Applications of Stochastic Processes in Asset Price Modeling
Preetam D’Souza
Introduction
Stock market forecasting
Investment management
Financial Derivatives Options
Mathematical modeling
Purpose
Examine different stochastic (random) models
Test models against empirical data Ascertain accuracy and validity Suggest potential improvements
Hypothesis
Stochastic methods will be close to accurate Average several runs Calibrate models
Background
Mathematically-oriented articles Theoretical nature
Few examples of numerical evidence
Stochastic Processes?
Random or pseudorandom in nature Future based on probability distributions Sequence of random variables
Brownian Motion
Follows Markov chain Based on random walk Wiener Process (Wt)
Continuous time Draws values from
normal distribution
Brownian Motion SDE
St : stock price µ : drift (mean) σ : volatility (variance) Assumes stock price follows stochastic
process Notice any problems?
Stock price may go negative
t tdS dt dW
Geometric Brownian Motion (GBM)
No more negative values Assumes that stock price returns follow
stochastic process
t t t tdS S dt S dW
Procedure
Implement Brownian motion models in Java 3 Inputs to Model
Drift Volatility Time steps
Run models for 1 year Compare with empirical data
Testing
Blue chip: IBM Historical data freely available
Yahoo ! Finance Compare simulated run with historical data
Accuracy tests Root Mean Squared Deviation
Simulated Run
IBM simulated run given initial price in January 2000
One year 255 trading days
Drift = 5% (risk-free rate)
Volatility = 0.2
Simulated Run (contd.)
IBM simulation with 3 simultaneous runs
Compare with empirical data (red, solid line)
Ending prices are very close
Note that this run is for January 1990-1991
What about predicting the future? IBM simulation for bear
session for January 1991-1992
Note how the drift rate is still positive
All runs deviate from mean line and follow empirical price
Ending prices are within $10 of closing price
Accuracy?
RMSD test Large vs. small
values RMSD = 22.735 vs.
9.457 for the run on the previous page
Coincidence?
Google shares from April 2008-2009
Simulation 3 (purple) shows uncanny accuracy
Other simulations throw off averaged run
More Examples (HMC)
More Examples (WMT)
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
Pros of Stochastic Models
Inputs for stochastic models can readily be gathered from empirical data
GBM model seems to fit stock price data well Risk incorporation as time increases Surprisingly accurate results
Within ~$10 after one year for IBM
Cons of Stochastic Models NO guarantee of convergence Past data plays a vital role in model
performance Do stock prices always follow historical trends?
There is no incorporation of current events Earnings reports Executive changes
Further development
Correlation statistics Comprehensive simulation runs Model calibration
Different probability distributions? Different stochastic models
Jump Diffusion
So, can stochastic processes predict the stock market?
Unfortunately, no. Inherent unreliability Stochastic models should be only a part of
the investment decision process Useful when used with traditional equity
analysis Powerful tool for complex option pricing
strategies