fail. The problem is not a simple one, but onethat should be resolved by each system developer, because it will influence-the type of sys-tematic approaches that willbe studied. The strongest argument against the random move-ment supporters is one of price anticipation. One can argue academically that allparticipants (the market) know exactly where prices should move following the release ofnews. However practical or unlikely this is, it is not as important as market movementbased on anticipation of further movement. For example, if the prime rate was raised twicein two months, would you expect it to be increased in the third month? Do you think thatothers will have mixed opinions, or that they assess the likelihood of another increase atdifferent levels (i.e., one might see a 25% chance of an increase and another see a 60%chance). Unless the whole market view expectations the same way, then the price willmove to reflect the majority opinion. As news alters that opinion the market will fluctuate.Is this random movement? No. Can this appear similar to random movement? Yes.
Excluding anticipation, the apparent random movement of prices depends on boththe time interval and the frequency of data used. When a long time span is used, from 1 to4 INTRODUCTION
 
20 years, and the data averaged to increase the smoothing process, the trending character-istics will change, along with seasonal and cyclic variations. Technical methods, such asmoving averages, are often used to isolate these price characteristics. The averaging of datainto quarterly prices smooths out the irregular daily movements and results in noticeablypositive correlations between successive prices. The use of daily data over a long timeinterval introduces noise and obscures uniform patterns.
In the long run, most futures prices find a level of equilibrium (with the exception ofthe stock index, which has had an upward bias) and, over some time period, show thecharacteristics of being mean reverting (returning to a local average price); however,short-term price movement can be very different from a random series of numbers. It oftencontains two unique properties: exceptionally long runs of price in a single direction, andasymmetry, the unequal size of moves in different directions. These are the qualities thatallow traders to profit. Although the long-term trends that reflect economic policy, easilyseen in the quarterly data, are not of great interest to futures traders, short-term pricemovements—caused by anticipation rather than actual events, extreme volatility, pricesthat are seen as far from value, countertrend systems that rely on mean reversion, andthose that attempt to capture trends of less duration—have been successful.
It is always worthwhile to understand the theoretical aspects of price movement,because it does paint a picture of the way prices move. Many traders have been challengedby trying to identify the difference between an actual daily price chart and one created bya random number generator. There are differences, but they will seem more subtle thanyou would expect. The ability to identify those differences is the same as finding a way toprofit from actual price movements. A trading program seeks to find ways to operatewithin the theoretical framework, looking for exceptions, selecting a different time frameand capture profits—and all without ignoring the fact that the theory accounts for most ofthe price movements. -
BACKGROUND MATERIAL
The contents of this book assume an understanding of speculative markets, particularly thefutures markets. Ideally the reader should have read one or more of the available tradingguides, and understand the workings of a buy or sell order and the specifications of con-tracts. Experience in actual trading would be helpful. A professional trader, a broker, or apurchasing agent will already possess all the qualifications necessary. A farmer or rancherwith some hedging experience will be well qualified to understand the risks involved. Sois any investor who manages his or her own stock portfolio.
Literature on markets and trading systems has greatly expanded in the 11 years sincethe last edition of this book. During that time the most comprehensive and excellent workhas been Jack Schwager’s two-volume set, Schwager on Futures (Wiley, 1995), whichincludes one volume on fundamental analysis and the other on technical analysis. JohnMurphey’s Zechnical Analysis of the Futures Markets (New York Institute of Finance, 1986)and Intermarket Technical Analysis (Wiley, 1991) are highly recommended. Ralph Vincepublished a popular work, Portfolio Management Formulas (Wiley, 1990), and there isPeter L. Bernstein’s 7he Portable MBA in Investment (Wiley, 1995), which again providesvaluable background material in readable form. There have been quite a few books on spe-cific systems and some on the development of computerized trading methods. The onecomprehensive book of studies that stands out is The Encyclopedia of Technical MarketIndicators by Robert W. Colby and Thomas A. Meyers (Dow Jones-Irwin, 1988), whichoffers an intelligent description of the calculation and trading performance of most marketindicators oriented toward equities traders. Comparing the results of different indicators,side by side, can give you valuable insight into the practical differences in these techniques.RESEARCH SKILLS 5
 
The basic reference book for general contract information has always been the Com-modity Trading Manual (Chicago Board of Trade), but each year Futures magazine pub-lishes a Reference Guide, which gives the current futures and options markets tradedaround the world. No doubt, all of this information will be available through Internet. Forbeginning or reviewing the basics, there is Todd Lofton’s Getting Started in Futures (Wiley,1989); Little and Rhodes, Understanding Wall Street, Third Edition (McGraw-Hill, 1991);and The Stock Market, 6th Edition by Teweles, Bradley, and Teweles (Wiley, 1992). Theintroductory material is not repeated here.
A good understanding of the most popular charting method requires reading the classicby Edwards and Magee, Technical Analysis of Stock Trends John Magee), a comprehensivestudy of bar charting. Writings on other technical methods are more difficult to find. Themagazine Technical Analysis of Stocks & Commodities stands out as the best source of regu-lar information; Futures magazine has fewer technical articles, but many of value and manyother commodity books express only a specific technical approach. Current analysis of manymarket phenomena and relationships‘can be found in The Financial Analysts Journal.
On general market lore, and to provide motivation when trading is not going as wellas expected, the one book that stands out is Lefévre’s Reminiscences of a Stock Operator(originally published by Doran, reprinted by Wiley in 1994). Wyckoff mixes humor and phi-losophy in most of his books, but Wall Street Ventures and Adventures Through Forty Years(Harper & Brothers) may be of general interest. More recently, Jack Schwager’s MarketWizards (New York Institute of Finance, 1989) has been very popular.
A reader ‘with a good background in high school mathematics can follow most of thisbook, except in its more complex parts. An elementary course in statistics is ideal, but aknowledge of the type of probability found in Thorp’s Beat the Dealer (Vintage) is ade-quate. Fortunately, computer spreadsheet programs, such as Excel and Quattro, allow any-one to use statistical techniques immediately, and most of the formulas in this book arepresented in such a way that they can be easily adapted to spreadsheets. Having a com-puter with trading software (such as Omega’s SuperCharts, MetaStock, or any number ofproducts), or having a data feed (such as Telerate or CQG), which offers technical studies,you are well equipped to continue.
RESEARCH SKILLS
Before starting, a few guidelines may help make the task easier. They have been set downto help those who will use this book to develop a trading system.
1. Know what you want to do. Base your trading on a solid theory or observation, andkeep it in focus throughout development and testing. This is called the underlyingpremise of your program.
2. State your hypothesis or question in its simplest form. The more complex it is, themore difficult it will be to evaluate the-answer.
- 3. Do not assume anything. Many projects fail on basic assumptions that were incor-rect.
4. Do the simplest things first. Do not combine systems before each element of eachsystem is proven to work independently.
5. Build one step at a time. Go on to the next step only after the previous ones havebeen tested successfully. If you start with too many complex steps and fail, you will
. have to simplify to find out what went wrong.
6. Be careful of errors of omission. The most difficult part of research is identifying thecomponents to be selected and tested. Simply because all the questions asked weresatisfactorily answered does not mean that all the right questions were asked. Themost important may be missing.6 INTRODUCTION
 
7. Do not take shortcuts. It is sometimes convenient to use the work of others tospeed up the research. Check their work carefully; do not use it if it cannot be veri-fied. Check your spreadsheet calculations manually. Remember that your answer isonly as good as its weakest point.
8. Start at the end. Define your goal and work backward to find the required input. Inthis manner, you only work with information relevant to the results; otherwise, you
might spend a great deal of time on irrelevant items.
OBJECTIVES OFTHIS BOOK
This book is intended to give you a complete understanding of the tools and techniquesneeded to develop or choose a trading program that has a good chance of being success-ful. Execution skill and market psychology are not considered, but only the developmentofa system that has been carefully thought out and tested. This itself is an achievement ofno small magnitude.
Not everything can be covered in a single book; therefore, some guidelines wereneeded to control the material included here. Most important are techniques that are com-mon to most markets, such as trend and countertrend techniques, indicators, and testingmethods. Popular analytic techniques, such as charting, are only covered to the degree thatvarious patterns can be used in a computerized program to help identify support and resis-tance, channels, and so forth. There has been no attempt to provide a comprehensive texton charting. Various formations may offer very realistic profit objectives or provide reliableentry filters, even though they are not included.
“Some popular areas, such as options, are not covered at all. There are many goodbooks on options strategies, and to include them here would be.a duplication of effort.Also, those strategies that use statistics, such as price/earnings ratios, specific to equities,have not been included, although indicators that use volume, even the number of advanc-ing and declining issues, you will find in the section on volume because they fit into a big-ger picture. This remains a book on trading futures markets, yet it recognizes that manymethods can be used elsewhere.
This book will not attempt to prove that one system is better than another, because itis not possible to know what will happen in the future. It will try to evaluate the conditionsunder which certain methods are likely to do better and situations that will be harmful tospecific approaches. Most helpful should be the groupings of systems and techniques,which allow a comparison of features and possible results. Seeing how analysts have mod-ified existing ideas can help you decide how to proceed, and why you might choose onepath over another. By seeing a more complete picture, it is hoped that common sense willprevail, rather than computing power.
PROFILE OF ATRADING SYSTEM
‘There are quite a few steps to be considered when developing a trading program. Some ofthese are simply choices in style that must be made, while others are essential to the suc-cess of the results. They have been listed here and discussed briefly as items to bear inmind as you continue the process of creating a trading system.
Changing Markets and System Longevity
Markets are not static. They evolve because the world changes. Among those items thathave changed during the past 10 years are the market participants, the tools used to watchthe market, the tools used to develop trading models, the economies of countries such asJapan, the union of European countries, the globalization of markets, and the risk of par-ticipation. Under this changing situation, a trading system that works today might not workPROFILE OF ATRADING SYSTEM 7
 
far into the future. We must carefully consider how each feature of a trading program isaffected by change and try to create a method that is as robust as possible to increase itslongevity.
The Choice of Data
System decisions are limited by the data used in the analysis. Although price and volumefor the specific market may be the definitive criteria, there is a multitude of other valid sta-tistical information that might also be used. Some of this data is easily included, such asprice data from related markets; other statistical data, including the U.S. economic reportsand weekly energy inventories, may add a level of robustness to the results but are lessconvenient to obtain.
Diversification
Not all traders are interested in diversification, which tends to reduce returns at the sametime that it limits risk. Concentrating all of your resources on a single market that youunderstand may produce a specialized approach and much better results than using amore general technique over more markets. Diversification may be gained by trading morethan one method in addition to a broad set of markets, provided the programs are uniquein style. Proper diversification reduces risk more than returns.
Time Frame
The time frame of the data impacts both the type of system and the nature of the results.Using 5-minute bars introduces considerable noise to your program, making it difficult tofind the trend, while using only weekly data puts so much emphasis on the trend such thatyour trading style is already determined. A shorter time may guarantee faster response toprice changes, but it does not assure better results. Each technique must be applied prop-erly to the right data and time frame.
Choosing a Method of Analysis
Some methods of analyzing the market are more complex than others. This in itself has nobearing on the final success. All good trading methods begin with a sound premise. Youmust first know what you are trying to extract from the market before you select a tech-nique. If you want to capitalize on long interest rate trends or on the result of governmentpolicy, then a weekly moving average or trend system will be the place to start. If you seefalse breakouts whenever price penetrates the high of the day in the second haif of thetrading session, you'll want to look-at a momentum indicator based on 5-, 10-, or 15-minute data. First the idea, then the tool.
Trade Selection
Although a trading system produces signals regularly, it is not necessary to enter all ofthem. Selecting one over another canbe done by a method of filtering. This can vary froma. confirmation by another technique or system, a limitation on the amount of risk that canbe accepted on any one trade, the use of outside information, or the current volume. Manyof these add a touch of reality to an automated process. You may find, however, that toomany filters result in no trading.
Testing -
There has been a lot of emphasis on testing, and there is a complete discussion in thisbook; however, testing is most important to confirm, or validate, your ideas. It fails when8 INTRODUCTION
 
you use broad tests to find successful techniques. The purpose of testing is to showrobustness, that the method works over a wide range of situations in a similar manner. Arobust solution will not appear to be as good as the optimal result, but performed prop-erly, it will be a more realistic assessment of expectations.
Risk Control
Every system must control its risk, and most analysts believe that nearly any system can beprofitable with proper risk management. This also means that any system can lead to ruinwithout risk controls. Risk can be managed from the trade level, with individual stop-losses, to asset allocation, by varying the size of the position traded, and by leveraging anddeleveraging. Some form of management is necessary.
Order Entry
A system that performs well on paper may be dismal when actually traded. Part of a trad-ing program is to know the method of entering and exiting the market and the cost of eachtrade. Style and cost will have a greater impact on short-term systems, which have a smallerprofit per trade and are, therefore, more sensitive to transaction costs. There is equal dam-age in overestimating costs as there is in underestimating them. By burdening a systemwith unrealistic fees, it may show a loss when it should be a successful trading method.
Performance Monitoring and Feedback
A system is not done when you begin trading, it is only in a new phase. Actual tradingresults must be carefully monitored and compared with expectations to know if it is per-forming properly. It is very likely that slippage will result in some changes to the systemtules or to the size of the position traded. Performance monitoring provides the essentialfeedback needed to be successful. Even a well thought-out and tested program may startout badly, but proper monitoring can put it on track.
AWORD ON NOTATION USED IN THIS BOOK
In attempting to make the contents of this book more practical for many readers, there arethree types of notation that can be found mixed together. Of course, the standard mathe-matical formulas for most methods appear as they had in the previous editions. Addedto that are spreadsheet examples, using Corel’s Quattro code, which is very similar toMicrosoft’s Excel. Readers should have no trouble transferring the examples found here totheir own choice of spreadsheet.
Finally there-is extensive program code with examples in Omega’s Easy Language.Although these programs have been entered and tested on TradeStation, there are occa-sional errors introduced during final editing and in transferring the code into this book.Readers are advised to check over the code and test it thoroughly before using it. In addi-tion, there are times when only a single line of code is shown along with the standardmathematical formula to help the reader translate the technique into a more practicalform. Because of the many different forms of formulas, you may find that the standard devi-ation function takes the spreadsheet form of @std rather than the Easy Language notation@stddev, or that @avg appears instead of @average. Please check these formulas for nota-tion consistent with your needs. 
Basic Concepts
. +. economics is not an exact science: it consists merely of Laws of Probability. The mostprudent investor, therefore, is one who pursues only a general course of action which is“normally” right and who avoids acts and policies which are “normaily” wrong.
L.L.B. Angas
division, because miniature voice-activated computers will be everywhere. We might
not even need to be able to add; it will all be done for us. We will just assume that it iscorrect, because computers don’t make mistakes. In a small way this is happening now. Noteveryone checks their more complicated spreadsheet calculations by hand to be certain theyare correct before going further. Nor does everyone print the intermediate results of com-puter calculations to verify their accuracy. Computers don’t make mistakes, but people do.
With computer software rapidly making technical analysis easier, we no longer think ofthe steps involved in a moving average or linear regression. A few years ago we used cor-relations only when absolutely necessary, because they were too complicated and time-consuming to calculate. It would even be difficult to know if you had made a mistakewithout having someone else repeat the same calculations. Now we face a different prob-lem—if the computer does it all, we lose our understanding of why a moving averagetrendline differs from a linear regression. Without looking at the data, we don’t see anerroneous outlier. By not reviewing each hypothetical trade, we miss seeing that the slip-page can turn a profit into a loss.
To avoid losing the edge needed to create a profitable trading strategy, the basic toolsof the trade are explained in this chapter. Those of you already familiar with these methodsmay skip over it; others should consider it essential that they be able to perform these cal-culations manually.
Ts will come a time when we no longer will know how to do the calculation for long
ABOUT DATA AND AVERAGINGThe Law of Averages
The law of averages is a greatly misunderstood and misquoted principle. It’s most oftenreferred to when an abnormally long series of losses is expected to be offset by an equaland opposite run of profits. It is equally wrong to expect a market that is currently over-bought to next become oversold. That is not what is meant by the law of averages. Over alarge sample, the bulk of events will be scattered close to the average in such a way as tooverwhelm an abnormal set of events and cause them to be insignificant.
This principle is illustrated in Figure 2-1, where the addition of a small abnormalgrouping to one side of a balanced group of near-normal data does not affect the balance.A long run of profits, losses, or price movement is simply abnormal and will be offset over10 BASIC CONCEPTS
 
TEM Io) sree large number of normal events. Further discussion can be found in “The The-OFY Ut Kearns” (Chapter 22).
Bias i), Data
wren ‘“inpling is used to obtain data, it is common to divide entire subsets of data intothan farts and attempt a representative sampling of each portion. These samples are“) Yirluhted to reflect the perceived impact of each part on the whole. Such a weightingwill Hany ify or reduce the errors in each of the discrete sections. The result of such weight-D8 "Hy cause an error in bias. Even large numbers within a sample cannot overcomeINteN 4 jy 4tad bias introduced by weighting one or more parts.Matty analysis and trading techniques often introduce bias in both implicit and explicitways § iteighted average is an overt way of adding a positive bias (positive because it is inten-HORA tn the other hand, the use of two analytic methods acting together may unknowingly
rely ett sly on one statistical aspect of the data; at the same time, other data may be used onlyOnC*: Ht thay be eliminated by offsetting use. The daily high and low used in one part of a pro-STAN stu the daily range (high to low) in another section would introduce bias.
How Much Data Is Enough?
Teun al analysis is fortunate to be based on a perfect set of data. Each. price that isCON hea by the exchange is exact and reflects the netting out of all information at thatMONI, Most.other statistical data, although it might appear to be very specific, are nor-en YW average value, which can represent a broad range of numbers, all of them either3THUE iv smaller. The average price received by all farmers for corn on the 15th of theMONI \wtnnot be the exact number. The price of Eurodollars at 10:05 in Chicago is the4" Aid only price.hen an average is used, it is necessary to collect enough data to make that average2CC\tie, Because much statistical data is gathered by sampling, particular care is given toCCW lating a sufficient amount of representative data. This will hold true with prices asCU Averaging a few prices, or analyzing small market moves, will show more erraticTESMMts, t is difficult co draw an accurate picture from a very small sample. :ace \ hen using small, incomplete, or representative sets of data, the approximate error, ora wes of the sample should be known. This can be found by using the standard deviationte Ceassed in the previous section. A large standard deviation means an extremely scat-TNL svt of points, which in turn makes the average less representative of the data. ThisPLS ig called the testing of significance. The most basic of these tests is the error result-178 {yy a small amount of data. Accuracy usually increases as the number of items becomes"SM, aud the measurement of deviation or error will become proportionately smaller.
1 .
1vor = =Vy orkNor Vv number of items sampled N or V@sqrev)
PIQUE 2-1 The law of averages. The normal cases overwhelm thé unusual ones. It is notvy for the extreme cases to alternate—one higher, then the other lower—to create a
— ee, -
 
 
Nearnormal
Abnormal Center)ON THE AVERAGE- Wl
Therefore, using only one item has an error factor of 100%; with four items, the error is50%. The size of the error is important to the reliability of any trading system. If a systemhas had only 4 trades, whether profits or losses, it is very difficult to draw any conclusionsabout performance expectations. There must be sufficient trades to assure a comfortablysmall error factor. To reduce the error to 5%, there must be 400 trades, which presents adilemma for a very slow trend-following method that may only generate 2 or 3 trades eachyear. To compensate for this, the identical method can be applied to many markets and thesample of trades used collectively. By keeping the sample error small, the risk of tradingcan be better understood.
ON THE AVERAGE
In discussing numbers, it is often necessary to use representative values. The range of val-ues or the average may be substituted to change a single price into a general characteristicto solve a problem. The average (arithmetic mean) of many values can be a preferable sub-stitute for any one value. For example, the average retail price of one pound of coffee inthe northeast is more meaningful to a cost-of-living calculation than the price at any onestore. However, not all data can be combined or averaged and still have meaning. The aver-age of all futures prices taken on the same day would not say anything about an individualmarket that was part of the average. The price changes in copper, corn, and the GermanDAX index, for example, would have little to do with one another. The average of a groupof values must meaningfully represent the individual items.
The average can be misleading in other ways. Consider coffee, which rose from 40¢ to$2.00 per pound in one_year. The average price of this product may appear to be $1.40;however, this would not account for the time that coffee spent at various price levels. Table2-1 divides the coffee move into four equal price intervals, then shows that the time inter-vals spent at these levels were uniformly opposite to the price rise. That is, price remainedat lower levels longer, and at higher levels for shorter time periods, which is very normalprice behavior.
When the time spent at each price level is included, it can be seen that the averageprice should be lower than $1.40. One way to calculate this, knowing the specific num-ber of days in each interval, is by using a weighted average of the price and its respectiveinterval - .
_ ay + Gnd, + asd; + aidsd+, +dyt+dsW=105
_ 6000 + 8000 + 8400 + 7200
Ww280
 
TABLE 2-1 Weighting an Average
 
Prices Go Average Total Days
From To during Interval for Interval Weighted a40 80 a, = 60 d, = 100° 6,000 0166680 120 a, = 100 d, = 80 8,000 - 01000120 160 a, = 140 d, = 60 8,400 00714
160 200 a, = 180 d,= 40 7,200 .00555412 BASIC CONCEPTS
 
Although this is not exact because of the use of average prices for intervals, it does closelyrepresent the average brice relative to time. There are two other averages for which timeis an important element—the geometric and harmonic means,
Geometric Mean
The geometric mean represents a growth function in which a price change from 50 to 100is as important as a change from 100 to 200.
G= (4 X a2 X @ XX ay)" or @mult(list) * (1/length of list)
To solve this mathematically, rather than using a spreadsheet, the preceding equation canbe changed to either of two forms:
Ina,+Ina,+--+Ina,n
InG=
or
In (G; X @2 X @y X +7" X Gn)n
InG=
The two solutions are equivalent. Using the price levels in Table 2-1, disregarding the timeintervals, and substituting into the first equation:
In 40 + In 80 + In 120 + In 160 + In 200.5 : _
Had one of the periods been a loss, that value would simply be negative. We now performthe arithmetic to solve the equation:
3.689 + 4.382 + 4.787 + 5.075 + 5.298
Ing=
InG=InG 5In G = 4.6462
G= 104.19 -
The geometric mean has advantages in application to economics and prites. A classicexample is to compare a tenfold rise in price from 100 to 1,000 with a fall to one-tenthfrom 100 to 10. An arithmetic mean of 10 and 1,000 is 505, while the geometric mean gives
G = (10 x 1000) = 100which shows the relative distribution as a function of comparable growth. Due to thisproperty, the geometric mean is the best-choice when averaging ratios that can be eitherfractions or percentages.Quadratic MeanThe quadratic mean is as calculated: uc
Q= vf WO @sqrt(@sum (list of a”)/length of list)
The square root of the mean of the square of the items (root-mean-square) is most wellknown as the basis for the standard deviation. This will be discussed later, in the section“Dispersion and Skewness.”DISTRIBUTION - 13
 
Harmonic Mean
The harmonic mean is more of a time-weighted average, not biased toward higher or lowervalues as in the geometric mean. A simple example is to consider the average speed of a carthat travels 4 miles at 20 mph, then 4 miles at 30 mph. An arithmetic mean would result in25 mph, without considering that 12 minutes were spent at 20 mph and 8 minutes at 30mph. The weighted average would give
wn 12X20) + 8X 30) _- 12+8 -
24
The harmonic mean is
1 1 1
Spo peepeei a a anH- n
which can also be expressed asH= nly. (2) or length of list‘(@sum<(list of 1/a)T He
For two or three elements, thé simpler forms can be used:
 
__2ab _ 3abe2 atb 2 ab +ac+be
This allows the solution pattern to be seen. For the 20 and 30 mph rates of speed, the solu-tion is .2x20 x30=o = 242" 20+30
which is the same answer as the weighted average. Considering the original set of numbersagain, the basic form of harmonic mean can be applied:
1 1 1 1 1
 
32,4,4,4t,41 40" 80" 120 160 * 200H 505708 _ i1425 —H=87.59
We might apply.the harmonic mean to price swings, in which the first swing moved 20points over 12 days, and the second swing moved 30 points over 8 days.
DISTRIBUTION
The measurement of distribution is very important because it tells you generally what toexpect. We cannot know what tomorrow’s S&P trading range will be, but we have a highlevel of confidence that it will fall between 300 and 800 points. We have a slightly lowerconfidence that it will vary from 400 to 600 points. We have virtually no chance of pickingthe exact range. The following measurements of distribution allow you to put a value onthe chance of an event occurring.14 BASIC CONCEPTS
 
Frequency Distributions
The frequency distribution can give a good picture of the characteristics pf the data. Toknow how often sugar prices were at different price levels, divide prices into 10 increments(e.g., 5.01 to 6.00, 6.01 to 7.00, etc.), and count the number of times that prices fall intoeach interval. The result will be a distribution of prices as shown in Figure 2-2. It should beexpected that the distribution of prices for a physical commodity, interest rates (yield), orindex markets, will be skewed toward the left-hand side (lower prices or yields) and havea long tail toward higher prices on the right-hand side. This is because prices remain athigher levels for only a short time relative to their long-term characteristics. Commodityprices tend to be bounded on the lower end, limited in their downside movement, by pro-duction costs and resistance of the suppliers to sell at prices that represent a loss. On thehigher end, there is not such a clear