1. Applications of Stochastic Processes in Asset Price Modeling Preetam D’Souza

2. Introduction Stock market forecasting Investment management Financial Derivatives  Options Mathematical modeling

3. Purpose Examine different stochastic (random) models Test models against empirical data Ascertain accuracy and validity Suggest potential improvements

4. Hypothesis Stochastic methods will be close to accurate  Average several runs  Calibrate models

5. Background Mathematically-oriented articles  Theoretical nature Few examples of numerical evidence

6. Stochastic Processes? Random or pseudorandom in nature Future based on probability distributions Sequence of random variables

7. Brownian Motion Follows Markov chain Based on random walk Wiener Process (W t )  Continuous time  Draws values from normal distribution

8. Brownian Motion SDE dS t = µdt + σdW t S t : stock price µ : drift (mean) σ : volatility (variance) Assumes stock price follows stochastic process Notice any problems?  Stock price may go negative

9. Geometric Brownian Motion dS t = µS t dt + σS t dW t No more negative values Assumes that stock price returns follow stochastic process

10. Procedure Implement Brownian motion models in Java 3 Inputs to Model  Drift  Volatility  Time steps Run models for 1 year Compare with empirical data

11. Testing Blue chip: IBM Historical data freely available  Yahoo ! Finance Compare simulated run with historical data  Correlation tests Pearson product-moment correlation coefficient

12. Simulated Run IBM simulated run given initial price in January 2000 One year  255 trading days Drift = 5% (risk-free rate) Volatility = 0.2

13. Analysis & Conclusions Stochastic models generate price fluctuations very similar to actual data Uncertainty increases as time steps progress Further calibrations must be made to fine tune models

14. Further development Correlation statistics Comprehensive simulation runs Model calibration  Assume lognormal distribution Different stochastic models