" book is going to be
placed-no matter which book the title is placed, it will be
contradictory with the rules. And, if you leave it out, the two books
will be incomplete.
[9] [Art95] cites the "El Farol Bar" problem as an example. Assume one
hundred people must decide independently each week whether go to the
bar. The rule is that if a person predicts that more than, say, 60
will attend, it will be too crowded, and the person will stay home; if
less than 60 is predicted, the person will go to the bar. As trivial
as this seems, it destroys the possibility of long-run shared,
rational expectations. If all believe few will go, then all will go,
thus invalidating the expectations. And, if all believe many will go,
then none will go, likewise invalidating those expectations.
Predictions of how many will attend depend on others' predictions, and
others' predictions of others' predictions. Once again, there is no
rational means to arrive at deduced a-priori predictions. The
important concept is that expectation formation is a self-referential
process in systems involving many agents with incomplete information
about the future behavior of the other agents. The problem of
logically forming expectations then becomes ill-defined, and rational
deduction, can not be consistent or complete. This indeterminacy of
expectation-formation is by no means an anomaly within the real
economy. On the contrary, it pervades all of economics and game
theory.
[10] Interestingly, the system described is a stable system, ie., if
the players have a hypothesis that changing equity positions may be of
benefit, then the equity values will fluctuate-a self fulfilling
prophecy. Not all such systems are stable, however. Suppose that one
or both players suddenly discover that equity values can be "timed,"
ie., there are certain times when equities can be purchased, and
chances are that the equity values will increase in the very near
future. This means that at certain times, the equites would have more
value, which would soon be arbitrated away. Such a scenario would not
be stable.
[11] Note that in a time interval of sufficiently many tosses of the
coin, say N many, that there will be PN many wins, and (1 - P)N many
losses. In each toss, the gambler's capital, V, increased, or
decreased by an amount f = rms. So, after the first iteration, the
gambler's capital would be V(1) = V(0) (1 + rms F(1)), and after the
second would be V(2) = V(0) (1 + rms F(1)) (1 + rms F(2)), and after
the N'th, V(N) = V(0) (1 + rms F(1)) (1 + rms F(2)) ... (1 + rms
F(N)), where F is either plus or minus unity. Since the
multiplications are transitive, the terms may be rearranged, and there
will be PN many wins, and (1 - P) many losses, or V(N) = V(0) * (1 +
rms)^(P) * (1 - rms)^(1 - P). Dividing both sides by V(0), the
starting value of the gambler's capital, and taking the logarithm of
both sides, results in ln (V(N) / V(0)) = P ln (1 + rms) + (1 - P) ln
(1 - rms), which is the equation for G = ln (V(N) / V(0)), the average
exponential rate of growth over N many tosses, providing that N is
sufficiently large. Note that the "effective interest rate" as
expressed in Equation (1.1), is a = exp (G) - 1.
[12] If the plotting program "gnuplot" is available, then the
following commands will plot Equation (1.123) using the method of
computation for the Shannon probability from Equations (1.117) through
(1.122), (1.111) through (1.113), and, (1.114) through (1.116),
restively.
plot [1:50] ((1 + (0.02 / sqrt (x))) **
((((0.0004 / 0.02) * sqrt (x)) + 1) / 2)) *
((1 - (0.02 / sqrt (x))) **
((1 - ((0.0004 / 0.02) * sqrt (x))) / 2))
plot [1:50] ((1 + (0.02 / sqrt (x))) **
(((0.02 / sqrt (x)) + 1) / 2)) *
((1 - (0.02 / sqrt (x))) **
((1 - (0.02 / sqrt (x))) / 2))
plot [1:50] ((1 + (sqrt (0.0004) / sqrt (x))) **
((sqrt (0.0004) + 1) / 2)) *
((1 - (sqrt (0.0004) / sqrt (x))) **
((1 - sqrt (0.0004)) / 2))
BIBLIOGRAPHY
[Art95] W. Brian Arthur. "Complexity in Economic and Financial
Markets." Complexity, 1, pp. 20-25, 1995. Also available from
http://www.santafe.edu/arthur, February 1995.
[BdL95] William A. Brock and Pedro J. F. de Lima. "Nonlinear time
series, complexity theory, and finance." To appear in "Handbook of
Statistics Volume 14: Statistical Methods in Finance," edited by
G. Maddala and C. Rao. New York: North Holland, forthcoming. Also
available from http://www.santafe.edu/sfi/publications, March 1995.
[Cas90] John L. Casti. "Searching for Certainty." William Morrow, New
York, New York, 1990.
[Cas94] John L. Casti. "Complexification." HarperCollins, New York,
New York, 1994.
[Con78] John Conover. "An analog, discrete time, single pole filter."
Fairchild Journal of Semiconductor Progress, 6(4), July/August 1978.
[Cro95] Richard M. Crownover. "Introduction to Fractals and Chaos."
Jones and Bartlett Publishers International, London, England, 1995.
[Fed88] Jens Feder. "Fractals." Plenum Press, New York, New York,
1988.
[Mod92] Theodore Modis. "Predictions." Simon & Schuster, New York, New
York, 1992.
[Pen89] Roger Penrose. "The Emperor's New Mind." Oxford University
Press, New York, New York, 1989.
[Pet91] Edgar E. Peters. "Chaos and Order in the Capital Markets."
John Wiley & Sons, New York, New York, 1991.
[Rez94] Fazlollah M. Reza. "An Introduction to Information Theory."
Dover Publications, New York, New York, 1994.
[Sch91] Manfred Schroeder. "Fractals, Chaos, Power Laws."
W. H. Freeman and Company, New York, New York, 1991.
--
John Conover, john@email.johncon.com, http://www.johncon.com/