Contents

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 of the entire world financial system. The behavior of markets reflects a complex system and fractal mathematics.

If the FDA approves a new life-saving drug, for example, hordes of people will rush to buy pharma stock, causing the price to spike. Thus, changes and price can’t be considered normally distributed. Others, in contrast, are value investors, preferring instead to invest in companies that are already mature, relatively stable and slow-but-steady growers. According to dominant theories in finance, every investor can be seen as Data.

His broad interests ranged along many practical sciences from hydrology to finance, where he left very significant footprints worth exploring further on. Historically, there was a very large and disparate body of mathematics that were unified by Mandelbrot. He defined “fractal dimension” which is an exact measure of the change in detail to the change in scale of a given object.

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.

Sadly, in our times, fixing the models we use often takes a back seat to scoring partisan points. The 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. Mandelbrot warns us that fractal finance is not mature yet. However, it is superior to the mainstream theories, since they dangerously underestimate risk. Apparently this applies to trading too and it is due to the multifractal nature of time.

A sound model of stock prices over time, would NOT 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.

## Adaptive Markets

Some days price changes are dramatic and numerous, while on others such changes are slight and scarce. In other words, sometimes there’s a lot of change, or information, and other times there’s very little. But what does the power law have to do with fractal geometry? Think back on the quality of self-similarity found in fractal patterns, as is the case for Romanesco broccoli. The cotton market, far from being smooth, was extremely rough. In fact, financial records report enormous surges and plunges in price – far too many to be considered part of a normal distribution.

Fortunately the concerns that immediately sprang to mind are addressed in the academic paper (peer review sometimes works!). FIGARCH also has those properties; their argument for the method is based on scale invariance properties. That said, modern high frequency financial econometrics now makes extensive use of Lévy processes, a more basic development that is discussed in this book. 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.

## The Misbehavior of Markets Key Idea #6: The dynamics of the market are best described as a fractal phenomenon

Mainstream financial theories try their best to impose a smooth understanding of market dynamics, but can’t match up to reality. Theories that embrace the market’s roughness would be more accurate and helpful. A normal distribution in prices means that the majority of price changes will be small. Just like with extremely tall or short people, the correlation follows that the larger the spikes or plunges in price are, the less likely they are to occur. Even if investors made rational choices all the time, would that mean that the same choice is rational for every investor? Has The Misbehavior of Markets by Benoit Mandelbrot and Richard L. Hudson been sitting on your reading list?

### Are fractals infinite?

Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.

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 total money makeover worksheets in the text. 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.

## Lists with This Book

In the case of the Cauchy distribution the curve is still not close to zero at -5 and 5. George Soros, called “a superstar among money managers” by The New York Times, shares the investment strategies he uses to read the mind of the market. In other words, both the prices as well as the change in price saw immense variance. In the face of this data, Bachelier’s model leaves us few means to cope. Fractal geometry, however, allows us to make sense of this roughness. Not only did the prices change in huge leaps, the mean magnitude of those leaps also varied significantly from one period to another.

These anomalies are in direct conflict with current theory. 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.

In some years, cotton prices didn’t change much at all, while in others, they varied extremely. A trend might result, for instance, from major news leaks, like the FDA’s axi review approval of a novel drug. As people scramble to get in on the action before it’s too late, the availability of the stock decreases, thus further lifting its price.

What celebrated mathematician 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. Around the time Mandelbrot was doing his work on cotton prices the work of Louis Bachelier was being rediscovered and embraced by the academic economics community. Bachelier claimed that the change in market prices followed a Gaussian distribution.

Yet, these rigid models vastly underestimate the true risk of markets. Thankfully, we have begun to implement mathematical tools that have helped us to cope with these volatile market risks. Extreme price swings are the norm in financial markets—not aberrations that can be ignored. Price movements do not follow the well-mannered bell curve assumed by modern finance; they follow a more violent curve that makes an investor’s ride much bumpier. Bubbles develop and burst and individual stocks have market values totally out of line with their assets, revenue and profits.

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 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.

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.

## The Misbehavior of Markets

However, I was pleasantly surprised by the clearness of thought and good, structured writing style. To use the river dam analogy, our current investment dams are not sufficient to weather the actual storms that will 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 . Let us know what’s wrong with this preview of The Behavior of Markets by Benoît B. Mandelbrot. Goodreads helps you keep track of books you want to read.

So the market, itself, was doing the Markowitz calculations. It was the most powerful computer of all, producing tick-by-tick the optimum investment fund. 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 forex4you of market mechanics. An understanding so deep he proposes no significant market model and merely a direction. 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.

This is a limitation of this work, and one that Mandelbrot himself laments. Additionally, the charts of commodity prices, for example, will look the same as those of currency exchanges or the Dow Jones Industrial Average. Excellent book by Mandelbrot himself on markets and why they aren’t brownian and how fractals can be used to represent markets.

It is psychology, individual and mass—even harder to fathom than the paradoxes of quantum mechanics. He stands as the most cited author in the book and many of the references by other authors are sources that largely cite him. It introduced a new paradigm – which is otherwise omitted by several Economics programs, at least in India – to better understand how markets work. It introduces a measure called fractal dimension, which is similar to the normal dimension in geometry, but is not an integer. I have a feeling, the multifractal world is too important to be left aside.

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. 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.

## A Man for All Markets

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. This book demonstrates that this is a solvable problem yet remains unsolved (although Mandelbrot’s work narrows in on an accurate range).

(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). 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. The math for doing this seems very sophisticated, and variations on these approaches have served as the backbone for the financial industry. On the whole, the book is written for a non-technical audience. Mandelbrot helpfully summarizes basic modern portfolio theory (from Bachelier’s thesis on bonds, to Markowitz, CAPM , finally to Black Scholes).

Mandelbrot believes fractals can be used to explain many naturally occurring phenomenon as well as complex systems as the economy. 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. 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. He uses fractal geometry to propose a new, more accurate way of describing market behavior. The result is no less than the foundation for a new science of finance.

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. 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. SourceAn option is said to be in-the-money if the current market stock price exceeds the strike price. Such options have intrinsic value as it can be used to purchase the underlying stock at the lower strike price, before profiting from immediately selling it at the higher market price.

If someone discovers a sound model of stock prices over time, then it would be 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. Benoit Mandelbrot is one of the greatest mathematicians of our time. Most renowned for his work on fractals , he applies a complexity science lens to financial markets. As with most transdisciplinary endeavors, the findings are truly astounding.

## A Fractal View of Risk, Ruin, & Reward

On most days, the price action on some asset might be boring. On other days, say when news breaks out, the price moves erratically. Time seems to compress on these eventful days, and stretch out over the dull ones. But in practice, it’s evident that many other factors moves price. Even in highly liquid FX markets, 80% of quotes end in 0 or 5, skipping the intermediate digits.

Mandelbrot criticizes methods such as GARCH as building on a flawed foundation in fixing changing volatility. Instead Mandelbrot advocates the use of power laws, so-called Cauchy-levy functions and fractals in creating models. He claims that fractals, especially ones that accounting for trading time that stretch or compress can model price movements in a more accurate way. The work seems pretty technical, and Mandelbrot admits freely that research in using multi-fractal models is only developing. More interestingly is the crash course lesson in fractals and measures of roughness.

And the same thing again for the recent financial collapse. Value investors look for under-priced value based on fundamentals. Computer simulations have shown that the two groups interacting in a market gives rise to unexpected emergent behaviour. An out-of-the-money option is one where the market price has yet to reach the strike price. This notion of a beta is what alleviated the computational pain of MPT. Once you forecast the overall market returns , you can then just estimate the beta for each stock.

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