Applications of Stochastic Processes in Asset Price Modeling

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Transcript Applications of Stochastic Processes in Asset Price Modeling

Applications of Stochastic
Processes in Asset Price
Modeling
Preetam D’Souza
Introduction
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Stock market
forecasting
Investment
management
Financial Derivatives
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Options
Mathematical modeling
Purpose
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Examine different stochastic (random)
models
Test models against empirical data
Ascertain accuracy and validity
Suggest potential improvements
Hypothesis
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Stochastic methods will be close to accurate
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Average several runs
Calibrate models
Background
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Mathematically-oriented articles
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Theoretical nature
Few examples of numerical evidence
Stochastic Processes?
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Random or pseudorandom in nature
Future based on probability distributions
Sequence of random variables
Brownian Motion
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Follows Markov chain
Based on random walk
Wiener Process (Wt)
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Continuous time
Draws values from
normal distribution
Brownian Motion SDE
dSt   dt   dWt
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St : stock price
µ : drift (mean)
σ : volatility (variance)
Assumes stock price follows stochastic
process
Notice any problems?
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Stock price may go negative
Geometric Brownian Motion (GBM)
dSt   Stdt   StdWt
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No more negative values
Assumes that stock price returns follow
stochastic process
Procedure
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Implement Brownian motion models in Java
3 Inputs to Model
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Drift
Volatility
Time steps
Run models for 1 year
Compare with empirical data
Testing
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Blue chip: IBM
Historical data freely available
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Yahoo ! Finance
Compare simulated run with historical data
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Accuracy tests
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Root Mean Squared Deviation
Simulated Run
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IBM simulated run
given initial price in
January 2000
One year
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255 trading days
Drift = 5% (risk-free
rate)
Volatility = 0.2
Simulated Run (contd.)
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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?
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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?
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RMSD test
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Large vs. small
values
RMSD = 22.735 vs.
9.457 for the run on
the previous page
Coincidence?
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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
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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
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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
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Within ~$10 after one year for IBM
Cons of Stochastic Models
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NO guarantee of convergence
Past data plays a vital role in model
performance
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Do stock prices always follow historical trends?
There is no incorporation of current events
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Earnings reports
Executive changes
Further development
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Correlation statistics
Comprehensive simulation runs
Model calibration
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Different probability distributions?
Different stochastic models
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Jump Diffusion
So, can stochastic processes predict the
stock market?
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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