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ULO1 – assessed through student ability to apply knowledge of multivariate functions, data transformations and data distributions to summarise data sets.

ULO2 – assessed through the student ability to analyse datasets by interpreting summary statistics, model and function parameters.

ULO4 - assessed through student ability to develop software codes to solve computational problems for real world analytics.

ULO5 – assessed through student ability to demonstrate professional ethics and responsibility for working with real world data

State the condition that share prices have to satisfy in order to be represented by a geometric Brownian motion? Does your data satisfy this condition? Check using appropriate statistical test and present your evidence in a suitable tabular format.

Compute and for your data assuming that they are both constant. You are encouraged to research the literature about and and find appropriate formulas to compute their values from the three months historic data.

Find information publicly available for your company, e.g. report and news releases. Using this information about the company, critically appraise the computed values.

Estimate the expected value of the share price on 16 November 2018 and compare with published data.

In the light of your findings, state the possible restrictions of your model

## Conditions for successful application of Geometric Brownian Motion

Question 1

The quantity   – in the percentage drift and  – is the percentage volatility. These parameters are considered constant when solving the differential equation to obtain the Geometric Brownian Motion which is the solution for the differential. The parameters in time series analysis of stock returns represents the continuous compounded expected return on the stock () and stock volatility () (Yang and Aldous 2015).

The assumptions of Geometric Brownian Motion (GBM) model include stock return data is assumed to follow a normal distribution with constant mean and variance. Second the observation at time t is independent of the observations at time t-1 or t + 1 (Yang and Aldous 2015). Also, the data is assumed to be continuous which is not the case with stock data.

Question 2

The company chosen for this study is the Australian and New Zealand Banking Group (ANZ.AX). Using the r-code presented in the appendix the closing share price for ANZ.AX was obtained for the period 1 august 2018 to 31 October 2018 from Yahoo Finance. The table 1 shows the stock prices.

 Table 1: ANZ.AX Daily Closing Stock Price (1/08/2018 to 31/10/2018) Date Closing Price Date Closing Price Date Closing Price Date Closing Price 2018-08-01 29.07 2018-08-24 28.50 2018-09-18 28.39 2018-10-10 26.01 2018-08-02 28.91 2018-08-27 28.52 2018-09-19 28.66 2018-10-11 25.91 2018-08-03 28.64 2018-08-28 28.78 2018-09-20 28.42 2018-10-14 25.43 2018-08-06 28.92 2018-08-29 29.43 2018-09-21 28.46 2018-10-15 25.55 2018-08-07 29.09 2018-08-30 29.39 2018-09-24 28.58 2018-10-16 25.81 2018-08-08 28.82 2018-08-31 29.50 2018-09-25 28.22 2018-10-17 25.88 2018-08-09 29.17 2018-09-03 29.21 2018-09-26 27.96 2018-10-18 26.04 2018-08-10 29.31 2018-09-04 28.80 2018-09-27 27.79 2018-10-21 25.79 2018-08-13 29.17 2018-09-05 28.59 2018-09-28 28.18 2018-10-22 25.29 2018-08-14 29.64 2018-09-06 28.53 2018-10-01 27.73 2018-10-23 25.45 2018-08-15 30.15 2018-09-07 28.40 2018-10-02 27.49 2018-10-24 24.80 2018-08-16 30.06 2018-09-10 28.26 2018-10-03 27.44 2018-10-25 24.91 2018-08-17 30.28 2018-09-11 28.58 2018-10-04 27.61 2018-10-28 25.20 2018-08-20 30.05 2018-09-12 28.37 2018-10-05 27.72 2018-10-29 25.66 2018-08-21 29.51 2018-09-13 28.06 2018-10-07 26.99 2018-10-30 25.93 2018-08-22 29.22 2018-09-14 28.15 2018-10-08 26.83 2018-10-31 25.85 2018-08-23 28.72 2018-09-17 28.40 2018-10-09 26.88 Source: Yahoo Finance (2018)

The table 2 shows descriptive statistics for the closing prices for ANZ.AX stock for the period 1 August 2018 to 31 October 2018

 Table 2: Descriptive Statistics Parameter Value Minimum 24.80 Maximum 30.28 Range 5.48 Mean 27.90 Variance 2.21 Standard Deviation 1.4864 Source: Author (2018)

The currencies are in Australian dollar (AUD).

Question 3

The two conditions needed for the successful application of Geometric Brownian Motion are:

1. The close prices are normally distributed with constant mean and variance.
2. The stock prices are independent of the previous values (Reddy and Clinton 2016).

The assumption applies when all the seasonal variations in the data have been removed or modelled. However, the data used span for three months (from 1 August 2018 to 31 October 2018) which does not contain seasonality (less than a year) (Reddy and Clinton 2016).

The table 3 show Shapiro-Wilk test for normality and Chi-squared Test for independence with their corresponding p-values.

 Table 3: Test for Assumptions Test Statistic P-value Shapiro-Wilk W = 0.91459 0.0002134 Chi-Squared (df=66) = 5.2269 1.000 Source: Author (2018)

The df- are the degrees of freedom for the chi-squared test.

Since the sample size is less than 2000 Shapiro-Wilk Test for normality is appropriate for the data. The set of hypotheses under Shapiro-Wilk test is:

H0: The data was obtained from a normally distributed population,

Ha: The data was obtained from non-normally distributed population.

The P-value of 0.0002134 is less than both . Therefore, the null hypothesis is rejected with conclusion that our data does not obey the assumption (i).

Also, For the second assumption the following hypotheses are tested based on Chi-squared test:

H0: The stock prices are independent of the previous values

Ha: Some association does exist between the stock prices and previous values

The p-value = 1.000 is greater than  indicating that the data obeys the assumption of independence.

Question 4

With the assumptions in the previous section the data can be modelled mathematically by a stochastic differential equation

Whose solution gives the Geometric Brownian Motion represented as

The parameters are calculated from the sample means of the log-returns of the stock data used such that:

Where, 252 represent the trading period in a year.

The table 4 shows the descriptive statics for the log-return

 Table 4: Descriptive Statistic Parameter Value Mean 0.17787% Standard deviation (S) 1.13403% Sample size (N) 66 Source: Author (2018)

Therefore,  and

The X values are the percentage of log return of the daily closing stock price of ANZ.AX company.

## Data analysis of ANZ.AX stock prices

Question 5

In order to make sense of the news about Australian and New Zealand Banking Group available at Yahoo Finance website dated 06 December 2018. Below is a screenshot of the report on ANZ.AX.

The mu calculated above need to be converted to prices as follows

The information provided on the website indicate that the annual expected return from ANZ.AX stock ranges between 24.68 - 30.39. The news is consistent with the annual drift value obtained (). The volatility obtained 1.13403% is less than the reported annual volatility of 1.91%. However, the value is close to the news with the slight difference originating from the use of small sample size (3 months). The analysis of stock prices gives a more accurate data if the data used span a period of at least 5 years. Given that assumption (i) was not met the variations might be attributed to the violation of the normality assumption.

Question 6

The table 5 shows the forecasted closing stock prices for ANZ.AX based on the GBM model.

 Table 5: Estimated ANZ.AX Daily Closing Stock Price (1/08/2018 to 31/10/2018) Date Closing Price Date Closing Price Date Closing Price Date Closing Price 2018-08-01 21.63 2018-08-24 3.13 2018-09-18 60.51 2018-10-10 6.42 2018-08-02 592.86 2018-08-27 12.90 2018-09-19 3.58 2018-10-11 104.91 2018-08-03 177.05 2018-08-28 9.12 2018-09-20 31.35 2018-10-14 7.79 2018-08-06 7.66 2018-08-29 86.89 2018-09-21 62.08 2018-10-15 26.37 2018-08-07 12.34 2018-08-30 31.89 2018-09-24 27.78 2018-10-16 2.88 2018-08-08 5.75 2018-08-31 145.70 2018-09-25 21.90 2018-10-17 19.21 2018-08-09 16.42 2018-09-03 88.93 2018-09-26 63.05 2018-10-18 9.30 2018-08-10 38.38 2018-09-04 14.22 2018-09-27 67.45 2018-10-21 20.93 2018-08-13 88.97 2018-09-05 18.44 2018-09-28 3.44 2018-10-22 23.60 2018-08-14 41.02 2018-09-06 29.30 2018-10-01 36.35 2018-10-23 123.21 2018-08-15 25.02 2018-09-07 27.08 2018-10-02 78.54 2018-10-24 45.76 2018-08-16 104.34 2018-09-10 116.63 2018-10-03 5.07 2018-10-25 76.06 2018-08-17 45.84 2018-09-11 31.07 2018-10-04 56.95 2018-10-28 11.58 2018-08-20 156.42 2018-09-12 103.80 2018-10-05 68.85 2018-10-29 31.04 2018-08-21 54.59 2018-09-13 11.64 2018-10-07 22.60 2018-10-30 15.27 2018-08-22 31.59 2018-09-14 29.93 2018-10-08 15.78 2018-10-31 14.20 2018-08-23 17.74 2018-09-17 15.59 2018-10-09 29.33 2018-11-16 22.41 Source: Author

From table 5, the expected value of the share price on 16 November 2018 is 22.41. However, the actual closing price on 16 November 2018 was 25.36.  Therefore, the estimation has an error of -2.95, which is beyond the allowed 0.05.

Question 7

Australia and New Zealand Banking Group Limited (ANZ.AX) is a banking company based in Australia. It provides financial and banking and financial services and products to both local and international community (Brown 2006, p.18). The descriptive statistic of the raw data for the period of three months indicate that the company has an average daily closing stock price of 27.90 AUD with the average ranging at plus or minus 1.49 AUD. The minimum price that the asset had registered in the period was 24.80AUD while the maximum was 30.28AUD. The company estimated target is at 29.09AUD which is within the range of the average close price.

In the analysis the Geometric Brownian Motion model was used to forecast the stock prices of ANZ.AX. The method takes care of the randomness of the stock market by using a standard normal error weight denoted as w(t). Using the weights and the stock price on 01 August 2018 as the initial value the estimates are presented in table 5. Comparing the randomized close price with the observed prices in Yahoo finance indicate that the GBM model under its main assumptions approximate the stock prices with minimal error. However, the normality test did not hold therefore some values generated by GBM lie outside the range estimated by the company.  In summary, the study explored the Geometric Brownian motion model for simulating stock price from Australian Exchange market (AX). The company under study is Australia and New Zealand Banking Group Limited (ANZ.AX)

References

Australia and New Zealand Banking Group Limited (ANZ), 2018. Profile, business summary, historical data. Yahoo! Finance. Retrieved from https://au.finance.yahoo.com/?utm_source=Marketing&utm_medium=SEM&utm_term=SEM_News&utm_content=non_branded_%2Byahoo%20%2Bfinance&ncid=googlesem_semnews_cc33emalx9k.

Brown, G., 2006. Australia and New Zealand Banking Group (ANZ): Aligning community strategy with business strategy. The Journal of Corporate Citizenship, (22), p.18.

Reddy, K. and Clinton, V., 2016. Simulating Stock Prices Using Geometric Brownian Motion: Evidence from Australian Companies. Australasian Accounting, Business and Finance Journal, 10(3), pp.23-47.

Yang, Z. and Aldous, D., 2015. Geometric Brownian motion model in financial market. University of California, Berkeley.

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