Benoit Mandelbrot - The MisBehaviour of Markets

Benoit Mandelbrot is regarded by many as the father of fractal mathematics.

He is highly critical of what can be described as the naive use of the normal distribution in much modern quantitative financial theory and practical modelling.

Mandelbrot considers that the normal distribution underestimates the actual frequency of extreme events.

In his excellent book THE MISBEHAVIOUR OF MARKETS he makes the case for the use of fractal based tools in the realm of quantitative financial analysis.

ABSTRACT
Randomness has distinct phases, namely slow/med/fast - analogous to the physical phases of matter - solid/liquid/gas.

[chapter 1]
Risk Ruin & Reward

RULES

1 Markets Are Risky

2 Trouble Runs in Streaks
--- turbulence tends to cluster

3 Markets Have a Personality
--- sum greater than the parts
--- prices largely determined endogenously
--- are (mathematically speaking) stationary(?)

4 Markets Mislead
--- pseudo-patterns are observable in fundamentally unpredictable processes
--- charting is alchemy

5 Market Time is Relative
-- markets move to a beat that has phases of different tempos
-- fast/slow 'trading time'
-- price movement graphs look the same over different time scales

[chapter 2]
By The Toss of a Coin or the Flight of an Arrow ?

fat cauchy tails vs. normal bell bottoms
the blind zen archer vs the coin tosser

TYPICAL EFFECT OF EXTREME OUTLIER ON AVERAGE/MEAN
Gaussian - minor - e.g. coin toss game wih sets
Cauchy - significant - e.g. distance from target of blind archer

in the markets extreme events happen more frequently than the normal distribution would have you believe

[chapter 3]
Bachelier and His Legacy
[the establishment of the canon]

1900 Louis Bachelier - price movements follow a random walk.
- each new change is independent of the past history
- changes are normally distributed (very few extreme changes)
- equal probability of up or down movement

the EFFICIENT MARKET HYPOTHESIS

In an ideal market (perfect information) all current information relevant to the price of a security will already be priced in.

=> today's price is independent of yesterday's

FLAWS
- price changes are not independent
- price changes not normally distributed
--- in reality there are more rare extreme values than the normal curve predicts

VOLATILITY IS VOLATILE
H = exponent of price dependence
alpha = volatility

[chapter 5]
THE CASE AGAINST MODERN FINANCE

FLAWED ASSUMPTIONS
1 People are rational and aim only to maximize profits
2 The marketplace consist of equivalent/identical actors
3 Price changes can safely be regarded as continuous
4 Price changes follow a Brownian motion

KURTOSIS
fat-tailed curve = higher
normal curve = 3
even more squashed = 1

STATISTICAL MOMENTS
1 Mean (centre)
2 Variance (spread)
3 Skewness (symmetry)
4 Kurtosis (curviness)

FRACTAL PROCESSES = self-similar
self-affine => multiple axes of self-similarity => multi-fractal

FRACTAL DIMENSION
quantifies 'roughness'
e.g. degree of space-filling of 2-d plane by 1-d line curve

H = LONG-TERM PRICE DEPENDENCE
[H : HE Hurst (Abu Nil), Ludwig Otto Holder]
H < 0.5 => anti-dependence (mean reversion)
H = 0.5 => independence, Brownian motion
H > 0.5 => long-term dependence / momentum

[chapter 12]
10 HERESIES OF FINANCE
1 Markets are turbulent
2 Markets are very risky, and the standard models generally significantly underestimate the actual degree of risk
3 Market timing matters greatly - big gains and small losses are clustered
4 Prices often leap, not glide. That adds to the risk.
5 In markets, time is flexible
6 Markets in all places and all ages work alike
7 Markets are inherently uncertain, and price bubbles are inevitable
8 Markets are deceptive
9 Forecasting prices can be perilous, but you _can_ estimate the odds of future volatility
10 In financial markets, the idea of 'value' has limited value

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