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

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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 (W t ) Continuous time Draws values from normal distribution

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Brownian Motion SDE S t : stock price µ : drift (mean) σ : 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

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