Real World Analytics: Using Geometric Brownian Motion to Predict Future Share Prices

This text discusses the use of geometric Brownian motion (GBM) to predict future share prices. It covers the assumptions and limitations of GBM, how to calculate drift and annual volatility, and includes a time series plot for GSK share price. The text also explores how to simulate expected share price values and compares them to published data. The Real World Analytics course is offered at a university and covers financial analysis and modeling using Excel and VBA.

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

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Number 1
σ t is the variance per unit time which gives more information on the order of random noise
μt is the expected return per unit time therefore it controls drift
σ t and μt are not constant. They vary with time.
Assumptions
i) The stock price follows ds= μSdt +σ SdW
ii) It is assumed that there are no dividends
iii) No short selling and that trading is continuous
iv) It is assumed that no taxes or costs of transaction paid
v) All securities are infinitely divisible
vi) No risk associated risks for lending or borrowing unlimited times.
Number 2
(Attached in excel file)
7/19/2018 8/8/2018 8/28/2018 9/17/2018 10/7/2018 10/27/2018 11/16/2018
36
37
38
39
40
41
42
A time series plot for Closing price,S(t) for GSK share price
Date
share price

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Number 3
For a share price to be represented as geometric Brownian motion must meet the following
condition:
S(t) = S0 eW t
Where W t = W0 + σ Bt + μt
From the above equation, S(t) is lognormally distributed with W 0 + σ Bt and variance σ 2 t
r(t) = S ( t )−S (t−1)
S (t−1) ~ N( μ,σ )
Where
S ( t ) is the closing price for time, t
S ¿) is the closing price for period t-1
r(t) = S ( t )−S (t−1)
S (t−1) ~ μδt +σ ∅ √δ t
δt = 1 day = 1
30 month
μ is the mean
∅ ~ N(0,1)
σ is the annualized volatility
Simplifying further 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)
Thus mean, μ = 1
n ∑
i=1
n
r (t)

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Number 4
In excel the drift and annual volatility of share prices were calculated using the following
formula:
μ = 1
n ∑
i=1
n
r (t)
Standard Deviation/Volatility, δ =
√ 365 ∑
i=1
n
r (t)2
n
Excel output
Value
mean , μ -0.00097
Daily variance 0.00011
annualized variance 0.039987
annualized standard
deviation, δ 0.199967
Number 5
I chose GlaxSmithKline plc (GSK) closing share prices. The annual volatility of the share prices
is (δ) is 0.2. The drift (μ) of the share prices was found to be -0.0001. According to
(MarketChameleon.com, 2018), the volatility of GSK in 2018 is 0.40 which show is two time the
calculated volatility. The margin between the calculated volatility and the volatility could be as a
result of errors and share prices fluctuation between 31st October 2018 and 23rd November 2018.
Number 6

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The expected share price value on 16th November 2017 through simulation is 34.655 while the
share price on the same date from a published data (yahoo finance) is 35.19. There is little
difference in the prices of the simulated share price and the published price. This suggests that
geometric Brownian motion can be used to predict future share prices with accuracy.
Number 7
Geometric Brownian motion (GBM) is a stochastic differential equation or Ito process model
used to define continuous-time stochastic process. GBM has and is widely used to predict future
share prices in stock markets. Geometric Brownian motion is a product of a stock volatility and
Weiner process which takes into consideration random volatility and time (Brewer, Feng, and
Kwan, 2018; Sengupta, 2010). Despite GBM ability to incorporate time and volatility in models,
it has the following limitations:
i. GBM assumes that the company whose share prices are under consideration is a
going concern and its stock prices are continuous in time
ii. GBM also assume that all stocks/shares follow a Markov process where the only
current price can be used to predict future share prices.
iii. The distribution of continuous compound returns follow a normal distribution
iv. The returns of shares are log-normally distributed
According to the GlaxSmithKline plc (GSK), share prices from August 2018 to 31 October 2018
show that the drift ( μ) is -0.0001 and the annual volatility (δ) was found to be 0.2. Using GBM
to predict share price on 16th November 2017 proved to be reliable since it was very close to the
share price on published papers on the same data. This suggests that geometric Brownian motion
can be used to predict future share prices with accuracy.

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Reference
Brewer, K., Feng, Y. and Kwan, C. (2018). Geometric Brownian motion, option pricing, and
simulation: some spreadsheet-based exercises in financial modeling. [online]
Epublications.bond.edu.au. Available at: http://epublications.bond.edu.au/cgi/viewcontent.cgi?
article=1131&context=ejsie [Accessed 24 Nov. 2018].
MarketChameleon.com. (2018). Glaxosmithkline PLC (GSK) Stock Quote | Price Chart | Volume
Chart. [online] Available at: https://marketchameleon.com/Overview/GSK/ [Accessed 24 Nov.
2018].
Sengupta, C. (2010). Financial analysis and modeling using Excel and VBA. Hoboken, N.J.:
Wiley.

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Appendix
Share price on 16th November 2017
Source: Yahoo finance