{Exit without volatility)
if option-0 or option=1 then beginif mavgcmavgl1} then exitlong on close;if mavg>mavgl1] then exitshort on close; - . .end; -
(Exit on high volatility and profitable direction)
1f option=2 or option=3 then begin 
 
402 ADVANCED TECHNIQUESif navgémavgll) or (close>etoseL1] and change>xup)imit) then exitiong an close:HF navg>mavgli} or (closecclose(1] and change>Xuplimit) then exitshort on close;end; ‘(Exit on high volatility and not a profitable direction?$f option=2 or option=4 or option=5 then beginsf mavgémavg(i] or (closec=close(1) and change>kuplimit) then exitiong on close;jf mavgrmavgll] or (eloser=close(1} and change>Xuplimit) then exitshort on close:
end:
IReplace high volatility days because they distort standard deviation!{ if change > vavg+2*factor*vsd then change = vavg;!
Note that parts of the program have comment braces around one or more lines, ‘These rep-resent other options that may be tested. It will be necessary to remove the comment bracesfrom one section and add them to another.
TRADE SELECTION
Once you have a basic system, the next step is to decide whether you can eliminate someof the losing trades without eliminating the profitable ones. In the extreme, we want toCliminate alf of the losing trades, even if we reduce some of the profits; of course, that isimpossible. Every system has a risk, even so-called “riskless” trades. Arbitrage, when doneproperly, has virtually no risk; however, it may be so competitive that the opportunities aretare and the margin of profit small, You might find, after months of trading a guaranteedirbitrage, that your returns are disappointing compared with the cost of doing business
Putting your money into U.S. bonds would have been nearly as good with a lot less work.
Nothing is free. Good trading is not a matter of hitting it rich on a single short position;it is grinding out a profit day by day and week by week. Occasionally, if you follow new mar-kets, you might find an opportunity that others have not yet seen. Ifyou act fast, you cancapitalize on this for a short time, until others see the same situation. Then bigger playerscome in and push you out. In turn, they are followed by other investors who combine toremove all the profit opportunity, even for themselves.
When selecting trades to eliminate, the easiest place to begin is by associating perfor-mance with volatility or price level. While some systems perform better in an environmentof higher volatility, it may be that the best return relative to risk occurs when there is lessvolatility, In general, markets with extremely low volatility do not perform well because theyhave less direction. Under moderate-volatility conditions, systems will return both lowerprofits and smaller losses; with high volatility we can expect large profits and high risk.
 
 
Predicting Volatility with Trading Ranges
“william Brower? performed a thorough study of trading ranges, looking to find a higherrange for tomorrow. This would be important for day traders and for any system in whichhigher volatility is an advantage. Specifically, high volatility is desirable for breakout sys-tens, A summary of his fesults is shown in Table 20-3 for-the S&P front 12/23/87 through
12/15/95.
Thomas Bierovic, On-Balance True Range .To visualize the change in volatility, Thomas Bierovic has created On-Balance True Rangeby following the same rules as On-Balance Volume, but using the True Range calculationinstead of Volume. He then calculates a 9-day exponential smoothing of the On-Balance‘True Range and uses the crossovers of the osciftator and smoothed oscillator to confirmsignals. Although the highs and lows may come at nearly the same time as other oscillators,
s william Brower, Inside Edge (Inside Edge Systems, 10 Fresenius Road, Westport, CT 06880, March/April 96).
 
 
 
tres Setiee set 8 hte in ee
PRICE-VOLUME DIS: xIBUTIONS.
 
TABLE 20-3 Predicting the Trading Range of the S&P 500*
 
 
 
Rule Conditions Tested Conclusion! ° < a =x Very good predictor, but few cases (161 at2 > HUI] +x Modest predictor, but few cases (101 at x3 Cll} #x>0>Cfl] Range got smaller when x = .35,4 alox<0< cy Range got smaller when x = 30<C[l]-x Range tended to increase as x increased§ HQ] <= HI] and [2] >= {{1] No significance7 HD >H{I] and {2} < t[1] Modest predictor of lower volatility8 ay of week Monday had lowest volatility, Tuesday and Friday the highest
Average(TrueRange,3)[1] > x Higher avg true range for 3 days was a verygood predictor of higher volatility
10 RSi(close.3)[1] < x When RSI < 40 good predictor of higher volatility
* Source: William Brower.
 
the positioning of the relative¢ positioni peaks and valleys may offer new patterns. For many traders,this simple interpretation can help separate high- and low-volatility conditions. Y "
PRICE-VOLUME DISTRIBUTION
Many systems. that have excellent historic simulations fail when
they are actually waded.These include models based on both intraday and daily data. One reason for this disap-pointing performance is the lack of understanding of market liquidity. Consider two systems:
Atrend-following method, which si ‘7 , ignals a buy or sell order whrelative to a specific price during the day en prices rise or fal
A countertrend system, which sells and buys at relative intraday highs or lows
Although each system intends to it i° n profit with their opposing philosophies, both actwhen prices make a relatively unusual move. Figure 20-4 shows the normal distribution ofintraday volume and the profitability associated with this distribution.‘
* Gary Ginter and Joseph J. Ritchie, “Data5 . . Errors and Profit Distortions,” in Perry J. Kaufm:Arialysts in Commodities (John Wiley & Sons, New York, 1980). m Feary J. Nauman, (2a), ebreal
 
FIGURE 20-4 Daily price/volume distribution.
©
1Volume Traded
   
Price
  
484 ADVANCED TECHNIQUES
‘The solid line bb’ is the,actual volume on a theoretically normal day in which pricesremained in a trading range. Volume is greatest at the median M and declines sharply to thehigh and low endpoints H and L, where only one trade may have occurred. The'dotted linecc’ represents the apparent profit for a countertrend system that makes the assumption ofa straight-line volume distribution, aa’. The endpoints are shown to contribute the largestpart of the profits when, in reality, no executions may have been possible near those levels.Assuming the ability to execute at all points on the distribution 6b’, the approximate profitcontribution is shown as da’. Trades are most likely to be filled at points that limit imme-diate profits.
For trend-following systems, no profits should be expected when buy or sell orders areplaced at the extremes of the day. The actual price distribution 66’ is the maximum thatcould be expected from such a system; in reality, the first day is usually a loss.
TRENDS AND NOISE
‘Throughout this book there are techniques that try to find the underlying trend when theprice direction is not clear. The investor who buys sooner is more profitable, but only if themarket moves higher. What makes it so difficult to identify the trend is noise. Noise is theerratic movement of price; by definition it is unpredictable, yet it has very definite statisti-cal attributes. '
Noise is the product of the market participants buying and selling at different times fordifferent purposes. Each has their own objectives and time framé. Noise can also be a priceshock—an unexpected event that may or may not cause a lasting price change. Noise hasmost of the qualities of a sequence of random numbers. Nearly half of all price moveschange direction from the previous move; about 25% of all price moves continue in thesame direction for two periods, and so on. These patterns should already be familiar toanalysts. The size of a price move, or price shock, also appears to be random; that is, 50%are quite small, 25% are twice as large, 12.5% are four times as large, and a few are verylarge.
Then how can you tell ifa price move up is an indication of a new direction or simplymore noise? The answer is that you can’t tell by studying only price until after the directionhas continued. There are situations, of course, where the news will give us information toshow that the price jump was a structural change (for example, when the Federal Reservelowers interest rates). But price alone won’t tell us.
We can improve our chances of selecting the trend direction based on a net price moveby choosing only the largest moves. This is the reason why a long-term trend is more reli-able than a short one. To see this clearly, consider Figure 20-5 to be random numbers, ormarket noise. Most of the time prices remain below the level P1, but a price move thatreaches P2 (twice as large) is still fairly common. Less often, price will jump to level P3 butonly rarely will it reach P8. Let's say that, out of 1,000 price moves, P1 is reached 500 times,P2 occurs 250 times, P4 occurs 64% times and P8 about 4 times. If we use a breakout sys-tem to decide the price trend, and our breakout criteria is less than P2, then there wouldhave been 250 total signals (both long and short) in which tg find the right trend direction.If there were 10 confirmed trends dusing this period, youneeded to reject or filter 240, or96%, of ali moves. Suppose your criteria was P8 instead of P2, and because the size of thesubsequent price trend must be bigger due to the larger entry criteria, there was only onegood trend during this period. Then you have a 1-in-4 chance of being correct, far berterthan the 1-in-25 using a very small trend criteria.
It is the magnitude of the market noise that determines the criteria needed for enter-ing a wend. For a broadly traded market, the noise pattern is very predictable, while mar-kets that have low volume might not conform closely to a traditional statistical profile.
 
 
TRENDS AND NOISE 485
FIGURE 20-5 Random occurrence of price levels.
 
Noise can be measured using the same calculations as volatility, gi inni0 tility, given at the beginning ofthis chapter, or as Kaufman’s efficiency ratio found in Chapter 17 (“Adaptive Techniques’).
EXPERT SYSTEMS
An expert system is one in which you draw conclusions based on an accumulated know!-edge base, which has been stored as data, facts, and relationships in the form of if-then-elserules. For our purposes, these would include price and economic data as well as theknowledge that, for example, bighly correlated markets move together, and high volatil-ity means high risk. When the first expert systems were developed, the core informationwas actually gathered by interviewing experts, which is how the name expert system was
derived. The success of this method depend: th iknowledge base. pends upon the quality and completeness of the
The knowledge base is used by an inference engi ich i ip igine, which is able to draw conclusionsfrom the facts stored in the knowledge base. Therefore, if we have the following information,
FACT 1: U.S. bonds are more volatile than Eurodollars.FACT 2: The S&P 500 is more volatile than U.S. bonds.FACT 3: Volatility is directly proportional to risk.then the inference engine can create the new factFACT 4: The S&P is riskier than Eurodollars.The inference engine provides a straightforward, logical process; however, there may
be many relationships to resolve, not all of which may apply to the problem you would liketo solve. For example, if we also have the following facts:
FACT 5: The S&P has a high degree of noise.
FACT 6: Eurodollars have a low degree of noise.
FACT 7: The S&P is currently trading below its 200-day moving average.
FACT 8: The S&P is currently trading above its 20-day moving average.9:
FACT The S&P has been above its 200-day moving average for 80% of the past 20years.
FACT 10: Eurodollar rates are driven by monetary bank policy.FACT 11: Current monetary policy is dominated by concerns of inflation. ©FACT 12: Inflation results in higher interest rates.486 ADVANCED TECHNIQUESFACT 13: Net trading returns for the S&P have been 40% per annum.FACT 14: Net trading returns for the Eurodollar have been 20% per annum.
then human logic might conclude that Eurodollar yields are likely to rise because of con-terns over inflation. In addition, the S&P has greater risk, erratic behavior, and is currentlynot 29 strong as it has been on average over the past 20 years, although it is now rising.Compared with Eurodollars, there is greater risk trading the S&R, but its returns have been
higher.
tow can this expert conclusion be duplicated by a computer? By adding 2 set of rulesthat parallels the thinking of experts, a computer can theoretically arrive at the same con-clusions. For example,
RULE la: IF the Producer Price Index rises by more than the annualized rate of 4%,
THEN we have inflation.
RULE 2a: IF there is high noise, THEN there is less chance of a trend.
RULE 3a: IF there is high noise, THEN there is greater risk.
RULE 4a: IF the faster trend is above the slower trend, THEN prices are trending up.
For each positive rule 1 through 4, there should also be a negative rule:
RULE 1b: IF the Producer Price Index does not rise by more than 4% annualized,
THEN we do not have inflation.RULE 2b: . IF there is low noise, THEN there is a greater chance of a trend.RULE 3b: IF there is low noise, THEN there is lower risk.RULE 4b: “IF the faster trend is below the slower trend, THEN prices are trendingdown,ambiguous cases. For example, in Rule 4 thereis the case in which the two trends are in conflict. In other situations, the positive rulemight be true, but the negative rule may not be as strong, In Rule ib, we see that the effectis that “we do not have inflation” rather than “we have deflation.”
Even with the negative rules there are some
Forward Chaining X
‘The process of combining the rules and facts to yield an expert opinion is called forwardchaining. For example, beginning with FACT 6, “Eurodollars have a low degree of noise,”ce“tnd she relevant rule, RULE 3b: “IF there is low noise, THEN there is lower risk,” andvreate a new fact: “Eurodollars have (relatively) low risk.” Note that in each case, the termslow, bigh, and faster are all relative
Once we have this new fact, that Eurodollars have relatively low risk, and we similarlyb risk, we can also conclude that Eurodollars have
conclude that the S&P has relatively higMonet risk than the S&B The process of following the path of each fact as it is handled by
various rules is called forward chaining. It will lead to other rules and other facts; it maybe that the expert opinion will be found along this route,-or that the combination of newfacts, such as the relative risk between the Eurodollars and the S&R will provide the expertopinion.‘This example shows only a few facts that can be easily summarized; however, there arethousands of pieces of information about performance characteristics, relationships toother markets, and fundamental factors that might alter expectations. If accumulated byasking experts, it is also likely that there will be conflicting information. While the humanbrain has a remarkable ability to sort through these items and select the information it con-siders most relevant, some important items can be overlooked when there is too much to
 
EXPERT SYSTEMS -EXPERT SYSTEMS 7
consider. An expert system is expected to use all of the data and reduce it to a single deci-sion In doing this, it must also select the most significant facts and resolve conflicts asso-ciated with the proper order of events and the time horizon of the investor.
A Technical Expert System
‘An expert system can treat indicator values as expert opinions and create systems withoutthe fundamental relationships shown in the previous section. An example by FishmanBarr, and Loick,* applied to the DJIA, defines rules as the relationships between the vari-ous indicators and calculations, and the facis as the values of those items. The purpose ofthis expert system is (0 inspect trending and nontrending characteristics of price move-ment to give the probability of a continued trend. The following example is adapted from
 
IF ADX > 18 AND ADX >= ADX[2}(AND nontrending is false or undefined),THEN there is a 95% chance the market is trending.Rule 2: IF ADX < 30 AND ADX <= ADX[2](AND trending is false or undefined),THEN there is a 90% chance the market is nontrending.Rule 3: JF the market is trending,THEN MACD = probability of .8 AND SD = probability of .5 (85%).Rule 4: IF the market is nontrending, :THEN MACD = probability of .55 AND SD = probability of .75 (75%).Rule 5: IF stochastic AND SK[1]} < 30 AND SD[1] < 30 AND SK[1] < SD[1}AND SK < SD, THEN sell with 80% confidence.‘Rule 6: IF stochastic AND SK[1] > 70 AND SD[1] > 70 AND SK[1] >AND SK> SD, THEN buy with 80% confidence (sp)Rule 7: IF MACD AND MACD value > signal line value,THEN buy with 75% confidence.Rule 8: IF MACD AND MACD value < signal line value,THEN sell with 75% confidence.
where the following periods are assumed:
ADX' isan 18-period ADX calculationADX[2] is the ADX 2 periods agoMACD is the Moving Average Convergence/Divergence with smoothing constants
 
-15 and .075SK is a 9-period, %K fast stochasticSD is a 9-period, %D slow stochastic -
and undefined means that there is no [information about the values. The facts are theFACT 1: ADX=19FACT 2: ADX[2] = 19FACT 3: SK=90FACT 4: SD=80
> Mark B. Fishman, Dean S. Barr, and Walter J. Loick, 7 CAs dl Analman, . Barr, . “Artificial Intelligence And Market Analysis,”sis of Stocks & Commodities (Bonus Issue 1993). " Sedna ecbneal nal 
488 ADVANCED TECHNIQUES
 
FACT 5: SK[1] —SK=68
FACT 6: SD[1]-SD=92 :FACT 7: signal line value = -70
FACT 8: MACD value = -68
FACT 9: the market is trending
‘The results, expressed as probabilities, offer greater insight into the likelihood of suc-cess using this method. Arriving at these probabilities, however, requires additional deci-Sions. Without any further information, we can assume that there is only a 50% chance thatthe stochastic is a correct trend indicator. If we were to test the number of times the sto-chastic indicated an upward move with the number of days that the subsequent pricemoved higher, we could get a much better indication of the chance of success.
FUZZY LOGIC
In the second example of expert systems, the results were expressed as probabilities.Because trend and indicator calculations are estimations of market movement, very little isstrictly true or false; therefore, probabilities are a more realistic way to view the results.The probability of being right can be represented by the reliability (percentage of prof-itable trades) of a system using these calculations. When right, the average payout is theaverage profits per trade, and when wrong, the loss is the average loss per trade. Whenyou can assign a probability to results, the variable has often been called fuzzy. But this wayof looking at values is really a twist on basic probability analysis.
The idea of fuzziness is intended to describe the lack of precision in normal humanconversation and thought. In its pure form, the use of fuzziness allows human uncertaintyto be introduced into methods of artificial intelligence. For example, we often say,
  
“There were a lot of people in line at the show.”
“| had to wait a long time.”
“It was really cold while 1 was waiting.”
“The stock market was strong yesterday.” -
“Unemployment dropped sharply.” x
Inall these cases, we understand what is being said although there are no specific val-ues associated with “a lot of people,” “a long time,” “cold,” “strong,” and “sharply.” In fuzzylogic, all is not true or false, 0 or 1, there or not there. True fuzzy logic will answer the ques-tion “If a haifeaten apple is still an apple, how much do you have to eat before it stopsbeing an apple?”
‘The concept of fuzziness includes fuzzy numbers, such as “small,” “about 8,” “close to5,” and “much larger than 10," as well as fuzzy quantifiers, such as “almost,” “several,” and“nost.” Phrases such as “Unexpected results of Government reports cause big moves" is acommon fuzzy expression. -
Fuzzy Reasoning
Fuzzy events and fuzzy statistics are combined into fuzzy reasoning, Iris a remarkable phe-nomenon that the answers to the following examples are clear to the human brain, but notto a machine (the answers can be found at the bottom of this page).”
6 parts of this section are drawn from Perry Kaufman, Smarter Trading (McGraw-Hill, NY, 1995).
‘pog (6) pue Ysou (2) Yyoras Caan (1) ‘are sojdurexa ay 03 sI9AASUE 941,
 
FUZZY LOGIC 489
 
Example 1; X is a small price move.Y is much smaller than X.How small is ¥?Example 2: Most price moves are small.Most small price moves are up.How many price moves are up?Example 3: Itis not quite true that the quarterly earnings were very bad.It is not true that the quarterly earnings were goodHow bad were the quarterly earnings?
Common Approach to Fuzzy Solutions
In the terminology of fuzzy logic, there are three aspects of problem solving: membership,to show how the data are related to each other; fuzzy rules, to draw conclusions; anddefuzzifiers, to turn the fuzzy answers back into useable results."
For example, we would like to predict whether a price move will be big enough to cap-ture a profit needed by our trading system. To have a robust system, we always use a 20-day trend but would like to predict which price moves will be above our minimum needsbefore we enter the trade. In general, we have found that, given commissions and slippage,we need to net $500 per trade. Because we are using a trend-following system that givesUp profits before exiting, we would like to see an $800 profit to capture a net of $500,Based on our needs, we can define our limits as:
1. Average peak move between $300 and $800 is acceptable2, Average peak move over $800 is desirable.3, Average peak move under $300 is unacceptable.
To decide whether current price movement is likely to give us enough profit to enter atrade, we measure the average price move over every 20-day period for the past 1 year (longterm) and for the past 1 month (short-term). Based on this approach, we create the rules:
1. If both the long-term and short-term profit potentials are unacceptable, then thecurrent profit potential is unacceptable.
2. Ifboth the long-term and short-term profit potentials are desirable, then the currentprofit potential is desirable.
3, If both the long-term and short-term profit potentials are acceptable, then the cur-rent profit potential is acceptable.
These three cases are very clear, but what if the long-term and short-term expectations aredifferent? For example, if the long-term average peak profits are $250 and the short-termare $600,svhat can be expected from the current move? In the current application of fuzzylogic, this converts to a problem in probability for which we need more information aboutthe history of these price moves. Through testing, we find that the l-year average pricemove has a standard deviation of $75 and the past 1 month has a standard deviation of$200. We could then construct a diagram that shows the expected results from combiningthe two measurements (see Figure 20-6). The average returns are shown with descendinglines based on their standard deviations. These cross at about $350 per trade, giving a rea-sonable answer to how to combine the two values. -
One additional measurement that cannot be overlooked is the potential error in eachstatistic. The longer-term measure used 250 trading days, while the short-term used only
® Murray A. Ruggiero, Jc., “Artificial trader jumps candlesticks,” Futures (February 1995), and “Lighting candle-sticks with fuzzy logic,” Futures (March 1996). 
 
490 ADVANCED TECHNIQUES
SSeSFIGURE 20-6 Expectations of a price move using two time periods.
  
Long-term expectations
   
Avg
 
te Short-term
expectations
\
       
 
20
3é3385==
30
100 200 300 400 500 600 700Profit expectations
——
about 20 days. The standard error for each calculation is, 6.3% and 22.3%, respectively. Inthe final interpretation of results, we can say that the longer-term data are more importantthan the short-term values by about 3.5 : 1. Then we could write additional rules:
4. If the long-term potential is unacceptable and the short-term is acceptable, then thecurrent potential is unacceptable. :5. If the long-term potential is unacceptable and the short-term is desirable, then the
current potential is acceptable.
Using just the rules, we can claim that we are applying fuzzy logic to our trading; however,the state of the art indicates that fuzzy logic is still hanging onto the coattails of simple prob-ability. When it lets go, itis likely to be the most significant breakthrough in market analysis.
FRACTALS AND CHAOS
Another new area that has captured the interest of market analysts is chaos. Chaos theoryis a way to describe the behavior of nonlinear systems, those that cannot be described by astraight line. Chaotic systems are not necessarily without any form or methdd, as theexpression is commonly used, but those phenomena that are not a simple variation of alinear relationship. Some examples of this behavior will be given later in this section.
‘One method of measuring chaotic systems is with various geometric shapes. This efforthas resulted in an area of mathematics now called fractal geometry; its approach strikes atrue note about how the real world of numbers actually works.® Ait of us have been taughtEuclidean geometry in school; it is the world of straight lines and clean edges in which wecan measure the length of a line or the area of a rectangle very casily: In the real world,however, there are no straight lines; if you look closely enough, they all have ragged edges
and all may be described as chaotic.
 
 
Fractal Dimension
In fractal geometry we find that there is a way of representing the irregularity of numbers,and the formations seen in nature. We first must accept the notion that there are no wholenumbers in-nature, that real-world objects are more likely to be described as fractional, orhaving a fractal dimension. The classic example of this is the algebra of coastline dimen-
° An excellent discussion of this topic can be found in Edgar Peters, Chaos and Order in the Capital MarketsGohan Wiley & Sons, New York, 1991). The coastline example is originally credited to Benoit Mandelbrot.
 
i
 
 
FRACTALS AND CHWS 491
 
sion. We will sce that the questions “How long is the coastline?” and “How far did pricesmove?” are very similar.
The answer to both these questions is “That depends on how it is measured.” Considerthe problem of measuring the coastline of Australia using a large wall map. If we take a 12-inch ruler, we might find that the coastline is about 10 feet (perhaps 10,000 miles accord-ing to the scale). Had we taken a slightly smaller ruler we would have been more accurateand perhaps have found the coastline to be 11,000 miles; even a smaller ruler would havefollowed the contours better and found 12,000 miles of coast. As the ruler gets smaller thecoastline appears to get longer. If we had an infinitely smaller ruler, the coast would be infi-nitely long. There is really no correct answer to the question, “How long is the coastline?”
Fractal dimension is the degree of roughness or irregularity of a structure or system.In many chaotic systems, there is a constant fractal dimension; that is, the interval used formeasuring will have a predictable impact on the resulting values in a manner similar to anormal distribution. Therefore, using a 12-inch rulec, we measure the coastline and get10,000 miles:
24-inch ruler 5,000-mile coastline
- 12-inch ruler 10,000-mile coastline
6-inch ruler 20,000-mile coastline
Note that the large 24-inch ruler returns a value that is actually smaller than what webelieve is a reasonable answer. This is because, when you place a long ruler from one pointto another on the map, it cuts across part of the land mass.
Using Fractal Efficiency
In Chapter 17 (“Adaptive Techniques”), there is a discussion of Kaufman's efficiency ratio.This ratio is formed by dividing the net change in price movement over periods by thesum of all component moves, taken as positive numbers, over the same 7 periods. If theratio approaches the value 1.0, then the movement is smooth (not chaotic); if the ratioapproaches 0, then there is great inefficiency or chaos. This same measurement has morerecently been called fractal efficiency.
Fractal efficiency measures the amount of chaotic movement in prices; this can also beconsidered similar to market noise. In Chapter 17, Kaufman related this to trending andnontrending patterns, when the ratio approached 1.0 and 0, respectively. While each mar-ket has its unique underlying level of noise, the measurement of fractal efficiency is con-sistent over all markets. Markets may vary in volatility, although their chaotic behavior istechnically the same; therefore, the characteristics of one market may be compared withothers by matching both the fractal efficiency and volatility.
Interpreting fractal efficiency as noise allows other trading rules to develop.