In the following a few highlights from Mandelbrot’s book:
“To me the greatest charm of the multifractal models is the economy. One simple set of rules can produce a great variety of behaviour, depending on the circumstances. By contrast, most financial academics are going through a love affair with another way of modelling market volatility. Its main inventor, Robert F. Engle shared a Nobel in 2003 for its development. It starts from the same facts I have been advancing in this book: Volatility clusters, due to dependence. To model that, it as already been mentioned that a set of statistical tools were developed; it is called GARCH, short for a model the cluster its starts with conventional Brownian model price variation. When the volatility jumps, it plugs in new parameters to make the bell curve grow; when the volatility falls; it plugs in new parameters to shrink the curve. You might say the bell curve vibrates, to fit the circumstances. GARCH is, certainly, a handy abacus now used by many option traders and financial directors to model risk. But it begs the question of what makes the bell curve vibrate. And as you try to work with the model, it becomes increasingly complicated. To say much with little: Such is the goal of good science. But most financial models say little with much. They input endless data, require many parameters, take long calculations. When the fail they are “fixed”. They are amended, qualified, particularized and complicated. Bit by bid, from a bad seed a big but sickly tree is build, with glue nails, screws, and scaffolding. That people lose money on these models should come as no great surprise.
The multifractal model, by contrast, begins with the unchanging, mathematicians would call them. Its economical and flexible and mimics the real thing […] My hope is that, someday, the small seed of multifractal analysis can grow into a fruitful new way of managing the world’s money and economy.” Despite the message of power laws one of Mandelbrot main points is the importance of time.
Hey explains the general dependence of three underlying functions with a metaphor: “ The family starts with the parent. The father takes clock time and transforms it into trading time. The mother takes clock time and changes it into a price. Merged together, the baby takes the father’s trading-time and coverts it into a price by the rules the mother provides. Last step: Use the new baby generator to make a full fractal price chart.”
If one read carefully he is still critical towards the practical applications and a main argument is to invest more into fundamental research on markets. Fractal theory needs more researchers working on it.
-Volatility comes in cluster and can be mild, normal and wild. Standard models only account for the normal periods (stationary issue).
-Fractal models can be scaled and do not depend on finite higher moments.
-Contrary most money can be made in wild periods, timing matters prices cluster.
-Risk mgmt and portfolio models could use multifractal model for monte carlo simulations. (Or one could do it Taleb’s way and build “antifragile” portfolios benefiting from randomness.)
To conclude: “Since my youth I have been shamelessly disrespectful of received wisdom […] My understanding of economics comes not from abstract theory but from observation.”
Mandelbrot concludes his book with the following:
“One night of February 1, 1953, a very bad storm lashed the Dutch coast. It broke the famous see dikes, the country’s ancient and proud bulwark against disaster. More than 1,800 died. Dutch hydrologists found the flooding had pushed the benchmark water-level indicators, in Amsterdam to 3.85 over the average seemingly impossible. The dikes has had been thought to be safe enough from such a calamity; the conventional odds of so high a flood were thought to have been less than one in ten thousand. And yet, further research showed, an even greater inundation of four meters had been recorded only a few centuries earlier, in 1570. Naturally the pragmatic Dutch did not waste time arguing about the math. The cleaned up the damage and rebuild the dikes higher and stronger.
Such pragmatism is needed in financial theory. It is the Hippocratic Oath to “do no harm”. In finance, I believe the conventional models and their recent “fixes” violate that oath. They are not merely wrong; they are dangerously wrong. They are like a shipbuilder who assumes that gales are rare and hurricanes myth; so he builds his vessel for speed, capacity and comfort- giving little thought to stability and strength. To launch such ship across the ocean in typhoon season is to do serious harm. Like the weather markets are turbulent. We must learn to recognize that, and better cope."