(Revised March 11, 2013)

Robert Murray, Ph.D.

*Omicron Research Institute*

(Copyright Ó 2013
*Omicron Research Institute*. All rights reserved.)

**Financial Econometrics and***QuanTek***Signal Processing & Linear Prediction****Wavelet Analysis of Time Series****Predictors of Future Returns****Optimal Trading Strategies****Efficient Market Hypothesis and Trading Strategies****Autocorrelation & Power Spectrum****Trading Time Scales****Law of Large Numbers****Overall Investment Strategy****Financial Returns as a Stochastic Process****Noise Reduction and Smoothing****Financial Risk & GARCH****Stochastic vs. Deterministic Process****Predictability of Market Processes****Correlations in Financial Time Series****Non-linear Behavior and Market Crashes****Portfolio Optimization and Rebalancing****References**

The ** QuanTek**
Econometric program is designed to utilize the most state-of-the-art ideas from
the rapidly evolving field of

The
prediction of future returns of financial assets is tricky business, of course,
and according to the **Efficient Market Hypothesis**, it cannot be done at
all. But some estimate of future
returns must be made, otherwise how does an investor know what investments to
make? An excellent book on the **Kalman**
filter, with applications to Finance, is __Forecasting, structural time series
models, and the Kalman filter__ (1989), by Andrew C. Harvey [H]. (The Kalman filter will be incorporated in a
future version of ** QuanTek**.)
Here is a quote from the preface to this book (p.xi), which illustrates
the situation perfectly:

“The inclusion of ‘forecasting’ in the title of the book is perhaps a little rash. It is always very difficult to predict the future on the basis of the past. Indeed it has been likened to driving a car blindfolded while following directions given by a person looking out of the back window. Nevertheless, if this is the best we can do, it is important that it should be done properly, with an appreciation of the potential errors involved. In this way it should at least be possible to negotiate straight stretches of road without a major disaster. Too many forecasting procedures seem to attribute the person in the back seat with supernatural powers when in fact his behaviour is more consistent with that of someone who is mildly inebriated.”

With this in
mind, it becomes very important to be able to discern the limits of our ability
to forecast expected returns, and to develop methods for separating the small
‘signal’ present in the data from the much larger stochastic noise. In particular, it is very easy to confuse
the stochastic noise with a ‘signal’, and end up developing trading rules that
merely follow the random patterns of the stochastic noise. The methods of ** Technical Analysis**
are particularly vulnerable to this malady, and these ‘intuitive’ methods end
up in the similar situation to someone looking up at the sky and perceiving
faces and other patterns in the clouds that are not, in fact, there at
all. This fallacy is due to the nature
of the human mind to perceive patterns in random data, and one must therefore
be careful to try to use rigorous methods of Statistics to discern the patterns
that are really there from the apparent patterns that are due merely to
stochastic noise.

What
is the problem that the ** QuanTek** program attempts to solve? There are many different trading strategies
and methods of estimating future price behavior of stocks and other
securities. Foremost among these is the
study of

On
the other hand, the traditional **Efficient Market Hypothesis** and **Random
Walk** model [PB] stipulate that there is no “signal” in the financial data,
and future price action cannot be predicted based on past price action. We take the point of view that the markets are,
indeed, very efficient, but not 100% efficient. There are small inefficiencies that can be uncovered and
exploited to make a profit, because traders and investors are not completely
rational, not completely well informed, and do not react instantaneously as the
Efficient Market Hypothesis supposes.
So it should be possible for a knowledgeable trader or investor with a
sound trading strategy to “beat the market”.
On the other hand, it is well known by now that the Random Walk model is
only a crude approximation to market behavior.
It supposes that price returns are independent random variables that
satisfy a Gaussian distribution. But it
is clear that the actual distribution is not Gaussian and exhibits “fat tails”,
such as the stable Paretian or Levy distributions [M, PB]. These fat-tailed distributions give rise to **risk**
that is far greater than the Gaussian distribution would predict, especially
when it comes to market “bubbles” and the subsequent “crashes” which inevitably
result. So the upside and downside risk
needs to be accurately estimated, taking into account the fact that the
distribution of returns is non-Gaussian.
This then involves the **GARCH** models of time-varying volatility
[G]. The ** QuanTek** program
will implement (in a future version) this strategy using Wavelet techniques to
estimate both the future time-varying returns and volatility, based on the fact
that these quantities are not exactly independent random variables, but may
exhibit correlation with the past.

One of the main features of the ** QuanTek**
program is the use of Wavelet analysis [PW] of the financial time series
data. This is an alternative to
analyzing the time series directly in the time domain as in a conventional
regression approach, on the one hand, and using the Fourier analysis or
frequency domain, on the other hand.
Here, the time domain means dividing the whole time series up into its
components with respect to time, whereas the frequency domain means taking
linear sums of the time series and decomposing it into signals with a definite
frequency.

In
the conventional **Fourier** analysis, the time series data is separated
into pure sine waves of a single frequency, extending over all time. Usually the Fast Fourier Transform (FFT) is
used, which requires that the data satisfy circular boundary conditions. So the data must be a fixed length, in our
case 2048 or 4096 units long, and the data are arranged to form a circular data
set. Then in the Fourier analysis the
circular data are decomposed into component frequency signals that extend over
the whole circle. This decomposition is
ideal for periodic signals such as waveforms in an electronic circuit, but not
so good for financial data. For one
thing, the Fourier transform assumes the time series to obey **stationary
statistics**, whereas we expect financial time series to obey **non-stationary
statistics**. The non-stationarity
means that the statistical properties such as the mean and covariance matrix
vary with time. So due to the “infinite”
time extent of the component waves in the Fourier analysis, it is not well
adapted to non-stationary time series such as financial time series.

It
turns out that the **Wavelet** analysis is much better suited to many types
of realistic signals than the Fourier analysis. In the Wavelet analysis the data are also circular with a fixed
length of 2048 or 4096 units, like the FFT.
But Wavelet analysis is halfway between the time domain of the signal as
a function of time, and the frequency (Fourier) domain of the signal as a
function of frequency. The basis
functions of the Wavelet analysis are waves that extend over a finite range of
both time and frequency. By comparison,
the basis functions of the frequency domain (Fourier) are pure sine waves of a
single frequency which extend over all time, while the basis functions of the
time domain are the signals at any specific time, but which extend over all
frequencies as a result. The Wavelet
basis functions are divided into octaves of frequency, and the high frequency
octave functions extend over short time intervals, while the low frequency
octave functions extend over long time intervals. This corresponds to what we expect in financial data – that the
low frequency fluctuations have an influence over long time periods, while the
high frequency fluctuations have an influence only over short time
periods. Thus using the Wavelet
analysis enables an immediate improvement in the signal to noise ratio, in that
the short-term fluctuations occurring in the distant past can be eliminated
right away, since they are unlikely to have any influence on the present or
future. Working in the Wavelet basis
enables half of the stochastic noise to be eliminated and a drastic reduction
in the number of degrees of freedom that have to be fitted in the regression,
eliminating (to some extent) the problem of “overfitting”.

In the Wavelet regression model, therefore, each wavelet level, corresponding to an octave of frequencies, is approximately independent and is fitted independently. Then the low-frequency octaves make a long-term prediction that extends for many time units into the future, while the high-frequency octaves make only a short-term prediction that extends only a few time units into the future. Thus, in effect, for the returns the long-term trends are extended into the future a number of days corresponding to the length of the trend, while the shorter-term trends are extended into the future a correspondingly shorter number of days. So this is a simple and natural way to look at the Wavelet analysis. The extension of the trends on each wavelet level is also dependent on the time-dependent correlation in the data, which in turn weights the trends on each wavelet level differently. In fact, if the trend on each wavelet level is extended with equal weight, the future projected return adds up to zero, corresponding to no correlation. The correlation enters due to the difference of the weights of the projected return on the different wavelet levels. These different weights are determined by the regression procedure, which is greatly simplified because the regression is performed only on the different wavelet levels, not on each past data day separately as in the usual time-domain approach. So this separation of the data into wavelets results in a vast simplification of the regression problem, and helps separate the important degrees of freedom in the financial time series from the rest, which make up the “stochastic noise”.

In
the usual approach in Econometrics, a stochastic model is postulated and then
this model is “estimated” or fit to the past time series data, and a
statistical analysis of the fit is performed.
However, in the ** QuanTek** program we use a more pragmatic
method, which can also be computed within a reasonable time. This method is to search for a set of

By
default, a set of three predictors is used as a starting point, which
correspond to different types of smoothings or filterings of the past data of
the same security. These we call the **Relative
Price**, **Velocity**, and **Acceleration**. They are different smoothings of the past data, to which a
simplified Wavelet Linear Prediction has been added, resulting in a future **trial
function** that can be fitted to the future data. The smoothing has the effect of eliminating the high-frequency
stochastic noise so the correlation of the lower-frequency components is
revealed more clearly. Actually, in the
Wavelet approach, on each wavelet level these three predictors are really one
and the same, only shifted from each other by a time lag corresponding to a
quarter cycle. So all that is needed is
to fit the single trial function on each wavelet level to future returns, using
a recursion algorithm on the past data to “train” the WLP filter for an optimum
estimation of future returns. Once this
fit is made, the three predictors together can be used to define and display
the optimal **buy/sell points**, with the three predictors corresponding to
the three basic correlations or **trading rules**: **Buy Low – Sell High
(Relative Price)**, **Trend Persistence (Velocity)**, and **Turning
Points (Acceleration)**.

The
**Relative Price** is just a smoothing of the logarithmic price itself,
relative to a long-term smoothing. For
example, we can take the *M*-day Wavelet smoothing of the log prices
minus, say, the 128-day smoothing as a reference. We expect an anti-correlation between the Relative Price (at some
point in the past) and the future returns due to the **buy low – sell high **mechanism. In other words, if the *M*-day Wavelet
smoothing yields frequencies, or “cycles” with period of *M* days, then we
expect a low point a quarter cycle in the past to be correlated with positive
returns at the present, and a high point a quarter cycle in the past to be
correlated with negative returns at the present. This correlation between the *smoothed* Relative Prices and *N*-day
future returns is revealed in the ** QuanTek** Correlation Test. We expect the Relative Price predictor to be
the best one because the price is the sum over returns, leading to noise
reduction, and on top of that the Relative Price is smoothed.

The
**Velocity** indicator is just the smoothed returns themselves. We expect a correlation between past returns
and future returns at the immediate present on a smoothing scale of *M*
days due to the mechanism of **trend persistence** or **buy high – sell
higher***.* In this case it
helps in defining trial functions and measuring correlation to project the
predictor into the future using, perhaps, the basic Wavelet Linear Prediction
(WLP) filter, or as an alternative, Ordinary Linear Prediction (OLP). This is perfectly acceptable, with regard to
measuring correlation between past and future returns, so long as only *past*
data are used in the projection. This
predictor also shows respectable correlation in the Correlation Test, depending
on the method used to make the future projection.

Finally,
we have the **Acceleration** predictor, which is the smoothed
first-difference of the returns. This
predictor should have the opposite correlation from the Relative Price, and may
be thought of as marking the **turning points**, or minima-maxima of the
prices. This indicator is less
effective than the other two, since it emphasizes the higher frequencies more
than the lower frequencies, but the Correlation Test sometimes shows a correlation
here as well.

In
a more sophisticated approach, using nonlinear predictors, we might for example
suppose that the past price action has a *memory*. This is the same idea used in Technical
Analysis to determine, for example, support and resistance levels. A *potential* function could be defined
to incorporate this memory, with a damping factor to represent the fading of
the memory. The potential function
could measure the number of trades, for example, at different price levels, and
define a “potential barrier” at the levels where many trades took place in the
past, to represent the **support** and **resistance** levels. This is an idea for the future, but it
represents the method by which predictors can be defined to represent the
nonlinear behavior of the price action.
Another nonlinear predictor that could be defined is a potential
centered on a definite level of *return*, which becomes deeper the longer
that return is maintained. This would
then represent what is known in Technical Analysis as an **uptrend** or **downtrend**. But the simple *linear* predictors
described above should give a reasonable account of market dynamics, to first
approximation, ignoring the nonlinear effects.

In
a future version of ** QuanTek**, we plan to generalize the predictor
approach to a regression of each security over predictors defined in terms of
the whole portfolio of securities and in addition various market indexes and
fundamental or economic data (such as interest rates). Generally, a simplified approach will be
adequate, such as an approach similar to the

The
question arises: “What is the optimal trading strategy?” The thing many people may not realize is
that any trading strategy relies on the presumption of some kind of **correlation
between past price action and future returns**. On the one hand, short-term traders may think that the shorter
the time scale that they can trade, the more of the up-down swings or volatility
they can utilize to make a profit.
However, this is only true if there is a **correlation** between the
technical indicators or trading rules they are using, and future price
swings. In fact, if the Random Walk
model is true, then there is no such correlation, and the **expected return**
is always zero no matter what kind of trading is done. However, the more rapid the trading is, the
more the **risk** is increased, resulting in a wider **variance** in the
eventual outcome. In addition, the
transaction costs are much greater. At
the other extreme, a buy-and-hold investor ignores the short-term swings in the
market, and only invests to take advantage of a long-term up-trend in prices
that they believe to exist. This
corresponds to an assumption that a **Random Walk model with drift**
applies, so the returns have a non-zero **mean** value, although the
de-trended returns will have zero correlation.
Thus the buy-and-hold investor is assuming a particular stochastic model
for the price action, which it turns out, is a gross oversimplification of the
true situation and is not really correct.
This assumption of the existence of a long-term trend seems dangerous,
because of the clear existence of market crashes and long periods of flat markets. Also it implicitly relies on the presumption
of a trending Random Walk, which is a **stationary** time series model, but
it is clear that the actual financial time series are much better explained in
terms of a **non-stationary** stochastic model.

Our
point of view, then, is that the optimal trading strategy must start with a
determination of the potential **correlation** in the returns data. This appears to vary from one security to
the next, so the correlation must be measured between a set of chosen **predictors**
and the future returns, for each security, and then a Linear Prediction model
is set up for each security using the predictors. The Wavelet LP filter then predicts the future returns and
volatility for each time scale, and trades are made based on this predicted
price action. The trader may choose
what time scale to measure the correlation and set up trading rules. But this must take place within the context
of a **diversified portfolio** to **control risk**. This may also be viewed in the context of
maintaining an **optimal portfolio**.
In standard portfolio construction the portfolio is *rebalanced* at
periodic intervals, say every month, to maintain the computed optimal mixture
of securities as the prices change [FFK].
But instead of rebalancing at fixed time intervals, our active trading
method can be thought of as a rebalancing for each security as predicted
trading opportunities arise in that security, for any chosen time scale *N*. In this way it is hoped that the **maximum
returns** may be realized by short-term trading, while at the same time incurring
**minimum risk** by doing it within a diversified portfolio. Also if the market appears overextended and
vulnerable to a crash, our **active trading strategy** gives the best hope
of getting out of the market with minimum loss. This is achieved by gradually decreasing positions in those
securities that become “overextended”.
Clearly this is a complicated problem, which is why a sophisticated
approach such as utilized by the ** QuanTek** program is the only
adequate way to address the problem.

According
to the **Efficient Market Hypothesis** [FFK], the market is 100% efficient
and it is impossible to predict future price action on the basis of any past
information. Most people agree that
this hypothesis is too extreme, and that small inefficiencies do exist which
can enable the knowledgeable trader or investor to make a profit. Indeed, this is the case with all economic
activity – it is frequently true that assets or property are mispriced, and
those individuals who have a deep understanding of the true value of these
assets or properties can make a profit by buying or selling them. On the other hand, with financial securities
the price series are well approximated by the **Random Walk** model, and it
is very difficult to forecast the future price action based on the past price
series alone. At least there are no
clearly measurable correlations in the price data that can be used for a
forecast – the returns series appears to be random white noise in almost all
cases. However, upon closer inspection
it does appear that small correlations in the data do exist, although these are
buried in the “white noise” and are also no doubt time dependent. The question, then, is how to utilize these
small, time-dependent correlations, assuming they exist, to formulate a
profitable set of trading rules.

I
like to think of stock trading as similar to gambling. If you play a game of Craps in a casino with
fair dice, then over the long run (Law of Large Numbers) you will slowly lose, because
the odds of winning or losing are even except for a small “percentage” that
goes to the house. However, suppose the
dice are slightly “loaded”. Then, even
though the advantage might not be noticeable over short time periods, over long
time periods you can win big money because the odds are now slightly in your
favor. The trading of financial
securities is the same way. If you can
find a set of trading rules that puts the odds of winning slightly in your
favor, then over the long run you can make large profits, at least
potentially. This is the goal of **Technical
Analysis** – to find the profitable trading rules, buried deep in the
stochastic noise, which can lead to consistent trading profits when the
stochastic noise is averaged out over the long run. This may also be viewed from a **Signal Processing** point of
view. There is a small “signal” with
significant correlation with future returns, buried in the stochastic noise,
and the problem is how to isolate or “estimate” this small signal buried in the
noise, and use it to formulate a profitable trading strategy. Then Technical Analysis may be viewed as a
“heuristic” attempt to estimate this small correlated signal buried in the
stochastic noise of the Random Walk model.
But perhaps now that we have powerful desktop computers at our disposal,
a more scientific approach to financial Signal Processing is called for.

Then,
given that the financial returns closely follow a Random Walk, except perhaps
for a small “signal” buried in the noise, how is a trading or investment
portfolio to be constructed? The choice
of portfolio requires an estimate of both **future returns** and **future
risk**. How are these to be
estimated? One idea is to estimate the
future returns or trend as equal to some long-term average of past returns,
utilizing the concept of **trend persistence**. This implies some kind of correlation between past returns and
future returns on some time scale. Now
here is a mathematical fact from the theory of *stationary* time series:
The autocovariance series is the Fourier transform of the power spectrum. This is called the Wiener-Khinchin theorem
[NR, p.498]. So if the Fourier
transform of the time series is taken and the Fourier components (corresponding
to each “frequency”) are squared to give the power spectrum, this is the
Fourier transform of the autocovariance series. (The autocovariance is the autocorrelation of the returns times
the variance.) So, the result is that
if the power spectrum is “flat”, corresponding to a “white noise” spectrum,
then there are no correlations in the returns series, and the Random Walk model
applies. Conversely, for there to be
correlations in the returns series, there must be a non-constant power
spectrum, so some of the “frequencies” have more “power” than others. So the search for correlations and
profitable trading rules is equivalent to the search for those time scales or
“frequencies” that have more “power” than the average. This can be extended to the *non-stationary*
case if we suppose that over short time periods the stationary case holds
approximately, and over longer time periods the power spectrum and
autocovariance change with time. Then
the **trend persistence** results from identifying those frequencies or time
scales with a higher power spectrum than average, at any given time, which then
results in a positive correlation between the past trend and future trend *on
those time scales at any given time*.

There
is a caveat to this. If we measure the
power spectrum directly, in the form of the Periodogram for a stationary time
series or its Wavelet counterpart for a non-stationary time series, then this
power spectrum itself is dominated by stochastic noise. Smoothing techniques can reduce this noise,
but in general a direct measurement of the power spectrum from the returns data
is not an effective method of detecting the underlying correlation. So instead of this, the parameters
corresponding to the power spectrum are fit directly to the future returns data
in a recursive algorithm which uses the past data in a **training** period,
which adapts the Wavelet Linear Prediction filter to make the optimal
prediction of future returns. This then
may be interpreted as a method of determining the “true” wavelet variance or
power spectrum, and hence the correlation, more accurately than a direct
measurement would yield.

The
Wiener-Khinchin theorem may be visualized in terms of the **Efficient Market
Hypothesis** by making the idealization that different traders or investors trade
on *cycles* of varying time scales.
The trading activity on various time scales is measured by the power
spectrum of the returns time series on those time scales. The market is efficient when the reaction to
new information is instantaneous and future returns are uncorrelated with past
returns. This corresponds to equal
trading activity on all time scales and a constant or “white noise” power
spectrum. Because, giving an arbitrage
argument, if any cycles were to predominate over the others, traders would take
advantage of these apparent cycles to make a profit, and then the trading
opportunity would vanish. So if the
market is perfectly efficient, the trading activity on all time scales has
equal intensity, the power spectrum is perfectly flat, and the returns consist
of random “white noise”.

However,
in reality this condition may be violated, especially if the activity is
measured over localized time intervals rather than taking an average. In the past, it used to be said that there
were strong correlations in the returns over time intervals of tick data
shorter than about 15 minutes. This was
no doubt due to most traders being without real-time quotes, so the trading
activity on these very short time scales was below the average. But in recent years the availability of
real-time data has been more prevalent, so these very short-term correlations
are no doubt largely eliminated by now.
But some traders still tend to be “behind the curve” with regard to new
information, so they do not react immediately, and this should still result in
a deficit of trading activity on the shorter time scales. This will result in positive correlations of
returns or **trend persistence**.
However, it may also happen that traders over-react and trade too
strongly on shorter time scales in response to news, and in particular in
response to the perception of a falling market. This will lead to a surplus of trading activity on the shorter
time scales, which results in negative correlations of returns or **trend
anti-persistence**. The price moves
tend to be too large in this case and then must correct back to their “true”
values, as traders correct for their initial over-reaction. This correction mechanism for over-reaction
to past prices and anti-persistence then leads to the **return-to-the-mean**
mechanism in the price series. (There
can be more than one return-to-the-mean mechanism at work at the same time, on
different time scales. Then, for
example, over a certain time scale the prices may be reverting to a strongly
up-trending mean price, then suddenly switch and revert to a longer-term trend,
resulting in a sudden crash back to this longer-term mean price. This, however, undoubtedly involves *non-linear
or higher-order correlations* in the data, such as perhaps a simultaneous
correlation with the entire market.
Another name for this is **critical phenomenon**, which is an active
field of research in EconoPhysics [PB] – a market crash can be thought of as a
“phase transition” in the market dynamics.)
In any case, the response of the market to changing conditions has not
been completely **efficient**, correlations are introduced between past and
future returns, and this results in further trading opportunities by the
arbitrage argument. This correction
mechanism also implies that any short-term trading imbalances are then
corrected by the market over the longer term, so although the power spectrum
imbalances and the resulting correlations may exist over the short-term, taking
a long-term average of the power spectrum leads to almost no visible
correlation in the returns data. In
particular, in the *stationary* approximation it is difficult to see
correlation in the returns series – these short-term trading imbalances are *non-stationary*
phenomena.

So
it is due to this property of the autocovariance sequence and the *non-stationarity*
of the time series that it doesn’t make much sense to presuppose the existence
of *permanent* correlations over particular time scales, such as very long
time scales. The very long-term trend
is just one particular range of frequencies or time scales, and there is no
more reason to assume the existence of a positive trend on these time scales
than on any other. However, it is
believed that in the generic case for financial time series (as well as other
time series occurring in Nature) long-range positive autocorrelation does exist
in the data, due to the occurrence of **fractal statistics**, so the series
is of the type known as a **Long Memory** or **Fractionally Differenced
(FD)** process, as advocated by Mandelbrot [M]. The Wavelet techniques are particularly well adapted to the
analysis of such processes [PW]. Such
an FD stationary process may indeed be a good approximation for the actual non-stationary
process of financial time series.

Rather
than *assume* the existence of a certain type of correlation in the price
data, a more plausible approach would be to *measure* correlations of
various types, on any chosen time scale, and then base trading rules on these
measured correlations, with the understanding that they are constantly changing
with time as the market dynamics changes.
This makes the statistical problem more difficult, however, as there is
now no longer a “Law of Large Numbers” as in the case of stationary time
series.

In
fact, it should be noted that, when modeling financial time series as
stochastic time series, there is the problem that there is only *one instance
of each time series*. If the
stochastic time evolution of the price of a security could be “played” over and
over, and the results averaged, then the Law of Large Numbers could be applied
to uncover the true “signal” in the stochastic noise, and the true statistical
properties of the time series could be measured. As it happens, there is only one instance of each time series to
work with, and since the statistics are non-stationary, there is no Law of
Large Numbers for any individual time series.

Thus
in the case of financial time series, the *ergodic hypothesis* [Hay] must
be used. This means that, instead of
applying the Law of Large Numbers to a large ensemble of “instances” of the
financial time series, all with identical statistical properties (which of
course does not exist), the averaging is instead applied to the single time
series over the time index of the time series.
This necessarily means that the statistics must be in some sense *stationary*. However, we do not have to assume that the
statistics are stationary in the narrow sense of constant mean and covariance
matrix over the whole time series. We
may instead look for more complicated, nonlinear or time dependent, functions
of the past data, or *predictors*, and test these for correlation with
future returns. However, in order to
perform the test, the assumption must be made that these nonlinear correlations
are stationary, at least over the duration of the correlation test, which will
usually take up most or all of the given time series. So the generalized notion of stationarity is that the *formula*
for computing these complex *predictors* is constant, at least over the
duration of the given time series. (The
formula itself could conceivably change from one block of data to the next, for
example if it uses some input parameter such as interest rates that change
slowly with time.)

Actually
the complex formula that remains constant with time can be identified with the
trading strategy itself or components of it.
Then this trading strategy will necessarily involve the whole portfolio
of securities, not just a single security.
The only way to apply the Law of Large Numbers and try to judge the
effectiveness of a trading strategy is to do it within a **diversified optimal
portfolio**. Then, with a large
enough portfolio, the stochastic noise can be averaged out and the growth (or
otherwise) of the whole portfolio can be measured and trading strategies
compared. This is why it is crucial to
devise the trading rules within the context of the whole portfolio, and not
just individual securities – otherwise there is no real way to measure how well
it works, and also no way to control **risk**.

Of
course, along with **Technical Analysis**, there is also **Fundamental
Analysis**. In order to achieve
superior returns and “beat the market”, it is necessary to make use of *all*
available information. This includes
studying each company or security that you want to invest in, to determine
whether it will be a favorable investment over the long-term. For stocks the **earnings** are very
important, as well as the price in relation to the earnings or P/E ratio. A good understanding of the macroeconomic
factors and how they affect each market sector is also crucial. This information is separate from the
information about the price action. In
fact the price action can be viewed as a *response* to these fundamental
and macroeconomic factors, with, however, a time delay and possible
over-response due to the fact that traders and investors are not completely
well-informed and not completely rational in their decisions. In the absence of this information, the
approach of **Technical Analysis** is to try to take this information into
account by means of its effect on the price action itself. Then you can try to spot the initial effect
of the new information on the price action, and then try to respond quickly
before others are able to respond.
However, this always puts you “behind the curve” to some extent, and it
is much better to be able to take into account both the **Fundamental** and
the **Technical** information together.
One way to do this would be to construct a long-term investment portfolio
based on the known Fundamental and macroeconomic data. Then doing short-term trading in the
portfolio based on Technical signals and data can enhance the investment
returns. Another way of looking at this
is that any long-term investment portfolio must periodically be *rebalanced*
[FFK], and the short-term trading based on Technical signals can be viewed as a
way of rebalancing the portfolio.
Another consideration is that if something goes wrong and the individual
security or the whole market has a sudden decline or “crash”, then it is
important to act quickly and get out of the position. If this can be done then it can result in avoiding potentially
large losses in such a situation, which would result if the “Buy and Hold”
strategy were followed.

According to the **Random Walk Model**, as first
proposed by Bachelier in his Ph.D. dissertation in 1900 [B], financial returns
(price differences) may be described by a *stochastic process* which is a
simple Random Walk. This stochastic
process is also known as **Brownian motion**. More precisely, this has been refined to the statement that the *logarithmic*
returns, or the differences of the *logarithms* of the prices (logarithm
of the price *ratios*), follow a Random Walk process that is called **Geometric
Brownian motion**. Unless the price ratios are large, these two processes are
nearly the same. This process may be
further generalized by considering the **Random Walk with drift**. This is the model many long-term
buy-and-hold investors have in mind, when they consider averaging out the
short-term random price fluctuations in favor of reaping a return on the
long-term trend (deterministic drift). (Unfortunately, it does not seem to be
that simple in reality.)

At any rate, if the returns follow a Random Walk, then
they are serially uncorrelated. In
fact, the probability distribution of returns is also usually assumed to be
Gaussian, in which case the (daily, let’s say) returns are *independent*
random variables. Thus they appear as random
white noise, with a flat power spectrum.
In fact, this is how they actually appear (in most cases) when viewed
using the **Periodogram** display or the graph of the raw returns in ** QuanTek**. Thus the search for correlation between the

In
order to extract the *signal* buried in the stochastic noise, the main
technique that is used in ** QuanTek** is

_{}

The correlation
between the two variables is, by definition, the covariance between them
divided by the standard deviation (square root of the variance) of each
variable. So this small correlation *ε*
between each past daily return and future daily return is too small to measure
compared to the variance of the daily returns.
Now, to illustrate the principle of *noise reduction* via **smoothing**,
let us consider a simple situation where the correlation is constant over a
range of *M* values of the past returns and *N* values of the future
returns, and the variance of all the daily returns is assumed to be constant,
given by _{}. Now, we can defined
*smoothed* past and future returns, over a time scale *M* and *N*,
respectively, by:

_{}

Now, it is a property of independent random variables that the variance of a sum of such variables is given by the sum of the variance. Hence we can write (ignoring the small correlation) for the variance of each of the above smoothed variables:

_{}

Now the correlation of the smoothed variables is (approximately) given by:

_{}

Thus it can be
seen that the correlation between the smoothed *M*-day past returns and
the *N*-day future return has been increased by a factor of _{} compared to the
correlation between the individual past and future returns. So the smoothing has *amplified* the
correlation and reduced the level of stochastic noise. We call the sum of the past returns a **predictor**
because it is a function of the past returns data. Note that the correlation cannot be greater than unity, so the
product _{} must be less than
unity. Otherwise, it would be
impossible for the small correlation *ε* to persist over the interval
of *M* past returns and *N* future returns, as was assumed in the
above simple calculation. So the
smoothing is a means of separating out the *low-frequency correlation*
from the *high-frequency noise*.

In
general, a **predictor** can be *any* function of the past price data
or any other economic data. The only
condition is that all the data that goes into the predictor be in the *past*
of the *future* returns that we are trying to predict. Then the projection of the future returns
can be computed by doing a **linear regression** on all the predictors that
are chosen. This regression has a much
greater chance of success over a simple regression over the raw daily returns,
because as shown above the correlation between each predictor and the future
returns is *amplified* by means of the smoothing. Even though a linear regression is used, the
model can actually be highly nonlinear because any *nonlinear* function of
the past prices or other data can be used for the predictor. So, for example, we can look for a
correlation between the past volatility (smoothed variance) and the future
returns. In fact it has been shown that
the variance is much more highly predictable than the returns themselves, and
this fact has been utilized in recent years in the **GARCH** (Generalized
AutoRegressive Conditional Heteroskedasticity) models. (Heteroskedasticity is another word for
time-dependent variance – the future variance is conditional on the past
variance and the dependence is modeled as an autoregression.)

To
devise a profitable trading strategy and at the same time control risk is a
difficult mathematical problem and is still not completely solved within the
field of Econometrics. An excellent
recent text explaining the latest research in this area is __Financial
Modeling of the Equity Market: From CAPM to Cointegration__ (2006), by Frank
J. Fabozzi, Sergio M. Focardi, & Petter N. Kolm [FFK]. In particular, the nature of financial risk
is still not completely understood, and according to Benoit Mandelbrot [M], the
financial markets are far riskier than is taken into account in the standard
Modern Portfolio Theory of the past decades.
The ** QuanTek** program uses Wavelet techniques [PW] to try to
estimate both the optimal trading strategy and the future downside risk, both
of which are time varying, so the Wavelet approach takes into account the

It is widely believed that financial
returns follow a stochastic process – indeed, this was Bachelier’s hypothesis
in his Ph.D. dissertation of 1900 [B] and was followed up with the development
of Modern Portfolio Theory in the 1950’s.
But what does this really mean?
It is important to clarify the concept of Stochastic (or Random) process
vs. a Deterministic process. A more
precise statement would be that financial returns *resemble* a stochastic
process. Because, it could be argued
that *ultimately*, there are no stochastic processes in the real world,
pure randomness is an abstract mathematical concept, and everything in the real
world is deterministic on the “exact” level.
The ultimate laws that govern the Universe are believed to be
deterministic at the most fundamental level, and hence if it were possible to
comprehend the Universe in the minutest detail, it would be seen that all human
activity is deterministic and stock prices in particular are
“predetermined”. Of course, this is not
a very good way of looking at it from a practical point of view. The practical point of view is that human
activity, such as making investment decisions and trading in the stock market, is
so fantastically complex that it can never be comprehended on the “fundamental”
level. All anybody can hope to do with
regard to financial markets is to comprehend a tiny subset of all the possible
aspects and degrees of freedoms. Then
the definition of randomness from the practical point of view, as for example
in the tossing of a coin, is that the real process is far to complex to
comprehend, our knowledge is incomplete, and therefore the outcome is random
just because of our lack of knowledge of all the details of the system.

Another example of this is the
random number generator in your computer.
If you start the random number generator, it generates a series of
numbers that appear random, according to most statistical tests. They are distributed randomly within some
interval, such as the interval from 0 to 1.
However, the random number generator is actually a *deterministic*
algorithm in your computer. If you
start the random number generator exactly the same way each time, it generates
exactly the same sequence of “random” numbers over and over. And if you knew the exact algorithm by which
it generates these numbers, you could *predict* the exact sequence of
numbers that it generates each time. It
is only the lack of knowledge of this algorithm that makes the sequence of
“random” numbers *appear* random.

So a more accurate statement is that
the financial markets *resemble* a stochastic system and financial returns
*resemble* a stochastic process.
Then the question arises, “What kind of stochastic process best
describes the returns series of a financial asset?” Bachelier believed that it was a simple Gaussian Random Walk, but
we have come to understand [M, PB] that it is far more complex than this. But this is the crucial question when it
comes to designing the optimal trading strategy. For example, if the markets are described by the simple Random
Walk with drift, then short-term trading is fruitless, the expected return from
trading is zero, and the only sensible strategy is Buy-and-Hold to take advantage
of the long-term drift. However, all
evidence suggest that the financial markets are far more unstable and
unpredictable than the simple Random Walk model would suggest, so to simply
trust in the long-term upward drift in the markets might be dangerous. An active trading strategy should be
adopted, and securities bought or sold on a relatively long time scale in order
to adjust the portfolio to changes in the market, the economy, and the
situation with respect to each individual security such as changes in projected
earnings or the market position of the company.

That having been said, it should also be pointed out that
short-term trading greatly increases the variance or **risk** of the
investment strategy, and it also greatly increases the transaction cost, so
unless the trader has a very good understanding of the underlying correlation
in the market (that a given investment is undervalued or overvalued), the best
strategy is a long-term one. But my own
view is that the middle ground is best, and depends on the frequency of the
data that is available. From the above
analysis of the amplification of correlation by smoothing, it would appear that
to ferret out shorter-term correlation requires shorter-term data. As a rule of thumb, I would suggest trading
no more frequently than 10 times the smallest time unit of the data that is
available. So for daily data, the
shortest trading time scale should be 10 trading days or two weeks. For real-time tick data, on the other hand,
it might be possible to take advantage of much shorter-term correlations that
might be present, but the trading time scale should still be long relative to
the frequency of the incoming data.

Here is another subtle point with regard to randomness. Mathematically, it is a theorem by
Kolmogorov [PW] that (considering an infinite, stationary time series) if the
power spectrum is positive everywhere (zero only on a set of “measure zero”)
then the time series is *stochastic*.
But if the power spectrum is discrete or has any finite gaps where it is
zero, then the time series must be *deterministic*. Now consider the Fourier transform or
Wavelet transform for a series. Both of
these utilize data that are “cyclic”, which form a closed loop, in our case of
2048 or 4096 data points. These transforms
are actually designed for *periodic* signals that repeat after every such
cycle – if the data does not repeat then it is being approximated as
cyclic. This cyclic data has a *discrete*
Fourier or Wavelet power spectrum. But
this cyclic, periodic data, like a periodic signal in Electronics, is actually
a *deterministic* signal.
Rigorously, it must be deterministic if the power spectrum consists of
discrete frequencies. When all these
discrete frequencies are mixed together, and the (discrete) power spectrum is
flat, then the result is a signal that resembles white noise, even though in
reality it is *deterministic*. In
such a signal each individual frequency is independent and may be continued
indefinitely into the future, resulting in complete predictability if the
amplitudes and phases of each frequency component are known. But these amplitudes and phases are just
what the Fourier or Wavelet transforms of the (periodic) signal give.

So here is an amusing thought: What if the returns series,
which appears to be stochastic white noise, is in reality at least partly an
admixture of deterministic cycles that exist independently of one another? Then if a trader picks out any one of these
deterministic cycles and trades on that cycle, the other cycles will average
out and the trader will make a profit from short-term trading, because that
cycle will be *predictable* since it is deterministic. This would be the hallmark of a *linear*
system, but in reality we expect the financial market to be an extremely
complex *non-linear* system. Then
all the individual cycles will *mix* together and predictability will be
lost. If the system is completely
random, with complete mixing of the component cycles, then there can be no
predictability. However, in reality we
might hope that the markets are not exactly efficient, the returns series is
not exactly random, and due to the slight inefficiency there could be a small
deterministic component to the signal.
Then, using the Wavelet decomposition, we could hope that on each wavelet
level the signal is partially predictable out to the time scale of that wavelet
level. This could be possible even
though the data appear to be random white noise, if there is a small
deterministic component due to slight inefficiency in the market. Indeed, this is also the fundamental
assumption underlying Technical Analysis.

It must be concluded that, given the available statistical tests on a finite data set, it cannot be determined conclusively whether the data are random or deterministic, or some combination of the two. If the time series were infinitely long and stationary, and somehow an infinitely fine Fourier analysis, with a continuous Fourier spectrum, could be computed, and from this a continuous Periodogram (display of the Fourier power spectrum, the square of the Fourier coefficients), then one could determine from this Periodogram whether the data are random white noise, a correlated stochastic process, or a deterministic process. If the (continuous) Periodogram were non-zero everywhere and had a constant power spectrum, then the (stationary) stochastic process would be completely random white noise. If the Periodogram were non-zero everywhere but non-constant, then it would be random but correlated, and hence partially predictable. But by Kolmogorov’s theorem, if the Periodogram had any finite gaps in which it were zero, and in particular if it consisted only of discrete frequencies, then it would be completely deterministic and perfectly predictable.

However, in real situations we have a finite data set, such as 2048 daily data units corresponding to 8 years of daily data. Then we may use the Fast Fourier Transform on the data of 2048 units, arranged as a circular data set. The Periodogram is then also discrete with 2048 values. Then it is difficult to tell absolutely whether the data are random or deterministic just by looking at the Periodogram. The (smoothed) Periodogram may appear to be that of random white noise, with a constant average value, hence with no predictability. On the other hand, if the Periodogram really consisted of discrete values, then it would be completely deterministic, and perfectly predictable. This would correspond to the case of something like a periodic electronic signal, with a fundamental period of 2048 time units, which repeats exactly every period. Then the signal is perfectly predictable – it simply repeats every fundamental cycle. So the actual financial data could be somewhere in between random white noise and a perfectly deterministic signal, and it would be impossible to tell from standard statistical tests alone which is the case.

In reality, the situation is more
complex than the cases of stationary random or deterministic behavior, because
we also suppose that the financial time series are **non-stationary**, with
statistical properties that depend on time.
This concept can be defined precisely only if we are given an infinite
ensemble of infinitely long time series, all with the same (non-stationary)
statistical properties, for then we can measure the statistical properties by
averaging over the infinite ensemble of “instances” of the series. For the case of real data, there is only one
instance of each financial time series, and we must rely on sample averages
over finite sections of the data, or **smoothings** of the data, to
approximately measure the statistical properties. Ultimately, we must define and measure the predictability of the
time series by computing various smoothings and indicators based on the past
data, and measuring the average **correlation** of these indicators with
future returns. When these correlations
are averaged over many different time series and many different securities in a
portfolio, they become a measure of the validity of the indicators and the
trading rules that are based upon them.

According to the standard Modern
Portfolio Theory of the ‘70s, a stochastic process consisting of just Gaussian
white noise described financial returns.
A famous hedge fund based its trading strategy on this assumption, which
also underlies the Black-Sholes options pricing formula, with disastrous
results, and the rest his history.
Finally it was realized that the statistical distribution of returns is
not Gaussian, but in fact has “fat tails” which means that the likelihood of
extreme events such as the market crash of 1987 is much more likely than
Gaussian statistics would predict. Also
it is known now that the volatility is not constant but is time-varying. Mandelbrot [M] has explained this behavior
as being due to **fractal statistics**.
The time series is described as a **fractionally differenced**
process with an **anomalous fractal dimension**. These types of process were originally discovered by studying the
statistical distribution of the Nile river levels. What this means for us is that there should exist **long-range
correlations** in the financial returns, but whereas in ordinary correlated
processes these correlations decay away exponentially with the time interval,
for the case of fractal statistics the correlations decay more slowly, as a
power law. This then gives rise to the
observed erratic behavior of financial markets, with a much greater likelihood
of extreme “outlier” events such as market crashes than would be expected on
the basis of Gaussian statistics.

Thus, fundamentally, due to this
long-range correlation of returns due to fractal statistics, we can expect to
find underlying **trend persistence** in financial data. This trend persistence is also a cornerstone
of **Technical Analysis** [Pr]. When
viewed in terms of the Periodogram, this fractal trend persistence would take
the form of a sharp spike at the very low-frequency end of the power
spectrum. Unfortunately, this is very
difficult to see in the actual computed Periodogram because there are very few
degrees of freedom at these low frequencies, and the spike is obscured by
stochastic noise in the Periodogram.
The Wavelet analysis is better suited to this type of spectrum due to
long-range fractal statistics.

However, we also know from
experience that there is a definite **return to the mean** mechanism in
financial markets. This means that when
prices are low, they tend to rise and when prices are high, they tend to
fall. So this correlation is between past
*price levels* and future *returns*.
However, this type of behavior could also be explained as a persistence
on individual Wavelet levels of the returns.
Each wavelet level represents an octave of Fourier frequencies, and so
both the (relative) prices and the returns have an approximately cyclical
behavior corresponding to the average frequency of the wavelet level. In other words, on each wavelet level the **Relative
Price**, **Velocity**, **Acceleration** indicators are each shifted
relative to each other by one-quarter cycle.
So on an individual wavelet level, the **trend persistence**
correlation of past and future returns, and the **return to the mean**
(anti-)correlation of past prices a quarter cycle in the past and future
returns, are really two aspects of the same thing. So the conclusion is, instead of just postulating a long-range
persistence of returns due to fractal statistics, we could instead generalize
this and postulate persistence on each wavelet level of the returns, with a
time scale roughly the same as that of the wavelet level. The persistence due to fractal statistics
would then appear as just the persistence on the highest or “scaling” level of
the wavelet analysis.

One particular situation deserves further comment. This is the situation such as occurred in 1929 and 1987, where the market enters a phase with a strong uptrend, where the Velocity predictor and Relative Price predictor are both strongly positive. Then due to the nonlinear effect, the longer the uptrend stays in place, the more positively correlated the future returns are to the Velocity predictor, but at the same time the Relative Price gets higher and higher resulting in it being more negatively correlated with future returns. Then the market enters a delicate balance, where the slightest fluctuation downward can break the uptrend and result in prices crashing due to the highly inflated Relative Price. But the dangerous thing about this situation is that the exact break point is in principle unknown and unpredictable. This phenomenon has been compared to avalanches, earthquakes, and cascading piles of sand. In each case the probability of the sudden occurrence can be estimated, but the exact point in time when it occurs is unpredictable. Using the linear Relative Price and Velocity predictors, their correlation with future returns would cancel, and the projected return would be somewhere around the neutral point. However, the time-dependent volatility could also be estimated by the linear regression technique, and it could yield a strong probability for a large downward move in prices, due to the large (positive) absolute values of the two predictors.

An
interesting point to note about market crashes, for example the one in 1987, is
that in many cases they seem to illustrate a **return to the mean**
mechanism. If you look at the long-term
(say, 10 years) price graph (say, the S&P 500) of the 1987 crash, you will
note that the market entered a strong uptrend a couple of years before the
crash, from a more moderate longer-term uptrend. Then when the market crashed, the price levels fell very rapidly
from the level of the shorter-term strong uptrend back to the lower level
longer-term uptrend. It appears to be a
jump from the shorter-term trending mean to a longer-term trending mean. This seems to imply that if we can identify
the trends on various time scales, they might have predictive power because we
can suppose that prices will revert from the shorter-term trends back to the
levels of the longer-term trends on time scales comparable to those of the
trends. The sudden jumps are no doubt
nonlinear phenomena, but a linear model in which the trends on various time
scales can be identified can approximate this behavior. But this is just what the Wavelet analysis
is designed to accomplish.

The
objective of the ** QuanTek** program is to provide a trading strategy
that will yield

The
way to accomplish the risk control is to trade within an overall **optimal
portfolio** strategy rather than in each security individually. Typically one computes the optimal portfolio
that yields the overall maximum return with minimum risk, and then one must do **portfolio
rebalancing** [FFK] periodically in order to maintain the optimum weights of
all securities in the portfolio. This
optimal weight is determined, typically, as in the classical Markowitz method
[SAB], using the **expected return** of each security along with the **covariance
matrix** of the returns of all the securities in the portfolio. As a simplification of this, a **Factor
Model** [SAB], utilizing only a few combinations of the securities (factors)
instead of all the securities, or the **Capital Asset Pricing Model (CAPM)**
[SAB], utilizing only the *market portfolio*, may be used. Then, as the better performing securities
rise in price and become over-weighted, they are sold at periodic intervals,
and the under-weighted securities are bought.
This mechanism automatically incorporates the *buy low – sell high*
strategy, thereby taking advantage of this important correlation with future
returns. Thus the trading strategy for each
security is linked to its performance and optimal weight within the whole
portfolio.

This
begs the question of how the securities are to be chosen in the first
place. Ultimately, it is best to take
fundamentals into account when selecting the securities for the portfolio. Also to achieve diversification, securities
from different sectors should be chosen, since the securities within each
sector tend to move in tandem and be correlated with each other. More generally, if the covariance matrix is
used, securities are chosen to be *anti-correlated* in order to achieve
the lowest possible portfolio risk for a given return. There is a problem with this, however, since
it has been shown that the covariance matrix of securities is in reality mostly
stochastic noise [LCBP]. Only a few
combinations of the securities correspond to statistically significant
eigenvalues of the covariance matrix, and one of these is simply the sum of the
securities in equal weights. So one
easy rule of thumb, rather than having to diagonalize the covariance matrix to
compute the portfolio and arrive at a solution which is an awkward mixture of
the securites [FFK], is to just maintain equal weights (price per share times
number of shares) of each security in the portfolio. However, better results might be obtained by utilizing more
sophisticated methods such as Value-At-Risk (VAR) [BP].

The
remaining question is when to buy and sell.
In general, it is difficult to predict optimal buy/sell points – the
apparent buy/sell points are generally just due to stochastic fluctuations of
price. In the portfolio-rebalancing
scheme, the portfolio is rebalanced back to its optimal weighting at fixed time
intervals, say every month. However,
the ** QuanTek** program does attempt to identify optimal buy/sell
points, in the hope that if the portfolio is rebalanced at these points, then
there is a chance to obtain a little better price. First, a time scale

The caveat here is that the values of the smoothed predictors at the present time depend on the Price Projection, so the predictors must incorporate this future projection. Then, the optimal portfolio usually also takes into account the Price Projection in the form of expected returns. So, consistent with the buy/sell points, a security with a positive expected return should be bought or held, and a security with a negative expected return should be sold or held, depending on the price level. But the expected return derived from the Price Projection depends on time, which in turn leads to the active trading strategy, continually rebalancing the portfolio in response to the ever-changing expected return and the specified optimal weighting.

It
should be pointed out that the expected return is an important element of the
estimation of the optimal weighting of the portfolio. Even though returns are hard to predict, they must be predicted
in order to arrive at some estimate for the expected return for the optimal
portfolio calculation. So this is the
difficult challenge, making full use of the latest results from the science of
Financial Econometrics (and EconoPhysics) that the ** QuanTek**
program attempts to meet.

[B] Louis Bachelier, __Théorie de la Spéculation__ (doctorial
dissertation),

*Annales
Scientifiques de l’Éciole Normale Supérieure* (iii),

Vol.17, pp.21-86 (1900) Translation: Cootner (1964)

[BP] Jean-Philippe Bouchaud & Marc Potters,

__Theory of Financial Risks: From
Statistical Physics to Risk Management, 2 ^{nd} ed.,__

Cambridge University Press, Cambridge, UK (2000)

[FFK] Frank J. Fabozzi, Sergio M. Focardi, & Petter N. Kolm,

__Financial
Modeling of the Equity Market: From CAPM to Cointegration__,

John Wiley & Sons, Hoboken, New Jersey (2006)

[G] Christian
Gourieroux, __ARCH Models and Financial Applications__,

Springer-Verlag, New York (1997)

[H] Andrew C. Harvey,

__Forecasting,
structural time series models and the Kalman filter__,

Cambridge University Press (1989)

[Hay] Simon Haykin, __Adaptive Filter Theory, 4 ^{th} ed.__,

Prentice Hall, Upper Saddle River, NJ (2002)

[LCBP]Laurent Laloux, Pierre Cizeau, Jean-Philippe Bouchaud, & Marc Potters,

“Noise Dressing of Financial Correlation Matrices”,

Physical Review Letters, Vol.83, No.7 (16 August 1999)

[M] Benoit Mandelbrot, __The (Mis)behavior of Markets__,

Basic Books, New York (2004)

[NR] William H. Press, Saul A. Teukolsky, William T. Vetterling, & Brian Flannery,

__Numerical Recipes in C: The Art
of Scientific Computing, 2 ^{nd} ed.__,

Cambridge University Press (1992)

[Pr] Martin J.
Pring, __Technical Analysis Explained, 3 ^{rd} ed.__,

McGraw-Hill, New York (1991)

[PB] Wolfgang Paul & Jörg Baschnagel,

__Stochastic
Processes: From Physics to Finance__,

Springer-Verlag, New York (1999)

[PW] Donald B. Percival & Andrew T. Walden,

__Wavelet Methods for Time Series Analysis__,

Cambridge University Press, Cambridge, UK (2000)

[SAB] W.F. Sharpe, G.J.
Alexander, & J.V. Bailey, __Investments,
5 ^{th} ed.__,

Prentice Hall, Englewood Cliffs, NJ (1995)