## The (Mis) Behavior of Markets: A Fractal View of Risk, Ruin and Reward

by Benoit B. Mandelbrot and Richard L. Hudson

After reading Nassim Nicholas Taleb’s book “The Black Swan: The Impact of the Highly Improbable” I was really interested in reading more about Mandelbrot and the application of his invention of fractal geometry to financial markets. To find this book in my boss’ library therefore was like stumbling upon hidden treasure.

This book falls into the category of MUST READ especially for financial market participants, risk managers and those responsible for risk assessment. A page turner no doubt for its clear and non-mathematical narrative, but a thought provoking book too whose objective appears to be the fostering of a healthy debate on the relevance of existing market behavior models. It advocates deeper thought about their use rather than the blind faith currently accorded to them.

Mandelbrot is skeptical on the over reliance on models based on the mathematically and computationally easy bell-curve of the standard normal model such as the Black Scholes model for valuing options; the Value-at-Risk model for measuring risk; the Sharpe Ratio and Beta for selecting investments and building portfolios. These models are based on unrealistic simplifying assumptions which do not reflect the true nature and pattern of the markets that they are meant to be modeling. Assumptions such as

- rational market participants;
- identical nature of investors with similar investment goals and holding periods;
- smoothness and continuity of price changes;
- independent price changes which are not influenced by how prices have varied in the past;
- the unchanging nature of price changes;
- the normal distribution of these changes where small changes are the norm and large changes are few and can be predicted with a degree of certainty;

are clearly patterns we do not see in financial markets in the past or today.

Through concrete examples he assesses the currently used standard tools and models of modern financial market theory and shows that market behavior is much more turbulent and wild than what these models postulate. By using the existing models risk tends to be grossly understated and market participants subsequently pay the price dearly for inadequate protection against dangers inherent in financial systems. He states that it is time for more thorough research into financial market patterns and the development of more realistic models that better reflect this market turbulence so as to protect against possible future market crisis.

He writes that his aim is not to understand why markets behave in the manner they do (though he does gives some pretty convincing reasons and arguments in this regard too) but to model how markets work based on the data available and mathematical tools. His extensive research into the subject has led to the development of his model for market behavior, the Multifractal Model of Asset Returns that brings together the patterns that he has discovered in market behavior, some of which are mentioned below:

- Price changes do not fit a bell curve. Extreme loss events that would be highly improbable based on the standard normal distribution have occurred with greater frequency and severity than expected. This shows that markets are much more risky than depicted by standard models and theories in use today.
- Markets tend to be turbulent subject to abrupt changes, jumps and discontinuities.
- The price changes tend to scale with time. Prices charts for a day, a month, a year tend to be indistinguishable from each other if unlabelled.
- Markets patterns reveal memory of the past, a long-term dependence on past price changes which most likely results in the development of price bubbles.
- Price volatility tends to occur in clusters, i.e. large price changes are more likely to be followed by further large changes whereas small price changes tend to follow prior small changes.
- Risk factors for price variations are the same for each time scale. However there is a difference between normal clock time and trading time and the time scale mentioned pertains to trading time. When modeling financial markets normal clock time would need to be deformed, shrunk or stretched, to model trading time, so as to accurately scale between periods of frantic trading and others when trading is thin.
- Arbitrage, i.e. the differences in prices, plays a greater role in financial markets, particularly in wild and turbulent markets, than averages or means and standard deviations (i.e. parameters of existing standard normal models).

Besides pressing for greater research and development in this area, Mandelbrot encourages and advocates the use of extensive stress testing as a standard tool for portfolio construction and risk assessment and management. He suggests a Monte Carlo simulation approach, with a random number generator that describes price variation, for generating thousands of series of hypothetical market price data. Based on the results a frequency distribution of possible outcomes is developed and then based on these likelihoods decisions are made on whether to stick with the existing strategy or change it.

The book provides valuable insights into the deficiencies and limitations present in models being widely used today. It is a good starting point in understanding these shortfalls, in finding the right questions to ask about standard tools, in looking at financial markets with a fresh new perspective.