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247 of 263 people found the following review helpful
on August 11, 2004
"...forty years after I started battle on the subject, most economists now acknowledge that prices do not follow the bell curve, and do not move independently. But for many, after acknowledging those points, their next comment is: So what? Independence and normality are, they argue, just assumptions that help simplify the math of modern financial theory. What matters are the results. Do the standard models correctly predict how the market behaves over all? Can an investor use Modern Portfolio Theory to build a safe, profitable investment strategy? Will the Capital Asset Pricing Model help a financial analyst, or a corporate financial officer, make the right decision? If so, then stop arguing about it. This is the so called positivist argument, first advanced by University of Chicago economist Milton Friedman."

Isn't it this positivism that the majority of practitioners of finance exhibit? I myself, though not a practitioner, held such thoughts. My reasoning had been based however more upon majority's acceptance -- if everyone else is acting upon the assumptions of normality and independence, I thought, what good will there be adopting a new theory? Isn't finance more akin to social sciences than to natural sciences after all?

It is these beliefs that Mandelbrot sets out to dispel with this monograph. He does so convincingly with great confidence and tenacity. The book consists of three parts, first the examination of the current theories (CAPM, MPT, Black-Scholes), next explanation of his methodology (fractal analysis), and finally of posing questions that should be answered (Mandelbrot asserts that virtually all the current theories should be reexamined under more realistic assumptions). To readers who have followed Mandelbrot's findings even remotely, there are no new advancements recorded in this book per se. He explains with concepts he developed throughout his entire career -- fractals; more specifically self-similarity, long-range dependence (via the Hurst exponent), and fractal decomposition of [trading] time. Mandelbrot's original research without doubt launched an entirely new field of study in science and engineering. Here his objective seems to be persuasion of the general public that an overhaul of existing methods is due. This may be evidenced by the absence of equations in the main text (some are included in the notes/appendix), and by the existence of the second author of this book.

The book is also a trajectory of Mandelbrot's intellectual development. He explains, with characteristic detail, why, how, and when he has become interested in the problems as he did. The result is interesting accounts of historical figures (Bachelier, Hurst, Markowitz, etc) and records of encounters with eminent figures in mathematics and economics (Lévy, Poincaré, Sharpe, and Fama (his student) to name a few).

There has long been a need for mathematical models that reflect the market more accurately. Should the new models be in form of incremental modifications of existing models or should they be based on an overhaul of the foundation as Mandelbrot proposes? Be prepared to be challenged, if not altogether persuaded, by Mandelbrot's arguments.
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724 of 790 people found the following review helpful
on October 4, 2004
The author renders a brilliant critique of modern finance theory. He criticizes all its components, including CAPM, the Efficient Market Hypothesis, and the Black Scholes model as being flawed. All these theories rely on two main assumptions. The first one is that market prices are normally distributed. The author, using price charts, demonstrates that market prices do not follow a normal distribution; but instead a Cauchy distribution. Such a distribution is associated with fatter tails. This means that catastrophic drop in market prices happen more frequently than a normal distribution suggests. The second assumption of modern finance is that market prices are independent of each other. Yesterday's prices have no influence on today's. The author makes a case that even if prices are not correlated, their volatility is correlated over time. Thus, big price swings tend to cluster. If a stock moved by 10% yesterday, it is likely it will move by an above average amount today even if we don't know the direction of that change. He calls this correlation of volatility (instead of price) long-term dependence.

Because the two main assumptions of modern finance are flawed, all related models are flawed as they understate risk. If such models understate risk, they actually overprice stocks and underprice options, and also understate the capital financial institutions should hold to withstand market risk.

If the author had stopped there, I would have given him a 5 rating. However, such a rebuttal of finance theory would make no more than a great essay. Instead, he attempts to build an entirely different edifice of modern finance over 300 pages. And, his theoretical foundation lacks any robustness. That's why I call it a castle of cards.

Mandelbrot builds his edifice of modern finance on two new parameters that would replace the mean return and volatility of return or standard deviation (mean and standard deviation being the parameters defining a normal distribution). His first parameter is Alpha, derived from Pareto's Law, is an exponent that measures how wildly prices vary. It defines how fat the tails of the price change curve are. The second one, the H Coefficient, borrowed from a hydrologist named Hurst, is an exponent that measures the dependence of price changes upon past changes.

Well, what is wrong with these two measures? He confesses at the end of the book that no two individuals calculate the same Alpha and H Coefficient when using the exact same historical data! Apparently, there is no one established way to calculate these two parameters. The divergence between the various methodologies can be huge. Using one method, you could derive Alpha and H coefficients that suggest a stock is not risky, using another method you would reach the opposite conclusion. So, after reading nearly 300 pages of intense theories you get that their own foundations are at this stage nonexistent. If Alpha and H are mathematically not replicable and well defined, you can't apply his multifractal geometry model in any meaningful way.

It will be up to someone else to build upon Mandelbrot's work and render it applicable to investment management by firming up the algorithms to calculate Alpha and the H Coefficient. Only then, will fractal geometry maybe turn out into a feasible challenge to the foundation of Modern Finance. But, at this stage contrary to what Mandelbrot pretends, it is not.

If you are interested in investment and finance theory, I strongly recommend other books such as: Robert Shiller's "Irrational Exuberance" and "Market Volatility." Also, Nicholas Taleb's "Fooled by Randomness" is very good. Also, Roger Lowenstein's "When Genius Failed: The Rise and Fall of Long Term Capital Management." This last book is a fascinating account of why a hedge fund failed because it relied excessively on the normal distribution, and used time series that were way too short when building its pricing models.
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109 of 122 people found the following review helpful
on January 2, 2006
A few months ago, I found almost casually an editorial by Nassim Nicholas Taleb, introducing this essay by Benoit Mandelbrot (you can find it on Wilmot Magazine,2005 pag.50-59 - downloadable from his webpage)
As most readers, I vaguely knew about Mandelbrot and his studies on fractal geometry - but simply it was not my peculiar field of interest, so when I saw the ad of his new book, it went ignored.
*
Taleb's editorial aroused my curiosity.
He was stressing the significance of this essay in challenging the current orthodoxies on finance and in recommending new tools for risk management.
*
In a sense Taleb's recommendation represents a guarantee.
He is a famous edge fund manager and the author of "Fooled by Randomness - The hidden role of chance in the Markets and in life", a book that impressed me with the wide culture, multi-disciplinary approach and the sheer acumen.
*
"The (mis)Behavior of the Market" was up to my expectations.
The book is interesting, and not just for the economic views it advances. Mandelbrot is extremely learned - not just in his field of expertise - and his approach is challenging while retaining great plainness of exposition.
*
The book is organized in three parts.
The first part deals with the old theories of finance and with the state of the art, to show how all of the old tools are mostly inadequate to control investment risk and how they leave investors with a false sense of safety.
In the second part - the most specific and technical - Mandelbrot proposes his view of how the markets behave, suggesting a multi-fractal approach as a substitute for the random walk/efficient market theory.
The third part proposes some conclusion based both on Mandelbrot's views and common sense.
*
The first part is probably the most interesting and also the most cogent.
Modern orthodoxy of finance (Capital Asset Pricing Model or CAPM) is based on the shaky assumption that financial phenomena can be described according to a Gaussian normal distribution, that is they claim to be able to eliminate the possibility of extreme un-forecasted events with a 99 percent probability and to indicate for each level of risk an efficient portfolio maximizing return.
Mandelbrot demonstrates rather conclusively that Gaussian normal distribution of financial prices has been subject to oversimplification to make the data fit in the model, because of "fat tails", concentration and extreme events.
This means that CAPM is useful only when there is less need of it - that is when markets are calm, while is of no utility with extreme events.
Exposing weaknesses in the orthodoxy is not an intellectual pastime, since everyone can still remember the crash of 1987, the many financial crises from 1992 (the disruption of the European exchange rate mechanism, the crises of Mexico, South East Asia, Russia, Argentina,...), the disaster of LTCM in 1998 (it employed 25 PhD and 2 Nobel medalist in economics for their works in finance) and lastly the financial crises after the buoyant markets and high tech bubbles of 2000.
*
By mismanagement and ignorance great fortunes are created but many more are wasted, so the need of a new more efficient tools is imperative.
*
The second part is much more specific and technical.
It explains Mandelbrot's multi-fractal approach, but is still a series of proposals for further inquiries, more than a comprehensive theory of market behavior.
*
In this second part, I sensed some minor inconsistencies, but since I'm not a mathematician and even less a scientist, I can be - and probably am - mistaken.
*
Mandelbrot seems to overlook that finance, unlike natural phenomena, is subject to a very specific epistemological problem. A theory of market behavior - if widely used - can cause alteration in that same behavior previously described and now forecasted (it happened - for example - in 1987 with portfolio insurance strategies and again with the January effect, that disappeared once it was discovered).
*
A relevant problem is also the interpretation of the financial data. His analysis of the prices of cotton over more than a hundred years is using raw data, apparently without taking any consideration for the underlying changes in production, distribution, commerce and use of financial instruments (options and futures). This is even more complex for stocks, since we have different markets, different regulation and different economic situations.
*
While in the first part Mandelbrot cites studies indicating that shares with low p/e and low p/b have shown higher returns over long time-spans, in the second part he declares that returns appear to be independent from time spans. Now this is a classic case of either .. or...
*
Also rejection of the concept of value ("In financial markets, the idea of "value" has limited value") in part three can be misleading.
True, volatility may be high and prices may be swinging wildly, but none the less a theory refusing to guess a fair value based on conservative estimates, can be extremely dangerous.
*
Previous remarks converge on the main weakness of the essay: pretension to describe financial phenomena ex post, that is to find an elegant mathematical model that easily explains the "(mis)behavior", with scarce attention to the underlying causes.
But if underlying causal events should be working no more, we can theoretically believe also effects will be different and we will be left with a new obsolete theory of market efficiency.
*
The third part is rather average.
There are chapters with sensible advice ("markets are turbulent", "more risky than the standard theories imagine ", "market timing matters greatly" ... and so on)
Chapter XIII is a rather gratuitous ad to financial wizards who are said to be using multi-fractal models. No proof is given of this instance - since most of these models are reputedly secret - and sincerely I cannot understand the relevance to mention them in the essay.
*
Because of my work and personal curiosity, I'm fascinated by financial risk and risk management.
If you happen to be fond of these themes, you may be interested in other works I chanced to read about the same topic:
"Against the Gods" by Peter Bernstein - very entertaining history of the human struggle in confronting chaos and randomness. He is also the author of "Capital Ideas. The Improbable Origins of Modern Wall Street".
"Fooled by Randomness - The hidden role of chance in the Markets and in life" by Nassim Nicholas Taleb, one of the most interesting essay on these argument.
"Randomness" by Deborah J. Bennett -intelligent small book whose thesis is that human mind has not evolved to cope instinctively with probability: the same market volatility could be ascribed to this evolutionary incapacity. She is also the author of an other small book - with a rather repugnant title: "Logic Made Easy", that is a serious and fascinating excursus in the history of Logics and an attempt to analyze how the mind works.
"Irrational Exuberance" by Robert Shiller. One of the best books published in the last years: behavioral finance is not of my taste, yet the first 50 pages (a good example of sensible fundamental analysis written before the great bust of the year 2000) are well worth many times the book price.
"The intelligent Investor" by Benjamin Graham. This is an evergreen. I insert it here because of his interesting remarks about variation of share-prices and return of shares during his long life.
"A Random Walk down Wall Street" by Burton J. Malkiel - probably the best non academic introduction to financial theories on the market.
*
You are most welcome if you can suggest other books about the same theme or just share ideas and comments!
Thanks for reading.
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43 of 48 people found the following review helpful
What celebrated mathematician (inventor of fractal geometry and the famous Mandelbrot Sets) Benoit Mandelbrot discovered when analyzing market behavior is that the markets tend to go to extremes. Instead of deviating from an average in a well-mannered linear way (as one might see in a Gaussian bell-shaped distribution) prices tend to rocket up and down according to a power law. In other words the variance in price movements is greater than economists realized, which means that the chance of ruin for any investor is significantly higher than was generally believed.

Furthermore, Mandelbrot discovered that market price distributions have a fractal quality to them in the sense that a chart of price movements on an hourly basis looks pretty much the same as a chart of price movements on a daily or a monthly or even a yearly basis. Additionally, the charts of commodity prices, for example, will look the same as those of currency exchanges or the Dow Jones Industrial Average. Just as a coast line has a ragged edge when viewed from the perspective of someone looking at a map, and a very similar ragged edge when viewed from an airplane, as well as seen on foot, down to the smallest of crags and crannies, so too do stock market prices.

This is an interesting discovery, but, as Mandelbrot warns in an abstract "to the scientific reader," it is one that "will NOT bring personal wealth." (My emphasis.)

Well, what a disappointment. But not a surprise. What this knowledge does do, Mandelbrot hopes, is to better inform investors of the underrated risks of investment and the greater chance of financial ruin should the downside "fat tail" of the fractal curve come to pass.

I have no doubt that Mandelbrot knows what he is talking about; however I wonder if the significance of his discovery is as important as he thinks it is. Perhaps the academics underrated risk, and maybe the same is true of many investors, but I suspect the practical players knew and know the truth. One doesn't have to look further back than October 19, 1987 to see a one-day drop in the stock market of truly gargantuan proportions, a drop so great that the probability of it actually happening was, as Mandelbrot observes, near the edge of the impossible.

Yes, markets do go to extremes. Bubbles develop and burst and individual stocks have market values totally out of line with their assets, revenue and profits. One had only to live through the go-go high tech market of the 1990s to know that. Mandelbrot claims that part of this inexplicably erratic behavior is due to the markets having a memory of sorts. He calls it "dependence," an hitherto underappreciated quality. The standard model of market behavior insisted that today's price movement is an independent event. At least that is Mandelbrot's assertion. Personally, I think most experienced traders know that today's price is affected by price movements in the past if only because traders themselves have memories. But more than that market prices seem influenced by the past because some of the same mechanisms, phenomena and conditions still prevail.

For example, on pages 184-185 Mandelbrot recalls that in 1982 IBM hired then small Intel to make its microprocessors and a company headed by the unknown Bill Gates to provide its software. He then observes "the fates of these three companies are still intertwined." He calls this "long dependence" and "a pillar of fractal geometry." However most investors would merely note that IBM, Intel, and Microsoft are in similar businesses whose stock prices rise and fall more or less together because of that. If IBM had started up a pretzel factory in 1982 would their stock prices be correlated? Mandelbrot seems to imply that they would; but I think it may be that he is so enamored of the magic of his fractals that he sees what he wants to see.

But maybe he is right. Maybe there is some ghost of influence in the past that would to some extent intertwine the market movements of the pretzel company started by IBM with that of IBM itself. If so, I would like to know the mechanism at work. Mandelbrot allows that he doesn't know what that might be. Again I think it is in the memory of the traders. Mandelbrot acknowledges as much on pages 185-186 when he mentions the old traders who had experienced the crash of 1929 and were therefore more cautious than they might have been without such a memory.

And this is really the bottom line about market behavior: the extremes to which markets go is largely the direct result of the extremes to which the minds (and hearts and souls) of the traders go. Human emotion is why some high tech stocks had greater market caps than Dow Jones blue chip companies even though the upstarts had no earnings. Human emotion is why tulip bulbs were once worth more than gold. And human emotion is why markets crash so suddenly, seemingly without rhyme or reason. And human emotion is why the distribution curves of market prices have fat tails, Mandelbrot's fractal discoveries notwithstanding.

Bottom line: interesting and well written (co-author and professional journalist Richard L. Hudson had a lot to do with that, I suspect) but of dubious utility for the practical investor.
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41 of 46 people found the following review helpful
on September 1, 2004
Mind expanding book.

If you manage money for a living you probably already know that academic finance is plagued with shortcomings.

The most skilled money managers in the world earn hundreds of millions dollars a year. If anybody could learn to manage money going to school we wouldn't observe those compensation levels.

Mandelbrot points out correctly that the standard mathematical tools are insufficient to manage financial risks.

Consistently outperforming money managers know this and either design patches or use proprietary tools.

Mandelbrot proposes his own mathematical framework as an alternative to mainstream finance. I won't tell how good it is. It is responsability of the reader to check if these tools work. There are supposedly successful hedge funds using this math (e.g. Olsen) but the smart reader will not believe anything until he verifies it through his own research.

While I recommend this book to anyone managing money, I don't believe Mandelbrot's ideas will be widely accepted in academia. In any case being a finance researcher has a curse. In the markets, if someone outperforms the indices someone else has to underperform. Not everybody can outperform. Therefore if your ideas are accepted and everybody uses your stuff then obviously it won't generate outsized returns and you won't be in the billionaire money manager club. Your only hope to see a million dollars in your lifetime will be the Nobel Prize. If you choose instead to make money with your stuff and keep it to yourself, you will be anonymous and lonely but stinking rich.

Therefore the only drawback I see is that Mandelbrot does not ask himself what would happen with his model if everybody adopts it. Can it change the way prices move? Financial advances change the markets and due to the curse I mentioned, they tend to nullify themselves: remember portfolio insurance, convertible pricing models etc. etc.

Mainstream finance (indexing in particular) "works" because it basically says "it's OK to be average, just mimic the indices". So its wide acceptance does not nullify itself.

I reach the conclusion that only a select few can take advantage of Mandelbrot's thinking. One thing is for sure: if anybody finds a way to make money with Mandelbrot's stuff he won't tell anyone about it.
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26 of 28 people found the following review helpful
on February 27, 2005
If you only buy books in the series "For when you only have time for answers", then this book is not for you.

Mandelbrot's book gives you strong hints about a possible financial model of the markets, but doesn't claim anything beyond providing you with an intellectual framework to start attacking the problem from an entirely different perspective to the "Efficient Markets" or the "Black-Sholes" current theology.

If you feel stuck in your current market practice, or find that what you do every day doesn't reconcile with the current theory, then this book might be a first attempt at reconciling your daily exerience with a strong theoretical model.

If your current practice is that you do it "because it works" but you feel uncomfortable with doing things for which you have no rational explanation, then Mandelbrot's book will set you on a path of deeper understanding. By the end of the book, you still won't know the answer, but you will have learnt a tremendous amount on the way actual price series might work.

Definitely not for casual reading.
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16 of 16 people found the following review helpful
on June 19, 2007
This book should be required reading for all of Wall Street's money managers. It would have saved billions of their clients' money.

Let's look at the Long Term Capital disaster. Intelligent people, Nobel laureates, so what went wrong? The answer is simple: fat tails! (The concept that supposedly one-in-a-billion events are, in fact, not unusual). Clearly, it was not lack of intelligence that caused their downfall, but a childish reliance on flawed financial theories.

Mandelbrot identifies and explains the flaws. True, he does not provide an alternative theory, but perhaps the whole point is that financial markets, being chaotic systems, are not predictable.

Let's look at another chaotic system, the weather. Here the parameters such as temperature, wind speed, humidity, are easily measurable. Yet, we still cannot predict the weather accurately for more than a day or two. How then can the stock market, which is far more complex than the weather, be predictable, when most of the parameters that affect it are not measurable, and some are not even known?

Perhaps the weakest part of the book is the beginning of a theory that Mandelbrot tries to found. He suggests that fractal equations produce charts that look and feel like real stock price charts, and that there might be some connection that can be exploited to predict or describe financial markets. He does not, however, go beyond this suggestion and hopes that someone else would develop his theory.

Bottom line: Mandelbrot's "fat tail" theory explains the financial disasters suffered by many "brilliant" money managers. It does not predict the market, but explains the risks of conventional capital market theories. It saves you money, and after all a penny saved is a dollar earned.
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46 of 53 people found the following review helpful
on September 14, 2004
This book details one of this generation's finest mathematical minds offers "obvious" observations he calls his "Ten Heresies of Finance."

Benoit Mandelbrot is known for making mathematical sense of facts everybody accepts but that geometers never assimilated. Clouds are not round. Mountains are not cones. Coastlines are not smooth. Add another: financial markets are not the safe bet your broker claims. In his first general audience book, Mandelbrot, with co-author Richard L. Hudson, reveal today's assumptions about the behavior of markets simply do not work.

"What passes for orthodoxy in economics and finance," the authors conclude, "proves on closer examination to be shaky business."

Among the book's observations:

1. Markets are turbulent. After spending a lifetime studying wind and ocean currents, he applies his multi-fractal math to analyze financial markets. "The tell-tale traces of turbulence are plainly there, in the price charts," he writes. The bell curve does not capture its changes.

2. Markets are inherently risky. Turbulence is dangerous. Market swings are wild and sudden. They are difficult to predict, more difficult to hedge and even more difficult from which to profit.

3. Marketing timing matters. Big gains and losses are concentrated into small time periods. News events such as earnings or economic announcements drive stock market prices.

4. Prices leap suddenly. This adds to risk. News announcements compel investors to act simultaneously and instantaneously.

Using his fractal tools, Mandelbrot describes how financial markets work. He describes the volatile, dangerous and in a unique way, strangely beautiful properties that for which few financial experts account.

This book is a must read for any serious investor. By pin-pointing flaws in accepted market wisdom, it provides a platform for a serious re-consideration of finance.
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23 of 26 people found the following review helpful
on November 20, 2004
This is another in a series of books written by Benoit Mandelbrot that deserves to have a 6 star rating.The current foundation of practically all financial analyses ,excepting the "safety-first"approach of Roy and Charnes and Cooper,is the mean-variance(standard deviation)approach which is incorporated into more advanced versions such as the capital asset pricing model(capm)and the Black-Scholes options pricing model for puts and calls.All of these models assume that all price movements in all financial markets are approximately the same small size,are independent of each other,are homogeneous and satisfy the law of large numbers and the central limit theorem.This leads directly to the assumption that all price movements can be modeled as(as if)being normally distributed.The reader should note that practically all classical and neoclassical economic theory is currently based on subjective expected utility(SEU)theory which bases much of its practical results on the applicability of the normal probability distribution.The entire argument made by Milton Friedman and Robert Lucas,jr.,that all false trading(contracting)at disequilibrium(nonequilibrium)prices cancels out in the long run over time ,is based on the claim that such price movements are normally distributed around the market clearing equilibrium price.This equilibrium market clearing price is automatically interpreted as being the mean of a normal probability distributiion.Such a price is thus an optimal price since the average of a normal probability distribution is also the maximum outcome possible.The entire claim that price adjustments lead to an optimal outcome in all(private sector) markets means that the socalled"INVISIBLE HAND"of the market is nothing more than a normal probability distribution.Any type of skewed and/or nonnormal distribution of price movements means automatically that such adjustments do not lead to an optimal outcome.Mandelbrot has thrown down the gauntlet,not only to the current purveyors of basic risk management,portfolio analysis,but to much of the economics profession as well.Mandelbrot does this by simply presenting massive amounts of empirical evidence showing that the commodity,futures,money,stock,and other markets price movements are not generally normally distributed.All of Mandelbrot's research has been replicated and duplicated by many other researchers in many other countries besides the USA.Since the mid-fifties,Mandelbrot has carefully and patiently presented his results in a series of widely cited scholarly articles and books.In this book,Mandelbrot has decided to take his case directly to the general public.The book is straightforward and easy to read and absorb.The only technical knowledge needed by a reader is some basic familiarity with the normal distribution and an understanding of the basics of calculating a Z-score.I can't recommend this book too highly.
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36 of 44 people found the following review helpful
on December 23, 2004
I was disappointed in"The (Mis)behavior of Markets" by the renowned mathematician Benoit Mandelbrot. His last two chapters are a must read by anyone interested or invested in financial markets. These final chapters explain ten of Dr. Mandelbrot's concise market observations. The first eleven chapters are a meandering memoir on Dr. Mandelbrot's market research, but would be a better read if condensed and less polemic.

The mathematical problem that Dr. Mandelbrot tackles can be likened to comparing the probability of getting a number on a six sided fair die, and a race with six horses. Mathematical theory explains perfectly outcomes of throwing the fair die; theory does not illuminate the horse race, which depends on facts that are unknown and unknowable. Empirically in races, the favorite has an extreme chance of winning while the long shot an unduly low possibility. Some field horses may have "normal probabilities" similar to a number on the die.

As many great academic minds and Nobel laureates have found, Bachelier distributions can model financial markets' probabilities. Dr. Mandelbrot rightly states that the Bachelier theory does not always fit market reality; markets show erratic volatility.

Dr. Mandelbrot misses opportunities to show his readers the points at which risk and reward are outside the Bachelier derived probabilities, dismissing prevalent theory in toto. His fractal model of volatility replaces the Bachelier derived probabilities by a power function. Both models attempt estimates on the "normal market" (analogous to handicapping a "field" horse) and fail to describe market extremes (favorite/longshot odds). It is the abnormal market misbehavior that causes great gains or losses for investors.

Illustrating this oversight, the cotton market at times moves to fresh high or low prices. At high cotton prices, the probability of the price reversing approaches one and the possible gains by being long cotton approaches zero. At some unknown point the price will reverse. At market lows, we observe the inverse. At market limits, Dr. Mandelbrot's model does not help readers identify market risk/reward probability. Hopefully, new researchers can use Dr. Mandelbrot's work to improve our understanding of market risk and reward.
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