Finance, Markets and Valuation
DOI:
10.46503/VUTL1758
Corresponding author
Jawad Saleemi
Received: 27 Jun 2020
Revised: 14 Jul 2020
Accepted: 21 Jul 2020
Finance, Markets and
Valuation
ISSN 2530-3163.
Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
An estimation of cost-based market liquidity from daily
high, low and close prices
Una estimación de la liquidez de mercado basada en los
costes a partir de los precios máximo, mínimo y de cierre
Jawad Saleemi
ID
1,2
1
Department of Management Sciences, University of Gujrat, Gujrat, Pakistan. Email:
j.saleemi@yahoo.com
2
Economics and Social Sciences Department, Universitat Politècnica De Valéncia. Valencia, Spain.
Email: Jasa1@doctor.upv.es
JEL:
Abstract
In the literature of asset pricing, this paper introduces a new method to estimate the cost-based market
liquidity (CBML), that is, the bid-ask spread. The proposed model of spread proxy positively correlates
with the examined low-frequenc y spread proxies for a larger dataset. The introduced approach provides
potential implications in important aspects. Unlike in the Roll bid-ask spread model and the CHL bid-ask
estimator, the CBML model consistently estimates market liquidity and trading cost for the entire dataset.
Additionally, the CBML estimator steadily measures positive spreads, unlike in the CS bid-ask spread
model. The construction of the proposed approach is not computationally intensive and can be considered
for distinct studies at both market and firm levels.
Keywords: Market Microstructure; Asset Pricing; Bid-Ask Spread; Market Liquidity; Trading Cost
Abstract
Este documento presenta un nuevo método, en la literatura sobre fijación de precios de activos, para
estimar la liquidez del mercado basada en el coste (CBML), es decir, el diferencial entre oferta y demanda.
El modelo propuesto con un proxy del diferencial (spread) se correlaciona positivamente con los proxy del
diferencial de baja frecuencia examinados para un conjunto de datos más grande. El enfoque introducido
proporciona potenciales implicaciones en aspectos importantes. A diferencia del modelo de diferencial
de oferta y demanda y el estimador CHL, el modelo CBML estima constantemente la liquidez del mercado
y el costo comercial para todo el conjunto de datos. Además, el estimador CBML mide constantemente los
diferenciales positivos, a diferencia del modelo de diferencial de oferta y demanda CS. La construcción
del enfoque propuesto es asumible computacionalmente y puede considerarse para estudios distintos
tanto a nivel de mercado como de empresa.
Keywords: Microestructura del mercado; Fijación de precios de activos; Diferencial de oferta y demanda;
Liquidez del mercado; Coste de negociación
How to cite this paper: Saleemi, J. (2020) An estimation of cost-based market liquidity from daily high,
low and close prices. Finance, Markets and Valuation 6(2), pp. 1–11.
1
Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
1 Introduction
This paper provides new insights of estimating the cost-based market liquidity in financial
markets. Starting with the Roll bid-ask spread model, a spread has been widely considered for
the estimation of cost-based market liquidity at the time of trade and future trading sessions.
The bid-ask spread is a useful indicator of executing cost faced by investors, and thus, a proxy
for market liquidity (Corwin & Schultz, 2012). In the literature of asset pricing, distinct models
have been proposed to framework the bid-ask spread. However, some shortcomings have been
identified in various bid-ask spread models (Goyenko, Holden, & Trzcinka, 2009).
In this paper, the CBML model is another version of low-frequency spread proxies. The
rationale of the introduced estimator is based on simple foundation, that is, a spread can be de-
termined by various components, including information asymmetry cost, immediacy cost, and
order processing cost. The proposed approach considers a wider set of information consisted in
daily high, low, and close prices, while constructing the possible presence of an informed trader.
Additionally, the CBML model looks at pre-and-post-trade prices by a logic that the liquidity
providers would be compensated against the provision of riskier liquidity and administration
expenses. Most importantly, the CBML estimator is not computationally intensive and provides
comprehensive implications for both academics and those who participate in trading.
To construct the CBML estimator, the model considers distinct theoretical assumptions: (a)
high prices and low prices are always initiated by buyers and sellers, respectively (Corwin &
Schultz, 2012); (b) an informed trader, either from buyer-side or seller-side, is always present
with equal probability in the market (Glosten & Milgrom, 1985); (c) a transaction discloses in-
ventory holding cost that liquidity providers demand against the provision of price fluctuations
(Amihud & Mendelson, 1980); and (d) liquidity providers would also be compensated for the
order processing cost at the time of trade (Roll, 1984). Based on these theoretical assumptions,
the CBML model is constructed in the following analytical steps:
S
t
=
H
t −1
− L
t −1
C
t −1
−
v
H
t
−v
L
t
C
t
(1)
Where,
S
t
is the bid-ask spread, and derived by the dierence between the ratio of an asset’s
range to its close price on day
t − 1
and the ratio of an informed asset’s range to its close price
on day t.
S
t
reflects the ease and cost of trading on day t. In the proposed model,
H
t −1
and
L
t −1
denoted to the highest price asked by a seller and the lowest price that a buyer was willing to
pay against the asset on day
t −1
, respectively.
C
t −1
is the closed price of an asset on day
t −1
.
For the following trading session, it is assumed that an informed trader would impact prices.
Assuming risk neutrality, the asset is valued in the following trading session at:
η
t
=
H
t
+ L
t
2
(2)
Where,
η
t
is the mean of high and low prices on day
t
. Understanding the equal probability
of an informed trader, the expected highest value for which a seller would sell the security is
assumed conditional on a trade at:
ν
H
t
= H
t
π + η
t
π (3)
Where, the expected lowest price that a buyer would pay against the asset is assumed
conditional on a trade at:
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
ν
L
t
= L
t
π + η
t
π (4)
The model further encounters the relationship of an informed asset’s range to its closed
price,
C
t
. The CBML model assumes that a ratio of asset’s range to close price would be greater
in case of higher probability of trading with an informed trader on day
t
. This model looks
at past prices by a logic that providers of liquidity would be compensated in the following
trading session against the price fluctuations and administration expenses. This implies, the
CBML estimator reflects cost-based market liquidity for two-consecutive single days. Within
this framework, the model assumes the volatility factor by computing the variance of spread
and then taking the square root of it.
C BM L
t
=
q
S
2
t
(5)
C BM L
t
reflects market liquidity, trading cost, and volatility for the financial asset. In the
CBML model, the framework of spread volatility is similar to the mathematical modelling for
the expected returns’ volatility. Based on general foundations of asset pricing and simple
computation, the CBML model can be suitably considered for variety of research in the field of
asset pricing.
The paper is structured as follows. Section 2 describes the theoretical background of the
prior research. A description of the dataset and distinct low-frequency spread measures is given
in Section 3. Section 4 discusses the research findings, and these findings are concluded in
Section 5.
2 Review of the Literature
In the literature of market microstructure, the financial market liquidity is one of the important
disciplines. The microstructure of the financial market is concerned with details of how financial
securities are executed at the time of trade. In the financial market, investors are possibly
interested to anticipate costs associated with trading and eects of these costs on assets’ prices.
Liquidity influences market eiciency, trading cost, returns, and systemic financial stability
(Chordia, Roll, & Subrahmanyam, 2001, 2008). Market liquidity is a multidimensional concept
and it is described in distinct context. Lybek and Sarr (2002) argue that a liquid market reflects
various features: (a) low trading cost; (b) immediacy of transaction execution; (c) depth, in other
words, the existence of limit orders; (d) breadth, which means small market impact of large
orders; and (e) resiliency, indicates that new orders correct market imbalances.
Whilst liquid markets are described in distinct features, market liquidity can be defined in
a number of ways. Market liquidity, in general, is the ease of trading an asset in the financial
market. In other words, the immediacy of transaction execution with limited price impact and
low transaction cost can be referred to higher liquidity. Market liquidity tends to be highly
volatile in the financial market, which impose systemic liquidity risk (Guijarro, Moya-Clemente,
& Saleemi, 2019). Liquidity is a time-varying risk factor, which interrelates with the transparency
of information about assets’ value (Bernales, Cañón, & Verousis, 2018), the number of liquidity
providers and their access to capital (Brunnermeier & Pedersen, 2008), and an increased liquid-
ity uncertainty which induces liquidity providers to ask for a higher compensation, that is, a
higher executing cost (Ho & Stoll, 1981).
Distinct models, focused either on bid-ask spread proxies or volume-based liquidity mea-
sures, have been proposed to estimate the market liquidity in the financial market (Goyenko
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
et al., 2009). In general, the range of ask price for which a seller wants to sell an asset and bid
price that a buyer wants to pay for an asset is referred to the liquidity and the cost of an asset,
that is, the bid-ask spread (Cohen, Maier, Schwartz, & Whitcomb, 1981). A small size of spread
is an indicator of higher liquidity. Moreover, ask and bid prices have been traditionally used
as proxy of volatility (Garman & Klass, 1980; Parkinson, 1980). Starting with the Roll spread
model, the literature in bid-ask spread has been gained tremendous development and various
components, namely as, information asymmetry cost, immediacy cost, and order processing
cost visualized in order to estimate the true spread.
Lesmond, Ogden, and Trzcinka (1999) introduced distinct spread proxies: (i) Zeros; (ii) LOT
estimator of eective spread; and (iii) LOT Mixed estimator. Zeros estimator is the proportion of
days with zero returns. The dierence between percent buying cost and percent selling cost is
referred to LOT estimator of eective spread. The LOT Mixed method estimates cost parameters
based on maximizing the likelihood function of daily stock returns. Hasbrouck (2004) proposed
a half-spread using Gibbs sampler Bayesian estimation of the Roll’s model. Goyenko et al. (2009)
and Holden (2009) jointly proposed an Eective Tick estimator following the concept of price
clustering, which is the probability weighted average of each eective spread size divided by
average price. Corwin and Schultz (2012) introduced a high-low estimator of bid-ask spread,
which is based on daily high and low prices. Fong, Holden, and Trzcinka (2017) proposed a
measure of monthly spread proxy, which is a simplified version of the LOT Mixed estimator.
Most recently, Abdi and Ranaldo (2017) constructed a spread model from close, high, and low
prices (CHL), which is a modified version of Roll (1984).
3 Database and methodology
The scope of this study is to present a new estimation of the cost-based market liquidity, that
is, the bid-ask spread. Additionally, this paper provides comprehensive comparison of the
proposed strategy with distinct low-frequency spread proxies. The data used in this study
contains daily obser vations of high, low, and closing prices, related to the S&P500 Index, and
collected during the period 3 January 2001-30 October 2019. The analysis was executed on R
programming soware, where distinct econometric techniques, namely kernel density estima-
tion (KDE) of the numerical distributions for the liquidity variables, the time-series analysis of
the liquidity variables, and the correlation analysis, were applied. This section also looks at
theoretical assumptions behind the construction of each applied spread proxy.
3.1 Roll Bid-Ask spread model
Roll (1984) assumes, that the true value of asset is based on random walk and independent
of the order flow. Therefore, buy and sells orders are considered equally likely and serially
independent. Under assumptions, that market makers bear only order-processing cost, this
estimator is based on the serial covariance of change in prices.
R S
t
= 2
p
−C ov (4P
t
, 4P
t −1
) (6)
The shortcoming of the Roll bid-ask spread model is that the covariance of price changes
can be positive. Therefore, the function of square root cannot compute spreads. Goyenko et al.
(2009) set a default numerical value to zero, when the sample serial covariance is positive. In
this study, the Roll model is analysed as:
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
2
s
−
C ov (4P
t
, 4P
t −1
P
t
), when C ov (4P
t
, 4P
t −1
) < 0
0, when C ov (4P
t
, 4P
t −1
) ≥ 0
(7)
3.2 CS Bid-Ask spread model
Corwin and Schultz (2012) introduced a high-low estimator of spread. It is based on assumptions
that daily high prices,
H
t
, are always buyer-initiated trades and daily low prices,
L
t
, are always
seller-initiated trades. The CS model reflects market liquidity, trading cost, and volatility for
two-consecutive single days.
C S
t
=
2e
α
t
− 1
e
α
t
+ 1
(8)
where:
α
t
=
1 +
√
2
p
β
t
−
√
γ
t
(9)
β
t
=
l n
H
t
L
t
2
+
l n
H
t +1
L
t +1
2
(10)
γ
t
=
l n
max (H
t
, H
t +1
)
mi n(L
t
, L
t +1
)
2
(11)
The CS spread model further addresses overnight trading which may cause to produce
negative spreads, where
γ
t
is larger in value than
β
t
. The negative bid-ask spread is a drawback
of the CS spread measure. Whilst a spread is defined as ask-price minus bid-price, a spread
is assumed a positive number in the literature of asset pricing. In order to deal with negative
spread, researchers are suggested for various adjustments: (a) assume negative monthly esti-
mates to zero; (b) set negative two-day estimates to zero and then compute the average; or (c)
compute the spread for only positive estimates and take the average.
3.3 CHL Bid-Ask spread model
Most recently, Abdi and Ranaldo (2017) proposed a modified version of the Roll bid-ask spread
model from daily close, high, and low prices (CHL). This model assumes, that the midrange of
high and low prices on day t and day t + 1 rely at the common close price of day t .
S
t
= 2
s
l n (c
t
) − l n
H
t
+ L
t
2
l n (c
t
) − l n
H
t +1
+ L
t +1
2
(12)
The CHL model is undefined, when the relationship between prices of day t and day
t + 1
around the common close price,
c
t
, causes of negative estimates. This implies, that the function
of price variance fails to estimate spread for negative observed values. In such cases, Abdi and
Ranaldo (2017) set default numerical values to zero.
S
t
= 2
s
max
l n (c
t
) − l n
H
t
+ L
t
2
l n (c
t
) − l n
H
t +1
+ L
t +1
2
, 0
(13)
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
N Min Median Mean Max Std. Dev. Skewness Kurtosis
RS 4735 0.0000 0.0048 0.0081 0.1221 0.0112 3.35 23.01
CS 4735 -0.1098 0.0004 -0.0015 0.0468 0.0103 -1.66 13.09
CHL 4735 0.0000 0.0024 0.0032 0.0367 0.0037 2.77 16.51
CBML 4735 0.00000156 0.0048 0.0068 0.0766 0.0071 3.03 18.6
Table 1. The descriptive statistics of variables are computed from daily obser vations
Figure 1. Density plot illustrating skewness for RS variable
The CHL spread is derived as:
C H L
t
=
1
N
N
Õ
t =1
S
t
(14)
Where N is the number of days in the month.
4 Results
The descriptive statistics of liquidity variables for the data sample are presented in Table 1,
which vividly indicates the numerical dierences among the spread proxies. As discussed earlier,
each spread proxy is constructed under some specific conditions. The theoretical assumptions
behind construction of each model would possibly impact the measurement of liquidity. As
can be seen in Table 1, positive skewness indicates the right-skewed distributions of liquidity
variables with values to the right of their mean. However, negative skewness is seen of the CS
spread model, which indicates the le-skewed distributions of liquidity variable with values to
the le of its mean. The higher kurtosis of liquidity variables is an indicator of extreme values in
the dataset.
Figures 1–4 further provide an illustration of the numerical distributions for liquidity vari-
ables under the concept of kernel density estimation. Such non-parametric technique visualized
the probability density function for each liquidity variable, while providing important quantity
of information. Density plots clearly show dierences in the numerical distributions of variables.
The reason is, the Roll model and the CHL spread measure failed to compute around 32.65%
and 21.37% observations, respectively. As mentioned earlier, the Roll model fails to estimate
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
Figure 2. Density plot illustrating skewness for CS variable
Figure 3. Density plot illustrating skewness for CHL variable
Figure 4. Density plot illustrating skewness for CBML variable
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
Figure 5. Time-varying cost-based market liquidity computed by RS variable
RS CS CHL CBML
RS 1 0.56 0.67 0.65
CS 0.56 1 0.67 0.45
CHL 0.67 0.67 1 0.46
CBML 0.65 0.45 0.46 1
Table 2. Correlation values among the spread proxies
spread when the covariance of price changes is positive, and the function of price variance in the
CHL model cannot compute spread for negative estimates. In such cases, the study assumed
all default numerical values to zero.
However, the CS model computed around 47.98% negative spreads in the data sample,
which is clearly a violation of reality. As discussed earlier, a spread is always considered a
positive number in the literature of asset pricing. The CS model was seen to produce negative
spreads, when the maximum range of high-to-low price ratio for two-day period is larger than
the expectation sum of price ranges over two consecutive single days. Most importantly, the
CBML measure contains skewed shape of numerical distributions, while consistently estimating
the positive spreads for the entire dataset.
Figures 5–8 show the cost-based market liquidity, while excluding all default numerical
values of the Roll model and the CHL spread estimator, and negative spreads estimated in the
CS model. However, the CBML variable is representing the cost-based market liquidity for the
entire dataset. Despite shortcomings in Roll, CHL, and CS models, it was observed that the
market liquidity is a time-varying risk factor and can suddenly disappear, as seen during the
recent global financial crisis. Although, some noise in the liquidity has been occurring over
time, but it is not persistent.
Table 2 shows the correlation coeicients among the spread proxies. In order to execute the
analysis, the study considered only positive estimates in the CS model, and those observations
for which the Roll and CHL models compute spreads. The results importantly revealed, that the
CBML model has statistically strong correlation with the model proposed by Roll (1984), but
the relationship of the CBML model is seen statistically moderate with bid-ask spread models
proposed by Corwin and Schultz (2012) and Abdi and Ranaldo (2017).
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
Figure 6. Time-varying cost-based market liquidity computed by CS variable
Figure 7. Time-varying cost-based market liquidity computed by CHL variable
Figure 8. Time-varying cost-based market liquidity computed by CBML variable
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Finance, Markets and Valuation Vol. 6, Num. 2 (July-December 2020), 1–11
5 Conclusions
This work constructs a new proxy of the cost-based market liquidity from daily high, low, and
close prices. Compared with Roll and CHL spread proxies, the proposed method, CBML, con-
sistently estimated the bid-ask spreads for an entire dataset, and utilized a wider set of daily
information, namely high, low, and close prices. Unlike in the CS spread measure, the CBML
measure steadily computed positive spreads. Additionally, the CBML model encounters the
possible presence of an informed trading. Despite dierences, the CBML proxy positively corre-
lates with the applied spread proxies for the dataset, excluding the negative spreads of the CS
model and the default values of the Roll and CHL models.
This estimation method for the cost-based market liquidity is straightfor ward, computa-
tionally less-intensive, and based on general foundations of asset pricing studies. Therefore,
the proposed approach is suitable for variety of research. This research encourages researchers
to study the proposed CBML proxy with a larger sample of liquidity measures, including the
high-frequency spread measures. The future research would undoubtedly explore the signifi-
cance of the CBML model in the study of asset pricing, corporate financing, and risk portfolio
management.
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