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 Fluctuations 2.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

The Dow Jones

Short, Medium & Long Term Fluctuations in Returns Returns are usually defined as (p(t+dt)-p(t))/p(t).

Short term fluctations

Autocorrelation

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

Efficient market hypothesis (Samuelson 1965)

Example:.

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 Y t 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

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)

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.