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Archives of Business Research – Vol. 10, No. 5
Publication Date: May 25, 2022
DOI:10.14738/abr.105.12350. Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock
Exchange. Archives of Business Research, 10(05). 140-152.
Services for Science and Education – United Kingdom
The Nexus Between Black-Scholes-Merton Option Pricing and
Risk: A Case of Ghana Stock Exchange
Joseph Antwi Baafi
Lecturer, Akenten Appiah-Menka University of Skills
Training and Entrepreneurial Development, Kumasi, Ghana
Faculty of Business Education, Department of Accounting
Mr. Eric Effah Sarkodie
Senior Lecturer, Akenten Appiah-Menka University of Skills
Training and Entrepreneurial Development, Kumasi, Ghana
Faculty of Business Education, Department of Accounting
INTRODUCTION
In financial capital market trading, individuals who are involved has the ultimate aim of making
gains (Don and Brooks, 2008). The gains could be made in a number of ways, but one obvious
way is through the reduction of risk associated with the trade (Don and Brooks, 2008). An
individual who owns stocks could decide to use options to protect the stock. That is if the
individual decide to build a portfolio of stock and options. According to Chicago Board Option
Exchange Market (CBOC, 2021), percentage of option contract traded between Jan 2021 to June
2021 average about 80 percent of total trade. In 2020, the average daily volumes of option
contracts amount to 20,466,938 (Options Clearing Corporation, 2021).
The options used as protective mechanism against losses are valued or priced. The main models
used in option pricing area Binomial Model (Benninga & Wiener, 1997), Trinomial Option
Pricing model (Rubinstein, 2000), Black-Scholes-Merton Option Pricing Model (Macbeth and
Merville, 1979), Heston Model (Rouah, 2013), Log-Levy Process (Kleinert and Korbel (2016)
and Double-Fractional Option Pricing Model. The binomial option pricing model is a model
that uses an iterative procedure and allows for nodes specification, during the time span
between the valuation date and option's expiration date. Binomial model allows for two
outcomes. The Trinomial Model however allows for three outcomes. The possible outcomes in
a time period are greater than, the same as and less than the current value. Both binomial and
trinomial pricing model were written in discrete time. The Black-Scholes-Merton model
considers option pricing in continuous time. The Heston Model was developed by Steven
Heston to use to price European Options. This is a type of stochastic volatility (Aloe, 2012).
Among these methods, Black-Scholes-Merton model is widely used by players in derivative
market in pricing an option. This model has been developed in modern times so that there is a
Black-Scholes-Merton calculator used to calculate option prices. This calculator gives an
appropriate value of the price of an option. An appropriate value would also help to protect the
underlying assets on which the option is purchased.
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Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange. Archives of
Business Research, 10(05). 140-152.
URL: http://dx.doi.org/10.14738/abr.105.12350
Derivative market for that matter, option markets has been developed in modern times mostly
in developed countries. Even though history of derivatives market date back to the fourth
century BC (Weber, 2009), the development was quick in the 19th century. However, such rate
of development cannot be seen in developing countries especially Africa. Out of the 32 major
trading options exchange market in the world, only one is found in Africa, which is the South
Africa Futures Exchange. In West Africa, there have been a number of discussions about option
exchange market but certain challenges are not making it possible (Cedric et al, 2013). The
challenges include physical market structure, product quality, standardization and grading
issues, traceability and exchange trading, price transparency and price volatility (Cedric et al,
2013).
Such problems inhibiting the establishment of organized option exchange markets are also very
common in Ghana. The establishment of option market has been postponed because of these
challenges (Whitehouse, 2020). But these challenges do not prevent researchers from
considering investigation of possible option trading activities in Ghana. Again, since risk exist
on every stock exchange trading, it would be important to consider various ways by which risk
could be mitigated. One of the ways could be to develop an option market. Developing an option
market means pricing options or valuation of option on the Ghana stock exchange. As said
before one theory of option pricing Black-Scholes-Merton model.
The main objective of this study is therefore to know how Black-Scholes-Merton model could
be used to help in appropriate option value. Again a risk assessment of stocks on GES would be
done. This would enable us see how Black-Scholes valuation of option would help reduce the
riskiness of an assets. This correct option value could also be used to mitigate risk on Ghana
Stock Exchange.
LITERATURE REVIEW
The roots of Black-Scholes-Merton formula date back to the 19th century (Don and Brooks,
2008). In the 1820s, a Scottish scientist, Robert Brown, observed the motion of pollen
suspended in water and noticed that the movements followed no distinct pattern, moving
randomly, independent of any current in water. This was known as Brownian Motion (Don and
Brooks, 2008). After a series of development by French Mathematician, Louis Bachelier (1900),
a renowned academician Albert Einstein (1900) and a Japanese Mathematician Kiyoshi Ito
(1951), Fischer Black and Myron Scholes used these developments as a basis for pricing assets.
The first approach of pricing assets was Capital Assets Pricing Theory and the other approach
was the use of Stochastic Calculus (Merton and Scholes, 2013). At the same time, Robert Merton
developed option pricing separately but wrote the entire mathematical equation in continuous
time period.
Ever since its inception, a number of empirical studies have been conducted over the years by
researchers. Notable among them are Macbeth and Merville (1979), Lee et al (2005), Song and
Wang (2013), Chesney and Scott (1989) and Duan (1995).
Macbeth and Merville (1979) found that option prices predicted by Black-Scholes model are on
average less (greater) than market prices for in-the-money (out-of-the-money) options. Again,
with the exclusion of out-of-the-money options with less than ninety days to expiration, the
extent to which Black-Scholes model underprices out-of-the-money option increases with the
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extent to which the option is out-of-the-money, and decreases as time to expiration decreases.
Lastly, the prices given by Black-Scholes model with out-of-the-money options which are less
than 90 days to expiration are greater than the market prices. But this method involved lots of
work. Song and Wang (2013), used a modern computational technique known as time- fractional Black-Scholes equation. This technique uses differential equation to the pricing
theory of option and concluded that this method requires less computational work.
Kleinert and Korbel (2016) also improved upon Black-Scholes model and showed that double- fractional differential equation predicted options prices better. Option prices given by double- fractional differential equation provides a more reliable hedge when included in a portfolio of
stocks. However, Yavuz, and Özdemir (2018) also defined a new fractional operator by the use
of iterative method. The fractional Black-Scholes equation was redefined as conformable
fractional Adomian decomposition method (CFADM) and conformable fractional modified
homotopy perturbation method (CFMHPM). The fractional Black-Scholes was solved using
these two methods. It was found that the resulting models were efficient and powerful
techniques in predicting option prices. Kumar et al (2012) had earlier used Laplace homotopy
perturbation method, which is a combined form of Laplace transform and homotopy
perturbation method. This method gave a quick and accurate solution to fractional Black
Scholes equation with boundary condition for a European option pricing problem. Other
researchers such as Jena and Chakraverty (2019), Ghandehari and Ranjbar (2014) and Akrami
and Erjaee (2015) have all used fractional Black Scholes approach in option pricing.
A number of studies have sought to reduce biases in Black-Scholes model notably Mitra (2012)
who sought to combine artificial neural network approach with Black-Scholes. Mitra (2012),
acknowledged that Black and Scholes formula showed certain systematic biases and thus
amongst the non-parametric approaches, Artificial Neural Networks (ANN) is found to be more
appropriate. Mitra (2012), conclude that model used in ANN method gave a much superior
results compared to Black-Scholes Model. ANN approaches was able to accounts for difference
in option value by automatically adjusting changes in relation to elements such as volatility.
Contreras et al (2010) also used quantum model and sought to interpret Black-Scholes equation
from the viewpoint of quantum mechanics. Another means of reducing biases in option value
was suggested by Burkovska et al (2015). The main procedures are POD-Greedy and Angle- Greedy procedure for the construction of the primal and dual reduced spaces. Numerical
evidence of reduction in biases were provided to prove real world option price approximation
quality.
Option price valuation has been dominated by literature in developed countries. This is
basically because the development of derivative markets started a long time. The same volume
and amount of literature cannot be said to have been cited using the African context.
Nevertheless, some works still exist. Flint and Maré (2017) considered the price of an option
when an underlying is assumed to display long memory using Black-Scholes model in the
context of South Africa. The results shown that Black-Scholes approach always admits a non- constant implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year
implied volatility is independent of the Hurst exponent and equivalent to fractional volatility.
This means equity index can accurately be modeled and this could be useful in derivatives
trading and delta hedging. In Nigeria, Okaro et al (2018), employed Black-Scholes model in
pricing palm-oil futures. The procedure in determining the value involved creating the model’s
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Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange. Archives of
Business Research, 10(05). 140-152.
URL: http://dx.doi.org/10.14738/abr.105.12350
implied contract prices using historical prices at seasonal peak and/or dip periods for ten years.
These contracts prices were related with assessed unit profit/loss margins of the same
historical periods and correlated with Pearson’s Coefficient. Okaro et al (2018) concluded that
B-S model is appropriate for valuing unstructured over-the-counter seasonal contracts. Other
studies on option pricing that do not necessary used Black-Scholes model include Venter and
Maré (2020), Du Plooy and Venter (2021), Mpapalika (2019), Jordaan and Van Rooyen (2010)
and Holman et al (2011).
In Ghana few works on option pricing especially Black-Scholes exist. Notable study involving
Black-Scholes is Antwi and Oduro (2018). In this study, the authors found that option prices
could be determined using Ghanaian stocks. However when the option have low volatilities and
low initial stock price, the call value is zero in most cases but put prices do not tend to zero
given the same conditions. Another study which is not directly related to Black-Scholes but has
something to do with the derivative market is Obeng-Darko (2019). In all these studies, Black- Scholes determination of option prices did not give a sense of risk reduction on the Ghana Stock
Exchange market. But the essence of options is to help individual reduce risk in a given
portfolio. This study therefore seeks to calculate option prices for companies listed on Ghana
Stock Exchange and determine show such option prices could help individual investors reduce
risk and increase profitability.
Methodology
The binomial option pricing model is derived by forming a hedge portfolio consisting of a long
position in shares and a short position in calls. In a two period model, the relative number of
shares to calls changes so the investor must do some trading to maintain the riskless nature of
transaction (Don and Brooks, 2008). If this is done, the portfolio will earn a risk-free rate if and
only if the option is priced by the formula obtained. If the option trades at any other price, the
law of one price is violated and the investor can earn an arbitrage profit (Merton and Scholes,
2013). In Black-Scholes-Merton model, trading occurs continuously, but the general idea is the
same. A hedge portfolio is established and maintained by constantly adjusting the relative
proportions of stock and options, a process called dynamic trading. The end result is obtained
through complex mathematics. This study follows Hull (2015) derivation of Black-Scholes
formula. Starting from a binomial option pricing model, the payoff from European Call option
is
max(�!�"�#$" − �, 0)
where S0 is initial stock price, d is proportional down movement, u is proportional up
movement, j is an upward movement, n-j is downward movement, X is strike price and T is time.
For binomial distribution, the probability of up and down movement is
�!
(� − �)!�!
�"(1 − �)#$"
The expected payoff for call option is
2 �!
(� − �)!�!
�"(1 − �)#$"
#
"%!
max(�!�"�#$" − �, 0)
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Discounting this at a risk-free rate, the option price is
� = �$&' 2 �!
(� − �)!�!
�"(1 − �)#$"
#
"%!
max(�!�"�#$" − �, 0) ... ... ... ... ... ... ... ... ... ... ... ... . .1
The call value in equation 1 is nonzero when,
�!�"�#$" > � or �� ;
(!
) < > −���(�) − (� − �) ln(�)
Because � = � *'⁄# "
��� � = � *'⁄# #"
�h�� ��������� �������
�� G
�!
�H > ��J�
� − 2��J�
�
�� � >
�
2 − ln (�!⁄�)
2�O�⁄�
Therefore equation 1 is
� = �$&' ∑ #!
(#$")!"!
�"(1 − �)#$" "01 (�!�"�#$" − �)
Where
� = �
2 − ln (�!/�)
2�O�/�
For convenience we define
�2 = 2 �!
(� − �)!�!
�"(1 − �)#$"�"�#$"
"01
... ... ... ... ... ... ... ... ... ... .2
�3 = 2 �!
(� − �)!�!
�"(1 − �)#$"
"01
... ... ... ... ... ... ... ... ... 3
So that
� = �$&'(�!�2 − ��3) ... ... ... ... ... ... ... ... ... ... ... ... . .4
When there is n trials and p is probability of success, probability distribution of the number of
success is approximately normally distributed with mean � and standard deviationO�(1 − �).
For large samples n
�3 = � X � − �
O�(1 − �)
Y ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . .5
Where N = cumulative probability distribution function. Substituting for �, we get
�3 = � [ ln ;
�!
� <
2�√�O�(1 − �)
+
√� ;� − 1
2<
O�(1 − �)
^ ... ... ... ... ... ... ... ... ... .6
From the equation for p
� = �&'/# − �$5*'/#
�5*'/# − �$5*'/#
By expanding the exponential functions in a series we see that, as n tends to infinity, p(1-p)
tends to 1⁄4 and √� ;� − 2
3
< ����� ��
(� − �3/2)√�
2�
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Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange. Archives of
Business Research, 10(05). 140-152.
URL: http://dx.doi.org/10.14738/abr.105.12350
So that equation 6 becomes
�3 = � [
ln ;
�!
� < + (� − �3/2) �
�√� ^ ... ... ... ... ... ... ... ... ... ... ... ... ... ... 7
And
�2 = 2 �!
(� − �)!�!
(��)"
"01
[(1 − �)�]#$" ... ... ... ... ... ... ... ... ... ... ... ... 8
Define
�∗ = ��
�� + (1 − �)� ... ... ... ... ... ... ... ... ... ... ... . . ... ... 9
1 − �∗ = (1 − �)�
�� + (1 − �)�
Thus equation 8 could be written as
�2 = [�� + (1 − �)�)]# 2 �!
(� − �)!�!
(�∗)"(1 − �∗)#$"
"01
Since �&'/# = �� + (1 − �)�,�h��
�2 = �&' 2 �!
(� − �)!�!
(�∗)"(1 − �∗)#$"
"01
This indicates �2 involves a binomial distribution where the probability of an up movement is
p* rather than p. By approximation we get
�2 = �&'� X �∗ − �
O�∗(1 − �∗)
Y
And substituting for � we have
�2 = �&'� X ln (�!/�)
2�√�O�∗(1 − �∗)
+
√�(�∗ − 1/2)
O�∗(1 − �∗)
Y
Substituting for u and d in equation 9 we get
�∗ = e �&'/# − �$5*'/#
�5*'/# − �$5*'/#
f e
�5*'/#
�&'/# f
By expanding the exponential functions and n tends to infinity, p*(1-p*) tend to 1⁄4
and √� ;�∗ − 2
3
< ����� ��
7&85$/39√'
35
with the results that
�2 = �&'� [
ln ;
�!
�< + (� + �3/2)�
�√� ^ ... ... ... ... ... ... ... ... ... ... 10
From equations 4, 7 and 10
� = �!�(�2) − ��$&'�(�3) ... ... ... ... ... ... ... ... ... ... ... .11
Where
�2 = ln(�!/�) + (� + �3/2)�
�√�
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�3 = ln(�!/�) + (� + �3/2)�
�√� = �2 − �√�
N(d1), N(d2) = cumulative normal probabilities
� = annualized volatility (standard deviation) of the continuously compounded (log) return on
the stock
r = continuously compounded risk-free rate.
For Put Option where
� (�!, �, �) = �(�!, �, �) − �! + ��$&'
Letting P stand for � (�!, �, �) gives the Black-Scholes-Merton option pricing model:
� = ��$&'[1 − �(�3)] − �![1 − �(�2)],
Where d1 and d2 are the same as in call option pricing model. Based on the properties of
standard normal distribution, the put option can also be represented as
� = ��$&'�(−�3) − �!�(−�2) ... ... ... ... ... ... ... ... ... ... 12
The study thus would use equation 11 and 12 to compute option prices for stocks on Ghana
Stock Exchange.
Data
Date used for the study was from Ghana Stock Exchange January 2020 to December 2020.
Variables used to calculate the possible call and put prices are stock price, strike price, time to
expiration, volatility and risk free interest.
Results and Discussion
In determining the value of Call and Put Options, we shall consider two possible scenarios:
when the option would be in-the-money and out-of-the-money.
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Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange. Archives of
Business Research, 10(05). 140-152.
URL: http://dx.doi.org/10.14738/abr.105.12350
Table 1: Situation the Call Option is in-the-money
Companies Stock
Price
Strike
Price
Time to
Expiration
(Years)
Volatility
Risk Free
Interest
Rate
Call
Price
Put
Price
AngloGold Ashanti 37 30 1 0 0.169 11.66 0.00
Aluworks Limited 0.11 0.8 1 0.005 0.169 0 0.57
Benso Oil Palm
Plantation 2.00 1.4 1 0.329 0.169 0.82 0.00
CalBank PLC 0.69 0.3 1 0.106 0.169 0.44 0.00
Clydestone Ghana
Limited 0.3 0.1 1 0 0.169 0.22 0.00
Camelot Ghana 0.11 0.4 1 0.007 0.169 0 0.23
Cocoa Processing
Company 0.03 0.01 1 0.005 0.169 0.020 0.00
Enterprise Group PLC 1.4 0.8 1 0.117 0.169 0.02 0.14
Ecobank
Transnational
Incorporation
0.08 0.01 1 0.009 0.169 0.07 0.00
Fan Milk Limited 1.08 0.7 1 1.34 0.169 0.49 0.00
Ghana Commercial
Bank Limited 4.05 2.50 1 0.50 0.169 2.00 0.07
Guinness Ghana
Breweries PLC 0.9 0.2 1 0.28 0.169 0.73 0.00
Golden Star
Resources Limited 9.5 7.3 1 0 0.169 3.33 0.00
Produce Buying
Company limited 0.03 0.01 1 0 0.169 0.020 0.00
PZ Cussons Limited 0.38 0.20 1 0 0.169 0.21 0.00
Standard Chartered
Bank Ghana PLC 16.31 12.00 1 1.84 0.169 6.18 0.00
Sam Wood Limited 0.05 0.01 1 0 0.169 0.040 0.00
Total Petroleum
Ghana PLC 2.83 1.9 1 0.28 0.169 1.23 0.01
Unilever Ghana PLC 8.29 5 1 2.39 0.169 4.07 0.00
Ghana Oil Company
Limited 1.5 0.9 1 0.10 0.169 0.74 0.00
Tullow Oil PLC 11.92 9 1 0.007 0.169 4.32 0.00
NewGold Issuer
Limited 105 99 1 12.23 0.169 21.52 0.13
Mega African Capital 5.98 4 1 0 0.169 2.60 0.00
Meridian-Marshalls
Holdings 0.11 0.08 1 0 0.169 0.04 0.00
HORDS Limited 0.1 0.06 1 0 0.169 0.06 0.00
Ecobank Ghana
Limited 7.2 5.0 1 0.84 0.169 2.98 0.00
Agricultural
Development Bank 5.06 3.0 1 0 0.169 2.47 0.00
MTN Ghana 0.64 0.40 1 0.044 0.169 0.30 0.00
Source: Author Calculation, 2021
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Table 1 above shows the value of call and put option prices calculated using black-scholes
formular for selected companies listed on GSE. The table assumes a condition where the strike
price is above the stock price. With the exception of Aluworks and Camelot Ghana, the
remaining companies have a positive value of call options. This indicates that if these
companies’ stocks are underlying assets, investor exercising the right to buy a financial
derivative on these stocks would make gains. However, any attempt to sell would result in a
loss. The only move an investor could make to gain on the financial market is to buy a call
option. For Aluworks and Camelot, a move to buy a call option would results in a loss.
The table also shows that with the exception of Aluworks Limited, Camelot Ghana, Ghana
Commercial Bank, Total Petroleum PLC, NewGold Issuer and Enterprise Group PLC, the value
of Put Option for the remaining companies were zero.
A list of companies that would give an investor, the right to either buy or sell under the current
condition of strike price above stock price is Enterprise Group PLC, Ghana Commercial Bank,
Total Petroleum and NewGold Issuer. This invariably makes trading financial derivatives on the
stock market one-sided. For derivatives market to survive there should be investor on both
sides of the exchange. This raises huge questions on GSE to entertain derivatives market
activities. This one-sided trade could be due to low volatility on the Ghanaian market. Figure 1
shows a pictorial graph of volatility against call option value. The figure shows that volatility
for most part was low and stable. A higher spike seen quickly reverted to the low levels
previously observed. For certain parts of the diagram, volatility was almost zero.
Figure 1: Call Option value against Volatility
0
5
10
15
20
25
0 0.106 0.005 1.34 0 1.84 2.39 12.23 0 0.044
Call Price
Volatility
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Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange. Archives of
Business Research, 10(05). 140-152.
URL: http://dx.doi.org/10.14738/abr.105.12350
Table 2: Situation the Call Option is out-of the-money
Companies Stock
Price
Strike
Price
Time to
Expiration
(Years)
Volatility Risk Free
Interest
Rate
Call
Price
Put
Price
AngloGold Ashanti 37 44 1 0 0.169 0.00 0.16
Aluworks Limited 0.11 0.19 1 0.005 0.169 0.00 0.5
Benso Oil Palm
Plantation
2.00 2.6 1 0.329 0.169 0.00 0.2
CalBank PLC 0.69 0.90 1 0.106 0.169 0.00 0.07
Clydestone Ghana
Limited
0.3 0.6 1 0 0.169 0.00 0.21
Camelot Ghana 0.11 0.17 1 0.007 0.169 0.00 0.03
Cocoa Processing
Company
0.03 0.05 1 0.005 0.169 0.00 0.01
Enterprise Group
PLC
1.4 1.8 1 0.117 0.169 0.28 0.00
Ecobank
Transnational
Incorporation
0.08 0.13 1 0.009 0.169 0.00 0.03
Fan Milk Limited 1.08 1.65 1 1.34 0.169 0.00 0.32
Ghana Commercial
Bank Limited
4.05 5.26 1 0.50 0.169 0.66 1.05
Guinness Ghana
Breweries PLC
0.9 1.4 1 0.28 0.169 0.03 0.31
Golden Star
Resources Limited
9.5 11.2 1 0 0.169 0.04 0.00
Produce Buying
Company limited
0.03 0.07 1 0 0.169 0.00 0.03
PZ Cussons Limited 0.38 0.61 1 0 0.169 0.00 0.14
Standard Chartered
Bank Ghana PLC
16.31 20.51 1 1.84 0.169 0.77 1.78
Sam Wood Limited 0.05 0.04 1 0 0.169 0.02
Total Petroleum
Ghana PLC
2.83 3.21 1 0.28 0.169 0.37 0.25
Unilever Ghana PLC 8.29 10.68 1 2.39 0.169 0.51 1.24
Ghana Oil Company
Limited
1.5 3.41 1 0.10 0.169 0.00 1.38
Tullow Oil PLC 11.92 14 1 0.007 0.169 0.08 0.00
NewGold Issuer
Limited
105 129 1 12.23 0.169 3.48 7.42
Mega African Capital 5.98 7.56 1 0 0.169 0.00 0.40
Meridian-Marshalls
Holdings
0.11 0.20 1 0 0.169 0.00 0.16
HORDS Limited 0.1 0.9 1 0 0.169 0.00 0.66
Ecobank Ghana
Limited
7.2 8.47 1 0.84 0.169 0.27 0.22
Agricultural
Development Bank
5.06 8.94 1 0 0.169 0.00 2.49
MTN Ghana 0.64 0.98 1 0.044 0.169 0.00 0.19
Source: Authors Calculation, 2021
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Table 2 above shows the condition where the strike price is below the stock price. The value of
Put Option showed a positive side for all listed stocks with the exception of Enterprise Group
PLC, Golden Star Resources Limited and Tullow Oil PLC where the value is zero.
The price of call option for 18 out of 28 listed stocks showed a value of zero. Ten (10) listed
stocks showed a positive call value. The table also showed that seven (7) companies namely
Ghana Commercial Bank, Guinness Ghana Breweries PLC, Standard Chartered Bank Ghana PLC,
Total Petroleum Ghana PLC, Unilever Ghana PLC, NewGold Issuer Limited and Ecobank Ghana
have a value for both call and put options. This means stocks of 21 companies cannot be an
underlying asset for trading financial derivatives because it will only be a one sided trade under
the current condition. The reason for this one sided trade phenomena can be attributed to low
volatility on the stock market. Figure 2 shows volatility rate against Put option value. The point
of low volatility is also shown with some part almost equal zero. As seen previously, a higher
spike in volatility quickly reverted to its lower position again.
Figure 2: Put Option Value against Volatility
Taking the two conditions of stock price above strike price and strike price above stock price
together, three companies namely Ghana Commercial Bank, Total Petroleum Ghana and
NewGold Issuer Limited can be underlying assets for trading financial derivatives on the stock
market.
Volatility on GSE
The issue of low volatility has come up strongly in previous discussion of the value of Options.
To this end, the authors decided to consider the concept. Figure 3 shows a line graph of volatility
rate of stocks on GES. The figure shows that the volatility rate is almost close to zero. The
calculated volatility rate for listed companies is 2.3. If an outline of 12.23 is dropped from the
data, the volatility rate is 0.6. A radar graph is also shown in figure 4 for a much clearly view
Figure 3: Volatility Rate of stocks on GES
0
1
2
3
4
5
6
7
8
0
0.005
0.329
0.106
0
0.007
0.005
0.117
0.009
1.34
0.5
0.28
0
0
0
1.84
0
0.28
2.39
0.1
0.007
12.23
0
0
0
0.84
0
0.044
Put Option Value
Volatility
Page 12 of 13
151
Baafi, A. J., & Sarkodie, E. E. (2022). The Nexus Between Black-Scholes-Merton Option Pricing and Risk: A Case of Ghana Stock Exchange. Archives of
Business Research, 10(05). 140-152.
URL: http://dx.doi.org/10.14738/abr.105.12350
Figure 4: Radar Graph of Volatility
Source: Authors Construct, 2021
A comparison of the Ghanaian rate with other rate elsewhere reveals a sharp difference. The
table below shows a sample of stock market and volatility around the world. Table 3 shows that
the highest volatility was 42 from US: S&P and lowest was 18.80 from Asia Dow. The average
volatility for these 8 listed sample markets was 26.68. This is at complete variance from
situation of GSE.
Table 3: Stock Markets and Volatility for sample market
Market Volatility
Asia Dow 18.80
Australia S&P 23.18
Japan: Nikkei 225 26.95
Singapore: Straits Times 22.85
US: Dow Jones 28
US: S&P 42
New York Stock Exchange 32.02
Chicago Board Option Exchange 19.66
Source: Wall Street Journal, 2021
-5
0
5
10
15
0
0.005
0.329
0.106
0
0.007
0.005
0.117
0.009
1.34
0.5
0.28
0
0
0
1.84
0
0.28
2.39
0.1
0.007
12.23
0
0
0
0.84
0
0.044
Volatility
Page 13 of 13
152
Archives of Business Research (ABR) Vol. 10, Issue 5, May-2022
Services for Science and Education – United Kingdom
It could be realized that for there to be two-sided and possible gains from trading a financial
derivative, volatility should be high in other to increase the riskiness of an underlying assets
and thus make gains. The lower the volatility, the lower the riskiness and the higher the
volatility, the higher the riskiness. Since Ghana Stock Exchange has lower volatility, traders
should worry less about riskiness. This is because almost zero volatility would lead to a gain
which is risk free.
CONCLUSION AND RECOMMENDATION
This study set out to use Black-Scholes-Merton option pricing model to calculate appropriate
option value for companies listed on the GES and to undertake an assessment of the riskiness
of stocks. The study found that with the exception of Aluworks and Camelot Ghana, the
remaining companies have a positive value of call options. A list of companies that would give
an investor, the right to either buy or sell under the current condition of strike price above stock
price is Enterprise Group PLC, Ghana Commercial Bank, Total Petroleum and NewGold Issuer.
This invariably makes trading financial derivatives on the stock market one-sided. This raises
huge questions on GSE to entertain derivatives market activities. This one-sided trade could be
due to low volatility on the Ghanaian market. The results also found that the price of call option
for 18 out of 28 listed stocks showed a value of zero. Ten (10) listed stocks showed a positive
call value. The table also showed that seven (7) companies namely Ghana Commercial Bank,
Guinness Ghana Breweries PLC, Standard Chartered Bank Ghana PLC, Total Petroleum Ghana
PLC, Unilever Ghana PLC, NewGold Issuer Limited and Ecobank Ghana have a value for both call
and put options. The above indicates that Ghana Stock Exchange market is not ready for option
pricing and financial derivatives activities in general.