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:

Jawad Saleemi 2

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)

Jawad Saleemi 5

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

Jawad Saleemi 7

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

Jawad Saleemi 9

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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