discussion
of this method.ADAPTIVE TREND CALCULATIONS 437a
sc, = [ER, x (fastest — slowest) + slowest]*
andfastest = 2 / (fastest moving average period + 1)slowest = 2 / (slowest moving average period + 1)ER,= __[price, ~ price,_,.|
> | price; - price;_,| -
The fastest and slowest values represent the range of periods over which KAMA can vary,each converted to its exponential smoothing constant equivalent. Nominally, these valuesare set to 2 and 30. By squaring the smoothing constant components, the slow endbecomes 30 x 30, equal to an extremely unresponsive 900-period trend. The effect of thisis to stop the movement of.the trendline when the market becomes very noisy. On theother end, when the market goes up steadily, without a reversal for m days, the speed of thetrend can reach a very fast 4-period equivalent.
The éfficiency ratio, ER, has more recently been called fractal efficiency. It has thevalue of 1 when prices move in the same direction for the full n periods, and a value of 0when prices are unchanged over the n periods. When prices move in wide swings withinthe interval, the sum of the denominator becomes very large compared with the numera-tor and ER approaches 0. Smaller values of ER result in a smaller smoothing constant anda slower trend.
If you choose to use the default values for KAMA, where ER is calculated over 10 peri-ods and the smoothing constant ranges from 2 to 30 (squared), its calculation can beexpressed in Omega code as
KAMA = KAMA[1] + ((@absvalue(close-close[10])/@summation(@absvalue(close~close(1]),.10)* .6022) + .0645)*2*(close-KAMA[1])
Time Period for ER
In Kaufman’s Adaptive Moving Average, the time period only affects the calculation of theefficiency ratio. The purpose is to keep that period, 7, as small as possible so that the eval-uation of a few days will allow the speed of the trend to change from fast to slow and backagain. User’s may begin testing in the range of 8 to 10 days.
Choice of Volatility
The use of-the volatility function, @summation(@absva}ue(close-close{1],10)) inOmega code, adds all the individual price changes rather than looking only at the high-low range. Noise is considered more active when prices jump up and down. It is not asimportant that prices moved from a particular low to a certain high, but the pattern ofhow it was done. A market in which prices moved from low to high then back to the lowand high again is clearly more volatile and noisier than one that made a single move fromlow to high. - -
Small Moves in One Direction
Occasionally there will be a series of price moves that progress in very small increments inone direction without a reversal, causing ER = 1. Although this is thearetically correct, itmay prove best to include a filter for those cases where the direction, D = |price,- price;:,|< min, or a scaling function that does not allow the smoothing constant to produce theusual fast trend as the direction D moves below min.438 ADAPTIVE TECHNIQUES
 
Trading KAMA .The Adaptive Moving Average can be traded as any trend, and has the same characteristicsof exponential smoothing. That is, when prices cross the trendline from up to down, thetrendline must turn down, Although KAMA may use a trend with the equivalent of 900days, the value of the current KAMA will still turn down a very small amount. Trading rulesthat base signals on the direction of the trendline should include a small band around theKAMA to prevent price penetrations during a sideways market to cause a change in thetrading signal. Without this band, the benefit of a sideways KAMA trendline is eliminated.When applying KAMA to a trading program, it can be seen that higher values of the effi-ciency ratio, ER, are rarely reached using a calculation period greater than 14, This wouldrequire an extremely directional move for 2 weeks of daily data. As longer periods areused, the pattern of ER becomes very uniform, based on the underlying level of noise ineach market. Because peak levels in the ER cannot be sustained, they are valuable candi-dates for profit taking.
Programming KAMAThe following Easy Language user function can be used by a system or indicator to calcu-late Kaufman's Adaptive Moving Average.
{KAMA : Kaufman:’s Adaptive Moving AverageCopyright 1990-1998, PJ Kaufman. Al} rights reserved}
inputs: period(numericsimple);vars: efratio(0), smooth(1), fastend(.666), siowend(.0645),AMA(O), diff (0), signal(0), noise(0);
{calculate efficiency ratio}
efratio = 1; :
diff = @absvalue(close - close[1]);
if currentbar:> period then beginsignal = @absvalue(ciose - close(period]);noise = @summation(diff,period);if noise <> 0 then efratio = signal / noise;end;
if currentbar <= period then AMA = close;
smooth = @power(efratio*(fastend - slowend) + slowend,2);
AMA = AMAC1] + smooth*(close - AMAL1]);
KAMA = AMA;
N
Chande’s Variable Index Dynamic Average (VIDYA)?
Tushar Chande’s Variable Index Dynamic Average (VIDYA) first appeared in TechnicalAnalysis of Stocks & Commodities in March 1992. Both VIDYA and Kaufman’s KAMA use anexponential smoothing as a base for varying the speed of the trend each day. VIDYA, how-ever, uses a pivotal smoothing constant, which is fixed, and varies the speed by using a fac-tor based on the relative volatility to increase or decrease the value of s.
VIDYA, =k Xs XC, + (1-& Xs) X VIDYA,_1
-
where s is a fixed smoothing constant, suggested as a 9-day equivalent(ie., Ss = 2/(9 + 1) = .20)Cis the closing price
? Tushar S. Chande and Stanley Kroll, The New Technical Trader (John Wiley & Sons, New York, 1994).ADAPTIVETREND VALCULATIONS 439
 
t and ¢— 1 represent the current and previous periodskis the relative volatility, calculated as
k= @stdev(C,n) / @stdev(C,m)
where @stdev is the standard deviation functionn is the recent past n daysm is a longer historic interval in days, m>n
Volatility
Chande uses a very practical relative volatility measure based on the standard deviation, ordistribution of price moves over the past n days, compared with a longer historic pattern,taken over 4 days. When & > 1 the current volatility is greater than the historic volatility;when & < 1 the current volatility is lower. VIDYA can be written as an indicator in Omegacode as{VIDYA ; Variable Index Dynamic Average by Tushar ChandeCopyright 1997-1998, PJ Kaufman. Al? rights reserved.Period suggested at 9, histper suggested at > 9}input: period(9), histper(30);var: k(O), sc(O), VIDYA(O);{relative volatility}k = @stddev(ciose, period) / @stddev(close,histper);{smoothing constant}‘sc = 2 / (period + 1);{VIDYA}VIDYA = k*sc*close + (1 - k*sc)*VIDYA[1];plotl (VIDYA, "VIDYA");
Alternate Choices for Varying the Trend Period
The use of an exponential formula, either the simplified one shown by Kaufman or the tra-ditional one of Chande, allows the smoothing constant to vary according to the design ofthe user. There are indicators with which we are already familiar that might be substitutedfor the smoothing constant.
Correlation Coefficient, 7The most likely choice, suggested by Chande, is the correlation coefficient, r?. This value isthe result of measuring the residuals from a linear regression calculation. When the regres-sion fit is very strong, the value of r? is near 1; when there is no apparent market direction,7 is near 0. This fits neatly into the same pattern of smoothing constants as the KAMA andVIDYA.
When programming this technique, the calculation of the correlation of the residuals,7, requires the use of arrays and your own code for the correlation. This can be done usingthe formulas found in Chapter 3. To avoid that complication, the following program cal-culates a similar technique (Method 1) by evaluating the correlation of the closing pricesand a time series containing the sequential values 1, 2, 3, ... . It then calculates the currentvalue of the residual based on a linear regression of closing prices (Method 2) and finds thecorrelation coefficient, r*, of the series of current residuals. In the following programmingcode Method 2 is plotted, and the result of Method 1 is commented out.
{Adaptive R2 : Adaptive Correlation Coefficient IndicatorCopyright-1997-1998, PJ Kaufman. Ati rights reserved.
Plot smoothing function based on linear regression residuals andon simple time series function}440 ADAPTIVE TECHNIQUES
 
input: period(97;var: R20}, AR2(0), timeseries(0), resid(O0), RR2(0), ARR2(0);timeseries = currentbar; ‘{Method 1: correlation of price and timeseries]if currentbar <= period then AR2 = closeelse beginR2 = (@correiation(close, timeseries, period) + 1) / 2;AR2 = AR2[1] + R2*(close - AR2(1]);end;{ plot] (AR2,"R2");1}{Method 2: correlation with current regression residual }resid = close ~ @linearregvaiue(ciase,period,O);RR2 = (@correlation(resid, timeseries, period) + 1) / 2;ARR2 = ARR2(1] + RR2*(close - ARR2C1]);Plot2 (ARR2,"ARR2"};
Momentum Calculations .
There are many momentum calculations, such as RSI, that vary from 0 to 1, or from -1 to+1. In most cases the trend is considered strongest when the momentum value is closestto the extremes. The momentum values can easily be changed to fit the 0-to-1 pattern ofthe smoothing constant by using the transformations
(1) For a range of 0 to 1 sc = @absvalue(M-,5)*2(2) Fora range of ~1 to +1 sc = @absvalue(M)
where sc is the smoothing constant to be used for the current periodM is the current momentum value
By reflecting the lower extreme around the midpoint of the calculation, then taking theabsolute value, the transformed indicator shows a stronger trend near 1 and less directionnear 0.
Programming the Adaptive RSIThe following Omega code adapts the RSI to a smoothing constant:
{Adaptive RSI : Adaptive Relative Strength Indicator \Smoothing function based on RSI}input: perioc(20);var: sc(O), ARSI(O);__ if currentbar <= period then ARS! = closeelse beginsc = (@absvalue(@RSI(close, period) / 100 - .5)*2);ARSI = ARSI[1] + Sc*(close - ARSIC1]);end;plotl (ARSI, "ARSI"); -Comparison of Adaptive Trends oo -Figure 17-1 shows three of the adaptive trending methods compared with a moving aver-age, all based on a base calculation period of 20. Kaufman’s KAMA appears to be slightlycloser to prices than Chande’s VIDYA and flattens out faster during sideways periods; other-wise, they have similar characteristics because they both use the structure of the exponen-tial smoothing calculation. The adaptive RSI (ARSD), used as a-substitute for the smoothingconstant, is most responsive to price and appears as though it moves throughthe center ofmany of the sideways periods. Because the ARSI is much faster, it tends to get caught with aADAPTIVE TREND-~ ALCULATIONS 44)
 
false signal at the end of each sustained price move. The 20-period moving average is rea-sonably smooth during this market segment and shows typical characteristics of lag. Whenconstructing trading rules for any of these methods, the buy and sell signals are best if takenas the direction of the trendline and not a price penetration of the trendline.
McGinley Dynamics’
Among the work produced by John McGinley is a list of popular adaptive methods. Hisown New McGinley Dynamics is included:
MD = MDC[2] + (close - MBC1]) / (k*p*@power({close/MD£1]),4))
where MO is the current McGinley DynamicMDC1) is the previous McGinley Dynamick is a constant = .60 (60% of a selected moving average period p)p is the moving average periodclose is the closing price
The Parabolic Time/Price System
The first well-known adaptive technique was the Parabolic Time/Price System,‘ whichattempts to reduce the lag intrinsic to a trend system. To do this, Wilder increased thespeed of the trend by shortening the number of days in the calculation, whenever pricesreached new profit levels. The philosophy of the Parabolic System is that time is an enemy.Once a position is entered, it must continue to be profitable or it will be liquidated.
3 John McGinley writes Technical Trends (2.0. Box 792, Wilton, CT 06897), and has compiled a selection of adap-tive techniques as well as developed the ones discussed here.
4J. Welles Wilder, Jr, New Concepts in Technical Trading Systems (Trend Research, Greensboro, NC, 1978).
 
FIGURE 17-1 Comparison of moving average, KAMA, VIDYA, AR2, and ARSI.
 
ey ere sre022096 C=85.0300 +5500 O=85,2500 H=85.2800 L=84.4800 AMA 83.4018 Mov Avg i line . ~~
 
 
Source: Chart created with TradeStatior® by Omega Research, Inc.442 ADAPTIVE TECHNIQUES
 
The Parabolic Time/Price System is always in the market; whenever a position is closedout, it is also reversed. The point at which this occurs is called the Stop and Reverse (SAR).When plotted, the Stop and Reverse point seems similar to a trendline, although it has adecreasing lag (the distance between the current price and the trendline, or SAR point, getcloser each period), as shown in Figure 17-2. During periods of short, consistent trending,the Parabolic SAR converges on the price trend, extracting excellent profits.
To calculate the SAR value, first assume a long or short position. If the market hasrecently moved lower and is now above the lows of that move, assume a long. Call the low-est point of the previous trade the SAR initial point (SIP) because it will be the startingPoint for the SAR calculation (SARI = SIP). Calculate each following SAR as
SAR(today) = SAR(prior) + AF(today) x [High(today) — SAR(prior)]
This is an exponential smoothing formula using the high price for a long position and thelow when a short is held. For this method the smoothing constant is called the accelerationfactor (AF), and it is initially set to .02 at the beginning of each trade. After a day in whicha new extreme occurs (a new high when long, or a new low when short), the AF isincreased by .02. In terms of moving average days, the AF begins at 99 days and increasesspeed to a maximum of a 9-day moving average, but not in a linear fashion. The accelera-tion factor, AF, cannot be increased above the value .20. .
" In the SAR calculation, the highest high of the current move is used when a long posi-tion is held, and the lowest low is used when a short is held. This feature keeps the SAR atits highest possible level during an upward move. An additional rule compensates for thisstrength and prevents premature reversal by not allowing the SAR to get any closer thanthe price range of the most recent 2 days:
If long, the SAR may never be greater than the low of today or the prior day. If it isgreater than this low, set the SAR to that low value. A reversal will occur on a newintraday low that penetrates the SAR.
 
FIGURE 17-2 . Parabolic Time/Price System, Feb 96 S&P 500, 60-minute chart.
 
eae1123196 C#72015 1460 0#731.75 He734.75 L=719.75 Parabolic 738.80~
 
 
Source: Chart created with TradeStation® by Omega Research, Inc.ADAPTIVE TREND tc... CULATIONS 443
 
For short positions, the SAR is initialized as the high of the recent move, AF = .02, andthe daily calculation uses the lowest low of the current price move. To allow someprice fluctuation, the SAR may never be below the high of today or the prior day.
Wilder's Parabolic Time/Price System was unique in varying the smoothing constant to sat-isfy the notion that time worked against the trade. The idea of increasing the speed of thetrend as the trade becomes more profitable is still a concept that has validity and has notbeen adopted in other adaptive methods. One weak point may be that AF always begins at.02. Although this allows the trade latitude to develop without being stopped out, a mar-ket that is moving quickly through the buy or sell signal would be better traded with afaster trend. This method also requires fairly consistent price swings to produce profits.
A strong point is that the initial Stop and Reversal is the market extreme high or lowrather than an arbitrary, computer-generated point. Also, the SAR does not get closer thanthe previous day’s high or low, which prevents a reversal when the market is movingstrongly in a profitable direction. For those readers interested in the further developmentof this method, the Parabolic Time/Price System is combined with Directional Movement(described in Chapter 23) to form the Directional Parabolic System discussed in Chapter 6.
The following Easy Language user function allows the user to specify the starting valueof the smoothing constant AFmin, the increment AFinc, and the maximum value Afmax.This provides for more flexible testing. Optimization of these values has been reported toproduce good results.
(Parabolic. ¥RT ~
This version of the Parabolic system was written by
Sam Tennis, VISTA Research and Trading, Inc.,
8103 Camino Reai, South Miami, FL 33143, Email: skt@vista.comand reformatted by PJ Kaufman}
{Input values were formerly AFmin = AFactor, AFinc = AFactor, and AFmax = ALimit}(Sample usage: Plotl (Paraboiic.VRT (0.02, 0.02, 0.20)[1], “Parabolic")};Function returns SAR (stop & reverse} price objective}
input: Afmin(numericsimple), AFinc(numericsimple}, AFmax(numericsimple);vars: Position(1), AFCAFmin), SAR(low[0]); Hi(high(01), LofLow(01);
if @currentbar <= 1 then Position = 1;if @currentbar > 1 then beginif High > Hi then Hi = High;if Low < Lo then Lo = Low;
if Position = 1 then beginif Low <= SARCi] then begin
  
Position = -i;SAR = Hi;AF + 3 = AFmin;Lo = Low; -Hi = High;end; {Reverse position long to short}endelse begin
if High >= SAR[i] then beginPosition = 1;SAR = Lo;AF = AFmin;Lo = Low;Hi = High;end; {Reverse position short to long}444 ADAPTIVE TECHNIQUES
 
send;end;if Position = i then beginif Position{1] = 1 then beginSAR = SAR[1] + AF * CHigh{l] - SAR(1]);if Ht > Hi[l] and AF < AFmax then AF = AF + AFinc;end; {we have been "long" for at least one bar}if SAR > Low then SAR = Low;if SAR > Lowli] then SAR = Low[1];end {long}else beginif Position{1] = -1 then beginSAR = SAR[1] + AF * (Low[l] - SAR[1]);if Lo < Lof1] and AF < AFmax then AF = AF + AFinc;end; {we have been “short” for at least one bar}if SAR < High then SAR = High;if SAR < High{1] then SAR = High[1};end; {short}
Parabolic.VRT = SAR;
The Master Trading Formula
Another landmark method of creating a variable-speed approach is Mart’s Master TradingFormula.” It is also based on an exponential smoothing formula where the smoothing con-stant and band around the trendline are calculated daily, based on market volatility. Mart’sgeneral technique is to combine the average true range (the maximum of yesterday’s closeand today’s high or low) for the past 15 days with the net change over the same timeperiod, Each of these volatility elements is ranked from 1 to 21 by defining a maximumvalue for each one and dividing that maximum into 20 equal zones (the 21st being any-thing higher). The two rankings are then averaged to get a correlated volatility factor(CVE). Another ranking of 21 zones is created by dividing the highest CVF into 20 equalparts. The CVF then relates linearly to a range of smoothing constants from .084 to .330,representing the approximate exponential equivalents of the moving average days from 5to 23. The trendline value is then created daily using the formula -XN
TREND (today) = TREND (prior) + CVF (today) x (HLC(today) - TREND(prior)]
where HLC (today) is the average of the high, low, and closing price. As with the other adap-tive methods, the speed of the trend increases as volatility increases.
A band is also placed around the trendline, based on an inverse relationship to the cor-related volatility factor. The band is widest when there is low volatility and the smoothingconstant produces a slow trend; it is narrowest in fast, highly volatile markets. Rules fortrading Mart’s Master Formula follow the basic band-trading rules. The system is always inthe market, taking long positions when the upper band is broken and short positionswhen the lower band is crossed.
’
ADAPTIVE MOMENTUM CALCULATIONS
Varying the period of calculation does not have to be limited to a trend formula. Theperiod for any technique can also be adaptive. The following techniques apply the sameconcept to indicators. -
> Donald Mart, The Master Trading Formula (Winsor Books, Brightwaters, NY, 1981).ADAPTIVE MOMEN . JM CALCULATIONS 445
 
Variable-Length Stochastic
Written by Frank Key,° this adaptive technique uses Kaufman's efficiency ratio, ER. Itchanges the stochastic period within the default range of 5 to 30, using only integer values.The effect is that, when prices are trending with respect to the stochastic, the stochasticperiod is lengthened to allow the trading signal to stay with the trade. At turning points inthe market, the stochastic period is shortened causing the reversal signal to be moreresponsive.
Stochastic periodNSP = @IntPortion(ER x (Maxperiod - Minperiod) + Minperiod)
Adaptive stochasticASTOK = @slowK{NSP)
where ASTOK is the adaptive stochastic value today (based on NSP days)Maxperiod is the maximum period, suggested as 30Minperiod is minimum period, suggested as 5ER is Kaufman’s efficiency ratio
The variable-length stochastic uses the efficiency ratio in a manner opposite to Kaufman'sAMA. While the AMA increases in speed when the trend gets stronger, Key’s variable-léngthstochastic decreases in speed. This technique gives price more room to fluctuate duringperiods in which a strong trend is also associated with high volatility.
Trend-Adjusted Oscillator
One problem that confronts many traders who use stochastics or other oscillators is thatthe oscillator values tend to cluster at the bottom of the chart when the trend is down, andat the top when the trend is up. For a countertrend trader, this results in many oversoldconditions during a downtrend, which can give false buy signals. A trend-adjusted oscilla-tor’ can help this bias by correcting the value of the oscillator by the amount that the trendof the oscillator varies from its midpoint (usually 50%). Therefore, if a 10-day oscillator hasbeen steadily declining and its 5-day moving average has a value of 40, then the currentoscillator value is raised by 10% (the midpoint of 50 minus the oscillator moving averageof 40). The calculations for this are:
> (Close; - Low,)/(High; - Low,)n-day oscillator = 0 =n
- yo
N.day moving average of oscillator = MAg = af
Trend-adjusted oscillator = Midpoint — (MAg — O)
§ Frank Key, Structured Software Systems, Mt. Holly, NJ. Copyright 1996.7B, Marshall Wall, “Rolling with the punches,” Futures (July 1996).446 ADAPTIVE TECHNIQUES,
 
Dynamic Momentum Index
This approach to creating a dynamic, variable-length RSI, which changes the number ofdays based on market volatility, can be used to change the calculation period of any tech-nique.® In the Dynamic Momentum Index (DMD), the length of the calculation periodincreases as the volatility declines. You may also select a pivotal period, one around whichthe calculation period will vary; for the RSI this will be 14 days. When the volatilityincreases, the periods will be less than 14 days and when volatility declines it will begreater. Based on this,
DMI = @in{ |volatility,
@stdev(close,5)@avg(@stdev(close,5),10)
The DMI is 14, the pivotal value, when the volatility is normal, that is, when the cur-rent 5-day standard deviation is the same as the 10-day average of the 5-day standard devi-ation, When the current volatility increases, the ratio is greater than 1 and thé value of theDMI becomes larger than 14. The function @Int takes the integer value of the calculationto be used as the number of days in the RSI calculation.
A few changes in the DMI approach may improve results and should be consideredwhen generalizing this technique. First, the standard deviation should always be applied tothe changes in price, rather than the prices themselves. This removes the trend compo-nent. Alternately, the volatility ratio may be more robust if the sum of the absolute value ofthe price changes is simply compared against the average of that sum,
@sum(@absvalue(close, — close, - :),5)@avg(@sum(@absvalue(close, — close, -,),5),10)
volatility, =
volatitity, =
AN ADAPTIVE PROCESS
An adaptive method can be a process as well as a formula. Rather than using an index or ratioto change the smoothing constant, which in turn alters the trend speed to be in tune with thecurrent market, you can retest your system regularly using more recent data. The periodbeing retested can always be a fixed number of days, or it can be selected visually, beginningwhen the market changed its pattern, became more volatile, underwent a price shock, ormoved to new highs or lows. The problem is having enough data to be satisfied that theresults are dependable; the faster you react, the less data there is to make a decision. Thisapproach, along with other testing methods, are covered in detail in Chapter 21 (“Testirig”).
A Development Example
The size of price changes can also be used to vary the period of a moving average. Whenprices become more volatile, as measured by the standard deviation of the price changes,a shorter period can be used to follow the market more closely. When there are smallerprice changes, or stable volatility, the preference is to slow the trend by increasing the cal-culation period.’ The following steps can be used to build and personalize a variable-speedmoving average:
* Tushar Chande and Stanley Kroll, The New Market Technician (John Wiley & Sons, New York, 1994).> George R. Arrington, “Building a Variable-Lengch Moving Average,” Technical Analysis of Stocks & CommoditiesGune 1991).CONSIDERING AD>.. (IVE METHODS 447
 
1. Select the range over which the period of the moving average may fluctuate. For ex-ample, a medium to fast program might range from 5 to 30 days.
2. Calculate the mean and standard deviation of price changes over a separate, fixedtime period, recommended as the length of the longest trend period (in this case,30 days), but enough days to have a reasonable sample (no less than 10 days). Usinga smaller period reduces the response time needed to switch from one moving av-erage speed to another, but causes the results to be more erratic.
3. Select zones, or threshhold levels, at which the market is seen to be relatively moreactive or less active. For example, within the boundary of the mean + .25 standarddeviations, we will define a low-volatility, less active area where the longest periodwould be used. Outside the boundary of the mean + 1.75 standard deviationswould be an unusual, highly active market requiring a fast trend.
4. Establish the rate at which the moving average period would change as volatility(price change) increases from the inside to outside boundary. This could be the lin-ear relationship
n, = (30-5) * ((P, — P,_,)/stdev ~ .25)/(1.75 ~ .25) +5,for 1.75 > (P, — P;..)/stdev > .25
when (P,— P,_ ;)/stdev is between 1.75 and .25. Otherwise, it is the max and min val-ues, 30 and 5, respectively. The general form is
today’s moving average period = (max ma period ~.min ma period) *(today’s change/stdev of changes ~ min stdev boundary) / (max stdevboundary — min stdev boundary) + min moving average period
CONSIDERING ADAPTIVE METHODS
At first glance, there appears to be a conflict between the sound statistical approach thatencourages the choice of a simple set of rules applied to a long period of test data andadaptive methods. The classic result is a statistically robust model, but one that might showconsiderable variation in performance over the test period. To stabilize returns we moveaway from fixed values in trading systems, such as a $500 stop-loss, or a 50-point breakout,and substitute risk and entry criteria that are based on volatility. The simplest of thesemethods uses a percentage of price; the most complex can be very intricate functions ofvolatility and cycles.
When a variable feature is incorporated into the strategy, the values should smoothlyadjust to the market patterns, but each case, such as extreme high or low volatility, hasfewer occurrences;