Abstract: The Black-Scholes-Merton model is one of the most popular option pricing models used in a market practice. This model is based on the unrealistic assumptions, including for example lack of transaction costs. While it is not possible to satisfy all the conditions of the model, it is logical to assume that perfectly liquid markets will meet them better, which will help to reduce the risk of error. The aim of the article is to measure the impact of liquidity to divergence of Black-Scholes-Merton model in comparison to real market closing prices. The result of research demonstrates moderate dependence between the volume of the WIG20 index, the volume of option transactions and a negative correlation with ILLIQ illiquidity indicator introduced by Amihud (2002). The results of the research lead to the conclusion that there is a positive correlation between the liquidity and the divergency between BSM model and the market prices.

<a href="https://dx.doi.org/10.15611/fins.2022.1.05">DOI: 10.15611/fins.2022.1.05</a>
<p>JEL Classification: C52, G12</p>
<p>Keywords: options, Warsaw Stock Exchange, WIG20,Black-Scholes-Merton model, pricing models, derivatives, index
option</p>
<h2>1. Introduction </h2>
<p>Options are risk transfer financial instruments with hedging and
speculative functions. Knowledge of option pricing is crucial for
professional investors and financial institutions, participating in
derivatives markets. One of the most popular methods of options
valuation is the Black-Scholes-Merton model. Due to its universality,
the model is widely used, and helps with the measuring the options
value. Some simplifications have been applied to build the valuation
model. These include the ability to trade any number of shares and no
transaction costs. While these assumptions are unrealistic, it can be
assumed that more liquid financial markets will reflect them better.</p>
<p>The aim of the article was to investigate the impact of liquidity on
the effectiveness of the Black-Scholes-Merton model based on the example
of the WIG20 index option. This knowledge can be essential to understand
the divergence between the BSM and market price and therefore,
facilitate reducing the risk of incorrect valuation. From the practical
standpoint, this could help with a more accurate estimation of
volatility as an input parameter in the model.</p>
<p>The subject of the effectiveness of option pricing models is not a
new concept in the literature, however the research so far has
concentrated on a comparison of individual models, including BSM to
indicate the most effective in the given period and market (Bates, 1995;
Rastogi, 2019). This paper focused on the BSM model. Thus, in regard to
the market liquidity impact on option pricing, in the literature there
are positions that indicate a more attractive valuation of options in
less liquid markets (Brenner, 2001) or indicate higher expected rates of
return in the case of less liquid options, called illiquidity premia
(Christoffersen, Goyenko, Jacobs, & Karoui, 2018). This article
examines this impact for the only options traded the on Warsaw Stock
Exchange – WIG20 index options.</p>
<p>The author examined the effectiveness of the option pricing model
from the perspective of the model risk, understood as a situation when
the model used is not able to correctly estimate the market value of the
financial instrument. Model risk may manifest itself as the result of
errors in estimating model parameters, adopting an incorrect model form
or incorrect model application. In the BSM model at the time of
valuation, all parameters except volatility are known. Therefore, it
could be assumed that the correct estimation of the variability would
allow to eliminate the model risk to a minimum. However, it should be
noted that even with the same input parameters, transaction prices may
differ from each other. To most accurately approximate the volatility
used in the study, the volatility published by the Warsaw Stock Exchange
was used.</p>
<p>Capital market liquidity is a complex concept. Market liquidity can
be understood as the ease of entering transactions. This ease may result
from many factors, such as low transaction costs. In this article the
basic methods of measuring market liquidity are reviewed, then two
selected methods are used as the basis for examining the impact of
liquidity on the effectiveness of option pricing models on the Warsaw
Stock Exchange.</p>
<p>The article consists of four sections. The first one presents the
Black-Scholes-Merton model and its assumptions, while the second one is
focused on the selected methods of measuring market liquidity. In the
third part, the research methods and results are described. The
conclusion of the study are presented in the final section.</p>
<h2>2. The
Black-Scholes-Merton option valuation model</h2>
<p>An
option is a contract which gives a one-party right to purchase or sell
the underlying asset at a predefined time in future at a specific price,
called the strike price. This is a risk-transfer financial instrument,
traded across global exchanges and over-the-counter markets. The
valuation of the option is an important topic for professional investors
and banks, due to various reasons, for example supporting the investment
decision-making process and the requirements of financial
reporting.</p>
<p>"Prices
for options and corporate obligations", a study published by F. Black
and M. Scholes in cooperation with R. Merton in 1973, was a significant
achievement in the option valuation field. It introduced a widely used
option valuation model named the Black-Scholes model. One year later, R.
Merton published his “Theory of rational option pricing” which improved
the model by adding the dividend element. The improved model, including
the dividend value, was named the Black-Scholes-Merton model and is
based on the following assumptions:</p>
<ol><li>
<p>The short-term interest rate is fixed and constant.</p>
</li>
<li>
<p>The price of the underlying asset follows a random walk, with a
variance rate proportional to the square of the share price. Therefore,
the distribution of the rates of return of the underlying asset has a
log-normal nature.</p>
</li>
<li>
<p>The dividend yield is continuous.</p>
</li>
<li>
<p>The option is European so it can be exercised only at the expiry
date.</p>
</li>
<li>
<p>There are no transaction costs.</p>
</li>
<li>
<p>There is an elevated share rate, paying only the short-term interest
rate.</p>
</li>
<li>
<p>There are no transaction limits for short sale.</p>
</li>
<li>
<p>There is no arbitrage.</p>
</li>
</ol>
<p>It should be noted that the above assumptions are difficult to meet
in the real market. For example, WIG 20 rate of returns for the analysed
period did not meet the criteria of a log-normal nature for the
Shapiro-Wilk or Lilliefors tests.</p>
<p>However, the application of the above assumptions allows for the
construction of a model in which the option price will depend on the
option price, time to expiry and other parameters, which will be
constant over time. Then, assuming there is no possibility of arbitrage,
one can create a position consisting of a long position on the
underlying and a short position in the option, the value of which will
not depend on the price of the underlying, but only on fixed parameters
and price.</p>
<p>According to the assumptions of the model, at time zero, at price
S0, strike price <em>X</em>, time to expiry <em>T</em>, price
of European call (<em>c</em>) or put (<em>p</em>) option, can be
calculated as:</p>
<p>
<span>\[c = S_{0}e^{- qT}N\left( d_{1} \right) -
{Xe}^{- rT}N\left( d_{2} \right)\]</span>
</p>
<p>
<span>\[p = {Xe}^{- rT}N\left( {- d}_{2}
\right){- S}_{0}e^{- qT}N\left( {- d}_{1} \right)\]</span>
</p>
<p>where</p>
<p>
<span>\[d_{1} = \ \frac{\ln\left(
\frac{S_{0}}{X} \right) + \left( r - q + \frac{\sigma^{2}}{2}
\right)T}{\sigma\sqrt{T}}\]</span>
</p>
<p>
<span>\[d_{2} = \ \frac{\ln\left(
\frac{S_{0}}{X} \right) + \left( r - q - \frac{\sigma^{2}}{2}
\right)T}{\sigma\sqrt{T}}\]</span>
</p>
<p><em>N (x)</em> is the cumulative probability distribution function
for a variable that is normally distributed with a mean of zero and a
standard deviation of 1.0 (Hull, 2000, p. 250).</p>
<p>The above formulas were used for the calculation of theoretical
option values in this paper. As input parameters for the valuation,
option indicators data published by the Warsaw Stock Exchange were used.
Each option series had own set of the indicators, and this fact was
considered for all the calculations. The input parameters can be defined
as:</p>
<ol><li>
<p>The <em>S0</em> price is the closing price of the WIG20
index for a given hour on a given trading day,</p>
</li>
<li>
<p>The risk-free rate is determined for each option expiry date in
accordance with the following steps:</p>
<ol><li>
<p>WIBOR capitalisation_continuous = ln (1 + WIBOR capitalisation annual
x (t / 365)) / (t / 365)</p>
</li>
<li>
<p>WIBID capitalisation_continuous = ln (1 + WIBID annual capitalisation
x (t / 365)) / (t / 365)</p>
</li>
</ol></li>
</ol>
<p>where: t – date for which a given interest rate is determined (e.g.
for WIBOR 1 week = 7 days, for WIBOR 1 month = 30 days).</p>
<ol><li>
<p>Then WIMEAN rates should be calculated. WIMEAN are the average of the
WIBOR and WIBID interest rates for the terms – 1 week, 2 weeks, 1 month,
3 months, 6 months, and 9 months, calculating the interest rates for
each expiry date by the linear interpolation of the available WIMEAN
rates.</p>
</li>
</ol>
<ol><li>
<p>Implied volatility calculated for each option series is based on the
arithmetic mean of the best buy and sell offers in the continuous
trading range in the period from 1 hour and 10 minutes before the end of
continuous trading, to 5 minutes before the end of continuous
trading.</p>
</li>
<li>
<p>The continuous dividend rate, with the capitalisation of the
portfolio of a given index for a given closing day <em>KAP_Index,</em>
the amount of dividend on KD shares and the number of shares of a given
company in the portfolio of a given index Package (Pakiet), were
determined using the following formula:</p>
</li>
</ol>
<p><span>\[DY_{Index} =
\frac{\sum_{}^{}{KDxPakiet}}{KAP\_ Index}\]</span></p>
<p>The detailed methodology for calculating the interest rate, implied
volatility and dividend rate, is presented in “Methodology for
calculating Greek coefficients for options on WIG20” (<a href="https://www.gpw.pl/pub/GPW/files/metodologia_wledzniki_opcje.pdf">https://www.gpw.pl/pub/GPW/files/metodologia_wledzniki_opcje.pdf</a>).</p>
<h2>3. Capital market liquidity
measurement</h2>
<p>The comprehensive concept of capital market liquidity is difficult to
define. Liquidity can be understood as the ease of trading in given
markets or financial instruments. The liquidity of a given financial
instrument is influenced by many factors, including transaction costs,
demand pressure, bid-ask spread, difficulty of locating counterparty and
trading volume (Amihud, 2002). The group of the most popular liquidity
measures includes (Wojtasiak, 2003; Porcenaluk 2013):</p>
<ol><li>
<p>Bid-ask spread – the difference in prices from the best sell and buy
offers, lower values may indicate greater liquidity.</p>
</li>
<li>
<p>Number of transactions – shows the activity of investors, higher
values may indicate greater liquidity of a given financial
instrument.</p>
</li>
<li>
<p>Trading volume – the number of financial instruments that have
changed owners, higher values may indicate greater liquidity.</p>
</li>
<li>
<p>Turnover value – the turnover volume multiplied by the prices in
concluded transactions, higher values may indicate greater
liquidity.</p>
</li>
<li>
<p>Amount of free-float financial instruments – the number of financial
instruments that are not held by long-term investors, higher values may
indicate greater liquidity.</p>
</li>
<li>
<p>Ratio of the volume of purchase offers to sale offers – a value close
to 1 is the potential balance between the liquidity of the demand and
supply sides.</p>
</li>
</ol>
<p>From the investor’s perspective, these measures should be compensated
by the expected rate of return. This fact should be considered in the
financial instrument valuation. Standard asset valuation theory assumes
the existence of perfectly liquid markets where any security can be
traded at no cost at all times (Amihud, 2002, p. 6). The assumptions of
the Black-Scholes-Merton model are also confirmed as it moves towards an
ideal market. It is therefore logical to assume that an increase in
market illiquidity will result in a discrepancy between the model's
valuation and the market price. The more illiquid the market, the
greater the divergence might be observed if the statement is true.</p>
<p>To measure the market liquidity influence on the Black-Scholes-Merton
model result, an application of appropriate market liquidity indicator
is required. The measurement of the liquidity is a complex task, due to
the difficulty in capturing of all the liquidity characteristics in one
tool or indicator. For instance, research on the bid-ask spreads
requires the availability of high-frequency data for the respective
period. Due to the unavailability of such data, researchers must use
substitutes, for example daily rate return or daily volume return. It
should be noted that measuring the market liquidity will always be
affected by error risk due to the following (Amihud, 2002, p. 37):</p>
<ol><li>
<p>A single indicator cannot capture all liquidity dimensions.</p>
</li>
<li>
<p>The result acquired on empirically obtained data may be distorted by
a single event occurring in the analysed period.</p>
</li>
<li>
<p>Using low-frequency data increases error risk.</p>
</li>
</ol>
<p>Market liquidity measures can be divided into groups based on
transaction costs or their estimation (Roll, 1984; Corwin & Schultz,
2012) and investor activity indicators based on volume, turnover value,
and rate of return (Amihud, 2002). One of the most popular liquidity
measures is bid-ask spread. However, this measure cannot be applied in
the Polish market due to the unavailability of transaction data. Even
one-minute intraday data will be insufficient to accurately determine
the amount of the spread.</p>
<p>The ILLIQ market illiquidity measure, introduced by Amihud, is one of
the most popular indicators used to measure liquidity and can be applied
to various time frames. Due to this fact and the universality of the
method, this indicator was selected for further research.</p>
<p>Amihud defines stock illiquidity as the average ratio of the daily
absolute return to the trading volume on that day, <em>| Riyd
| j = VOLDiyd</em>: <em>Riyd</em> is the return on
stock i on day d of year y and <em>VOLDiyd</em> is the
respective daily volume in dollars. This ratio gives the absolute
(percentage) price change per dollar of daily trading volume, or the
daily price impact of the order flow. The <em>ILLIQ</em> value can be
calculated using the following formula:</p>
<p>
<span>\[{ILLIQ}_{iy} = 1/D_{iy}\sum_{t =
1}^{Diy}{\left| R_{iyd} \right|/{VOLD}_{ivyd}}\]</span>
</p>
<p>where D_iy – number of days for which data are available for the
financial instrument <em>i</em> in year <em>y</em>.</p>
<p>The result should be interpreted as follows: the higher the index
value, the more illiquid the market. In the context of this research,
lower values will be interpreted as moments of greater market liquidity,
for which less divergence of option valuation should be expected.</p>
<h2>4. Impact
of liquidity on the divergence between the Black-Scholes-Merton model
and the transaction prices</h2>
<p>As demonstrated in the previous section, the accuracy of the data
allows to reduce the information noise and measurement error. It follows
that high frequency data at a single transaction level would allow for a
more accurate measurement. However, due to the unavailability of such
data for the Polish market and the difficulties related to the analysis
of such a volume of data for a wider range of dates, a substitute should
be employed. The Stooq.pl service provides historical stock market data
in the following intervals: 5 minutes, hourly, and daily, accessible on
the database website (https://stooq.pl/db/h/). The daily data were
rejected due to the high risk of making a mistake in estimating the
value of options using the Black-Sholes-Merton model. This risk comes
from the fact that when analysing the options’ daily closing prices,
information from which point during the session the price comes from is
unavailable. Hence, in the case of low liquidity on a given series of
options, the difference in the theoretical price resulted from the
Black-Scholes-Merton model and the market closing price may come from
the difference between the closing price of the WIG20 index and the
index price at the time of execution of the option transaction.
Therefore, hourly data were used to reduce this risk. The 5-minute data
contained too short a range of data to be considered reliable in the
context of the study.</p>
<p>Historical quotations of all (335) option series for the period from
7 June 2021 to 1 April 2022, presented in hourly intervals, were used to
determine the value of the options with the Black-Scholes-Merton model.
For each trading hour and option series, the difference between the
theoretical value and the closing price, interpreted as the market
value, was determined. Then, for each trading day the average daily
difference was calculated. The charts below show the dependence of daily
divergences of transaction prices obtained by the Black-Scholes-Merton
model depending on volatility, daily WIG20 index volume, daily option
volume and <em>ILLIQ</em> index.</p>
<p>
<img src="/articles/2022/Prymon/media/image1.png" />
</p>
<p><strong>Fig. 1.</strong> Average daily difference between the market
and the Black-Scholes-Merton model valuation and daily volatility.</p>
<p>Source: own elaboration based on Stooq.pl data (<a href="https://stooq.pl/db/h/">https://stooq.pl/db/h/</a>)</p>
<p>
<img src="/articles/2022/Prymon/media/image3.png" />
</p>
<p><strong>Fig. 2.</strong> Average daily difference between the market
and the Black-Scholes-Merton model valuation and ILLIQ.</p>
<p>Source: own elaboration based on Stooq.pl data
(https://stooq.pl/db/h/)</p>
<p>
<img src="/articles/2022/Prymon/media/image5.png" />
</p>
<p><strong>Fig. 3.</strong> Average daily difference between the market
and the Black-Scholes-Merton model valuation and daily volume of
underlying index.</p>
<p>Source: own elaboration based on Stooq.pl data
(https://stooq.pl/db/h/)</p>
<p>
<img src="/articles/2022/Prymon/media/image7.png" />
</p>
<p><strong>Fig. 4.</strong> Average daily difference between the market
and the Black-Scholes-Merton model valuation and daily volume of WIG20
Options</p>
<p>Source: own elaboration based on Stooq.pl data
(https://stooq.pl/db/h/)</p>
<p>The Pearson linear correlation coefficient was determined for the
daily WIG20 volume, WIG20 options and the <em>ILLIQ</em> indicator, see
the results below:</p>
<p>- for the WIG20 volume: 0.6443,</p>
<p>- for the volume of WIG20 options: 0.4536,</p>
<p>- for ILLIQ: -0.5273.</p>
<p>In the case of volume, a positive, moderate correlation can be
noticed, and the relationship is more important for the volume on the
base index than on the option transactions themselves.</p>
<p>For the <em>ILLIQ</em> index, a negative moderate correlation can be
observed. Hence, with an increase in market illiquidity measured by this
indicator, discrepancies between the valuations obtained by the
application of the Black-Scholes-Merton model decreases.</p>
<p>The above correlations were not statistically significant.</p>
<h2>5. Conclusion</h2>
<p>The Black-Scholes-Merton model of option pricing is based on
assumptions which are difficult to meet in the real market. An increase
in liquidity should bring the model to the real market. Therefore, one
can logically assume that an increase in liquidity increases the model
efficiency – understood as the ability to obtain results close to the
market prices. However, the results of the research indicate the
opposite relationship.</p>
<p>During the period under examination, the divergence between the
average daily difference between the BSM model results and transactional
prices increased as the volume on either the underlying index or options
increased. The use of the selected indicators measuring the market
liquidity presented in this paper, does not allow for a more precise
estimation of the input parameters. This is because the correlation is
moderate and not statistically significant for any of these indicators.
Additionally, results of the research show the opposite relationship
than the assumed logic would suggest.</p>
<p>The reason of such a phenomenon could be another market dependency,
namely that between market volatility and transaction volume. In periods
of increased volatility, the transaction volume is usually much higher.
The expectations for the future volatility may be more varied, which
turns into a more difficult estimation of volatility in the option
valuation model.</p>
<p>From the practical standpoint, volume and market liquidity indicators
data could be supportive for the estimation of volatility in the BSM
model, but not enough for their improvement. The only undisputable
conclusion is that in conditions of increased market volatility, the BSM
model will be characterised by the higher risk of incorrect result.</p>
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<ol><li>
</li></ol>