Download Applications of Stochastic Processes in Asset Price Modeling

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
dSt   dt   dWt





St : stock price
µ : drift (mean)
σ : volatility (variance)
Assumes stock price follows stochastic
process
Notice any problems?

Stock price may go negative
Geometric Brownian Motion (GBM)
dSt   Stdt   StdWt


No more negative values
Assumes that stock price returns follow
stochastic process
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