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