SIT718 Real World Analytics: Geometric Brownian Motion on NAB.AX

This assignment provides a comprehensive analysis of National Australian Bank Limited (NAB.AX) share prices using Geometric Brownian Motion. It begins by outlining the assumptions of stochastic processes and then proceeds to analyze the historical share prices, observing a decreasing trend in late 2018. The assignment details the application of Geometric Brownian Motion, including the formulas for calculating drift and volatility, and compares the estimated volatility with published data, discussing potential reasons for discrepancies. The simulated share price is compared to the actual price, assessing the accuracy of the Geometric Brownian Motion model. The assignment concludes with a discussion of the advantages and shortcomings of using Geometric Brownian Motion for stock price prediction, referencing relevant academic literature and CNBC data. Appendices include volatility data and historical share prices.

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REAL WORLD ANALYTICS
By Student Name
Course
Professor,
University Name
City, State
Date

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Question 1
In the stochastic process, σ t is defined as the variance per unit time. Actually, σ t is measure of
order of a random noise
μt is also known as drift measures expected return per unit time.
σ t and μt are time variant thus not constant
Assumptions
a) No transaction cost and taxes involved
b) All stock prices must satisfy the condition ds= μSdt +σ SdW.
c) No short selling
d) Trading is continuous
e) Securities are infinitely divisible
f) No dividends to shareholders at the end of trading periods
Question 2
7/19/2018 8/8/2018 8/28/2018 9/17/2018 10/7/2018 10/27/2018 11/16/2018
22
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25
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30
A time series plot for Closing prices(p(t)) for National Australian
Bank Limited (NAB.AX) share price
Date
share price

Real World Analytics 3
A time series plot for the share prices for National Australian bank Limited shows a decreasing
trend in share price from August 1st, 2018 to 31st, October 2018. Historical share prices for
National Australian Bank limited (NAB.AX) are attached in excel sheet 1. The share prices used
in this question were downloaded from www.yahoofinance.com.
Question 3
The condition “S(t) = S0 eWt ” must be satisfied by a share price in order to be considered a
Geometric Brownian Motion.
Suppose that S(t) = S0 eW t
Where W t = W0 + σ Bt + μt
It is worth noting that S(t) follows a lognormal distribution with : W 0 + σ Bt and Variance= σ 2 t
Thus rewriting returns( r(t) inform of S(t) yields;
r(t) = S ( t )−S (t−1)
S (t−1) ~ N( μ,σ )
Where:
S ( t ) = closing share price of a stock at time, t
S ¿) = closing share price of a stock at time , t-1
simplifying r(t) further gives :
r(t) = S ( t )−S (t−1)
S (t−1) ~ μδt +σ ∅ √δ t
δt = 1 day ⇒ δ t= 1
30
μ =drift or mean
∅ ~ N(0,1) (normally distributed with a mean of 0 and standard deviation of 1)
σ = annualized volatility

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Simplifying r(t) gives :
S ( t )−S (t−1)
S (t−1) = μδt +σ ∅ √ δ t
S ( t )−S (t−1) = S(t−1) μδt +S(t−1)σ ∅ √δ t
S(t) = S(t-1)(1+ μδt +σ ∅ √ δ t)
Therefore,
Mean /drift, μ = 1
n ∑
i=1
n
r (t )
Question 4
The formulas to compute drift and volatility (standard deviation) are:
μ = ∑
i=1
n
r (t)
n
σ =
√ 365 ∑
i=1
n
[r (t)]2
n
These formulas were used in the calculation of National Australian bank Limited drift and
volatility in excel (excel sheet 2).
Value
mean , μ -0.0016
Daily variance 7.55E-05
annualized variance 0.027553
annualized standard deviation, δ 0.165992
Question 5
In this assignment, I utilized historical stock prices of National Australian Bank Limited
(NAB.AX). The estimated annual volatility (σ ) of NAB.AX share prices was found to be at
0.166 with a drift of -0.0016 (see appendix 1). According to the CNBC (2018), the annual

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volatility ( σ ) of NAB.AX share prices is 0.06 which is lower than the estimated volatility. The
difference in the between the published and estimated volatility values could be as a result of
market valuations, liquidity, and structural imbalances. For instance, in developed countries,
ultra-low interest rates have hiked the share prices of property shares beyond their actual values.
Question 6
The estimated share price through simulation on 16th November 2017 is 31.95 while the
published share price on yahoo finance is estimated at 30.08. Comparing the simulated and the
published share prices, it is evident that no much difference between them thus Geometric
Brownian Motion can be a good predictor of future share prices with high accuracy level.
Question 7
Reddy and Clinton (2009) defined Geometric Brownian motion (GBM) as a stochastic differential
equation used to for continuous-time stochastic process. Application of GBM in financial field
cannot be overlooked.GBM has been used to predict with accuracy the share price of different
stocks because it takes time into consideration during the prediction process. Sengupta (2010)
found out that GBM factors volatility and time in prediction of share prices. In addition,
Sengupta (2010) argued that GBM is a product of Wiener process and volatility of a stock.
Despite the fact that GBM has advantage over other stock price forecasting techniques, it has
several shortcomings:
a) The assumption that r(t) (returns) are always lognormally distributed is not always true
b) The assumption that only current share price can be used to predict future stock prices
(Markov process) is not always true.
c) Not all continuous compound returns follow normal distribution as GBM assumes.

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d) The assumptions that company’s share price of interest is a going concern and continuous
over time is not always correct.
According to the CNBC (2018) volatility results, the annualized volatility (δ) was found to be
0.06 whereas the simulated volatility of NAB share prices is 0.166. A comparison between the
two values show no great difference thus a conclusion that Geometric Brownian motion is a
good predictor of future stock prices is valid.

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References
CNBC, N. (2018). NAB-AU: National Australia Bank Ltd - Stock Quote and News - CNBC.
[online] Cnbc.com. Available at: https://www.cnbc.com/quotes/?symbol=NAB-AU [Accessed 7
Dec. 2018].
Finance.yahoo.com. (2018). Yahoo is now part of Oath. [online] Available at:
https://finance.yahoo.com/quote/NAB.AX/history?p=NAB.AX [Accessed 7 Dec. 2018].
Krishna Reddy and Vaughan Clinton (2016).Simulating Stock Prices Using Geometric Brownian
Motion: Evidence from Australian Companies’, Australasian Accounting, Business, and Finance
Journal, Vol 10, Iss 3, Pp 23-47 (2016), (3), p. 23. doi: 10.14453/aabfj.v8i3.3.
Sengupta, C. (2010). Financial analysis and modeling using Excel and VBA. Hoboken, N.J.:
Wiley.

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Appendices
Appendix 1: Volatility of National Australian Bank
Source: CNBC 2018
Appendix 2: Share prices for NAB-AX

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Source: Yahoo finance