Why Stock Markets Crash

Why stock markets crash?Sornette’s argument in his book/article is as follows:1. The motion of stock markets are not entirely random

in the ’normal’ sense.2. Crashes in particular are ’abnormal’ and have a certain

statistical signature.3. A plausible model of trader behaviour during crashes is

based on ’copying’ or ’herd mentality’.4. The statistical signature produced by such models is

close to that seen in the markets.5. Fitting parameters of copying models to stock market

data gives a reasonable fit.6. Sornette and his colleagues have predicted the

occurance of particular crashes.

Mathematics applied to social sciences

Sornette’s argument in his book is as follows:1. The motion of stock markets are not entirely random

in the ’normal’ sense (observation).2. Crashes in particular are ’abnormal’ and have a certain

statistical signature (observation/statistics).3. A plausible model of trader behaviour during crashes is

based on ’copying’ or ’herd mentality’ (model).4. The statistical signature produced by such models is

close to that seen in the markets (solution).5. Fitting parameters of copying models to stock market

data gives a reasonable fit (data fitting).6. Sornette and his colleagues have predicted the

occurance of particular crashes (prediction).

Mathematics applied to social sciences

Sornette’s argument in his book is as follows:1. The motion of stock markets are not entirely random

in the ’normal’ sense (observation).2. Crashes in particular are ’abnormal’ and have a certain

statistical signature (observation/statistics).3. A plausible model of trader behaviour during crashes is

based on ’copying’ or ’herd mentality’ (model).4. The statistical signature produced by such models is

close to that seen in the markets (solution).5. Fitting parameters of copying models to stock market

data gives a reasonable fit (data fitting).6. Sornette and his colleagues have predicted the

occurance of particular crashes (prediction).

Course Outline

1. Short, Medium and Long Term Fluctuations2. Pricing Derivatives (Johan Tysk)3. Positive feedbacks, negative feedbacks and

herd behaviour.4. Networks and phase transitions. (Andreas

Grönlund)5. Log-periodicity and predicting crashes. 6. Stock Market Crash Day.

The Dow Jones 1790-2000

The Dow Jones 1980-1987

Short, Medium & Long Term Fluctuations in Returns

Returns are usually defined as (p(t+dt)-p(t))/p(t).

Short term fluctations

Autocorrelation

2

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tt XXER

Trading strategy

• Can use correlation with past to predict the expected future.

• Profit is determined by standard deviation of return fluctuations (say approx 0.03%).

• Invest $10,000, 20 trades a day, 250 days a year: 10000*(1.0003)5000 =$44,806 (!).

• But transaction cost must be less than $3 per $10,000.

Medium term fluctations

Medium term fluctations

Efficient market hypothesis

(Samuelson 1965)

Example: .

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Y(0,4)

Frequency

ttt aXX 1

Efficient market hypothesis

• Axiom of expected price formation based on rational, all-knowing agents.

• Noise generated by underlying noise in the value of the world (similar variance).

• Any irrational, ill-informed agents will generate more noise, but will over time be pushed out the market by rational agents.

• Relies on agents not using Yt in their pricing of futures (no copying each other).

Long time scale patterns

Hidden patterns?

• Autocorrelation does not detect all patterns.

Hidden patterns?

• Autocorrelation does not detect all patterns.• Look at drawdowns instead.

Drawdown distribution

Drawdown distribution

Largest drawdowns

Constructing a confidence interval

• Take all days of time series and reshuffle them.

• Find the distribution of resulting drawdowns.

Confidence interval

Stretched exponential model

Power laws (Mantegna & Stanley, 1995)

Power laws (Mantegna & Stanley, 1995)

Summary

• Costs too high to gain from short term correlations.

• Medium term fluctations are usually exponentially distributed.

• In the long term there are occasional drawdowns (crashes) which are inconsistent with the exponential model.

• Other apparent structures in the market.