1. Modeling the Behavior of the S&P 500 Index Mary Malliaris Loyola University Chicago 10 th IEEE Conference on Artificial Intelligence for Applications

2. 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

3. 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

4. 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

5. Backpropagation Neural Network Input layer Hidden layer Output layer Each node applies a function to the sum of weighted inputs and computes one output

6. 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

7. Network Structure One input layer Two hidden layers – 24 nodes in the first hidden layer – 8 nodes in the second hidden layer One output

8. Comparison 10 periods MAD MSE Correlation between expected and actual

9. 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.