(B) Fractal Geometry is at the heart of Financial Engineering. Pioneered in early 1960 by French American Mathematician Benoit Mandelbrot, a retired Professor from Yale University and an IBM Scientist and Fellow, Fractal Geometry provides an elegant model to analyze the behaviors of a financial time series, in particular, the price moves of stocks or any other trading asset and the return of a portfolio. Studying the prices of cotton in early 1960, Mandelbrot concluded that prices can have large deviations also called outliers or fat tails in statistics or Gray Swans by Nassim Taleb. Because of the Gray Swans, the distribution of asset prices and asset returns is far from normal and is the product of Multifractality.

The Volatility of Stock Prices
Stock prices bounce a lot and move in irregular trends. Simply put, the stock market can go to three states:

• Mild volatility when stock prices have average moves and commonly observed over any time series. In that case, market returns converge toward a mean let say 7%.
• Wild volatility when stock prices have big moves and only observed during rare events (such as Black Monday in October 1987 when stock prices lost 10% in one day), market irrational exuberances (such as the 2000 Internet bubble) or market crashes (such as the 2008 credit crisis).
• Slow volatility when prices are not changing much and observed when there is no market news to move the market or sometime in August when traders are on vacations.

The Behaviors of Stock Prices
Stock prices over any time series (intra-day, day, week, year…) exhibit three major behaviors:

• Discontinuity: stocks can have large price variations. Up and down. They are “discontinue”. Analysis of any chart of stock prices over sufficient time periods shows a substantial number of “spikes” that stand out clearly.
• Concentration: most of the time, price changes are a small relative percentage. They are concentrated. In that case, market returns seem to converge to a mean. Analysis of any chart of stock prices confirms that those relative changes other than the spikes merge into a strip.
• Dependence: price moves have a long-term dependency. An event here and now can affect every other event in the distant future or an event elsewhere. Analysis of any chart of stock prices demonstrates that the spikes tend to cluster and occur during periods when the strip is broad. This process is very similar to the number of aftershocks after a major earthquake. The volatility of stock prices tends to cluster. That was very much the case during the 2008 credit crisis when in October 2008 the DJI traded in a 1,000 range pushing the S&P 500 Volatility Index (VIX) to 75 while its normal range is between 12 and 25.

The Old Bachelier Model or Brownian Motion of the Stock Market
The Bachelier model or Brownian Motion pioneered in early 1900 by Louis Bachelier made two important assumptions:

• Stock prices moves are independent of one day to the next one
• Stock prices moves are continuous and so the probability of large price deviations is extremely small or likely improbable.

The Bachelier model leads to a Gaussian or Normal distribution of stock prices moves where market returns converge toward an average return and the risk of a deviation in a portfolio is in a (or few) standard deviation(s).

The normal distribution of asset returns has been a major assumption of Finance Theory in particular for asset pricing in the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT), and for option pricing in the Black-Scholes formula.

The Fractality of Stock Prices
Fractals provide a representation of statistical measures that are preserved across scales. In a nutshell, Fractals establish the repetition of price moves and market returns that can be 7%, -7%, 20%, -20%, 40% or -40% over any time series: intra-day, day, month or year. A normal distribution of market returns fails to take into account the occurrence of large deviations. The fundamental assumption of Fractal Geometry is that there is no qualitative change when the scale of an object changes. A property called “scale invariance”.

A fractal can be represented by a power law , f(x) = ax^k + O(x^k), and its distribution of stock prices or market returns follows a power law distribution P(X>x) ~ O (x^-α) where α is the scale of the fractal.

When the market moves are mild α>2, the distribution of the stock prices is a Gaussian distribution. When the market moves are wild if α<2, the distribution of the stock prices is a Cauchy distribution. The mathematician Paul Levy demonstrated that both the Gaussian and the Cauchy distribution can be expressed by the same power law distribution, a theorem that Mr. Mandelbrot, a Ph.D. student from Mr. Levy, called L-stable for a Levy stable distribution.

Long-Term Dependency of Stock Price Moves:
Mandelbrot called initially the measure of long-term dependency of stock prices across different time scales the Hurst Coefficient or H in memory of Harrold Hurst.

Harrold Hurst was an exceptional hydrologist of Sa Majeste Britanique who understood that to design the right dam for the Nile in Egypt, it was paramount to establish the precise sequence of floods and droughts. How high shall be a dam for the inhabitants of Cairo is a function of long-term dependence between data in a series of measurements.

Similarly, dependency in the stock market is the relationship between the effect of a particular event that has affected the stock market and another event that has affected the market at a given time or will affect the market in the future.

There are three important variations of H:

• When 1/2 < H < 1, the market has “long-memory in returns”. Data are persistent and trending of prices takes the same direction.
• When H = α = 1/2, the market has a Brownian Motion and the distribution is normal.
• When 0 < H < 1/2, data are anti-persistent. Successive changes in the market tend to cancel each other out and stand out as seemingly unrelated.

Multifractal Trading Time θ (t)

Fortunes are won or lost when stock prices are going to the sky or going to the abyss. At that time, volatility is extreme. And, time is flying. When no news seems to move the market or all traders seem to be on vacation, time is dead. The trading time of the stock market can expand or reduce the clock time.

If the clock time t is considered an initiator and the stock market a generator, the trading time becomes a multifractal function θ (t) the time.

Fractal Brownian Motion BH(t)
A Fractal Brownian Motion BH(t) for a financial time series is a fractal function specified by a distribution P(X>x) ~ O (x^-α) with a scale α such as 1 < α < ∞ and a Hurst coefficient H.

BH(t) is a Brownian motion when H = 1/2, is antipersistent when 0 < H < 1/2 and persistent with long-memory returns when 1/2< H < 1.

Multifractal Model of Asset Returns (MMAR)
Given t the clock time, and θ (t) a Multifractal Trading Time, and BH(t) a Fractal Brownian Motion of a time series of logarithmic stock prices P(t).

P(t) is the compound process or Multifractal Model of Asset Returns of θ (t) and BH(t):

P(t) ≡ BH [θ (t)]

Both θ (t) and BH(t) are self-affine and self-affinity is preserved when they are compounded.

Mandelbrot with two of his Ph.D. students provided a confirmation of the MMAR applied to the Deutschemark/US Dollar exchange rates.

References
Books from Benoit Mandelbrot:

• Mandelbrot, Benoit and Hudson, Richard (2004). The (Mis) Behavior of Markets. A Fractal View of Risk, Ruin and Reward. Profile Books.
• Mandelbrot Benoit (1997), Fractal and Scaling in Finance, Springer

Publications from Benoit Mandelbrot:

• Mandelbrot, Benoit (2003) Heavy tails in finance for independent or multifractal price increments.  Handbook of Heavy Tailed Distributions in Finance.  Edited by Svetlozar T. Rachev.
• Mandelbrot, Benoit (2001) Scaling in financial prices, I: Tails and dependence. Quantitative Finance: 1, 113-123.
• Mandelbrot, Benoit (2001) Scaling in financial prices, II: Multifractals and the star equation. Quantitative Finance: 1, 124-130.
• Mandelbrot, Benoit (2001) Scaling in financial prices, III: Cartoon Brownian motions in multifractal time.  Quantitative Finance: 1, 427-440.
• Mandelbrot, Benoit (2001) Scaling in financial prices, IV: Multifractal concentration. Quantitative Finance: 1, 641-649.
• Mandelbrot, Benoit with Calvet Laurent, & Fisher Adlai (1997). The multifractal model of asset returns.  Cowles Foundation Discussion Papers.
• Mandelbrot, Benoit (1963) The Variation of Certain Speculative Prices. Journal of Business, 36, 394-419.

Note: The picture about is the Mont Saint-Michel in France which is born from the fractality of a rock and a fractal architecture that started in the 10th century.

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