Modeling the Behavior of the S&P 500 Index

Mary MalliarisLoyola University Chicago

10th IEEE Conference on Artificial Intelligence for Applications

Structure of the S&P

• Random or Chaotic?• If a neural network can determine market

prices better than the random walk model, it would challenge the efficient market hypotheses and support a chaotic dynamics structure for the market

Random Walk Model

P(t+1) = P(t) + e(t+1)Where e(t+1) is from a distribution with mean

mu and variance sigma-squared

Chaotic Dynamics

• A chaotic function must satisfy three requirements:– It must sample infinitely many values– It is sensitive to initial conditions– The periodic points of the function are dense in R

Backpropagation Neural Network

• Input layer• Hidden layer• Output layer• Each node applies a function to the sum of

weighted inputs and computes one output

Data• Weekly data from each Friday for two years• 1989 and 1990• 10 variables:– S&P 500 closing Index– 3 month treasury bill interest rate– 30 year T. Bond interest rate– Weekly New York Stock Exchange volumn– M1, M2– Price/earnings ratio– Gold price, Crude Oil price– CBOE put/call ratio

Network Structure

• One input layer• Two hidden layers – 24 nodes in the first hidden layer– 8 nodes in the second hidden layer

• One output

Comparison

• 10 periods• MAD• MSE• Correlation between expected and actual

Results

• Neural network outperformed the random walk model in each period

• This is supportive of the deterministic structure of the stock market returns

• This is encouraging to researchers who wish to develop deterministic theories that may eventually replace the existing probabilistic paradigm.