Capital stocks present a profitable investment opportunity. This is accompanied with the satisfaction of owning a piece of the company or business whose shares are in question (Anthony & Johnson, 2008; Ireland, et al., 2008). Capital stocks are however exposed to a lot of uncertainty with regards to the fluctuations in the stock prices. Although the capital stocks have huge potentials for return and profit, this aspect of uncertainty makes it a highly risky investment undertaking (Kiechel, 2010; French, 2017).
In order to account for the uncertainty in capital stock prices and stock trading, statistical techniques are applied. These techniques are aimed at predicting the stock prices at a future time. Such information on the future prices is valuable for prospective investors in the capital stocks trading. The information informs them on when the prices will be sufficiently low for them to buys the stocks cheaply, as well as when to sell when the prices are predicted to reach their peak over a given period. The information also gives estimates on whether a particular companies stocks are performing and likely to continue performing or not.
A statistical model applied for the analysis and prediction of stocks is the Stochastic Process Model and its modified form, the Geometric Brownian Motion model. Through this model, the returns and the volatility of the stocks of a given company can be determined. With these two parameters determined we are then able to predict the price of a stock at a future time.
This paper aims at analyzing the application of the Geometric Brownian Motion, the modified form of the Stochastic Process Model, to the analysis and prediction of share prices. As a case study, we will consider data on the capital stock prices of the British Petroleum plc company for the period between 1st of August 2018 to 31st of October 2018.
The British Petroleum plc Company is a company that deals with oil and gas exploration and supply, and is headquartered in London, United Kingdom (BP plc, 2018). It is among the largest energy companies and operates in 70 countries around the world (Safina, 2011). The primary and secondary listings of the company are in the London Stock Exchange and, the New York Stock Exchange and Frankfurt Stock Exchange respectively.
Then the μt represents the drift (Klebaner, 2012). The drift is the expected returns on the stocks at a given time t. This is determined by observing the price of the stock at times t and t-1. The two time difference gives the information on the difference in the stock prices per unit time change.
The σt, given that the variance = σ2, gives the value for the standard deviation of the expected returns of the stocks at a given time t (Klebaner, 2012). In terms of the modeling of stock prices using the Geometric Brownian Motion model, the value of the standard deviation (denoted as σt) gives the value for the volatility of the stock prices. In this context, we can define volatility as the dispersion measure for the expected returns of a stock.
These two quantities, μt and σt can be considered as constant with time. This can only be achieved if a significantly large amount of data and subsequently data points are used for the analysis. The analysis of the large dataset will allow for some degree of consistency in the data characteristics hence providing the possibility of the μt and σt being constant with change in time t.
The following assumptions are considered for the explanation of the μt and σt (Klebaner, 2012):
We assume that the data used for the analysis and prediction using the Geometric Brownian Motion model is significantly large.
We assume that the following condition on the μt is met:
We assume that the following condition on the σt is met:
This paper uses the data on the stock prices of the British Petroleum plc company. The data is considered for the period between August 1st 2018 and October 31st 2018. The data was collected from (Yahoo Finance, 2018).
The data consist of 65 observations for the daily records during the period of interest with 7 variables: Date, Open, High, Low, Close, Adj.Close (Adjusted values for the Close) and Volume. For the purpose of the research in this paper, the Adj.Close (Adjusted values for the Close) variable was taken as the variable of interest and used for the modeling and, analysis and prediction. The data had no missing values or null entries.
In order for share prices to be represented by a Geometric Brownian Motion model, its data must satisfy the condition of normality. This implies that for the Geometric Brownian Motion model to be applied, the data is assumed to be normally distributed.
Visual Test for Normality above, we observe that not all the data points fall approximately on the reference line. This implies that the distribution of the Adjusted Close (Adj.Close) data variable is significantly different from the normal distribution. Hence the data fails the normality test.
The results of the Shapiro-Wilk Test are represented in the table below:
Table 1: Shapiro-Wilk Test
|
Shapiro-Wilk Normality Test |
|
|
data: BPdata$Adj.Close |
|
|
w = 0.95206352 |
p-value = 0.01344364 |
From the results in Table 1: Shapiro-Wilk Test, we observe the p-value = 0.01344364 < 0.05. Therefore we conclude that at significance level α = 0.05, the distribution of the Adjusted Close (Adj.Close) data variable is significantly different from the normal distribution. Hence the data fails the normality test.
However at significance level α = 0.01, we observe the p-value = 0.01344364 > 0.01. Therefore we conclude that at significance level α = 0.01, the distribution of the Adjusted Close (Adj.Close) data variable is not significantly different from the normal distribution. Hence the data passes the normality test.
Thus, in general we can say that the data considered for the research satisfies the normality condition for representation by a Geometric Brownian Motion model at significance level α = 0.01.
Through Maximum Likelihood Estimation method, the values of the μt and σt can be obtained using the formulas below (Osei, 2017):
The analysis in this papers gives the drift or mean returns of the stocks for British Petroleum plc Company as -0.0008. This indicates that the company’s stocks have been having negative returns for the investors, although by very low margins. This is similar to the report on share performance for the British Petroleum plc by (Barchart, 2018).
The analysis in this paper also gives the volatility of the stocks for British Petroleum plc Company as 0.0121. This represents a very small amount of volatility, implying that the stock prices for the British Petroleum plc are more stable and hence less risky. The same opinion is given on the stability on the shares of the company by (Motley Fool, 2018).
Using a global (user generated) GBM function in R for the Geometric Brownian Motion model, the estimated value of the stock price for British Petroleum plc on 16th November 2018 (at time t = 78) is given as:
The published stock price for British Petroleum plc on 16th November 2018 from (Yahoo Finance, 2018) is 40.84. Hence the difference between the estimated and actual values is 0.1114. The model can therefore be said to be reasonably accurate in the prediction of the stock price for British Petroleum plc.
Conclusion and Discussion
Despite the Geometric Brownian Motion model used in this paper being a significant improvement to the Stochastic Process Model, the model is restricted to normally distributed data. This implies that the Geometric Brownian Motion Model cannot be applicable for instances where the tests for normality of the data fails at both α = 0.05 and α = 0.01 levels of significance. This would, by extension limit the application of the Geometric Brownian Motion Model in the analysis and prediction of stock prices.
However, various transformation techniques can be applied to the data, such as the log transformation, which will in turn make the data normally distributed as is the case for the natural logarithm (ln) transformation applied for the analysis in this paper.
From the analysis in this paper, we observe that over the period of interest, the stock for British Petroleum plc have had a marginal negative return. The value for the negative return is very small, but could represent a significant amount of money in cases where the stock holder has a lot of shares.
The value of the volatility of the stock prices of British Petroleum plc is significantly low. Thus we can conclude that the stock prices for the company are stable are present a safe investment opportunity and subsequently profitable once the stocks returns are positive.
References
Anthony, S. D. & Johnson, M. W., 2008. Innovator’s Guide to Growth. 1st ed. New York: Havard Business School Press.
Barchart, 2018. Barchart. [Online]
Available at: https://www.barchart.com/stocks/quotes/BP..LS/performance
[Accessed 23 November 2018].
BP plc, 2018. who we are. [Online]
Available at: https://www.bp.com/en/global/corporate/who-we-are.html
[Accessed 27 November 2018].
French, J., 2017. Asset Pricing With Investor Sentiment: On The Use of Investor Group Behaviour to Forecast ASEAN Markets. Research in International Business and Finance, 5(1), pp. 124-148.
Ireland, R. D., Hoskisson, R. & Hitt, M., 2008. Understanding Business Strategy: Concepts and Cases. 2nd ed. London: Cengage Learning.
Kiechel, W., 2010. The Lords of Strategy. 2nd ed. New York: Havard Business Press.
Klebaner, F. C., 2012. Introduction to Stochastic Calculus with Application. 3rd ed. New York: World Scientific Publishing.
Motley Fool, 2018. NASDAQ. [Online]
Available at: https://www.nasdaq.com/article/better-buy-exxonmobil-vs-bp-cm1063450
[Accessed 24 November 2018].
Nornadiah, R. & Bee, W. Y., 2011. Power Comparisons of Shapiro-Wilk, Kolmogorov, Lilliefors and Anderson-Darling Tests. Journal of Statistical Modelling and Analytics , pp. 21-33. 2(1).
Osei, A., 2017. Stochastic Modeling of Stock Price Behavior on Ghana Stock Exchange. International Journal of Systems Science and Applied Mathematics, 2(6), pp. 116-125.
Safina, C., 2011. A Sea in Flames:The Deepwater Horizon Oil Blowout. 1st ed. New York: Crown Publishers.
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