Applications of Stochastic Processes in Asset Price Modeling

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Applications of Stochastic Processes in Asset
Price Modeling

• Preetam DSouza

2
Introduction

• Stock market forecasting
• Investment management
• Financial Derivatives
• Options
• Mathematical modeling

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Purpose

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

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Hypothesis

• Stochastic methods will be close to accurate
• Average several runs
• Calibrate models

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Background

• Mathematically-oriented articles
• Theoretical nature
• Few examples of numerical evidence

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Stochastic Processes?

• Random or pseudorandom in nature
• Future based on probability distributions
• Sequence of random variables

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Brownian Motion

• Follows Markov chain
• Based on random walk
• Wiener Process (Wt)
• Continuous time
• Draws values from normal distribution

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Brownian Motion SDE

• St stock price
• µ drift (mean)
• s volatility (variance)
• Assumes stock price follows stochastic process
• Notice any problems?
• Stock price may go negative

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Geometric Brownian Motion (GBM)

• No more negative values
• Assumes that stock price returns follow
  stochastic process

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Procedure

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

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Testing

• Blue chip IBM
• Historical data freely available
• Yahoo ! Finance
• Compare simulated run with historical data
• Accuracy tests
• Root Mean Squared Deviation

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Simulated Run

• IBM simulated run given initial price in January
  2000
• One year
• 255 trading days
• Drift 5 (risk-free rate)
• Volatility 0.2

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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

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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

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Accuracy?

• RMSD test
• Large vs. small values
• RMSD 22.735 vs. 9.457 for the run on the
  previous page

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Coincidence?

• Google shares from April 2008-2009
• Simulation 3 (purple) shows uncanny accuracy
• Other simulations throw off averaged run

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More Examples (HMC)
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More Examples (WMT)
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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

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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

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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

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Further development

• Correlation statistics
• Comprehensive simulation runs
• Model calibration
• Different probability distributions?
• Different stochastic models
• Jump Diffusion

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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