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The (Mis)Behavior of Markets
Benoit B. Mandelbrot, one of the century's most influential mathematicians, is world-famous for making mathematical sense of a fact everybody knows but that geometers from Euclid on down had never Clouds are not round, mountains are not cones, coastlines are not smooth. To these classic lines we can now add another Markets are not the safe bet your broker may claim. In his first book for a general audience, Mandelbrot, with co-author Richard L. Hudson, shows how the dominant way of thinking about the behavior of markets-a set of mathematical assumptions a century old and still learned by every MBA and financier in the world-simply does not work. As he did for the physical world in his classic The Fractal Geometry of Nature , Mandelbrot here uses fractal geometry to propose a new, more accurate way of describing market behavior. The complex gyrations of IBM's stock price and the dollar-euro exchange rate can now be reduced to straightforward formulae that yield a far better model of how risky they are. With his fractal tools, Mandelbrot has gotten to the bottom of how financial markets really work, and in doing so, he describes the volatile, dangerous (and strangely beautiful) properties that financial experts have never before accounted for. The result is no less than the foundation for a new science of finance.
352 pages, Hardcover
First published September 18, 1997
About the author
Benoît B. Mandelbrot
21 books287 followersBenoît B. Mandelbrot, O.L.H., Ph.D. (Mathematical Sciences, University of Paris, 1952; M.S., Aeronautics, California Institute of Technology, 1949) was a mathematician best known as the father of fractal geometry. He was Sterling Professor Emeritus of Mathematical Sciences at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory.
Mandelbrot was born in Poland, but his family moved to France when he was a child; he was a dual French and American citizen and was educated in France. He has been awarded with numerous honors, including induction into the Legion d'honneur, as well as the 1986 Franklin Medal for Physics, the 1993 Wolf Prize for Physics, the 2000 Lewis Fry Richardson Medal of the European Geophysical Society, and the 2003 Japan Prize "for the creation of universal concepts in complex systems."
Mandelbrot was born in Poland, but his family moved to France when he was a child; he was a dual French and American citizen and was educated in France. He has been awarded with numerous honors, including induction into the Legion d'honneur, as well as the 1986 Franklin Medal for Physics, the 1993 Wolf Prize for Physics, the 2000 Lewis Fry Richardson Medal of the European Geophysical Society, and the 2003 Japan Prize "for the creation of universal concepts in complex systems."
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August 24, 2010
I first heard about the efficient market theory in Law School. I remember thinking at the time what obvious bullshit it was. But it was academia, and it was pretty harmless bullshit, so let the economists play whatever games they want. What difference did it make?
The theory goes that the markets already consolidate all the information available to them, so that price already incorporates all the information available to the market. From there, we get the random walk theory -- that prices will move in a random fashion, so that each price move is basically the flip of a coin. And from these premises, we then get modern financial theory, including the idea that risk is the same as statistical variance, which leads to the Sharpe ratio for evaluating an assets value, and also leads to the Black Scholes method for pricing options, and to modern portfolio theory (by which a portfolio is designed around the appropriate level of risk for an individual investor). The math for doing this seems very sophisticated, and variations on these approaches have served as the backbone for the financial industry.
But there's a slight problem. The basic assumptions underlying all this theory is wrong. Prices don't vary like the flips of a coin. They are much wilder that tossing a coin. According to conventional theory, Black Monday in 1987 should have occurred perhaps once in an eon. In 1997, there were three days in a short period of time that had moves which the theory would predict should occur only once every 10,000 years or so. And the same thing again for the recent financial collapse.
Wall Street adopted these economic theories. They were relatively easy to use. They have an air of scientific knowledge to them. There are nobel prizes awarded to the developers of them. But, they don't fit the data at all. They grossly underestimate risk, largely because they insist that data should fit a bell curve, when it simply doesn't fit. And they then took risks, this time with their own money, based on these false economic engineering ideas. The result was the near collapse (and the jury may still be out on this) of the entire world financial system.
Mandelbrot, the author of this book, is a mathematician. He invented the field of fractal geometry, and is probably most notable for the Mandelbrot set, which yields incredibly intricate and beautiful fractal designs. As far back as the early 1960s, Mandelbrot did extensive study on Cotton markets. For some reason, there is good data on daily cotton prices going back to the mid 1800s. As a result of his studies, Mandelbrot concluded that markets are more fractal than continuous, and that the assumptions that economists used were simply wrong. He started complaining about this 40 years ago, and as recent history shows, people still are not listening. He claims to have written this book to bring his case directly to the public. (The same process worked pretty well for him in popularizing the idea of fractals in the first place, and ultimately getting them accepted in academia, where math departments were skeptical of geometry with practical application.)
The book is very well written, and easy to understand, especially since it deals with a field where people tend to be abstruse and to obfuscate whenever possible. (Have you ever read any statement by Alan Greenspan, for example?) The criticisms of standard financial theory seem perfectly sound.
However, when it comes to giving practical advice, the book seems on much shakier grounds. He calls for more open minded study of markets, which would not be a bad thing, especially since we now know that the market making companies are all too big to fail. On the last page, he calls for a "coordinated search for patterns in the financial markets." And this is fine. But at several places he makes fun of "chartists, " precisely because chartists think that they have found discernible patterns in the markets. He doesn't offer any evidence to show that they cannot have done so, and his call at the end shows that he thinks there may be such patterns.
Overall, I think this book is worth reading for anyone who is interested in the behavior of markets. And its fun for people, like me, who are deeply skeptical of the usefulness of anything that comes out of an economist's mouth.
The theory goes that the markets already consolidate all the information available to them, so that price already incorporates all the information available to the market. From there, we get the random walk theory -- that prices will move in a random fashion, so that each price move is basically the flip of a coin. And from these premises, we then get modern financial theory, including the idea that risk is the same as statistical variance, which leads to the Sharpe ratio for evaluating an assets value, and also leads to the Black Scholes method for pricing options, and to modern portfolio theory (by which a portfolio is designed around the appropriate level of risk for an individual investor). The math for doing this seems very sophisticated, and variations on these approaches have served as the backbone for the financial industry.
But there's a slight problem. The basic assumptions underlying all this theory is wrong. Prices don't vary like the flips of a coin. They are much wilder that tossing a coin. According to conventional theory, Black Monday in 1987 should have occurred perhaps once in an eon. In 1997, there were three days in a short period of time that had moves which the theory would predict should occur only once every 10,000 years or so. And the same thing again for the recent financial collapse.
Wall Street adopted these economic theories. They were relatively easy to use. They have an air of scientific knowledge to them. There are nobel prizes awarded to the developers of them. But, they don't fit the data at all. They grossly underestimate risk, largely because they insist that data should fit a bell curve, when it simply doesn't fit. And they then took risks, this time with their own money, based on these false economic engineering ideas. The result was the near collapse (and the jury may still be out on this) of the entire world financial system.
Mandelbrot, the author of this book, is a mathematician. He invented the field of fractal geometry, and is probably most notable for the Mandelbrot set, which yields incredibly intricate and beautiful fractal designs. As far back as the early 1960s, Mandelbrot did extensive study on Cotton markets. For some reason, there is good data on daily cotton prices going back to the mid 1800s. As a result of his studies, Mandelbrot concluded that markets are more fractal than continuous, and that the assumptions that economists used were simply wrong. He started complaining about this 40 years ago, and as recent history shows, people still are not listening. He claims to have written this book to bring his case directly to the public. (The same process worked pretty well for him in popularizing the idea of fractals in the first place, and ultimately getting them accepted in academia, where math departments were skeptical of geometry with practical application.)
The book is very well written, and easy to understand, especially since it deals with a field where people tend to be abstruse and to obfuscate whenever possible. (Have you ever read any statement by Alan Greenspan, for example?) The criticisms of standard financial theory seem perfectly sound.
However, when it comes to giving practical advice, the book seems on much shakier grounds. He calls for more open minded study of markets, which would not be a bad thing, especially since we now know that the market making companies are all too big to fail. On the last page, he calls for a "coordinated search for patterns in the financial markets." And this is fine. But at several places he makes fun of "chartists, " precisely because chartists think that they have found discernible patterns in the markets. He doesn't offer any evidence to show that they cannot have done so, and his call at the end shows that he thinks there may be such patterns.
Overall, I think this book is worth reading for anyone who is interested in the behavior of markets. And its fun for people, like me, who are deeply skeptical of the usefulness of anything that comes out of an economist's mouth.
January 27, 2012
Benoit Mandelbrot is the inventor of the mathematical concept of fractals. His earlier book The Fractal Geometry of Nature was a truly groundbreaking book about fractals and how they are seen in nature. In The Misbehavior of Markets he turns his attention to the application of fractal concepts to markets. Mandelbrot shows that price fluctuations:
1) are not independent from one time period to the next
2) appear to be the same, regardless of the time scale involved (hours/days/months/years)
3) do not obey a Gaussian (normal bell-like) distribution, but instead follow a power-law distribution.
These characteristics are exactly the opposite of the assumptions that are normally used in financial circles. As a result, most financial models severely under-estimate financial risk. Most financial models use certain parameters (like the beta factor) that purport to measure price volatility. Mandelbrot shows that many of these parameters are worse than useless; they are so wrong, they are dangerous and can lead to world-wide financial ruin.
This book is also somewhat of a biography; Mandelbrot details some of the fascinating aspects of his life, and that of his parents. One of the reasons contributing to his move from France to the U.S. is the disdain of French mathematicians to applied research. The only problem with this book is that Mandelbrot writes in a tone that is too strident for my taste. Nevertheless, I strongly recommend this book to anyone with an interest in applied mathematics or finance.
1) are not independent from one time period to the next
2) appear to be the same, regardless of the time scale involved (hours/days/months/years)
3) do not obey a Gaussian (normal bell-like) distribution, but instead follow a power-law distribution.
These characteristics are exactly the opposite of the assumptions that are normally used in financial circles. As a result, most financial models severely under-estimate financial risk. Most financial models use certain parameters (like the beta factor) that purport to measure price volatility. Mandelbrot shows that many of these parameters are worse than useless; they are so wrong, they are dangerous and can lead to world-wide financial ruin.
This book is also somewhat of a biography; Mandelbrot details some of the fascinating aspects of his life, and that of his parents. One of the reasons contributing to his move from France to the U.S. is the disdain of French mathematicians to applied research. The only problem with this book is that Mandelbrot writes in a tone that is too strident for my taste. Nevertheless, I strongly recommend this book to anyone with an interest in applied mathematics or finance.
December 3, 2020
A premodern take on time series data, the markets behavious and whether the natural concepts still apply to them. Affinity and walking through a whole maze of philosophical concepts. Uni- and multifractality. Recursive and non-recursive models of the market.
While I would've loved it to be more hands on (ie, supply the calcs where applicable, for instance), I can't really be too grumpy since this is a very old work that turned out to be fascinating in light of how far we have taken our market BS.
While I would've loved it to be more hands on (ie, supply the calcs where applicable, for instance), I can't really be too grumpy since this is a very old work that turned out to be fascinating in light of how far we have taken our market BS.
January 22, 2018
A very accessible book by the inventor of Fractals, Benoit B. Mandelbrot.
Read this book, you will not look at the world the same way again.
Read this book, you will not look at the world the same way again.
January 28, 2009
This book lays lots of groundwork before it finally gets to the point. I would recommend a reader read the first chapter of part III (10 Heresies of Finance) at the start to give yourself a grounding then read the rest of the book. It might help to know where he's going during part I and part II.
All in all, some interesting beginnings of theories and comparisons. There is almost no math involved. But if you're scared of math, this is a great glimpse into fractals and it starts to show glimpses of how they may be used in the future to assist in market theory (but he does not actually go into market theory in this book).
Decent book, but too much bragging and not as much directness as I wanted. I'm not a huge fan of this genera though, so it was good for what it was.
All in all, some interesting beginnings of theories and comparisons. There is almost no math involved. But if you're scared of math, this is a great glimpse into fractals and it starts to show glimpses of how they may be used in the future to assist in market theory (but he does not actually go into market theory in this book).
Decent book, but too much bragging and not as much directness as I wanted. I'm not a huge fan of this genera though, so it was good for what it was.
August 20, 2012
When I first encountered this book I did a slight doubletake, "wait, THE Benoit Mandelbrot?"
"Why is he writing about financial markets?" I wondered.
I knew of Mandelbrot in mathematics, computer science, and natural sciences -- I had no idea how deep his obsession with economics was till I read this book.
In a way, it's almost depressing, his biggest contributions were to fields he didn't seem to care about as much as economics (a field that in turn didn't seem to care about his work).
Mandelbrot's work in economics, and Taleb's after him, has now become widely accepted, especially after the fact of recent financial disasters. Investors and non-investors learned the hard way that the current risk models (relying on bell curves) were inaccurate. The math that was telling us this has been around since the 1970s (arguably prior, but well-formulated in the 1970s).
Despite him sounding slightly egocentric in this book, as many reviewers charge, he's actually being incredibly modest: there's a VERY large body of natural science that would be impossible today had Mandelbrot not created a unified "fractal geometry". Personally, I found his tone more whimsical and intellectually curious than egocentric.
Historically, there was a very large and disparate body of mathematics that were unified by Mandelbrot. He defined "fractal dimension" which (in simple terms) is an exact measure of the change in detail to the change in scale of a given object. Most financial data, such as stock price over time, for example, has a "fractal dimension" greater than 1 (sometimes only slightly). This measurement of "fractal dimension" is stable and well-defined mathematically.
Furthermore, for a given object, if its "fractal dimension" is greater than its "topographical dimension", then it is, by definition, a fractal. This book does a decent job providing an overview of this idea (especially the part about measuring coastlines). Mandelbrot's account of this work is extremely fascinating, any other writer would have simply lavished praise on Mandelbrot for his ideas; Mandelbrot in turn told a wonderful story of how these ideas came to fruition.
Fractal modeling within the natural sciences is extremely common if not the norm (and technically speaking, even the internals to a Monte Carlo simulation rely on fractals, albeit simple ones). The name though is a bit of a misnomer, mathematically it means something different than what most people consider a "fractal".
That stock prices over time are "fractal" is true, again by definition, of how we measure stock prices; but not all "fractals" are simple, some of them have more than one "fractal dimension", stock prices unfortunately fall into that category.
A multifractal object is an object where more than one "fractal dimension" variable is needed to describe the object. This includes magnetic fields, fluid dynamics, and stock prices over time.
A simple fractal model would have been easier to figure out, and economists (or at least Hedge Fund managers) likely would have figured out a good formula long ago, before any mathematician labeled it as a "fractal".
All that said, a multifractal equation for investment theory, is unfortunately not fully articulated-- in other words, there's not yet an agreed upon formula that would apply to all markets (or all stocks)-- much work is still needed to get from "yes, stock prices over time are multifractal", that part we know is true and should not be controversial, to "here is the exact formula", that part, the E=mc^2 moment in finance, has not been defined. And I'm not sure economists (let alone investors) fully appreciate the implications of what that would mean.
A sound model of stock prices over time, would NOT (importantly) be a forecasting model. In other words, knowing this formula would not guarantee you money. Full stop, this is the point where investors usually lose interest.
There's an important difference between modeling the behavior of a system versus predicting exactly what the system will do several years from now.
Forecasting models are not the same as risk models and stock option pricing. The latter two are absolutely essential to a sustainable market, the first one (forecasting) is wishful thinking for relative wealth (that is, making easy money when other investors didn't). Unfortately, most of the effort in finance is put into forecasting stock prices (wishful thinking) rather than risk modeling or option pricing.
If someone discovers a sound model of stock prices over time, then it would be (by mathematic definition) a multifractal model (i.e., expressible as a multifractal model). And this would, importantly, provide an accurate risk measure that would in turn allow for usable stock option pricing, as well as provide a clear definition within portfolio management of just how risky is risky.
That said, because most markets behave somewhere between almost-normalized (almost a pure random walk, a clean and symmetric bell curve) to chaotic (highly volatile, non-symmetric and non-normalized), the typical risk measure of any market is provably higher than those currently used in modern portfolios or option prices. The exact amount higher is what remains to be understood.
To use the river dam analogy, our current investment dams are not sufficient to weather the actual storms that will (and have) hit. River networks are useful metaphors, but markets are a thing onto themselves, if rivers behaved like markets then we'd experience more turbulent waters, things like flash draughts and jumping water levels (imagine an entire river dropping a few meters instantly).
So, how to invest to survive a future collapse? This book demonstrates that this is a solvable problem yet remains unsolved (although Mandelbrot's work narrows in on an accurate range). In practice, this type of work tends to get lost in the body of inarticulate economic theories as well as in the desperately greedy investment practices.
I suspect the reason this gets so easily confused is that most people are looking for forecasting models ("what's the price tomorrow? when can I retire?") and not accurate models of risk and option pricing (ironically, preventing any forecasting model from even having a chance). Personally, forecasting models seem stupidly impossible (they would negate themselves when everyone tried to capitalize on them), but an accurate risk model and option pricing would be extraordinarly beneficial to everyone -- at the very least, to know how high our dams should be...
"Why is he writing about financial markets?" I wondered.
I knew of Mandelbrot in mathematics, computer science, and natural sciences -- I had no idea how deep his obsession with economics was till I read this book.
In a way, it's almost depressing, his biggest contributions were to fields he didn't seem to care about as much as economics (a field that in turn didn't seem to care about his work).
Mandelbrot's work in economics, and Taleb's after him, has now become widely accepted, especially after the fact of recent financial disasters. Investors and non-investors learned the hard way that the current risk models (relying on bell curves) were inaccurate. The math that was telling us this has been around since the 1970s (arguably prior, but well-formulated in the 1970s).
Despite him sounding slightly egocentric in this book, as many reviewers charge, he's actually being incredibly modest: there's a VERY large body of natural science that would be impossible today had Mandelbrot not created a unified "fractal geometry". Personally, I found his tone more whimsical and intellectually curious than egocentric.
Historically, there was a very large and disparate body of mathematics that were unified by Mandelbrot. He defined "fractal dimension" which (in simple terms) is an exact measure of the change in detail to the change in scale of a given object. Most financial data, such as stock price over time, for example, has a "fractal dimension" greater than 1 (sometimes only slightly). This measurement of "fractal dimension" is stable and well-defined mathematically.
Furthermore, for a given object, if its "fractal dimension" is greater than its "topographical dimension", then it is, by definition, a fractal. This book does a decent job providing an overview of this idea (especially the part about measuring coastlines). Mandelbrot's account of this work is extremely fascinating, any other writer would have simply lavished praise on Mandelbrot for his ideas; Mandelbrot in turn told a wonderful story of how these ideas came to fruition.
Fractal modeling within the natural sciences is extremely common if not the norm (and technically speaking, even the internals to a Monte Carlo simulation rely on fractals, albeit simple ones). The name though is a bit of a misnomer, mathematically it means something different than what most people consider a "fractal".
That stock prices over time are "fractal" is true, again by definition, of how we measure stock prices; but not all "fractals" are simple, some of them have more than one "fractal dimension", stock prices unfortunately fall into that category.
A multifractal object is an object where more than one "fractal dimension" variable is needed to describe the object. This includes magnetic fields, fluid dynamics, and stock prices over time.
A simple fractal model would have been easier to figure out, and economists (or at least Hedge Fund managers) likely would have figured out a good formula long ago, before any mathematician labeled it as a "fractal".
All that said, a multifractal equation for investment theory, is unfortunately not fully articulated-- in other words, there's not yet an agreed upon formula that would apply to all markets (or all stocks)-- much work is still needed to get from "yes, stock prices over time are multifractal", that part we know is true and should not be controversial, to "here is the exact formula", that part, the E=mc^2 moment in finance, has not been defined. And I'm not sure economists (let alone investors) fully appreciate the implications of what that would mean.
A sound model of stock prices over time, would NOT (importantly) be a forecasting model. In other words, knowing this formula would not guarantee you money. Full stop, this is the point where investors usually lose interest.
There's an important difference between modeling the behavior of a system versus predicting exactly what the system will do several years from now.
Forecasting models are not the same as risk models and stock option pricing. The latter two are absolutely essential to a sustainable market, the first one (forecasting) is wishful thinking for relative wealth (that is, making easy money when other investors didn't). Unfortately, most of the effort in finance is put into forecasting stock prices (wishful thinking) rather than risk modeling or option pricing.
If someone discovers a sound model of stock prices over time, then it would be (by mathematic definition) a multifractal model (i.e., expressible as a multifractal model). And this would, importantly, provide an accurate risk measure that would in turn allow for usable stock option pricing, as well as provide a clear definition within portfolio management of just how risky is risky.
That said, because most markets behave somewhere between almost-normalized (almost a pure random walk, a clean and symmetric bell curve) to chaotic (highly volatile, non-symmetric and non-normalized), the typical risk measure of any market is provably higher than those currently used in modern portfolios or option prices. The exact amount higher is what remains to be understood.
To use the river dam analogy, our current investment dams are not sufficient to weather the actual storms that will (and have) hit. River networks are useful metaphors, but markets are a thing onto themselves, if rivers behaved like markets then we'd experience more turbulent waters, things like flash draughts and jumping water levels (imagine an entire river dropping a few meters instantly).
So, how to invest to survive a future collapse? This book demonstrates that this is a solvable problem yet remains unsolved (although Mandelbrot's work narrows in on an accurate range). In practice, this type of work tends to get lost in the body of inarticulate economic theories as well as in the desperately greedy investment practices.
I suspect the reason this gets so easily confused is that most people are looking for forecasting models ("what's the price tomorrow? when can I retire?") and not accurate models of risk and option pricing (ironically, preventing any forecasting model from even having a chance). Personally, forecasting models seem stupidly impossible (they would negate themselves when everyone tried to capitalize on them), but an accurate risk model and option pricing would be extraordinarly beneficial to everyone -- at the very least, to know how high our dams should be...
May 1, 2018
Financial markets have a very strange property. One would think they were entirely man-made, about as far removed as you could get from the laws that govern nature. Yet if you look closely enough at the kind of share price charts that you might see online or in newspapers - as Mandelbrot certainly has - you might be in for a surprise.
A trader will tell you that it can be impossible to tell the difference between a daily, weekly or monthly price chart, if the axis labels are removed. This is the essence of the sort of 'scaling' that is found in the fronds of a fern, for example, or bronchial tubes in the lung, or florets of a cauliflower. Zoom in on any part, and it is a smaller replica of its larger parent. Zoom in again, and the pattern repeats itself: not ad infinitum, but several times. Mandelbrot devoted much of his career to studying these patterns, modelling them using mathematical equations and illustrating them with abstract graphics he called 'fractals'. Using such concepts, he was able to produce simulated price charts that would have fooled any trader.
All this might have remained a fascinating academic cul-de-sac, but Mandelbrot's models proved to have a use - because strangely, they were better models of how markets actually worked than the 'modern portfolio theory' that was being worshipped at the time (and, unfortunately, is still regarded as sacred by many market participants). Modern portfolio theory, which is largely based on work by Markowitz and Sharpe, amended by Fama (one of Mandelbrot's pupils) and French, is based on the idea that the Gaussian bell curve is a good guide to expected investment returns. There are a host of other dodgy assumptions too, such as the independence of price changes - i.e. one price change has no impact on the next. (Oh yes, and then there's the assumption that people always behave rationally, which the slightest observation of human nature will dispel.)
It wouldn't have taken someone of Mandelbrot's intellect to blow these assumptions out of the water - a much lesser mind could have done it. But the key reason that modern portfolio theory took such hold in the 1960s and 1970s seems to be because there was nothing better on offer - or at least, nothing that offered easy answers. Mandelbrot didn't offer easy answers; but he did offer an explanation of how markets behaved (or misbehaved) that - unlike MPT - fit the facts. But his explanation was inconvenient, so it was largely ignored.
Mandelbrot's model of how markets work depends on a few observations. To begin with, what he calls 'the well-mannered bell curve' does not represent the distribution of market returns. The real curve is much more violent and the investor's ride much bumpier. We know this because events that ought to be astronomically rare, assuming the bell curve is a good model, happen fairly often (think Black Monday, the Russian debt crisis, subprime...) A 'power-law' distribution, not a bell curve, is at work - leading to 'fat tails' - more frequent occurrences of unlikely events than the bell curve would suggest. Such unlikely events can mean ruin for the investor.
Modern portfolio theory tries to deal with such uncertainty by measuring an investment's volatility (standard deviation). But not only is this approach flawed by its reliance on the bell curve, it is also flawed in other ways. Most obviously, the volatility of an investment is not a constant, but varies. Volatility is itself volatile!
Instead of 'volatility', Mandelbrot prefers to talk in terms of randomness. He believes markets are 'wildly' random, not 'mildly' random as MPT would have it. Developing this idea, he identifies two effects that different investments and markets possess in different degrees: a Noah effect, and a Joseph effect. The Noah effect is a tendency towards sudden, devastating change or discontinuity; the Joseph effect is a tendency towards sequential movement in the same direction (seven years of plenty, seven years of famine). Mandelbrot argues that the 'personality' of a particular market or investment can be described by the relationship between these two effects, which are mathematically measurable.
The final crucial ingredient in Mandelbrot's recipe for simulating a market is that of 'trading time'. He observes that market activity comes in bursts; sometimes prices move fast, sometimes slowly. Again, this is an intuitive concept for any trader.
Once he has explained all these concepts - fairly lucidly, with lots of intriguing and rather beautiful illustrations and not a single mathematical equation - Mandelbrot combines them together. It's at this stage it is easy to get lost - I had to read the crucial chapter a few times. What he does is marry together his notion of the way prices change with his notion of trading time to produce a realistic model of a price chart, where time travels sometimes slow, sometimes fast.
Mandelbrot (and his co-author, Richard Hudson) are fairly clear with the reader right from the start that nothing in this book will help you make money. But it might prevent you losing it. What Mandelbrot has done is produce a convincing description of how markets really work, which is a warning to anyone who believes they can be tamed, and a perfect antidote to the smug certainties of modern portfolio theory. He's also achieved the real feat of making the book readable (though there are equations in the notes for those who want them), putting some very high-powered thinking within the reach of the determined general reader.
A trader will tell you that it can be impossible to tell the difference between a daily, weekly or monthly price chart, if the axis labels are removed. This is the essence of the sort of 'scaling' that is found in the fronds of a fern, for example, or bronchial tubes in the lung, or florets of a cauliflower. Zoom in on any part, and it is a smaller replica of its larger parent. Zoom in again, and the pattern repeats itself: not ad infinitum, but several times. Mandelbrot devoted much of his career to studying these patterns, modelling them using mathematical equations and illustrating them with abstract graphics he called 'fractals'. Using such concepts, he was able to produce simulated price charts that would have fooled any trader.
All this might have remained a fascinating academic cul-de-sac, but Mandelbrot's models proved to have a use - because strangely, they were better models of how markets actually worked than the 'modern portfolio theory' that was being worshipped at the time (and, unfortunately, is still regarded as sacred by many market participants). Modern portfolio theory, which is largely based on work by Markowitz and Sharpe, amended by Fama (one of Mandelbrot's pupils) and French, is based on the idea that the Gaussian bell curve is a good guide to expected investment returns. There are a host of other dodgy assumptions too, such as the independence of price changes - i.e. one price change has no impact on the next. (Oh yes, and then there's the assumption that people always behave rationally, which the slightest observation of human nature will dispel.)
It wouldn't have taken someone of Mandelbrot's intellect to blow these assumptions out of the water - a much lesser mind could have done it. But the key reason that modern portfolio theory took such hold in the 1960s and 1970s seems to be because there was nothing better on offer - or at least, nothing that offered easy answers. Mandelbrot didn't offer easy answers; but he did offer an explanation of how markets behaved (or misbehaved) that - unlike MPT - fit the facts. But his explanation was inconvenient, so it was largely ignored.
Mandelbrot's model of how markets work depends on a few observations. To begin with, what he calls 'the well-mannered bell curve' does not represent the distribution of market returns. The real curve is much more violent and the investor's ride much bumpier. We know this because events that ought to be astronomically rare, assuming the bell curve is a good model, happen fairly often (think Black Monday, the Russian debt crisis, subprime...) A 'power-law' distribution, not a bell curve, is at work - leading to 'fat tails' - more frequent occurrences of unlikely events than the bell curve would suggest. Such unlikely events can mean ruin for the investor.
Modern portfolio theory tries to deal with such uncertainty by measuring an investment's volatility (standard deviation). But not only is this approach flawed by its reliance on the bell curve, it is also flawed in other ways. Most obviously, the volatility of an investment is not a constant, but varies. Volatility is itself volatile!
Instead of 'volatility', Mandelbrot prefers to talk in terms of randomness. He believes markets are 'wildly' random, not 'mildly' random as MPT would have it. Developing this idea, he identifies two effects that different investments and markets possess in different degrees: a Noah effect, and a Joseph effect. The Noah effect is a tendency towards sudden, devastating change or discontinuity; the Joseph effect is a tendency towards sequential movement in the same direction (seven years of plenty, seven years of famine). Mandelbrot argues that the 'personality' of a particular market or investment can be described by the relationship between these two effects, which are mathematically measurable.
The final crucial ingredient in Mandelbrot's recipe for simulating a market is that of 'trading time'. He observes that market activity comes in bursts; sometimes prices move fast, sometimes slowly. Again, this is an intuitive concept for any trader.
Once he has explained all these concepts - fairly lucidly, with lots of intriguing and rather beautiful illustrations and not a single mathematical equation - Mandelbrot combines them together. It's at this stage it is easy to get lost - I had to read the crucial chapter a few times. What he does is marry together his notion of the way prices change with his notion of trading time to produce a realistic model of a price chart, where time travels sometimes slow, sometimes fast.
Mandelbrot (and his co-author, Richard Hudson) are fairly clear with the reader right from the start that nothing in this book will help you make money. But it might prevent you losing it. What Mandelbrot has done is produce a convincing description of how markets really work, which is a warning to anyone who believes they can be tamed, and a perfect antidote to the smug certainties of modern portfolio theory. He's also achieved the real feat of making the book readable (though there are equations in the notes for those who want them), putting some very high-powered thinking within the reach of the determined general reader.
January 24, 2019
Benoît Mandelbrot, was a great mathematician, the inventor of fractal geometry. Who can be untouched by the beauty of a Mandelbrot set? Those sets is a manifestation of the vast aesthetic power of math – the langue of nature – or the nature itself as Max Tegmark argues in his book Our Mathematical Universe. Fractal geometry might be the evidence that math is not just a thing that only exists in the brain of humans. Fractals, are one of the great secrets of nature that geniuses like Mandelbrot revealed. His broad interests ranged along many practical sciences from hydrology to finance, where he left very significant footprints worth exploring further on.
In the financial market theory Mandelbrot is less well-know. Perhaps this is due to his valiant conquest against the establishment. For the proponents of the efficient market hypothesis, modern portfolio theory, Black-Sholes model, the Sharpe ratio measure, random walk and many other orthodox theories the author gives the opportunity to think it over. I like when somebody succeeds in illustrating that life is not always arranged by the bell-curve.
I loved this book, and it was definitely the one that spoke directly to me. I was expecting some dense jungle of concepts which is quite common when the author of that caliber tries describe some complex matters. However, I was pleasantly surprised by the clearness of thought and good, structured writing style.
Regarding applications in finance, there are some traders out there using fractal strategies, and this book brings some sense to it and a growing interest to continue exploring the area.
I have a feeling, the multifractal world is too important to be left aside. With its spectacular cleverness it is more than just gorgeous Mandelbrot set pictures.
In the financial market theory Mandelbrot is less well-know. Perhaps this is due to his valiant conquest against the establishment. For the proponents of the efficient market hypothesis, modern portfolio theory, Black-Sholes model, the Sharpe ratio measure, random walk and many other orthodox theories the author gives the opportunity to think it over. I like when somebody succeeds in illustrating that life is not always arranged by the bell-curve.
I loved this book, and it was definitely the one that spoke directly to me. I was expecting some dense jungle of concepts which is quite common when the author of that caliber tries describe some complex matters. However, I was pleasantly surprised by the clearness of thought and good, structured writing style.
Regarding applications in finance, there are some traders out there using fractal strategies, and this book brings some sense to it and a growing interest to continue exploring the area.
I have a feeling, the multifractal world is too important to be left aside. With its spectacular cleverness it is more than just gorgeous Mandelbrot set pictures.
January 28, 2021
So uh, talk about timing.
Pretty much the day I finished this book the whole Gamestop saga began kicking off. Perhaps the highest praise I could give this book is that after reading it no one should find those types of shenanigans surprising.
Mandelbrot uses a mixture of hardcore first order mathematics and philosophy to shred the theories of financial orthodoxy and replace them with much more useful (and less iatrogenic) concepts. If anyone wants to get into finance or side trading, I would strongly urge them to read this book before any others.
Pretty much the day I finished this book the whole Gamestop saga began kicking off. Perhaps the highest praise I could give this book is that after reading it no one should find those types of shenanigans surprising.
Mandelbrot uses a mixture of hardcore first order mathematics and philosophy to shred the theories of financial orthodoxy and replace them with much more useful (and less iatrogenic) concepts. If anyone wants to get into finance or side trading, I would strongly urge them to read this book before any others.
September 5, 2022
I like books about science by the scientists themselves, because then the description of the science is probably accurate and you get some insight into the development of the ideas. Mandelbrot's forays into finance are interesting because his ideas go against the body of work connected to the Efficient Market Hypothesis, which has garnered a bucket of "Nobel prizes" and is the basis for the conventional wisdom on how to invest. His conclusions seem obvious if you are familiar with Ben Graham and others who actually succeeded in the real world of finance. It is interesting though to see a detailed take-down of the assumptions underlying prevailing academic/Wall Street BS dogma.
The Intelligent Investor
Unknown Market Wizards: The Best Traders You've Never Heard of
The Intelligent Investor
Unknown Market Wizards: The Best Traders You've Never Heard of
February 8, 2012
In these turbulent economy we seem to be victims of the financial markets. Benoit Mandelbrot, famous mathematician and inventor of fractal geometry, joined forces with Richard Hudson, to write a book about financial theory. “The (Mis)behavior of Markets” falls in the popular science genre. It is low on formulas, instead you can find lots of historical anecdotes and opinions.
1. Risk, Ruin and Reward
We start with a brief history of finance. The author asks us to play a game. Out of 4 charts we need to select the ones that are real and the ones that are fake.
2. By the Toss of a Coin or the Flight of an Arrow?
Chance is important in finance. There is the mild form of chance, described by the bell curve. On the other hand, there is the more extreme Cauchy probability distribution. Financial theory follows the mild path, but Mandelbrot is convinced that this is wrong and a more wild variability is to be expected.
3. Bachelier and His Legacy
The third chapter is about Bachelier and his coin-tossing view of finance. His work led to the theory of the efficient market. According to this theory, the market is so efficient that all information is directly reflected in the price of financial assets.
4. The House of Modern Finance
People who helped build the house of modern finance and their theories are mentioned – Markowitz, Sharpe, Black and Scholes. Even though some received Nobel Prizes, they still lost a lot of money in the markets.
5. The Case Against the Modern Theory of Finance
Mandelbrot tries to demolish the house of modern finance starting with shaky assumptions. He tries to disprove these assumptions. More evidence is presented, such as the low price earnings and price book anomalies. These anomalies are in direct conflict with current theory.
6. Turbulent Markets: A Preview
Turbulence is a nice metaphor for trading. Mandelbrot tries to convince us, that we should be thinking of fractals, when we look at stock charts. He uses cartoons of stock charts to achieve that.
7. Studies in Roughness: A Fractal Primer
Fractal geometry deals with roughness. It introduces a measure called fractal dimension, which is similar to the normal dimension in geometry, but is not an integer.
8. The Mystery of Cotton
This chapter describes a research project of Mandelbrot, when he worked at an IBM laboratory. He discovered a power law in the log returns of cotton prices. The evidence pointed at a L-stable probability distribution with features somewhere between a normal and Cauchy distribution.
9. Long Memory, from the Nile to the Marketplace
Hurst, a famous hydrologist, faced the challenge of figuring out a pattern to the Nile river. Hurst discovered a long term dependence in his data set. It is suggested, that the so called Hurst exponent could be a new yardstick, that would explain better long memory effects in financial markets.
10. Noah, Joseph, and Market Bubbles
The author refers to characters in the Bible to describe different forms of wild variability. For people familiar with the Bible this is a good example. In my opinion we can call it a shaky assumption at best.
11. The Multifractal Nature of Trading Time
Some days are slow, some days just fly by. Apparently this applies to trading too and it is due to the multifractal nature of time.
12. Ten Heresies of Finance
A list of ten big errors in financial theory. Markets are riskier, than we thought. Timing matters. Prices often leap.
13. In the Lab
Mandelbrot warns us that fractal finance is not mature yet. However, it is superior to the mainstream theories, since they dangerously underestimate risk.
The book ends with notes containing formulas and bibliography listing scientific articles. A thrilling book, that I could not put down, until I read it cover to cover. It is the finance equivalent of “A Brief History of Time”. I give it 5 stars out of 5.
1. Risk, Ruin and Reward
We start with a brief history of finance. The author asks us to play a game. Out of 4 charts we need to select the ones that are real and the ones that are fake.
2. By the Toss of a Coin or the Flight of an Arrow?
Chance is important in finance. There is the mild form of chance, described by the bell curve. On the other hand, there is the more extreme Cauchy probability distribution. Financial theory follows the mild path, but Mandelbrot is convinced that this is wrong and a more wild variability is to be expected.
3. Bachelier and His Legacy
The third chapter is about Bachelier and his coin-tossing view of finance. His work led to the theory of the efficient market. According to this theory, the market is so efficient that all information is directly reflected in the price of financial assets.
4. The House of Modern Finance
People who helped build the house of modern finance and their theories are mentioned – Markowitz, Sharpe, Black and Scholes. Even though some received Nobel Prizes, they still lost a lot of money in the markets.
5. The Case Against the Modern Theory of Finance
Mandelbrot tries to demolish the house of modern finance starting with shaky assumptions. He tries to disprove these assumptions. More evidence is presented, such as the low price earnings and price book anomalies. These anomalies are in direct conflict with current theory.
6. Turbulent Markets: A Preview
Turbulence is a nice metaphor for trading. Mandelbrot tries to convince us, that we should be thinking of fractals, when we look at stock charts. He uses cartoons of stock charts to achieve that.
7. Studies in Roughness: A Fractal Primer
Fractal geometry deals with roughness. It introduces a measure called fractal dimension, which is similar to the normal dimension in geometry, but is not an integer.
8. The Mystery of Cotton
This chapter describes a research project of Mandelbrot, when he worked at an IBM laboratory. He discovered a power law in the log returns of cotton prices. The evidence pointed at a L-stable probability distribution with features somewhere between a normal and Cauchy distribution.
9. Long Memory, from the Nile to the Marketplace
Hurst, a famous hydrologist, faced the challenge of figuring out a pattern to the Nile river. Hurst discovered a long term dependence in his data set. It is suggested, that the so called Hurst exponent could be a new yardstick, that would explain better long memory effects in financial markets.
10. Noah, Joseph, and Market Bubbles
The author refers to characters in the Bible to describe different forms of wild variability. For people familiar with the Bible this is a good example. In my opinion we can call it a shaky assumption at best.
11. The Multifractal Nature of Trading Time
Some days are slow, some days just fly by. Apparently this applies to trading too and it is due to the multifractal nature of time.
12. Ten Heresies of Finance
A list of ten big errors in financial theory. Markets are riskier, than we thought. Timing matters. Prices often leap.
13. In the Lab
Mandelbrot warns us that fractal finance is not mature yet. However, it is superior to the mainstream theories, since they dangerously underestimate risk.
The book ends with notes containing formulas and bibliography listing scientific articles. A thrilling book, that I could not put down, until I read it cover to cover. It is the finance equivalent of “A Brief History of Time”. I give it 5 stars out of 5.
May 14, 2013
Excellent book by Mandelbrot himself on markets and why they aren't brownian (that is, move up and down in a continuous fashion) and how fractals can be used to represent markets. He goes into volatility and how a Gaussian distribution cannot properly describe markets. And yet for the most part the big models people use (Sharpe ratio, Black-Scholes volatility calculation) are all based on Gaussian distributions, greatly underestimating the risk of ruin.
Great companion to Nassim Taleb's books.
Great companion to Nassim Taleb's books.
July 26, 2022
Talebs daddy. Mulig dette blir masteren
December 20, 2015
The reason for it garnering a 5 star rating is not due to it's literary merit. This is not a novel, but a scientific book written for the layman. I loved it for the way that (when I had finished reading certain chapters) it helped me to visualize nature as an expression of a fractal/chaos set (known as a Mandelbrot set). Whereas before, whenever I went for a walk and looked up into the sky I would just see chaotic assemblies of clouds and leaf growth, now I am seeing some of the haunting images in this book being expressed by the growth and distribution in nature.
I thoroughly enjoyed the explanation of how the physics of wind turbulence were used to derive a blueprint for measuring the turbulence or volatility in Markets. The realism of Mandelbrot's philosophy is very enlightening. The way in which he challenged economists to go back to the drawing board in order to understand the devastating market crashes of 1987 and 1929 was very influential. The most important lesson I took from the book is that (in reality, although it may not seem like it) the mathematical tools that we currently have, are very limited, especially when it comes to measuring complexity and growth. Another great lesson I learned is that if you have an idea that challenges the status quo, then make sure you back it up with lots of mathematical proofs and real world examples, as well as reading the current intellectual atmosphere in order to gauge whether the establishment is ready to accept new groundbreaking ideas.
Overall a great read, especially if you are a finance major like myself, and also if you are interested in complexity studies and or the study of fractals.
I thoroughly enjoyed the explanation of how the physics of wind turbulence were used to derive a blueprint for measuring the turbulence or volatility in Markets. The realism of Mandelbrot's philosophy is very enlightening. The way in which he challenged economists to go back to the drawing board in order to understand the devastating market crashes of 1987 and 1929 was very influential. The most important lesson I took from the book is that (in reality, although it may not seem like it) the mathematical tools that we currently have, are very limited, especially when it comes to measuring complexity and growth. Another great lesson I learned is that if you have an idea that challenges the status quo, then make sure you back it up with lots of mathematical proofs and real world examples, as well as reading the current intellectual atmosphere in order to gauge whether the establishment is ready to accept new groundbreaking ideas.
Overall a great read, especially if you are a finance major like myself, and also if you are interested in complexity studies and or the study of fractals.
November 3, 2020
Mandelbrot’s thesis: Abandon the Gaussian and embrace power laws.
He’s the father of fractals, a very powerful tool in every discipline of science I have encountered. The problem with finance is that our mathematical tools and systems are ill-equipped to understand it, even though we have poured trillions into it. To paraphrase Mandelbrot, we know why earthquakes occur but we cannot explain what leads to a man’s financial ruin. His interest is not of a speculator, but a mathematical modeller.
Having read his student Taleb’s work first, I was aware of the ideas that Mandelbrot proposes. Reading the teacher’s first-hand account improved my understanding of it. And his humour keeps you engaged.
He’s the father of fractals, a very powerful tool in every discipline of science I have encountered. The problem with finance is that our mathematical tools and systems are ill-equipped to understand it, even though we have poured trillions into it. To paraphrase Mandelbrot, we know why earthquakes occur but we cannot explain what leads to a man’s financial ruin. His interest is not of a speculator, but a mathematical modeller.
Having read his student Taleb’s work first, I was aware of the ideas that Mandelbrot proposes. Reading the teacher’s first-hand account improved my understanding of it. And his humour keeps you engaged.
June 17, 2017
This book tells us that standard financial theory is wrong, price changes are not independent of each other, changes are wilder than the theory assumes and changes are not continuous.
The book is very interesting in parts, some of the explanations are very lucid, but in parts it is repetitive and some the layman explanations of don't make much sense. Overall I enjoyed the book and learned what's wrong with present theories of finance, but to go beyond that I need to learn the actual math used by Mandelbrot to reach his conclusions.
The book is very interesting in parts, some of the explanations are very lucid, but in parts it is repetitive and some the layman explanations of don't make much sense. Overall I enjoyed the book and learned what's wrong with present theories of finance, but to go beyond that I need to learn the actual math used by Mandelbrot to reach his conclusions.
January 29, 2019
I can very well see why Taleb considers this as the deepest and most realistic book on finance, since I occasionally felt that I was reading Taleb, not Mandelbrot. It is frightening to read books like these, since you become more and more convinced that what you learned in university was bunk. Financial markets do not follow bell curves, and standard risk measurements are plainly wrong. Markets are much more risky than we make them to be, and lives can be ruined as a result. What to make of this? As Mandelbrot put it, this book will not make you rich, but it may save you from ruin.
September 21, 2022
Bunnsolid av idolet til Taleb, mannen bak fraktalgeometri.
Skriver som Taleb (men uten arrogansen) om alt som er galt med antakelsene om normalfordelt avkastning, rasjonelle aktører og at kursutvikling er stokastisk.
Boka er velskrevet og pedagogisk, men det er en fordel å ha noen forkunnskaper om finans og statistikk.
Skriver som Taleb (men uten arrogansen) om alt som er galt med antakelsene om normalfordelt avkastning, rasjonelle aktører og at kursutvikling er stokastisk.
Boka er velskrevet og pedagogisk, men det er en fordel å ha noen forkunnskaper om finans og statistikk.
October 31, 2012
I read this several years ago, and I enjoyed it very much. I wish I could find my copy, but I loaned it to a former student, and never saw it again.
August 14, 2020
For me this was a very important book.
It introduced a new paradigm - which is otherwise omitted by several Economics programs, at least in India - to better understand how markets work.
The overall presentation of ideas was lovely and far from the usual stodgy manner of economics books. For instance, I loved how the author connected seemingly mundane movements in cotton prices to the mysteries of the universe; he takes a step back from the mire of details to marvel at the big picture every now and then.
It's a very approachable (sometimes a bit too approachable) book written for the layperson.
Overall I think this book is meant to inspire as much as it was written to inform.
It introduced a new paradigm - which is otherwise omitted by several Economics programs, at least in India - to better understand how markets work.
The overall presentation of ideas was lovely and far from the usual stodgy manner of economics books. For instance, I loved how the author connected seemingly mundane movements in cotton prices to the mysteries of the universe; he takes a step back from the mire of details to marvel at the big picture every now and then.
It's a very approachable (sometimes a bit too approachable) book written for the layperson.
Overall I think this book is meant to inspire as much as it was written to inform.
August 17, 2019
Not as valuable as might be expected
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 was 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.
--Dennis Littrell, author of “The World Is Not as We Think It Is”
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 was 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.
--Dennis Littrell, author of “The World Is Not as We Think It Is”
October 13, 2021
Denna bok innehåller en djupdykning i problemen med förenklade modeller, och heurestiker för vad som bör göras annorlunda. Den är välskriven, lättsam, och tänkvärd. Dess fokus är marknadsekonomins beteenden, och hur de skiljer sig från vad skolböckerna säger att de bör göra.
August 20, 2011
Like most good books about the markets, Benoît Mandelbrot's The mis Behavior of Markets is not really about trading or making money (although, if it helps you better understand risk, it could save you money--which is essentially the same as making money). In fact, one could almost say the book is about fractal processes, using the markets as a case study. In this way, it is reminiscent of Nassim Nicholas Taleb's Fooled by Randomness, which uses the markets largely as a basis to investigate logical fallacies (and it is not particularly surprising to find blurbs of praise from Taleb on both the front and back covers).
The standard models of finance have been soundly thrashed, and yet they continue to be used because they are easy and convenient. As clever as they may have been, Markowitz and Sharpe and Black and Scholes, et al, were basically financial alchemists. Part of Mandelbrot's point is that their models are not merely inadequate, they so severely underestimate risk as to be catastrophically dangerous--worse than useless, and not improved by adding epicycles. Even if no one except perhaps some hermetic academics believe in the efficient-market hypothesis, typical analysis continues to rely on tools that make assumptions like normal distribution of returns and the applicability of continuous mathematics.
Mandelbrot's work in the area of finance remained, to the time of his death in October 2010, purely descriptive; he never created a model suitable for predicting market behavior. This is a limitation of this work, and one that Mandelbrot himself laments. He writes, "It is beyond belief that we know so little about how people get rich or poor, about how it is they come to dwell in comfort and health or die in penury and disease. Financial markets are the machines in which much of human welfare is decided; yet we know more about how our car engines work than about how our global financial system functions.... So limited is our knowledge that we resort, not to science, but to shamans." This is a somewhat strange and emotional outburst, given how much simpler an engine is than a bunch of humans playing high-stakes guessing games. But it gives some indication of the difficulty involved.
One of Mandelbrot's more interesting propositions concerning the markets is that returns can show dependence without correlation, a seeming paradox. But as he explains it, "The key to this paradox lies in the distinction between size and direction of price changes. Suppose the direction is uncorrelated with the past: The fact that prices fell yesterday does not make them more likely to fall today. It remains possible for the absolute changes to be dependent: A 10 percent fall yesterday may well increase the odds of another 10 percent move today--but provide no advance way of telling whether it will be up or down. If so, the correlation vanishes, in spite of the strong dependence." Compellingly, he presents a multifractal model that combines Brownian price changes with fractal time, yielding clusters of volatility strikingly similar to real market behavior.
Clearly, this book is not the key to riches. That might be nice, but it would not be nearly as interesting. This book is more about how the same mathematics that applies to Nile flood levels or the measurement of coastlines can be used to describe the behavior of markets. Read it because you take pleasure in investigating recurring mathematical motifs in the world, not because you are on a single-minded quest to make money.
The standard models of finance have been soundly thrashed, and yet they continue to be used because they are easy and convenient. As clever as they may have been, Markowitz and Sharpe and Black and Scholes, et al, were basically financial alchemists. Part of Mandelbrot's point is that their models are not merely inadequate, they so severely underestimate risk as to be catastrophically dangerous--worse than useless, and not improved by adding epicycles. Even if no one except perhaps some hermetic academics believe in the efficient-market hypothesis, typical analysis continues to rely on tools that make assumptions like normal distribution of returns and the applicability of continuous mathematics.
Mandelbrot's work in the area of finance remained, to the time of his death in October 2010, purely descriptive; he never created a model suitable for predicting market behavior. This is a limitation of this work, and one that Mandelbrot himself laments. He writes, "It is beyond belief that we know so little about how people get rich or poor, about how it is they come to dwell in comfort and health or die in penury and disease. Financial markets are the machines in which much of human welfare is decided; yet we know more about how our car engines work than about how our global financial system functions.... So limited is our knowledge that we resort, not to science, but to shamans." This is a somewhat strange and emotional outburst, given how much simpler an engine is than a bunch of humans playing high-stakes guessing games. But it gives some indication of the difficulty involved.
One of Mandelbrot's more interesting propositions concerning the markets is that returns can show dependence without correlation, a seeming paradox. But as he explains it, "The key to this paradox lies in the distinction between size and direction of price changes. Suppose the direction is uncorrelated with the past: The fact that prices fell yesterday does not make them more likely to fall today. It remains possible for the absolute changes to be dependent: A 10 percent fall yesterday may well increase the odds of another 10 percent move today--but provide no advance way of telling whether it will be up or down. If so, the correlation vanishes, in spite of the strong dependence." Compellingly, he presents a multifractal model that combines Brownian price changes with fractal time, yielding clusters of volatility strikingly similar to real market behavior.
Clearly, this book is not the key to riches. That might be nice, but it would not be nearly as interesting. This book is more about how the same mathematics that applies to Nile flood levels or the measurement of coastlines can be used to describe the behavior of markets. Read it because you take pleasure in investigating recurring mathematical motifs in the world, not because you are on a single-minded quest to make money.
October 17, 2010
The (Mis)Behavior of Markets by Mandelbrot and Hudson is a pretty good book about a fascinating topic. Mandelbrot's thesis is that many common beliefs underpinning market modeling software are fundamentally incorrect, and that in using them we are exposing ourselves to massively more risk than we expect. This book was published in 2004.
To describe Mandelbrot as prescient in characterizing the inadequacy of market modeling is to understate the situation. Using very little serious math and very few equations, Mandelbrot shows how the existing models presume a Gaussian distribution of risk (i.e. that most changes are small, the frequency of changes is inversely proportional to their size), and thus lead to statements such as "this particular market condition had less than a 1 in 10^50 chance of occurring." Mandelbrot argues convincingly that the right model for understanding the markets is a fractal Pareto distribution (a power law-based series, displaying self similarity and extreme volatility).
I believe that this mis-modeling is endemic throughout those subjects which can get away without having to provide testable assertions: economics, market dynamics and securitization, climate science, and many others. The antidote to poor science is reminding ourselves of the scientific method:
observation -> hypothesis -> experiment -> validation or refutation -> repetition
Sadly, in our times, fixing the models we use often takes a back seat to scoring partisan points. Hopefully that will change in time.
The book would have rated higher but for the overly self-magnifying style taken by the author: that was grating, but the book was still worth reading anyway.
To describe Mandelbrot as prescient in characterizing the inadequacy of market modeling is to understate the situation. Using very little serious math and very few equations, Mandelbrot shows how the existing models presume a Gaussian distribution of risk (i.e. that most changes are small, the frequency of changes is inversely proportional to their size), and thus lead to statements such as "this particular market condition had less than a 1 in 10^50 chance of occurring." Mandelbrot argues convincingly that the right model for understanding the markets is a fractal Pareto distribution (a power law-based series, displaying self similarity and extreme volatility).
I believe that this mis-modeling is endemic throughout those subjects which can get away without having to provide testable assertions: economics, market dynamics and securitization, climate science, and many others. The antidote to poor science is reminding ourselves of the scientific method:
observation -> hypothesis -> experiment -> validation or refutation -> repetition
Sadly, in our times, fixing the models we use often takes a back seat to scoring partisan points. Hopefully that will change in time.
The book would have rated higher but for the overly self-magnifying style taken by the author: that was grating, but the book was still worth reading anyway.
March 14, 2021
Much ado over very little. Mandelbrot seems to derive much of his personality from being a contrarian, which itself is not of much interest — his ideas are interesting, but his stories about why nobody believes him because he’s such a revolutionary are much less so. He’s also got an ego on him, with constant name dropping (take a shot every time he mentions Harvard, MIT, IBM, or UChicago). It’s not a good look on anyone, and even less so on someone who is already popularly successful — see Feynman as an example of doing this kind of thing better.
A whole lot of time is dedicated to the existing thoughts in economics, which is nice, but not too exciting. The main purpose of this is to show how fractal models fit the situation better, but the problem is that while fractals can make compelling models, their pragmatic usefulness as anything past faking a realistic chart is questionable. They do make reasonable models with Monte Carlo simulations for risk assessment, but the problem remains the same as other models — they don’t really predict much in reality.
On the plus side, I learned how fractals work, and their application here is reasonable-ish. I suspect models with smarter distribution functions + momentum mechanisms could beat out a fractal approach, but maybe not? It’s an interesting idea, but I see why the establishment wasn’t ready to scrap everything and start from this instead. Mandelbrot seems baffled that this hasn’t happened, of course.
A whole lot of time is dedicated to the existing thoughts in economics, which is nice, but not too exciting. The main purpose of this is to show how fractal models fit the situation better, but the problem is that while fractals can make compelling models, their pragmatic usefulness as anything past faking a realistic chart is questionable. They do make reasonable models with Monte Carlo simulations for risk assessment, but the problem remains the same as other models — they don’t really predict much in reality.
On the plus side, I learned how fractals work, and their application here is reasonable-ish. I suspect models with smarter distribution functions + momentum mechanisms could beat out a fractal approach, but maybe not? It’s an interesting idea, but I see why the establishment wasn’t ready to scrap everything and start from this instead. Mandelbrot seems baffled that this hasn’t happened, of course.
July 25, 2009
This book has three characters in it:
-Benoit Mandelbrot, author
-The Market, the protagonist/antagonist/chorus as per Greek drama
-Benoit Mandelbrot's ego
Maybe it's a side effect of some incident as a child but the author has no reservations about promoting himself. Whole paragraphs are devoted to his "enlightened breakthroughs" and profound understanding of market mechanics. An understanding so deep he proposes no significant market model and merely a direction.
He stands as the most cited author in the book and many of the references by other authors are sources that largely cite him.
Anyone with a whiff of exposure to authors who lambaste the GARCH and GCF will be familiar with the arguments waged here and there is little reason to read it unless you want to supplicate totally to the church of Mandelbrot. The one area that was interesting was the description of the Cauchy distribution and "long memory". The former a distribution with no finite mean or variance and the latter a phenomenon where correlation is exhibited over long (centuries) periods of time.
I can't stand self-aggrandizing authors, yet I made it through the book. I don't know if that's merely my weakness as a completist or a testament to the author's otherwise clarion writing.
-Benoit Mandelbrot, author
-The Market, the protagonist/antagonist/chorus as per Greek drama
-Benoit Mandelbrot's ego
Maybe it's a side effect of some incident as a child but the author has no reservations about promoting himself. Whole paragraphs are devoted to his "enlightened breakthroughs" and profound understanding of market mechanics. An understanding so deep he proposes no significant market model and merely a direction.
He stands as the most cited author in the book and many of the references by other authors are sources that largely cite him.
Anyone with a whiff of exposure to authors who lambaste the GARCH and GCF will be familiar with the arguments waged here and there is little reason to read it unless you want to supplicate totally to the church of Mandelbrot. The one area that was interesting was the description of the Cauchy distribution and "long memory". The former a distribution with no finite mean or variance and the latter a phenomenon where correlation is exhibited over long (centuries) periods of time.
I can't stand self-aggrandizing authors, yet I made it through the book. I don't know if that's merely my weakness as a completist or a testament to the author's otherwise clarion writing.
December 21, 2017
Although Mandelbrot has a point with regards to some of the issues with modern portfolio theory, this book does not provide the solution.
The style of the book is similar to those by Taleb, who considers Mandelbrot his mentor. Like Taleb, Mandelbrot is full of himself and unfortunately the second author is in awe of Mandelbrot. The continued explanations why Mandelbrot is amazing and why he is right and everyone else is wrong get old fast.
While Mandelbrot explains that work performed by others is not scientific enough, it is interesting to note that some of his evidence is similarly unscientific. In one example to show that daily patterns scale to monthly patterns he changes the sample between the two investigations, so he does not proof his point.
The main scope of the book would have been clearer without the inclusion of Mandelbrot's lifestory.
The style of the book is similar to those by Taleb, who considers Mandelbrot his mentor. Like Taleb, Mandelbrot is full of himself and unfortunately the second author is in awe of Mandelbrot. The continued explanations why Mandelbrot is amazing and why he is right and everyone else is wrong get old fast.
While Mandelbrot explains that work performed by others is not scientific enough, it is interesting to note that some of his evidence is similarly unscientific. In one example to show that daily patterns scale to monthly patterns he changes the sample between the two investigations, so he does not proof his point.
The main scope of the book would have been clearer without the inclusion of Mandelbrot's lifestory.
October 20, 2019
An interesting, unassuming autobiographical account of fractals in finance.
It is a good introduction to various concepts of fractal econometrics, with special attention devoted to Hurst exponent concept.
It is a good introduction to various concepts of fractal econometrics, with special attention devoted to Hurst exponent concept.
March 27, 2022
A somewhat dated but well-written and reasonably accurate popular-science overview of some topics in mathematical finance that have otherwise received not much in the way of popular explanation makes this definitely worth a read if you are an econometrician, finance academic, or (a certain kind of) quant. Starting with a decent summary of MBA-level classic finance theory (Markowitz portfolios, CAPM, Black-Scholes), it then summarizes some of the empirical shortcomings of these models, focusing on tail behavior and dependence properties, then discusses Mandelbrot and his collaborators' work on heavy tails and long memory, and some of the origins of these and related ideas, including a bit on fractals. While heavy tails are now widely acknowledged and many popular explanations (of varying levels of quality) are out there, I am hard pressed to think of any other popular treatment of long memory covering both the structure of the idea and explaining the historical and conceptual motivations, which are often lost in abstraction in academic treatments, and this all is done well here. As an exposition of many of the ideas later taken up by exponents of "complexity science" by one of their main progenitors, it is a much more grounded discussion than you may get from secondary sources, which often lose both accuracy and modesty in the translation.
Now, onto the downsides.
A common weakness in popular books by senior academics is to give a view of the field which downplays others' work and overstates the relevance of whatever their current project is, and some of that is here. In the takedown of classical finance, there is a subtle but strongly misleading process between evidence and conclusions regarding non-Gaussianity. Essentially, in the original formulations of these concepts, Gaussianity was used in or inspired the derivation of the formula. Distributional tests of the data then showed fairly conclusively that the tail properties of the data differ from those of the Gaussian. What is intimated but not actually a logically valid conclusion from these results is that the formula is therefore no longer valid. The reason this doesn't work is that the features of the Gaussian that get used in the formula need not be the ones that are tested by the empirical test. And in fact, both Markowitz portfolio formula and CAPM can be and are regularly derived without relying on the exact tail behavior of the Gaussian. That said, Black-Scholes does depend sensitively on these properties, and there are much more severe tests, some of them discussed in the book, that do contradict the validity of these formulas. Part of the issue may be non-standard use of the word Gaussian in the text. In some places, it appears to be used to refer to the concept of being a process within the domain of attraction of a Gaussian limit law (or at least, would be unambiguously mathematically incorrect if used in the sense of actually having a finite sample Gaussian distribution), which is a much broader concept to which the evidence marshaled in other chapters does not speak nearly so clearly. This is an important point because a lot fo the more modern fixes in the academic finance literature, which he dismisses as still based on the Gaussian paradigm, are themselves designed precisely in order to match the evidence he shows refutes the (narrow sense) Gaussian assumption. Mandelbrot here is guilty primarily of sometimes using ambiguous language, but it irks me probably because the logical fallacy which is only intimated here is so loudly and frequently yelled at mainstream researchers, who are perfectly aware of the facts, by poorly informed complexity cranks who read a misleading airport book summary of related ideas. There are plenty of grounds on which to criticize mid-late 20th century academic finance, and Black-Scholes in particular, but precisely what parts of finance theory need to be consigned to ashes is a much harder question than it looks. (Though see Donald MacKenzie's excellent work on how financial practitioners used even Black-Scholes and other models based on Gaussian assumptions in "off-label" ways which accounted for some of the downsides).
On the more recent work, the chapter on multifractal processes, which are presented as a culmination of this research agenda, combining the work on tail behavior, memory, and self-similarity, is fine for a "here's my latest set of papers summary," but these ideas ended up having fairly limited impact on finance theory and practice. To oversimplify, this is basically a (fractional) Brownian motion composed with a time change which is the CDF of a random measure with certain regularity properties. The limitations and comparison with methods that did continue to get used are glossed over in this text (the phrase FIGARCH appears once in a brief note in an appendix), with the comparison to GARCH explained mostly as one of parsimony, rather than fit to any particular set of facts. Fortunately the concerns that immediately sprang to mind (for me, someone with a PhD in time series econometrics) are addressed in the academic paper (peer review sometimes works!). Specifically, it allows long memory in volatility but avoids the oft-stated reason (not really mentioned in this book) for dismissing fractional Brownian motion as a model for financial markets which is that it implies return predictability which fairly robust theories suggest should be fairly limited and is empirically hard to see. FIGARCH also has those properties; their argument for the method is based on scale invariance properties. I will need to think harder about how empirically relevant those are, but the summary in the book, which admits that they just cut out the high frequencies when testing this property is likely to explain why the method isn't particularly popular, as it is precisely in the high frequency regime, which has become more important to finance in the past decade or two, where "microstructure" features of processes for which exact time scaling is crucial become nontrivial, and the observed time series properties are very different, making scale invariant models not very useful. That said, modern high frequency financial econometrics now makes extensive use of Lévy processes, a more basic development that is discussed in this book.
Now, onto the downsides.
A common weakness in popular books by senior academics is to give a view of the field which downplays others' work and overstates the relevance of whatever their current project is, and some of that is here. In the takedown of classical finance, there is a subtle but strongly misleading process between evidence and conclusions regarding non-Gaussianity. Essentially, in the original formulations of these concepts, Gaussianity was used in or inspired the derivation of the formula. Distributional tests of the data then showed fairly conclusively that the tail properties of the data differ from those of the Gaussian. What is intimated but not actually a logically valid conclusion from these results is that the formula is therefore no longer valid. The reason this doesn't work is that the features of the Gaussian that get used in the formula need not be the ones that are tested by the empirical test. And in fact, both Markowitz portfolio formula and CAPM can be and are regularly derived without relying on the exact tail behavior of the Gaussian. That said, Black-Scholes does depend sensitively on these properties, and there are much more severe tests, some of them discussed in the book, that do contradict the validity of these formulas. Part of the issue may be non-standard use of the word Gaussian in the text. In some places, it appears to be used to refer to the concept of being a process within the domain of attraction of a Gaussian limit law (or at least, would be unambiguously mathematically incorrect if used in the sense of actually having a finite sample Gaussian distribution), which is a much broader concept to which the evidence marshaled in other chapters does not speak nearly so clearly. This is an important point because a lot fo the more modern fixes in the academic finance literature, which he dismisses as still based on the Gaussian paradigm, are themselves designed precisely in order to match the evidence he shows refutes the (narrow sense) Gaussian assumption. Mandelbrot here is guilty primarily of sometimes using ambiguous language, but it irks me probably because the logical fallacy which is only intimated here is so loudly and frequently yelled at mainstream researchers, who are perfectly aware of the facts, by poorly informed complexity cranks who read a misleading airport book summary of related ideas. There are plenty of grounds on which to criticize mid-late 20th century academic finance, and Black-Scholes in particular, but precisely what parts of finance theory need to be consigned to ashes is a much harder question than it looks. (Though see Donald MacKenzie's excellent work on how financial practitioners used even Black-Scholes and other models based on Gaussian assumptions in "off-label" ways which accounted for some of the downsides).
On the more recent work, the chapter on multifractal processes, which are presented as a culmination of this research agenda, combining the work on tail behavior, memory, and self-similarity, is fine for a "here's my latest set of papers summary," but these ideas ended up having fairly limited impact on finance theory and practice. To oversimplify, this is basically a (fractional) Brownian motion composed with a time change which is the CDF of a random measure with certain regularity properties. The limitations and comparison with methods that did continue to get used are glossed over in this text (the phrase FIGARCH appears once in a brief note in an appendix), with the comparison to GARCH explained mostly as one of parsimony, rather than fit to any particular set of facts. Fortunately the concerns that immediately sprang to mind (for me, someone with a PhD in time series econometrics) are addressed in the academic paper (peer review sometimes works!). Specifically, it allows long memory in volatility but avoids the oft-stated reason (not really mentioned in this book) for dismissing fractional Brownian motion as a model for financial markets which is that it implies return predictability which fairly robust theories suggest should be fairly limited and is empirically hard to see. FIGARCH also has those properties; their argument for the method is based on scale invariance properties. I will need to think harder about how empirically relevant those are, but the summary in the book, which admits that they just cut out the high frequencies when testing this property is likely to explain why the method isn't particularly popular, as it is precisely in the high frequency regime, which has become more important to finance in the past decade or two, where "microstructure" features of processes for which exact time scaling is crucial become nontrivial, and the observed time series properties are very different, making scale invariant models not very useful. That said, modern high frequency financial econometrics now makes extensive use of Lévy processes, a more basic development that is discussed in this book.
January 30, 2021
I once interned somewhere where I worked with a bright French guy, who had previously graduated from a grande école and worked as a quant. We chatted during breaks about the Efficient Markets Hypothesis, algotrading, and Harry Markowitz (he was not a fan). "Economics still needs its Galileo, its Newton," he said. It's possible that he had read this book, which contains a very similar idea.
Mandelbrot was a peripatetic and intellectually omnivorous scholar, who moved from Poland to France to the United States, and worked on meteorology, computer graphics (at a job at IBM), finance, and a host of other fields, in addition to his main focus of pure maths (where he is best known for inventing the concept of fractal geometry). It was his openness to practical problems that led him to the US; when he got his PhD French mathematics was in the grip of the Bourbaki (of which his uncle and brother were members), which was too much "math for math's sake" for his taste. His uncle gave him a book by George Zipf describing the distribution of word frequency in common corpuses, as Pareto had done earlier for income distributions. Neither Pareto nor Zipf were right about the specifics, but the basic idea - that seemingly chaotic data could be modeled buy surprisingly simple patterns - seemed promising. Thus a career was born.
Mandelbrot's hero in some ways is Louis Bachelier, the first mathematician to analyse financial data seriously, modelling it as a random process. He used the concept of Brownian motion (then a fairly edgy idea in statistical mechanics, recently revived by a young Albert Einstein) to model the chaos of options prices in the Third Republic. As with Mandelbrot, his interest in finance was seen as not comme il faut by the theory-oriented world of French maths, and - despite being championed by Poincaré - he struggled to find a teaching post. In fact one of Mandelbrot's central claims is that the randomness of financial markets is not Brownian: impossibly rare events seem to happen all too often. Like the phases of matter, there are different types of randomness, he claims. Mild randomness is Brownian motion, a coin toss, or a random walk (as in Malkiel's book), and modeled by the Gaussian bell curve and the Black-Scholes formula. By contrast, there is also slow and then "wild" randomness, found in turbulent flow and stock prices, and modeled by fractal geometry. Such information doesn't help make one rich - the mathematics don't allow simple prediction - but they do disprove some of the more egregious simplifications of financial maths. (Mandelbrot thinks there is some merit in fundamental analysis, but considers technical analysis akin to astrology.) Some of this may be less surprising nowadays when it is common to find Wall Street firms full of physics PhDs crunching out algorithms. But some of Mandelbrot's ideas are also still controversial. (His style, a bit long on self-promotion, can be off-putting.)
One of the best bits of this book is the frequent use of illustrations, something Mandelbrot became accustomed to while working at IBM. He notes that Lagrange and Laplace bequeathed to science a suspicion of illustrations, partly because of their desire for analytical proof, but also because the illustrations in their time were not so precise. However, computer graphics can be made with any degree of precision desired, and actually help the scientist apprehend connections which can be proven later. (Keep going, belief will follow, D'Alembert said of the calculus; here it is the opposite.) This is another case where today's explosion of "data vis" has caught up with Mandelbrot. He uses "cartoons" (in the art historical sense) to illustrate his ideas; one example is the Koch curve, ("this lamentable plague of a function with no derivatives" in the words of one contemporary mathematician).
Mandelbrot criticises the currently popular time-series prediction models like SARIMAX and GARCH ("a name only a statistician could love"), and suggests instead a "multifractal generative model", with some sketches and real-life examples. But the book is not exactly a prescription for financial modelling, more a critique of the current state, and a proposal that statistical tools for modelling options prices and portfolio risk are still in their infancy: Galileo still awaits.
Mandelbrot was a peripatetic and intellectually omnivorous scholar, who moved from Poland to France to the United States, and worked on meteorology, computer graphics (at a job at IBM), finance, and a host of other fields, in addition to his main focus of pure maths (where he is best known for inventing the concept of fractal geometry). It was his openness to practical problems that led him to the US; when he got his PhD French mathematics was in the grip of the Bourbaki (of which his uncle and brother were members), which was too much "math for math's sake" for his taste. His uncle gave him a book by George Zipf describing the distribution of word frequency in common corpuses, as Pareto had done earlier for income distributions. Neither Pareto nor Zipf were right about the specifics, but the basic idea - that seemingly chaotic data could be modeled buy surprisingly simple patterns - seemed promising. Thus a career was born.
Mandelbrot's hero in some ways is Louis Bachelier, the first mathematician to analyse financial data seriously, modelling it as a random process. He used the concept of Brownian motion (then a fairly edgy idea in statistical mechanics, recently revived by a young Albert Einstein) to model the chaos of options prices in the Third Republic. As with Mandelbrot, his interest in finance was seen as not comme il faut by the theory-oriented world of French maths, and - despite being championed by Poincaré - he struggled to find a teaching post. In fact one of Mandelbrot's central claims is that the randomness of financial markets is not Brownian: impossibly rare events seem to happen all too often. Like the phases of matter, there are different types of randomness, he claims. Mild randomness is Brownian motion, a coin toss, or a random walk (as in Malkiel's book), and modeled by the Gaussian bell curve and the Black-Scholes formula. By contrast, there is also slow and then "wild" randomness, found in turbulent flow and stock prices, and modeled by fractal geometry. Such information doesn't help make one rich - the mathematics don't allow simple prediction - but they do disprove some of the more egregious simplifications of financial maths. (Mandelbrot thinks there is some merit in fundamental analysis, but considers technical analysis akin to astrology.) Some of this may be less surprising nowadays when it is common to find Wall Street firms full of physics PhDs crunching out algorithms. But some of Mandelbrot's ideas are also still controversial. (His style, a bit long on self-promotion, can be off-putting.)
One of the best bits of this book is the frequent use of illustrations, something Mandelbrot became accustomed to while working at IBM. He notes that Lagrange and Laplace bequeathed to science a suspicion of illustrations, partly because of their desire for analytical proof, but also because the illustrations in their time were not so precise. However, computer graphics can be made with any degree of precision desired, and actually help the scientist apprehend connections which can be proven later. (Keep going, belief will follow, D'Alembert said of the calculus; here it is the opposite.) This is another case where today's explosion of "data vis" has caught up with Mandelbrot. He uses "cartoons" (in the art historical sense) to illustrate his ideas; one example is the Koch curve, ("this lamentable plague of a function with no derivatives" in the words of one contemporary mathematician).
Mandelbrot criticises the currently popular time-series prediction models like SARIMAX and GARCH ("a name only a statistician could love"), and suggests instead a "multifractal generative model", with some sketches and real-life examples. But the book is not exactly a prescription for financial modelling, more a critique of the current state, and a proposal that statistical tools for modelling options prices and portfolio risk are still in their infancy: Galileo still awaits.
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