# Geometric Brownian Motion

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UNIVERSITY OF NOTTINGHAM

Geometric Brownian Motion

An analysis of model applicability to stock market indices

Adam Goodwin, Jamie Baptiste, Alex Hirst, Vikesh Nathan and Chiu Tsan Leung

Recent financial crises have been strong motives for extended research into the financial modeling of stock pricing. Geometric Brownian Motion (GBM) is now a widely used process for stock price modeling. Our investigation into GBM tests the assumptions of this process to see whether it can be accepted as a good model for underlying assets in option pricing. Theoretically, GBM seems like a good model due to its Markov property; however our results produced an interesting outcome. GBM was found to be a good model for shorter time periods but for longer time periods, the assumptions of the model fail. We also tested GBM on different market indices and found the FTSE 100 held for an extended period of time whereas the Nikkei 225 and the Hang Seng did not. Given the testing used for our investigation, we were able to implement these tests through a real time updating software package in R. This was produced with the intention that given a set of data from the given indices, the user is shown all graphical and numerical results from our testing. The user is then able to see whether the process can be accepted as a good model for their chosen stock or index and time period.

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Contents 1 Introduction ......................................................................................................... 2

1.1 Background ..................................................................................................... 2

1.2 Objectives ....................................................................................................... 2

1.3 Report overview ................................................................................................ 3

2 Geometric Brownian Motion model ............................................................................... 4

2.1 Brownian Motion ............................................................................................... 4

2.2 The assumptions ............................................................................................... 4

2.3 Using the natural log of the returns .......................................................................... 5

3 Methodology ......................................................................................................... 6

3.1 Testing the assumptions of the Geometric Brownian Motion model ..................................... 6

3.1.1 The normality assumption ............................................................................... 6

3.1.2 The constant variance assumption ...................................................................... 7

3.1.3 The constant mean assumption .......................................................................... 8

3.1.4 The assumption of independence ....................................................................... 9

3.2 The data ......................................................................................................... 9

3.3 Sampling methods ........................................................................................... 10

4 Results and discussion ............................................................................................ 12

4.1 Summary Table of Analysis ................................................................................. 12

4.2 Key findings ................................................................................................... 12

4.2.1 All model assumptions held on a quarterly basis .................................................... 12

4.2.2 All assumptions held for FTSE 100 annual periods ................................................. 17

4.2.3 Constant variance and normality did not hold for all periods ...................................... 19

4.2.4 Constant mean and independence held for all periods ............................................. 20

4.2.5 10 year weekly log return data was consistent with daily log return data ....................... 21

4.3 Further discussion ........................................................................................... 23

5 Further development ............................................................................................. 25

6 Software ............................................................................................................ 26

7 Conclusions ........................................................................................................ 27

8 Appendix ........................................................................................................... 28

8.1 Software user guide .......................................................................................... 28

8.2 Derivation of Geometric Brownian Motion ................................................................ 29

8.3 P value tables ................................................................................................. 31

8.4 Additional figures ............................................................................................ 38

9 References ......................................................................................................... 41

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1 Introduction

1.1 Background

Real option valuation plays a key role in capital investment decisions. In simple terms an option is the right, but not the obligation, to undertake business initiatives such as expansion, contraction, deferring or abandoning a capital investment project. These projects may increase the wealth of the business shareholders and being able to value these potential money making projects is key. This firm believes that certain stock market indices are useful underlying assets to price real options and this highlights the importance of ensuring the selected valuation model is accurate in forecasting stock prices. Our study focuses on Geometric Brownian Motion (GBM) as the underlying model for predicting these price movements. Since the introduction of Brownian Motion by Osborne (1959) in the appraisal of common stocks, Geometric Brownian Motion has been considered one of the most important models in valuing the growth of a stock over time. Samuelson (1965) was the first to advocate the application of the GBM model to predict price behaviour. Within its vast array of applications it is perhaps most well-known for its use in modelling stock prices in the Black-Scholes options pricing formulae; arguably the most important concept in modern financial theory. First introduced by Black and Scholes in the 1970s within their conceptual framework, they assume that stock prices follow a Geometric Brownian Motion with constant drift and volatility. Here, drift can be defined by the overall trend the stock price has. The magnitude of the rise or fall in stock price increases as the magnitude of drift increases. The accuracy of GBM has come under some scrutiny with this investigation being a consequence. Many scholars have questioned the validity of the GBM models assumptions, which include:

1. Normality of returns 2. Independence the Markov property 3. Constant drift 4. Constant volatility

Arguments for the use of GBM include:

- The Markov property can be defined as a process in which predictions of future activity can be made solely on the present knowledge where historical activity is irrelevant. This is in line with the weak form of the Efficient Market Hypothesis.

- The process exhibits the same variation in its path as actual stock prices. - The process prohibits negative values, which is appropriate as stock prices can

never be negative. - The model is relatively simple to use in calculations.

1.2 Objectives

The main objective of this study is to confirm or refute the previously mentioned assumptions of the GBM process in predicting stock market movements. The study

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focuses on three indices: the FTSE 100, the Nikkei 225 and the Hang Seng. These indices are the most relevant to the shipping company concerned, who need a real option to be valued. A working piece of software will also be produced that enables the user to implement the statistical techniques used in the investigation, over user defined time horizons on the stated indices.

1.3 Report overview

In the subsequent pages a brief summary is given of the reviewed literature which led to

the testing of the previously mentioned assumptions. The methodology follows, starting

with the theoretical background of the GBM model, proceeding to discuss the

assumptions and the approach taken to test them. The methodology section also

elaborates on the choice of tests and sampling techniques of the data in question and is

followed by the analysis itself. The analysis is divided into five sections, with each section

corresponding to a key finding. For each key finding, an in-depth analysis and discussion

of the results is provided. The analysis section is completed with further discussion of the

findings. In the next section, a description of the software developed is given, including a

simple user guide in the appendix. The final sections contain an overall conclusion of the

investigation as well as suggestions for further developments that have been identified.

The developments include potential improvements of the GBM model and further testing.

An appendix and references are included which offer more