Why Stock Markets Crash

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Why Stock Markets Crash
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Why stock markets crash?

• Sornettes argument in his book/article is as
  follows
• The motion of stock markets are not entirely
  random in the normal sense.
• Crashes in particular are abnormal and have a
  certain statistical signature.
• A plausible model of trader behaviour during
  crashes is based on copying or herd
  mentality.
• The statistical signature produced by such models
  is close to that seen in the markets.
• Fitting parameters of copying models to stock
  market data gives a reasonable fit.
• Sornette and his colleagues have predicted the
  occurance of particular crashes.

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Mathematics applied to social sciences

• Sornettes argument in his book is as follows
• The motion of stock markets are not entirely
  random in the normal sense (observation).
• Crashes in particular are abnormal and have a
  certain statistical signature (observation/statis
  tics).
• A plausible model of trader behaviour during
  crashes is based on copying or herd mentality
  (model).
• The statistical signature produced by such models
  is close to that seen in the markets (solution).
• Fitting parameters of copying models to stock
  market data gives a reasonable fit (data
  fitting).
• Sornette and his colleagues have predicted the
  occurance of particular crashes (prediction).

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Mathematics applied to social sciences

• Sornettes argument in his book is as follows
• The motion of stock markets are not entirely
  random in the normal sense (observation).
• Crashes in particular are abnormal and have a
  certain statistical signature (observation/statis
  tics).
• A plausible model of trader behaviour during
  crashes is based on copying or herd mentality
  (model).
• The statistical signature produced by such models
  is close to that seen in the markets (solution).
• Fitting parameters of copying models to stock
  market data gives a reasonable fit (data
  fitting).
• Sornette and his colleagues have predicted the
  occurance of particular crashes (prediction).

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Course Outline

• Short, Medium and Long Term Fluctuations
• Pricing Derivatives (Johan Tysk)
• Positive feedbacks, negative feedbacks and herd
  behaviour.
• Networks and phase transitions. (Andreas
  Grönlund)
• Log-periodicity and predicting crashes.
• Stock Market Crash Day.

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The Dow Jones 1790-2000
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The Dow Jones 1980-1987
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Short, Medium Long Term Fluctuations in Returns

• Returns are usually defined as (p(tdt)-p(t))/p(t)
  .

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Short term fluctations
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Autocorrelation
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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.

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Medium term fluctations
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Medium term fluctations
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Efficient market hypothesis

• (Samuelson 1965)

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Example .

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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).

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Long time scale patterns
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Hidden patterns?

• Autocorrelation does not detect all patterns.

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Hidden patterns?

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

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Drawdown distribution
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Drawdown distribution
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Largest drawdowns
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Constructing a confidence interval

• Take all days of time series and reshuffle them.
• Find the distribution of resulting drawdowns.

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Confidence interval
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Stretched exponential model
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Power laws (Mantegna Stanley, 1995)
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Power laws (Mantegna Stanley, 1995)
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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.