Stochastic Processes and Geometric Brownian Motion in Business Analytics

This presentation covers the basics of Stochastic Processes and Geometric Brownian Motion (GBM) in Business Analytics. It explains the assumptions, conditions, limitations and applications of GBM in stock markets. The presentation also includes a time series plot for BP share price and a table for BP share price drift and volatility.

Business Analytics
welcome

Introduction to Stochastic Processes
• Continuous stochastic process can be written in
differential form ,
ds= μSdt + σ SdW
Where,
• σt is the variance per unit time and is usually used to
giver order of random noise
• μt is the expected return per unit time thus it
controls drift.
• Both σt and μt are not constant. They vary with time.

Assumptions of Stochastic Modelling
• The stock price follows ds= μSdt + σ SdW
• All securities are assumed to be infinitely
divisible.
• The assumption that no dividends at the end of
trading year is made.
• It is assumed that trading is continuous and no
short selling
• taxes and costs of transactions are assumed to be
zero

Conditions For Share Price To Qualify as a GBM

BP share price drift and volatility table
Value
mean , μ -0.0006
Daily variance 0.000164
annualized variance 0.059722
annualized standard deviation, δ 0.244381

A Time Series Plot for BP Share Price
7/19/2018 8/8/2018 8/28/2018 9/17/2018 10/7/2018 10/27/2018 11/16/2018
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A time series plot for Closing price,S(t) for British Petrolium p.l.c.
(BP) share price
Date
share price

Limitations of GBM
Geometric Brownian motion (GBM) is a stochastic differential equation used to define
continuous-time stochastic process (Reddy and Clinton, 2016; Kanniainen,
2009).Geometric Brownian motion has wide applications. One of the many
applications of GBM is prediction 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). Even though GBM takes care of time during calculation of drift
and volatility, it has four limitations:
• The returns of shares are log-normally distributed
• GBM assumes that the company whose share prices are under consideration is a
going concern and its stock prices are continuous in time
• The distribution of continuous compound returns follow a normal distribution
• GBM also assume that all stocks/shares follow a Markov process where the only
current price can be used to predict future share prices.

References
• 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].
• Kanniainen, J.(2009). Can properly discounted projects follow geometric Brownian
motion?’, Mathematical Methods of Operations Research, vol. 70, no. 3, pp. 435–450,
viewed 3 December 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.
• 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.
.

Appendix (share price on 16th November 2017)

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