Geometric Brownian Motion. The aim of this project is to gain an understanding of the Monte Carlo simulation method (Value-at-Risk measure) and to create a user-friendly piece of software that will allow investors to determine the potential risk a chosen

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Name: Tom Forward                                                          Year: 2010

Chapter 1: Introduction

A main factor that plays a big part in the price of a stock or shares is the economy, the economy was doing really well up until 2008 when the recession started, this caused share prices to drop dramatically and businesses having to close down. Customers began to spend less, this affected most companies as it caused a downturn in the economy. Investment levels dropped and businesses did not perform to the same standard. Investment became riskier as banks started to collapse, some even going bankrupt causing businesses to close down too. The recession in the past years (which is still ongoing) has led to fewer people investing and share prices decreasing. Due to this risk management has become important when investing.

1.1 Equity

Stocks are also known as equities or shares. An equity is when a stock or share is bought either privately or within a company, this can sometimes be risky because if the company does well in terms of valuation of the equity then the stock or shares bought would have been a good investment to the buyer and profit could be made, but if the company does badly, then the net worth of the stock or shares held is less than what its initial value was and a loss could be made through the returns generated. Risk measures need to be considered when investing in equities to avoid potential losses.

1.2 Aim of the project

The aim of this project is to gain an understanding of the Monte Carlo simulation method (Value-at-Risk measure) and to create a user-friendly piece of software that will allow investors to determine the potential risk a chosen stock has.  Geometric Brownian motion will be used to simulate the behaviour of stock prices using historical price data; the data used will be three years worth of daily share prices collected from DataStream. The user friendly software will be created to allow trouble-free navigation throughout the process of finding out how risky a stock can be. The software will also be designed to alert users of any errors that have been made through VBA coding and functions available.

Chapter 2: Background

A simulation model will be used in this project in order to generate an estimation of a future stock price. However, there are many simulation models used within the financial sector in order to determine potential future stock prices. Two simulation models will be discussed and researched throughout this chapter in order to gain an understanding of the ways future stock prices could be estimated.

2.1 Symmetric Garch Model

Garch stands for general autoregressive conditional heteroscedasticity (GARCH). They are a set of models where the returns of a stock are assumed to be generated by a stochastic process or with a time varying volatility. There are whole ranges of different Garch models, only certain models are used in practice. The symmetric Garch model gives a time dependant value of volatility and can be defined using the following formula. [3] 

Where ω > 0, αβ ≥ 0.  

ω is called the Garch constant

α is the Garch error coefficient and β is the Garch lag coefficient.

β is an indication of how long it takes for the periods of volatility to die out.

Where α is an indication of how sensitive the volatility is to the movements in the market.

The above constants have to be estimated using historical data.

εt is normally distributed with mean 0 and standard deviation σt, since  σt is dependent on the returns of the stock at time t, εt is said to be conditional on the set of data available. [3]

The distribution of εt will change at each time step and therefore needs to be recalculated for each time step for input into the above model. σt is then estimated and the stock price can then be calculated using the following formula. 

2.2 Geometric Brownian Motion (GBM)

Key facts:

  • Geometric Brownian Motion was originally used by Robert Brown to describe the movement of particles.
  • Geometric Brownian Motion is used to simulate the future price of stocks based on a stochastic model. It is one of the most popular stochastic processes being used in the financial sector.
  • Geometric Brownian Motion is not 100% accurate though it does give an approximation of future performance on stocks/shares.
  • Geometric Brownian Motion defines changes of a stock price in continuous time. Shown below.

A few of the variables in this equation are easy to identify and understand below indicates each variable.

 is the annual expected volatility of returns for the stock whose current price is represented by S.

t represents time.

μ is defined as the “annual expected rate of return for the stock,” or “drift”.

z is a random variable.

dt represents the very small changes in time (continuous time).

dz represents very small changes in z.

  • Geometric Brownian Motion can also forecast stock prices based on a discrete time version of the equation above. This model simplifies how the random walk is incorporated. Shown below.

dS has been replace by the term 

Which reflects the change in the price S of the stock between time t, and the next time increment, . Note that because the Wiener process is a type of Markov process, the price of the stock at time t is the only relevant value for forecasting the price of the stock at its next value at time (). [2]

dt, an infinitesimally small change in time, is replaced by a larger increment of time change called “delta t” or .

To define a stock price at a given time increment the discrete formula on the previous page will need to be re arranged to look like this:

  • Geometric Brownian motion can also be used to give an estimation of a future stock price by calculating the stock price at some point in time. Shown below.

 is the price of the stock today.

 Is the annual expected volatility of returns for the stock.

 Is the annual expected rate of return for the stock.

 The time increment

ε from the normal distribution with mean 0 and standard deviation 1.

  • Geometric Brownian motion formula using Ito’s lemma will be used in the software to estimate future stock prices.

(Equations taken from Claire Lines lecture notes, 2011 – MA3997 ‘GBM and Simulation’)

 reflects the change in the price S of the stock between time t (), and the next time increment ().

 is the time increment.

 is a random number from the normal distribution with mean 0 and standard                            deviation of 1.

 is the annual expected rate of return for the stock.

 is the annual expected volatility of returns for the stock.

2.3 Stochastic process

A stochastic process is based on successive draws of a random variable from a defined probability distribution. The values of the random variable cannot be known for certain at the start of the process, the value can only be characterised using the tools of probability such as defining the type of distribution and the key statistics describing the distribution. [2]

2.4 Markov process

A Markov process is a particular kind of stochastic process. One of its properties is that only the current value of the dependant variable (stock price) is relevant for predicting its next value. In other words, the stock price path leading to its current value or price history has absolutely no influence or effect on what the next stock price will be in a Markov process. This process does not use the historical stock prices as they are not useful. [2]

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2.5 Wiener process

A wiener process is a specific type of Markov process where the random variable z is drawn from a normal distribution with a mean of 0 and a standard deviation of 1. [2]

2.6 Random number (z)

There are a number of ways in which to generate random numbers. Below will explain how to generate random numbers in Microsoft excel.

N(0,1) is a particular case of the normal distribution where the mean, is 0, and the standard deviation, is 1, from the general form of the normal distribution N(.

In order to calculate a normally distributed random number with mean ...

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