New
Adaptive Method of Following the Tendency and Market Cycles
Present article is the first of the set of articles
dedicated to description of the new mechanical trading system worked out by the
author. The system is based on usage for exchange quotation analysis such
mathematical instruments as digital filtration and spectral estimate of the
intermittent (discontinuous) historical series. The system is realized in the computer software able to
generate trading signals. At testing the system demonstrated excellent
results – with yield 692.1% and profitfactor 17.88.


FOREWORD
The chart describing the dynamics (time history) of changes in
exchange quotations or real deals prices can have arbitrary form, namely:
even, uneven, broken or discontinuous (with gaps). In mathematics such
functions are named nonanalytic. But even such nonanalytic function can be
represented on the finite time horizon by infinite sum of sine functions.
Above given statement is the subject (content) of the
famous theorem by Fourier, that engineer Jean Batiste Josef Fourier proved at
the meeting of the French Academy in December 21, 1807.
It follows that any timing
functions (signals) can be unambiguously represented by the functions of
frequency that are named their frequency content. These functions describe
the signal frequency composition. For determined (nonrandom) signals the
transition from signal time description to the frequency description that is computation
of the frequency content is made through direct Fourier transform.
However random noise cannot be described already through
frequency content, as the Fourier transform from noise is also a random
process. Usually stochastic processes are presented with the process spectral
density of the power (SDP). SDP is the Fourier transformation of not the
random process itself but its autocorrelation function.
Filtration is a process of changing the signal
frequency spectrum in some desirable direction. This process can lead to
intensification or weakening the frequent components in some range of
frequency,

to suppression or separation some specific frequency component.
Digital filter is a digital system that can be used
for filtration the digital (discrete) signals defined only in discrete
moments of time. It can be realized with the software method on computer.
With computers and modern information systems
appearance methods of technical analysis using filtration and mathematical
approximation began to develop rapidly. Relative simplicity of their
calculation and integrability into information systems has made their effect.
Hundreds of indicators have been contrived.
And all of them beginning with Moving Averages, RSI, MACD, Momentum,
Stochastic, are digital filters because they change signal frequency spectrum
in some direction. In other words they have their transfer function with the
gain factor depending on frequency.
For example, for Moving Averages this factor is proportionate to
function sin (f)/f, where f – is a normalized frequency. However during the
technical indicators development the transfer function is neither calculated
by their founders at all nor is advised to the product users. Therefore
change in direction the signal frequency spectrum is remained unknown for the
most of the users and that fact drives them in obviously disadvantage. The
main issue roots here.
The second problem is in the following. Time history
of changing the currency rates and stock price for goods (commodity stock
prices) or equities is always represented as a digital signal.

Digital signals possess a number of properties known to only a narrow
circle of specialists that should be considered in developing technical indicators.
The most important property of the digital signals is the fact that spectrum
of the digital signals is a periodic function! Ignoring the properties of the
discrete signals leads to unavoidable distortions digital entering time
series such as aliasing – frequency superposition (overlay), ambiguity and
spectrum leakage.
The third problem is connected with the fact that
spectral density of the prices deviations, strongly differ with each other at
different markets. Therefore even if the developers of the technical
instruments have resolved the two of the problems, the user as a rule is not
presented with the distinct algorithm for their parameters setup. Instead of
that the user is offered the next scheme: to change at his own choosing the
indicator parameter and then test.
Unfortunately, besides the complexities as above
mentioned there is one more a very serious problem leading to the fact that
optimized technical instruments, that worked well in the past, can work bad
or cannot work at all in the future. This is non stationary state
(timedependence) of the historical series that the technical analyzers have
to deal with.
Non – stationary state of the stock exchange
processes leads to the fact that price fluctuations spectrum on the same
market will depend on the time of its computation (valuation). If to compare two prices of SDP computed
at different time periods, one can notice that spectral peaks slowly “float”
or split.




The concept of the right and left displacement (offsetting) for trend
markets known to us from the technical analysis is explained with this fact.
This concept is sequent featured for any waves (light, sound and etc.) effect
known in physics as the “Doppler effect”: change of wavelength of harmonic
(periodic) fluctuation (oscillation) watched at the movement of the wave
source with regard to the receiver.
In other words nonstationary state of the analyzed processes is not
the hindrance in specification the trend direction but is even a direct evidence
of the trend movement availability at the financial and commodity stock
exchange, which is prejudiced by orthodox sticklers or apologists of the
fundamental analysis.
Present publication is devoted to the new method of the
technical analysis based on filtration and spectral estimator of the
discontinuous historical series. Later on we will name it “Adaptive
Trend & Cycles Following Method”, or AT&CF method. The new method
effectiveness significantly surpasses all trading systems efficacy known to
the author.
The first article of the whole set of articles is
dedicated to description of general principles of the system construction,
its separate features and indicators. In the next articles such interesting
results got during the working process as spectral estimators of the currency
pair EUR/USD rate volatility and specific algorithm of the trading signals
generating will be given a thorough consideration.
The main purpose of the present publication is to draw the
attention of the trading systems developers and technical analysts to the new
possibilities of the digital methods of the financial and commodity stock
exchanges analysis.
PURPOSES AND TASKS OF THE METHOD
Main purpose of the AT&CF method is forming a minimal set of
technical instruments with set properties sufficient for algorithm
development that would provide maximal possible for the specific market yield
with minimal risk level.
For this purpose achievement the following tasks are
being solved:
 Frequency content
(spectral distribution) of the price oscillations is studied;

 Adaptive procedure of
setting or guiding non recursive digital filters is implemented with
result of creation a set of pulse transition features (PTF) optimized with
earlier received spectral estimators consideration;
 The procedure of the PTF and input
(entering) discrete time series is realized (usually consisting of close
prices of the week, day, hour and so on and so forth), that is digital
filtration resulted in defining a set of indicators as given below; the
filtration procedure is repeated with entry (appearance) of the new
value of the digital (discrete) input series;
 Trading algorithm is
worked out with main principles of construction stated below.
Area of AT&CFmethod application includes any
financial and commodity stock exchanges. However market capitalization and
market liquidity undoubtedly influence the maximal dimension of open position
without its distorting the results given below. For FOREX market this
threshold (barrier) is no less than 100 mio* USD for main currency pairs.
That is why AT&CFmethod is of interest for any category (kind) of
investors.
NEW INSTRUMENTS OF THE TECHNICAL ANALYSIS AND
THEIR INTEPRETATION
Main innovation of the AT&CFmethod is an adaptive trend line with
arbitrary form, the movement direction of which is the direction of the
prevailing trend on the market. The adaptive trend line is a low frequency
component of the entry time series marked with the help of the digital filter
of low frequency (FLF), letting low frequencies through and cutting high
frequencies of the price instability (fluctuations). The lower frequency
cutting f_{c} of FLF, the smoother the trend line becomes. Such
approach is in compliance with the trend notion adopted in all technical and
radio physical applications and

cannot shock both technical analysts and investors. The points lying on the adaptive trend
line possess a very strong inner relation (connection). Independent are only
values of the points being from each other in the distance equal to or more
than so called Nyquist interval T_{N}=1/2 f_{c}. The lower
frequency of the filter cutting, the stronger is this inner relation and
hence, the more time is required for the predominant tendency reversal.
The readers not experienced in theory and practices
of the digital filtration I recommend to spare more attention not to the ways
of computation the indicators used by the method, but to the issues of their
interpretation and received results. Actually it is more important for
investors that this or another method should work and give profit, and to
investigate and be experienced in the details of the method is a matter of
the specialists of another category. For the new instruments examination
let’s refer to the illustration in Fig.1.
FATL (Fast Adaptive Trend Line)– is formed
with the digital filter of the low frequency FLF  1. Filter FLF – 1serves to
suppress noises of high frequency and market cycles with very short periods
of oscillation that can be considered as noise.
SATL (Slow Adaptive Trend Line)– is formed
with the digital filter of the low frequency FLF  2. Filter FLF – 2 serves
to suppress noises and market cycles with longer periods of oscillation.
These filters parameters (frequency of cutting f_{c}
and fading A in the stop band) were calculated with spectral estimate of the
currency rate EUR/USD. Filters of low frequency FLW –1 and FLF – 2 provide
attenuation in the stop band with no less than 40 dB and absolutely don’t
distort the amplitude and phase of entry discontinuous price series in the
pass band (bandwidth). These properties of the digital filters provide
significantly improved (in comparison with simple moving average) noise
suppression that in its turn allows reducing sharply the probability of
appearance “false” signals for purchase and sell.




There are no analogues to FATL è SATL among
widely known technical instruments. These are not moving “average”, but just the
adaptive lines estimates of the short term and longterm trends. Unlike moving “average”, FATL and SATL
have no any phase delay with regard to current prices. FATL (k) value is a
mathematical close price expectation close (k), where k – is a number of
trading days. The value of N
punctual moving “average” MA (k), is strictly speaking, mathematical
expectation not close (k), but close (kN/2), where k – is the number of the
trading day. The value of SATL (k) is mathematical expectation of FATL (k)
for any k on preset time domain (slice) T.
RFTL (Reference Fast Trend Line)– support “fast” trend line and RSTL
(Reference Slow Trend Line) – support “slow” trend line are digital
filters response of FLF 1 and FLF 2 to the entry discontinuous series taken with
delays equal to corresponding Nyquist intervals T_{Ni}. Support lines
of RFTL and RSTL are analogues to simple moving “average” in the sense of
their delay in relation to the current prices. If instead of the pulse FLF
features with complicated forms to use pulse feature with 1/N balance
corresponding to the procedure of N – point wise moving average, the analogue
would be complete.
Indicators of FTLM (Fast Trend Line Momentum) and STLM (Slow Trend
Line Momentum) show the tempo of change (fall or growth) of FATL and SATL and are
calculated similarly to indicator Momentum by formulas:

Fig. 1. Adaptive trend lines.


FTLM (k) = FATL (k) – RFTL (k),
STLM (k) = SATL (k) – RSTL (k).
Main difference of FTLM and STLM from classical technical instrument Momentum
is that for its calculation not the close prices but smoothed (leveled) in
the result of filtration values of the trend line are used. In the result
FTLM and STLM turn out more leveled (smoothed) and regular functions than the
classical instrument Momentum, and therefore have more forecasting value.
FTLM and STLM lines were calculated with observance of all the rules of
discrete mathematics as the first differences between the two nearest
independent points of range bound channel processes.

During computation the classical indicators Momentum requirement is
often not fulfilled and it leads to unavoidable distortions in the spectrum
of the entering signal. Specialists in the digital processing of the signals
name these distortions aliasing that is frequencies overlay or ambiguity.
This ambiguity leads to strong irregularity and chaos in the classical
technical indicator Momentum.
A set of the technical instruments of the method contains
two more new oscillators. They are indexes of RBCI and PCCI, also shown in
Fig. 1.

RBCI (Range Bound Channel Index) – is calculated by means of
the channel (bandwidth) filter (CF). Channel filter simultaneously fulfills
two functions:
 Removes low frequent
trend formed by low frequent components of the spectrum with periods,
more T_{2}= 1/f_{c2};
 Removes high frequency
noise formed by the high frequent components of the spectrum with
periods, less T_{1}= 1/_{fc1}.
The periods T_{1} and T_{2} are chosen to comply with
condition T_{2}> T_{1}




Cutting frequencies f_{c1 }and _{fc2} are chosen so
that all prevailing market cycles should enter the frequencies range bound by
f_{c1} è f_{c2}.
Summarily RBCI (k) = FATL (k) –SATL (k).
Indeed, RBCI approaches its local maximum the prices
approach upper border of the trading channel and when RBCI approach its local
minimum the prices approach the lower border of the trading corridor.
Let’s mark main property of RBCI index. This is
quasistationary (that is almost stationary) process bound by the frequency
range both from above and below.
PCCI Index (Perfect Commodity Channel Index) –
is a perfect commodity channel index is calculated by the formula:
PCCI (k) = close (k) – FATL (k).
It has some outer similarity in the calculating
method with commodity channel index CCI by D. Lambert. Indeed, CCI index is calculated as
normalized difference between current price and its moving average and PCCI –
as the difference between closing price and its mathematical expectation
represented by the FATL value. Here lies more than in comparison with CCI the
perfection of PCCI. PCCI index– is a normalized for its standard deviation
high frequency component of the currency rate volatility.
Main Principles for the Trading Algorithm
Development and Interpretation Rules for New Technical Instruments
Main principles to be observed at the concrete trading algorithm
development are the following:
 Trade only in the
direction of the prevailing tendency the direction of which is specified
by “slow” adaptive trend line SATL;
 To consider dynamic
characteristics of the “fast” and “slow” trend represented by the FTLM
and STLM indicators;
 To use information on what
area of the values (neutral, overbought, oversold, local maximum and
local minimum) is the sum of prevailing market cycles (index of RBCI) in
chosen by means of frequency range spectral analysis;

TABLE 1. OVERALL PERFORMANCE OF THE TRADING
SYSTEM WORK BASED ON ADAPTIVE METHOD OF FOLLOWING THE TENDENCY AND MARKET
CYCLES




 To take oscillator signals
as secondary ones in cases when trend indicators are evidence of the
very marked bearish or bullish tendency availability;
 To take oscillator
signals as main ones in cases when trend indicators give signals about absence
of the very marked tendency;
 To use flexible system
of protective stop orders based on the values of RBCI, PCCI indexes and volatility
values of the “fast” market oscillations.
Main rules for the above mentioned instruments interpretation are the
following:
 Growing SATL line is
evidence of the bullish trend on the market. The point of the reversal
beginning of the bearish trend is considered the point of the local
minimum of SATL. The point of
finishing the reversal of the bearish trend is the point where the sign
of STLM changed from minus into plus.
 Falling SATL line is
evidence of the bearish trend on the market. The point of the bullish
trend reversal beginning is considered the point of the local SATL maximum. The point of finishing
the bullish trend reversal is the point where STLM sign changed from
plus into minus.
 Close to horizontal the
form of SATL is evidence of the neutral tendency.
 STLM interpretation requires special
attention. Positive value of STLM is evidence of the bullish trend and
the negative one testifies the bearish trend. STLM is an advance
indicator. Local minimum of STLM always precedes the local minimum of
SATL. Local maximum of STLM always precedes the local maximum of SATL.
Achievement by STLM its points of extremum is necessary but insufficient
condition for the achievement by the curve of SATL the top or the
bottom. Growing STLM at growing SATL is evidence of the bullish trend
acceleration. Horizontal and positive STLM at growing SATL is evidence
of the set bullish trend. The more absolute the value of STLM, the more
potential the bullish trend has. Falling STLM at falling SATL testifies
the bearish trend acceleration.

Fig. 2. Total value of the P&L trading system
based on the AT & CF method
Table
2. Time behavior characteristic of the trading system work based on
AT&CFmethod including P&L with one entry the market and total
P&L (the beginning).




Horizontal and negative STLM at growing SATL testifies the bearish
trend setting. The more absolute value of STLM, the more potential the
bearish trend has.
 Growing “fast” FATL
trend line at the growing “slow” SATL trend line is evidence of the
strong bullish trend on the market.
 Falling “fast” line of
FATL at the falling “slow” line of SATL is evidence of the strong
bearish trend on the market.
 Growing FATL line at
falling SATL line is evidence of either bullish correction at the
bearish trend or consolidation.
 Falling FATL line at
growing SATL line is evidence of either bearish correction at the
bullish trend or consolidation.
 The beginning or
resuming the movement in one direction of FATL and SATL lines give
signals either on the tendency reversal or finishing the correction and
resuming price movement in the SATL direction.
Characteristic of the System
Overall performance of the system functioning according to the
AT&CF method is given in Table 1. Here you can find the system’s work
characteristics on long (EUR/USD buy) and short (EUR/USD sell) positions separately.
Cumulative value of the P&L trading system
functioning on the base of the AT&CFmethod is shown in Fig.2. Dependence
of P&L on time has distinctly marked linear tendency to growth.
Table 2 shows time behavior characteristic of the
trading system work including P&L of one operation and cumulative value
of

Table
2. Time behavior characteristic of the trading system work based on
AT&CFmethod including P&L with one entry the market and total
P&L (ending).


P&L. The first column shows data of operations made, the second
one – the type of operation.
Here the following symbolic notations are used: Buy
– EUR/USD, Sell – EUR/USD, L Exit (Long Exit) – is long
position closing for EUR/USD, S Exit (Short Exit) – is short position closing
for EUR/USD, LSExit (Long StopExit) – exit from long position with stop
signal, SSExit (Short StopExit) – exit from short position with stop signal.
Table 2 shows that our system is not a system of
continuous action,

that leaves fairly big potential for its characteristics improvement.
Table 1 shows that average profit for one trading
operation (with profit or losses) is
$19,225, and the ratio of maximal sequence of gains to maximal
sequence of losses is 10 to 2. With this fact consideration the chosen
strategy brings to the conclusion: through limited number of the market
entries the total risk to lose the first (primary) margin tented to zero.
Direct confirmation of this paradoxical at first sight conclusion is the fact
that in 6 months trading

(February 8,1999.), the amount of money on the account was $212,700
($100,000 – primary margin, $112,700 –received profit). Thus the primary
margin can be simply withdrawn from the account (debited). After that the
trading begins with zero probability to lose it. If to choose another
strategy, for example, trading with permanent financial leverage, the curve
P&L in Fig. 2 would be almost parabolic. And in this case the risk would
be at the same level defined by average risk for one market entry.


© Vladimir
Kravchuk, 20002001

