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The Impact of Liquidity on Option Prices
Robin K. Chou, San-Lin Chung, Yu-Jen Hsiao and Yaw-Huei Wang
Abstract
This article illustrates the impact of both spot and option liquidity levels on option
prices. Using implied volatility to measure the option price structure, our empirical
results reveal that even after controlling for the systematic risk of Duan and Wei
(2009), a clear link remains between option prices and liquidity; with a reduction
(increase) in spot (option) liquidity, there is a corresponding increase in the level of
the implied volatility curve. The former is consistent with the explanation on hedging
costs provided by Cetin, Jarrow, Protter and Warachka (2006), whilst the latter is
consistent with the ‘illiquidity premium’ hypothesis of Amihud and Mendelson (1986a).
This study also shows that the slope of the implied volatility curve can be partially
explained by option liquidity.
Keywords: Liquidity; Option price; Implied volatility curve; Hedging cost
JEL Classification: G12; G13
Robin K. Chou ([email protected]) and Yu-Jen Hsiao ([email protected]) are collocated at the
Department of Finance, National Central University, 300 Jhongda Road, Jhongli 320, Taiwan; San-Lin Chung
([email protected]) and Yaw-Huei Wang ([email protected]) are collocated at
the Department of Finance, National Taiwan University, 85 Roosevelt Road, Section 4, Taipei 106, Taiwan. We
would like to thank Chuang-Chang Chang, Hsuan-Chi Chen, Ren-Raw Chen, Jin-Chuan Duan, Bing-Huei Lin,
Mark Shackleton, Chung-Ying Yeh and Shih-Kuo Yeh for their helpful comments. The authors are also grateful
to the National Science Council of Taiwan for the financial support provided for this study.
1
1. INTRODUCTION
Standard asset pricing theory assumes that the market is frictionless and competitive
and thus liquidity is not priced. However, once these assumptions are relaxed, the
standard theory may not be readily applicable. For instance, it has already been well
documented within the market microstructure literature that liquidity factors are
important determinants of stock and bond returns.1 Such returns have been found to
be affected by liquidity, as measured by the bid-ask spread (Amihud and Mendelson,
1986a, 1991; Kamara, 1994; Eleswarapu, 1997), the price impact of trades (Brennan
and Subrahmanyam, 1996), and volume or turnover ratio (Datar, Naik, and Radcliffe,
1998; Haugen and Baker, 1996).
In most of the prior studies examining the effects of liquidity on asset prices, the
focus is mainly placed on stocks and bonds. Studies within the extant literature
examining the effects of option liquidity on the pricing of options are something of a
rarity. In one example, however, Brenner, Eldor, and Hauser (2001) use a unique
dataset to explore the effects of liquidity on the pricing of currency options, examining
currency options issued by the Bank of Israel that are not traded until maturity. Their
hypothesis is tested by comparing these options to similar exchange-traded options;
however, they reject the hypothesis that liquidity has no effect on the pricing of
1 For more comprehensive details, see the survey articles of Easley and O’Hara (2003) and Amihud,
Mendelson, and Pedersen (2005).
2
options, and find that such non-tradable options are priced at about 21 per cent less
than the exchange-traded options.
Bollen and Whaley (2004) go on to suggest that changes in implied volatility
are directly related to net buying pressure from public order flows, finding that the
most significant changes in the implied volatility of S&P 500 options are attributable
to buying pressure for index put options, whereas changes in the implied volatility of
individual stock options are generally found to be dominated by the demand for call
options. Garleanu et al. (2009) note that ‘end users’ tend to have net long positions
in SPX options, particularly with regard to out-of-the-money puts, and net short
positions in individual stock options. They conclude that the demand patterns for
index options and single-stock options help to explain the overall expensiveness and
skew patterns of index options.
However, given that options are contingent assets, in addition to the liquidity of
options, arbitrage pricing theory asserts that the liquidity of the underlying asset is
also of relevance to the pricing of options. Thus, several theoretical models have
been developed aimed at incorporating the spot liquidity effect into the option
pricing formulae; for example, Frey (1998) studies how a large agent, whose trades
result in price moves, can replicate the payoff of a derivative security, thereby
deriving a non-linear partial differential equation for the hedging strategy.
3
As opposed to directly inferring the price impact of spot liquidity, Cho and
Engle (1999) use transaction data to examine the effects of spot market activity on
the percentage bid-ask spreads of S&P 100 index options, proposing a new theory
on market microstructure which they refer to as ‘derivative hedge theory’. They
argue that if market makers in the derivative markets can hedge their positions using
the underlying asset, then the liquidity and spread within the derivative markets will
be determined by the liquidity in the spot market, rather than by the activities of the
derivative market itself. Thus, they find that option market spreads are positively
related to spreads in the underlying market.
Frey (2000) and Liu and Yong (2005) also consider the costs involved in the
replication (or super-replication) of a European option in the presence of price impact.
Liquidity with a stochastic supply curve is modeled by Cetin, Jarrow and Protter
(2004) and Cetin et al. (2006), who obtain the pricing formulae for European call
options. Cetin et al. (2006) provide further empirical evidence that spot liquidity cost
is a significant component of the option price and that the impact of illiquidity is
dependent upon the moneyness of the option; that is, the impact is more (less)
significant for out-of-the-money (in-the-money) options.
Following on from these studies, we examine the liquidity effect on option
prices from both the spot and option markets. We use data on 30 component stocks
4
in the Dow Jones Industrial Average (DJIA) index as at 31 December 2004. The spot
liquidity measures used in this study fall into the two broad categories of
‘trade-based’ and ‘order-based’ measures. Trade-based measures include ‘cumulative
trading volume’ (VOL), ‘number of trades’ (NT), and ‘average trade size’ (ATS).
Order-based measures include ‘absolute order imbalance’ (AOI), ‘average
proportional quoted spread’ (AQS), and ‘average proportional effective spread’
(AES). The option liquidity measures include ‘trading volume’ based upon the
overall number of contracts (OVOL), ‘option proportional bid ask spread’ (OAQS),
‘dollar trading volume’ (DVOL), and total option ‘open interest’ (OI).
This study contributes to the extant literature not only by clarifying the roles of
spot and option liquidity in determining option prices, but also by indicating which
liquidity proxies are more informative for the pricing of options. Based upon our use
of implied volatility to represent the structure of option pricing, our empirical results
demonstrate that even after controlling for the systematic risk of Duan and Wei
(2009), as well as other control variables, a clear link remains between the pricing of
options and liquidity.
Specifically, we find strong evidence to show that options with a lower
proportional bid-ask spread and underlying stocks with a higher average
proportional quoted spread ultimately lead to a higher level of implied volatility (i.e.,
5
a higher option price). The former finding that options become more expensive
when the options market becomes less illiquid supports for the ‘illiquidity premium’
hypothesis proposed by Amihud and Mendelson (1986a), whilst the latter finding
that options become more expensive when the spot asset is less liquid is consistent
with the findings of Cetin et al. (2006), who note that the Black-Scholes hedging
strategy results in a positive spot liquidity cost.
We also find that the liquidity of options can partly explain the implied
volatility ‘smile’ documented by Rubinstein (1985) and others. Specifically, our
results indicate that when the option market becomes more liquid (i.e. when there is
a lower option proportional bid-ask spread), the implied volatility curve becomes
steeper (more negatively skewed).2 The results suggest that the liquidity of the
option market is positively related to demand pressure. Thus, the implied volatility
slope becomes more negative with an increase in option activity due to the increase
in demand pressure.3
The remainder of this paper is organized as follows. A description of the data
used in our study is provided in Section 2, along with a description of the spot and
option liquidity measures. Section 3 discusses the methodology adopted and the
empirical results obtained. Our concluding remarks are offered in Section 4.
2 However, we find little linkage between spot liquidity and the slope of the implied volatility curve.
3 Garleanu et al. (2009) show that since non-market makers mainly sell high-strike equity options,
these options are particularly cheap and thus the implied volatility slope is negative.
6
2. DATA AND VARIABLE MEASURES
2.1 Data
Component stocks of the Dow Jones Industrial Average Index (DJIA) are selected as
our study sample, with the sample period running from 1 January 2001 to 31
December 2004, and thus providing a total of 1,004 trading days. We select a total of
30 DJIA component firms, essentially because these stocks are actively traded and
have sufficient options trading data. The market prices of the options written on the
stocks of these firms are collected from Ivy DB OptionMetrics.
We use the standardized volatility surfaces, provided by OptionMetrics, with four
different time-to-maturity periods (30, 60, 91 and 182 days) for each stock and for each
trading day. Specifically, for each maturity period, OptionMetrics calculates the
Black-Scholes implied volatility surfaces made up of 13 strike prices reported as deltas
for both call and put options, with the delta for call (put) options ranging between 0.2
(–0.2) and 0.8 (–0.8), at intervals of 0.05.4 The OptionMetrics data also provide
end-of-the-day bid and ask quotes, open interest and trading volume for all options,
which are used to calculate the option liquidity measures adopted in this study.
The spot liquidity data is obtained from the Center for Research in Securities
Prices (CRSP) and the Trades and Quotes (TAQ) database. The CRSP database
4 The calculation of the implied volatility surfaces is based upon a kernel smoothing algorithm and
an interpolation technique.
7
provides the number of shares traded per day as well as the daily number of trades,
whilst the TAQ database provides time-stamped trades and quotes for stocks listed
on the NYSE, the AMEX and the NASDAQ National Market system.
2.2 Variable Measures
2.2.1 Dependent variables
As opposed to using the single-strike (e.g. at-the-money) option price, we use the
option prices of all out-of-the-money options, reported in the implied volatility
surfaces of OptionMetrics, as the means of measuring the overall price impact of
liquidity on the option market.
We begin by converting the implied volatility surface of each firm (with 13
strikes) into European option prices for each trading day and then calculating the
model-free implied volatility for each firm in the preliminary regression test. The
model-free implied volatility is computed using these option prices and the
methodology of Bakshi, Kapadia and Madan (2003). Britten-Jones and Neuberger
(2000) also show that under the assumption that the underlying asset price follows a
diffusion process, the risk-neutral integrated return variance between the current date
and a future date can be fully specified by the set of option prices expiring on the
future date. Jiang and Tian (2005) further extend their proof to the case where the
asset prices contain jumps.
8
Specifically, we calculate τ-period ‘model-free implied volatility’, ( , )MFIV t ,
as:
2 2
2
( , ) [ ( , ) ] [ ( , )]
( , ) ( , ) ,
Q Q
t t
r
MFIV t E R t E R t
e V t t
(1)
where R(t, τ) = ln S(t + τ) – ln S(t) is the τ-period log return; μ(t, τ) is the mean log
return; and V(t, τ) is the price of the volatility contract. Following Bakshi et al. (2003),
the mean log return is calculated using the prices of the volatility contract, the cubic
contract, and the quadratic contract;5 the prices of these contracts can be determined
from the market prices of the equity options.6
Following Duan and Wei (2009), we also use the level and slope of the implied
volatility curves to measure the price impact of liquidity on the options market.
Specifically, we run the following regression for each firm to obtain the time series of
the level and slope of the implied volatility curves for each maturity category:
0 1 ,IV his
jk j j j jk j jka a y y 1 , 2 , , jk I , (2)
where Ij is the number of options in a particular maturity category for the jth stock;
IV
jk (his
j ) denotes the implied (realized) volatility level; yjk is the moneyness
measured by the strike price divided by the underlying price (Kjk /Sjk); and jy is the
sample average of yjk . Therefore, the intercept α0 j (regression coefficient α1 j ) serves
5 Refer to Equation (39) in Bakshi et al. (2003).
6 The detailed formulae are provided in Equations (7) to (9) of Bakshi et al. (2003)
9
as the measure for the level (slope) of the implied volatility curve.
2.2.2 Spot liquidity measures
Liquidity represents price immediacy, with a market being regarded as liquid if
traders can buy and sell many shares quickly, with low transaction costs, and at a
price close to the previously prevailing price (Cho and Engle, 1999); however, spot
liquidity is not easy to measure since it is very difficult to distinguish between
normal price movements and those attributable to large orders.
It is suggested in the prior studies that no consensus has been reached on the
most appropriate proxy for spot liquidity; thus a broad range of measures are
adopted (Aitken and Carole, 2003), with the various measures used falling into two
broad categories, trade-based measures and order-based measures; both types of
measures are used in this study. Our trade-based measures include: (i) ‘cumulative
trading volume’ (VOL), defined as the number of shares traded per day; (ii) ‘daily
number of trades’ (NT); and (iii) ‘average trade size’ (ATS), defined as the number of
shares traded each day divided by the number of trades on the day.
These measures are very attractive, essentially because they are quite simple to
calculate using readily available data, and because they also have widespread
acceptance, particularly amongst market professionals. They are, however, ex-post
as opposed to ex-ante measures, insofar as they indicate what investors have
10
previously traded, but of course, this is not necessarily a good indication of what
they are likely to trade in the future (Aitken and Carole, 2003).
Aitken and Carole (2003) provide evidence to show that order-based liquidity
measures provide a better proxy for liquidity, since they can immediately and more
accurately capture the ability to trade and the associated trading costs. Three
order-based measures are included in this study: (i) ‘absolute order imbalance’ (AOI),
defined as the absolute difference between buy and sell orders; (ii) ‘average
proportional quoted spread’ (AQS); and (iii) ‘average proportional effective spread’
(AES).
For each trade, we search for the prevailing quotes using the standard Lee and
Ready (1991) algorithm, with the quotes required to be at least five seconds old and
within 30 minutes of trades in order to avoid those quotes that are recorded out of
sequence, or those that have become stale. The quoted proportional spread is defined
as the difference between the best ask and the best bid prices, divided by the quote
mid-point. The effective proportional spread is defined as twice the absolute difference
between the trade price and the prevailing quote midpoint, divided by the quote
mid-point. Bid-ask spreads represent the average cost of a round-trip transaction of a
normally traded quantity (Cho and Engle, 1999).
2.2.3 Option liquidity measures
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Similar to the spot liquidity measure, the bid-ask spread is also a popular measure of
option liquidity,7 essentially because the bid-ask spread can be viewed as the price
demanded by the market maker for providing liquidity services and the immediacy
of execution (Amihud and Mendelson, 1986b). Thus, we follow Cao and Wei (2009)
to calculate a volume-weighted ‘average of the proportional spread’ (OAQS) as an
option liquidity proxy for each day.
For completeness, we also include some of the alternative option liquidity
measures used in a number of the prior studies.8 In specific terms, we include
‘option trading volume’ (OVOL), ‘option dollar trading volume’ (DVOL) and total
option ‘open interest’ (OI) as transaction-based measures. Option trading volume is
the total number of contracts during the day; option dollar trading volume is the
mid-point of the bid and ask multiplied by the number of contracts; and OI is
summed over all of the moneyness levels of the options.
Since the implied volatility surfaces provided by OptionMetrics are standardized
at the four maturity periods of 30, 60, 91, and 182 days, we apply a linear
interpolation to obtain the option liquidity variables for each maturity period. For
example, if we assume that the near-term (next-term) option has 16 days (44 days) to
expiration, then the OVOL of options with 30 days to maturity is calculated as
7 See: Vijh (1990), Jameson and Wilhem (1992), Cho and Engle (1999), Elting and Miller (2000)
and Cao and Wei (2009). 8 Examples include: Elting and Miller (2000), Hasbrouck and Seppi (2001) and Cao and Wei (2009).
12
follows:
1 1( 30 days)= ( 16 days)+ ( 44 days)
2 2OVOL OVOL OVOL ,
where τ is the time to maturity of the option.
2.2.4 Control variables
In order to rule out the possible effects on option prices from other firm-specific and
market variables, we follow the methodology of Duan and Wei (2009) to add a
number of control variables, including the systematic risk proportion (SYS), firm
size (SIZE) and leverage ratio (LEV) of each stock, the risk-neutral skewness
(SKEW) and kurtosis (KURT) of individual stock options, and the model-free
implied volatility of S&P 500 index options (SPMFIV).
The straightforward definition of SYS is the ratio of systematic variance over
total variance. We follow Duan and Wei (2009) to run daily OLS regressions for
stock j with a one-year rolling-window, as follows:
jtmtjjjt RR . (3)
We then go on to estimate the systematic risk proportion, j
2 m
2 /j
2, which is
essentially the R2
of regression model (3). If we need a measure of systematic risk
proportion for a specific period (for example, one month), then it is calculated by
averaging the daily estimates over that period. Firm size is defined as the market
13
capitalization of each stock, and leverage ratio is defined as the book value of debt /
(book value of debt + market value of equity). The daily risk-neutral skewness and
kurtosis of individual stock options are computed using the Bakshi et al. (2003)
methodology, as is the daily model-free implied volatility of the S&P 500 index.9
2.3 Summary Statistics
All variables used in this study are summarized in Table 1 for ease of reference. Our
main analyses are based on options data with a time-to-maturity of 30 days,10
with
the summary statistics being reported in Table 2. For each measure, we first
calculate the time-series average for each stock and then calculate the
cross-sectional average, with the mean, median and standard deviation referring to
these cross-sectional averages.11
As we can see from the table, the mean implied
volatility of individual equity options (0.32952) is larger than that of the S&P 500
index option (0.24908).
<Tables 1 and 2 are inserted about here>
The correlations between the variables are reported in Table 3, from which we
9 Our calculations are identical to those of Duan and Wei (2009), Appendix B.
10 The results on data with other maturity periods are discussed later in our check for robustness.
11 The quoted and effective proportional spreads for our DJIA stocks might appear low. However,
quotes on NYSE and NASDAQ dropped substantially after decimalization on January 29, 2001 and
on April 9, 2001, respectively. Our sample period starts in January 2001, mostly after the
decimalization. Using the 100 NYSE firms whose market capitalization most closely matched those
of the 100 large NASDAQ firms as his large firm sample, Bessembinder (2003) reported an average
proportional quoted spread of 0.096% after decimalization for the large capitalization sample stocks,
which is similar to our results. It is plausible that his sample firms are smaller than our DJIA firms in
size.
14
can see that AQS and AES are both highly correlated with implied volatility; this
positive correlation appears to be consistent with the ‘derivative hedge theory’
proposed by Cho and Engle (1999), who demonstrate that the hedging activities of
option market makers through the underlying asset market leads to a positive
correlation of the spread between the two markets. With the exception of NT, all of
the other measures of spot liquidity are positively correlated with implied volatility,
with the preliminary results apparently revealing a significant correlation between
spot liquidity and option implied volatility.
<Table 3 is inserted about here>
Unsurprisingly, the highest correlation amongst all of the spot liquidity
measures is to be found between AQS and AES. The correlations between other spot
liquidity variables are generally low, indicating that the dynamics of these spot
liquidity measures, other than AQS and AES, could differ significantly. These spot
liquidity variables are treated separately in our subsequent analyses. We find similar
results to the high correlation between skewness and the slope of the implied
volatility curve reported by Bakshi et al. (2003), with a correlation of 0.55 between
SKEW and ‘model-free implied volatility’ (MFIV).
3. EMPIRICAL RESULTS
In our efforts to determine whether option prices are affected by liquidity, we begin
15
by adopting the testing procedure of Chan and Fong (2006) to investigate the
cross-sectional results of the time-series regressions for the 30 component stocks of
the DJIA index, and then follow the analysis framework of Duan and Wei (2009) to
further explore the liquidity impact on the levels and slopes of the implied volatility
curves. Essentially, while the first part is just to focus on whether there is any
relationship between option prices and liquidity, the second part is to figure out
whether liquidity is priced in options.
3.1 The Effects of Liquidity on Option Prices
A summary of the cross-sectional estimation results of the time-series regression
models is presented in Table 4, with the various spot and option liquidity proxies and
control variables being individually, or jointly, employed to explain the option prices
represented by MFIV. In their attempts to determine option prices, Dennis and
Mayhew (2002) and Duan and Wei (2009) investigate specific variables, including
systematic risk, option-implied moments and firm-specific characteristics; our
empirical analysis is motivated by these studies, and therefore begins with an
exploration of the same variables in Models (1) and (2) of Table 4.
Duan and Wei (2009) suggest that systematic risk is priced into options; thus,
we use only the systematic risk proportion, SYS, as the independent variable in
Model (1) of Table 4. Consistent with their finding, we find that the average of the
16
coefficient estimates is negatively significant at the 1 per cent level; however, the
average adjusted R2
is only 6.5 per cent.
Following Dennis and Mayhew (2002) and Duan and Wei (2009) and also
controlling for the aggregate market condition, we include SKEW, KURT, and
SPMFIV, along with two firm-specific characteristics, SIZE and LEV, as the
independent variables in Model (2). The averages of the coefficient estimates of
these variables are all significant at the 1 per cent level, with a substantial rise, to
86.9 per cent, in the average adjusted R2
. Our findings are basically the same as
those in the prior studies; thus, these six variables are suitable control variables for
our subsequent investigation of the impact of liquidity on option prices.
Since no general consensus has yet been reached in the prior studies on the
most appropriate liquidity measure, we first of all investigate all of the possible
candidate measures; thereafter, those with more information for determining option
prices are retained for further analysis, with the most informative proxy being
selected from each liquidity variable group, as explained in the following sections.
The criteria used to select a liquidity proxy include the significance of the regression
coefficient, the proportion of cross-sectional coefficients with both an expected sign
and significance at the 5 per cent level, and the adjusted R2 of the regression model.
Using the six variables noted above as the control variables, we investigate the
17
individual, incremental contribution of every measure of spot liquidity in determining
option prices in Models (3)-(8) of Table 4, where liquidity is proxied by VOL, NT, ATS,
AOI, AQS, and AES. By definition, VOL, ATS, and AQS are closely related to NT, AOI,
and AES, respectively; we therefore split the spot liquidity proxies into three groups.
Although both VOL and NT are volume-related variables, there are discernible
differences in their explanatory power. In Model (3) of Table 4, the average of VOL
coefficients is significantly positive at the 1 per cent level with 83.33 per cent of
which having a t-statistic greater than 1.96. In contrast, the NT coefficient with a
t-statistic greater than 1.96 is only 66.67 per cent in Model (4), although the average
of the coefficients is also significant at the 1 per cent level (with a slightly lower
t-statistic). In the model which includes VOL, the average adjusted R2 is also slightly
higher than that in which NT is included, although the difference is small; we
therefore use VOL as the volume-related liquidity proxy.
As we can see from Models (5) and (6) of Table 4, the average of the ATS (AOI)
coefficient is significant at the 1 per cent (5 per cent) level. The coefficients for ATS
with t-statistics greater than 1.96 are also much higher than those of AOI (93.33 per
cent vs. 53.33 per cent). Furthermore, the average adjusted R2 for Model (5) is slightly
larger than that for Model (6). Thus, although both the ATS and AOI measures are
related to trade size, the former does appear to be more informative than the latter in
18
determining option prices.
As shown in Model (7) of Table 4, the coefficient estimate for AQS has the
lowest p-value amongst the six spot liquidity proxy variables, whilst similar results
are found for AES in Model (8). The findings from Models (7) and (8) are consistent
with those obtained by Cho and Engle (1999) and Aitken and Carole (2003),
indicating that the bid-ask spread is a better proxy for spot liquidity than other
measures, in terms of explaining MFIV.
Although both AES and AQS have similar explanatory power, the average
adjusted R2 for Model (7), which includes AQS, is larger than that for Model (8),
which includes AES, and the proportion of the AQS coefficients with t-statistics
greater than 1.96 is also higher than that of the AES coefficients (100 per cent vs.
93.33 per cent). As a result, we select AQS as the proxy variable for the bid-ask
measures. Thus, from the six spot liquidity proxies, we select VOL, ATS and AQS
for our subsequent analysis, since these proxies appear to be more informative.
As noted by both Brenner et al. (2001) and Deuskar, Gupta, and Subrahmanyam
(2008), the price of an option is affected by its own liquidity; thus, in Models (9)-(12)
of Table 4, we investigate the incremental contribution of each option liquidity
variable in determining option prices, with the respective liquidity in these models
being proxied by the four measures OVOL, DVOL, OAQS, and OI, as proposed by
19
Cao and Wei (2009). Of these measures, OVOL and DVOL, which are volume-related
variables, are highly correlated.
As shown in Models (9) and (10) of Table 4, both OVOL and DVOL have a
positively significant impact on option prices at the 1 per cent level, with similar
explanatory power, albeit with a higher t-statistic for the DVOL coefficient. Whilst
DVOL takes into account the effect of moneyness, OVOL essentially allocates the
same weight to volume across different levels of moneyness. We therefore use
DVOL for our subsequent analysis, as opposed to OVOL, since it should prove to be
more informative, although it is noted that both measures do appear to be equally
effective in the regression analyses.
The regression results of Models (11) and (12) of Table 4 indicate that both
OAQS and OI are also important factors in determining option prices. As expected,
the coefficient of OAQS (OI ) is negatively (positively) significant at the 1 per cent
level; however, the proportion of the OAQS coefficients with t-statistics smaller
than –1.96 is much higher than that for OI with t-statistics greater than 1.96 (80 per
cent vs. 53.33 per cent). Amongst the four option liquidity measures, only OVOL
and DVOL are categorized in a group. According to the regression results reported
earlier, we select DVOL, OAQS, and OI for our subsequent analysis.
<Table 4 is inserted about here>
20
Based upon the results of Table 4, the three most significant spot liquidity
proxy variables in each liquidity variable category (VOL, ATS, and AQS) and the
three most significant option liquidity measures (DVOL, OAQS, and OI) are
simultaneously included into a regression model with the six control variables. The
summary estimates of the coefficients on these liquidity variables are reported in
Table 5. In order to confirm the robustness of our results, we also implement the
regression model on the data, by year.
As shown in the first column of Table 5 (relating to the full sample), amongst
all of the spot liquidity proxies, the average of the AQS coefficients is still found to
be highly and positively significant, whereas the averages of the VOL and ATS
coefficients are not significant. The proportion of AQS coefficients with t-statistics
greater than 1.96 remains at 100 per cent for the full sample period. For the option
liquidity measures, the cross-sectional proportion of OAQS with significance is
found to be the highest, at 70 per cent, although the coefficient averages of all option
liquidity proxies are significant at below the 1 per cent level, with signs that are
consistent with those reported in Table 4.
<Table 5 is inserted about here>
These findings, in conjunction with those reported in Table 4, imply that low
(high) spot liquidity leads to high (low) option prices, with options becoming less
21
expensive when the options market becomes more illiquid; this is consistent with the
hedging cost explanation provided by Cetin et al. (2006) and the ‘illiquidity
premium’ hypothesis of Amihud and Mendelson (1986a).
However, when running the regression year-by-year as a check for the robustness
of our results, as shown in the remaining columns of Table 5, AQS is the only liquidity
proxy which consistently has a highly and positively significant coefficient across
almost all of the sub-samples. Within these sub-samples, the proportions of the AQS
coefficients with t-statistics greater than 1.96 are generally not as high as those for the
full sample; however, they are consistently found to be the largest amongst all of the
liquidity proxies. Although some of the option liquidity measures do not retain their
significance in all of the sub-samples, the signs of their coefficients are generally
consistent across years.
Given that the time-to-maturity for all of the option-related variables thus far
examined has been fixed at 30 days, questions may arise as to whether our empirical
findings are dependent upon the selection of any specific maturity period. Thus, in
order to further validate our results, we also compile all of the option-related data for
the 60-, 91-, and 182-day maturity periods, and then rerun all of the tests.12
The
12
In order to match the MFIV maturity periods, the three control variables, SKEW, KURT and
SPMFIV, are computed using options with the same maturity periods.
22
results for the full sample across all maturity periods are reported in Table 6.13
<Table 6 is inserted about here>
To briefly summarize, the results reported in Tables 5 and 6 imply that the
impact on option prices from spot liquidity, proxied by AQS, is robust across both
years and maturity periods, whereas the impact of option liquidity, as measured by
OAQS, is robust across maturity periods only.
On the whole, our empirical findings are robust across maturity periods with
AQS and OAQS being found to be the most robust liquidity measures.14
Thus, our
empirical results strongly indicate that option prices are significantly influenced by
both spot and option liquidity, although the former is found to be more influential.
3.2 The Level and Slope Effect Tests
The commonality in liquidity demonstrated by Chordia, Roll, and Subrahamnyam
(2000) implies that liquidity is a systematic risk, since it cannot be diversified within
the market. Furthermore, Duan and Wei (2009) show that when controlling for the
total risk of the underlying asset, a greater level of systematic risk still leads to a
higher level of implied volatility and a steeper implied volatility curve slope; it
therefore seems reasonable to surmise that the liquidity, for both spot and option,
13
For the purpose of space saving, the results for the sub-samples, by year, are not reported here;
however, they are available upon request. 14
These results are consistent with those produced by the panel regressions, which are again
available upon request.
23
will be reflected in both the level and slope of the implied volatility curve.
Based on the Duan and Wei (2009) framework, we investigate whether the
liquidity effect is revealed in the implied volatility curves of individual stock options
by testing the following null hypotheses. The option prices are characterized by the
levels and slopes of the implied volatility curves, with the spot (option) liquidity
being measured by AQS (OAQS ) as the most significant liquidity proxy variable.
We propose the following four testable hypotheses:
Hypothesis 1a: The level of the implied volatility curve is unrelated to the
level of spot liquidity.
Hypothesis 1b: The level of the implied volatility curve is unrelated to the
level of option liquidity.
Hypothesis 2a: The slope of the implied volatility curve is unrelated to
the level of spot liquidity.
Hypothesis 2b: The slope of the implied volatility curve is unrelated to the
level of option liquidity.
The tests are implemented using the two-step regressions proposed by
Fama-MacBeth (1973). Specifically, we first of all obtain the time-series estimates
for the level and slope of the implied volatility curve, which are then applied in the
second step to run the cross-sectional regressions for our examination of whether
24
they are related to liquidity. The first-step regression is implemented on a monthly
basis, with all observations in a one-month period being collected for each firm; thus,
we end up with two time series of 48 observations on the intercept and slop
coefficients, α0 j and α1 j. In other words, the data serving as the inputs for the
second-step regression are the two 48-by-30 metrics (48 months x 30 firms). In the
second step, we perform the following month-by-month cross-sectional regression to
test Hypotheses 1a and 1b.
0 0 1 2 3
4 5 , 1, 2, , 30.
j j j j
j j j
a SYS SKEW KURT
AQS OAQS e j
(4)
The 48 monthly estimates of each regression coefficient are then averaged, with
the corresponding average t-statistic being calculated by ( ) 48 . .i imean S D .
This Wald test is a conditional test controlling for the effects of risk-neutral
skewness, kurtosis, and the systematic risk proportion. We expect to find that if the
spot (option) liquidity is unrelated to option prices, then γ4 = 0 (γ5 = 0).
The results for the level effect tests (Hypotheses 1a and 1b) are reported in Table 7.
The results reported in Panel A confirm those of Duan and Wei (2009), who note
that after controlling for stock-specific total volatility, the implied volatility level is
significantly and positively related to the systematic risk proportion of the
underlying stock. Panel B of Table 7 reveals strong evidence for the rejection of
25
Hypotheses 1a and 1b, with the effect of spot and option liquidity on the implied
volatility level remaining significant at the 1 per cent level, even after controlling for
the influence of risk-neutral skewness, kurtosis, and the systematic risk proportion.
These results are consistent with the findings of Cetin et al. (2006) who show
that with an increase in spot liquidity (i.e., a decline in AQS), there will be a decline
in the level of the implied volatility curve, which means that the coefficient γ4 should
be positive;15
and indeed, the majority of the γ4 estimates are found to be positive,
as indicated by the percentage under γ4 > 0. Since the results are consistent across all
maturity periods, the evidence for the rejection of Hypothesis 1a is extremely robust.
As regards the effect of option liquidity on the level of implied volatility, our
results are consistent with the ‘illiquidity premium’ hypothesis of Amihud and
Mendelson (1986a). Panel B of Table 7 shows that with an increase in OAQS, there
is a corresponding reduction in option prices (i.e., the implied volatility level). The
coefficient γ5 is significantly negative at the 1 per cent level, even after controlling
for the influence of risk-neutral skewness, kurtosis, and systematic risk proportion.
The majority of the γ5 estimates are found to be negative, as indicated by the
percentage under γ5 < 0. Thus Hypothesis 1b is also rejected, with the results
appearing to have no dependence on the maturity period selected.
15
For further details, refer to Figure 5 in Cetin et al. (2006).
26
The results are also of relevance to the recent findings of Bollen and Whaley
(2004) and Garleanu et al. (2009), both of which note the importance of buying (or
demand) pressure on option pricing. Specifically Garleanu et al. (2009) find that the
net demand for equity options by non-market makers, across various levels of
moneyness, is related to their cost and skew patterns. Combining our results with
theirs, one would expect that options become more expensive (cheaper) with greater
buying (selling) pressure, larger (smaller) option liquidity, and lower (higher) spot
liquidity.16
Regarding the explanatory power of the implied volatility level, there are
obvious improvements in the adjusted R2
with the inclusion in the regression of the
spot and option liquidity variables; for example, Table 7 reveals that with
consideration of the liquidity factors, the adjusted R2
for 30-day options is raised from
59.9 per cent to 70.2 per cent.
<Table 7 is inserted about here>
To test Hypotheses 2a and 2b, we replace the dependent variable in Equation (4)
with the slope α1 j and repeat the above procedure, as follows:
1 0 1 2 3
4 5 , 1, 2, , 30.
j j j j
j j j
a SYS SKEW KURT
AQS OAQS e j
(5)
16
Moreover, Garleanu et al. (2009) also suggest that the demand effect on option expensiveness is
weaker when there is more option liquidity.
27
The results for the slope effect tests (Hypotheses 2a and 2b) are reported in Table 8,
from which we find several points worth noting. Firstly, Panel A supports the
findings of Duan and Wei (2009) that the slope of the implied volatility curve is
related to the systematic risk proportion, although not all of the coefficients are
found to be statistically significant. Secondly, the results shown in Panel B provide
strong evidence for the rejection of Hypothesis 2b; that is, they demonstrate that the
slope of the implied volatility curve is also related to option liquidity. One possible
explanation for our results is because the liquidity of the option market is positively
related to demand pressure. Since Garleanu et al. (2009) suggest that the implied
volatility slope is due to demand pattern of equity options, thus the implied volatility
slope becomes more negative with an increase in option activity due to the increase
in demand pressure.
Thirdly, no conclusive relationships are discernible between the implied
volatility curve slope and spot liquidity (i.e., under most cases, γ′4 is not found to be
significant). This is a result which runs contrary to the findings of Cetin et al. (2006),
who report that spot liquidity can partly explain the implied volatility smile observed
in the equity option market.
Finally, the explanatory power of the implied volatility slope is enhanced after
the spot and option liquidity variables are taken into account, particularly for
28
short-term options. As shown in Table 8, when AQS and OAQS are included in the
regression, there is an increase in the adjusted R2 of 5.7 per cent (6.6 per cent) for
30-day (60-day) options.
<Table 8 is inserted about here>
4. CONCLUSIONS
The effects of liquidity on the value of financial assets have seen significant growth
in interest amongst academics and practitioners alike over recent years; we set out in
this study to examine the effects of liquidity on option prices. Our main findings
include: (i) the model-free implied volatility of equity options can be explained by
both spot and option liquidity; (ii) a higher level of spot illiquidity leads to a higher
implied volatility curve level, which is in line with the finding of Cetin et al. (2006),
that every option has an intrinsic, significant spot liquidity cost; (iii) a higher option
liquidity level leads to a higher implied volatility curve level, which is consistent
with the prediction of the ‘illiquidity premium’ hypothesis of Amihud and
Mendelson (1986a); and (iv) a higher option liquidity level also leads to more
pronounced implied volatility skewness.
29
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33
Table 1 Variable definitions This table reports the definitions of the variables used in the empirical analysis in this study; P denotes price, with the subscripts referring to the following: t is an
actual transaction, A refers to ‘Ask’, B refers to ‘Bid’, and M represents the Bid-Ask mid-point.
Measure Notation Definition
Dependent Variable
Model-free Implied volatility MFIV Daily model-free implied volatility of equity options with 30-day maturity computed using the Bakshi
et al. (2003) methodology.
Spot Liquidity Variables
Cumulative Trading Volume VOL Total number of shares traded per day.
Number of Trades NT Total number of daily trades.
Average Trade Size ATS Total number of shares traded each day ÷ the number of trades for the day.
Absolute Order Imbalance AOI Absolute value of the number of buyer-initiated minus seller-initiated trades for the day.
Average Proportional Quoted Spread AQS (PA – PB)/ (PM)
Average Proportional Effective Spread AES 2*│Pt – PM │/PM
Option Liquidity Variables
Option Trading Volume OVOL Trading volume of the total number of contracts.
Option Dollar Trading Volume DVOL ΣVOL* (PA + PB)/2 (Hasbrouck and Seppi, 2001). Option Proportional Bid Ask Spread OAQS [ΣVOL* (PA – PB)/PM]/ΣVOL (Cao and Wei, 2009).
Option Open Interest OI Total option open interest.
Control Variables
Systematic Risk Proportion SYS R2 of the systematic risk factor computed using a one-year (250-day) rolling window regression.
Risk-neutral Implied Skew SKEW Daily risk-neutral implied skewness of equity options with 30-day maturity computed using the Bakshi
et al. (2003) methodology.
Risk-neutral Implied Kurtosis KURT Daily risk-neutral implied kurtosis of equity options with 30-day maturity computed using the Bakshi
et al. (2003) methodology.
S&P500 Model-free Implied Volatility SPMFIV Daily model-free implied volatility of S&P500 index options with 30-day maturity computed using the
Bakshi et al. (2003) methodology.
Firm Size SIZE Market capitalization.
Leverage LEV Book value of debt ÷ (book value of debt + market value of equity).
34
Table 2 Cross-sectional summary statistics
This table presents the cross-sectional summary statistics of the variables used in this study, reporting
the mean, median and standard deviation (S.D.) of the time-series variables for 30 DJIA component
stocks over the sample period which runs from 1 January 2001 to 31 December 2004.
Measure Mean Median S.D.
Dependent Variable
MFIV 0.32952 0.32774 0.05682
Spot Liquidity Variables
VOL (in thousands) 7,780 5,892 8,030
NT 5,952 3,555 9,182
ATS 1,538 1,530 474
AOI 364 227 444
AQS (%) 0.07828 0.07588 0.02010
AES (%) 0.07048 0.06286 0.03092
Option Liquidity Variables
OVOL 5,955 3,808 6,515
DVOL 11,773 6,277 13,889
OAQS (%) 18.013 18.068 2.999
OI 93,395 67,978 91,250
Control Variables
SYS 0.32822 0.35722 0.12883
SKEW –0.77434 –0.79050 0.16177
KURT 2.81419 2.80831 0.13371
SPMFIV 0.24908 0.24909 0.00003
SIZE (millions) 1,191 1,008 909
LEV 0.38223 0.36403 0.25334
35
Table 3 Correlation matrix of the cross-sectional means of the time series variable coefficients
MFIV VOL NT ATS AOI AQS AES OVOL DVOL OAQS OI SYS SKEW KURT SPMFIV SIZE
VOL 0.32
NT –0.05 0.61
ATS 0.43 0.62 –0.14
AOI 0.05 0.40 0.41 0.08
AQS 0.75 0.33 –0.15 0.54 0.05
AES 0.62 0.30 –0.04 0.40 0.09 0.80
OVOL 0.08 0.47 0.38 0.20 0.24 0.07 0.08
DVOL 0.12 0.30 0.20 0.17 0.14 0.13 0.11 0.78
OAQS –0.25 –0.09 0.04 –0.16 –0.01 –0.19 –0.15 –0.08 –0.14
OI –0.11 0.05 0.19 –0.12 0.04 –0.16 –0.13 0.19 0.13 0.12
SYS –0.13 0.04 0.36 –0.30 0.06 –0.34 –0.20 0.07 0.00 0.08 0.14
SKEW 0.55 0.20 –0.08 0.34 0.04 0.48 0.37 0.06 0.07 –0.18 –0.07 –0.21
KURT 0.57 0.14 –0.05 0.21 0.03 0.40 0.33 0.05 0.06 –0.15 –0.01 –0.07 0.32
SPMFIV 0.86 0.23 –0.05 0.32 0.02 0.62 0.52 0.01 0.05 –0.21 –0.16 –0.01 0.41 0.51
SIZE –0.37 –0.19 –0.20 –0.04 –0.06 –0.18 –0.20 –0.04 0.01 –0.02 0.00 –0.32 –0.17 –0.17 –0.39
LEV 0.07 0.17 0.37 –0.14 0.07 –0.10 0.00 0.08 –0.02 0.13 0.10 0.49 –0.02 0.03 0.11 –0.76
36
Table 4 Regression results of model-free implied volatility on liquidity proxy variables
This table presents the OLS regression results for the 30 component stocks of the Dow Jones Industrial Average Stock Index over the period from 1 January 2001 to 31
December 2004; the variable definitions are detailed in Table 1. We run separate time-series regressions for each stock and then average the regression coefficients
across the stocks; S.E. refers to the averaged standard errors, and Adj. R2 refers to the cross-sectional means of the
R2 for the time-series regressions. Liquidity variable
t-value >1.96 (< –1.96) refers to the percentage of the regression coefficients of the liquidity variables with t-statistics of greater (less) than 1.96 (–1.96). *** indicates
significance at the 1% level and ** indicates significance at the 5% level.
Independent
Variables
Models
(1) (2) (3) (4) (5) (6)
Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E.
Intercept 0.363 0.008*** 0.288 0.067*** 0.308 0.065*** 0.351 0.069*** 0.215 0.066*** 0.275 0.067***
VOL (×10–9
)
3.497 0.477***
NT (×10–6
) 4.754 1.400***
ATS (×10–5
) 1.886 0.291***
AOI (×10–5
) 1.243 0.537**
SYS –0.135 0.024*** –0.107 0.015*** –0.103 0.015*** –0.121 0.015*** –0.078 0.015*** –0.108 0.015***
SKEW 0.048 0.005*** 0.043 0.005*** 0.046 0.005*** 0.042 0.005*** 0.047 0.005***
KURT 0.023 0.003*** 0.024 0.003*** 0.024 0.003*** 0.023 0.003*** 0.024 0.003***
SPMFIV 1.002 0.028*** 0.968 0.028*** 0.983 0.028*** 0.984 0.028*** 1.004 0.028***
SIZE (×10-12
) –1.555 0.350*** –1.603 0.337*** –1.830 0.360*** –1.311 0.339*** –1.485 0.348***
LEV –0.628 0.126*** –0.677 0.123*** –0.714 0.129*** –0.552 0.124*** –0.620 0.126***
Liquidity Variable
t-value > 1.96 (%) N.A. N.A. 83.33 66.67 93.33 53.33
Liquidity Variable
t-value < –1.96 (%) N.A. N.A. 0.00 16.67 3.33 10.00
Adj. R2
(%) 6.50 86.9 87.90 87.40 87.90 87.10
37
Table 4 (Contd.)
Independent
Variables
Models
(7) (8) (9) (10) (11) (12)
Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E.
Intercept 0.163 0.062*** 0.245 0.064*** 0.293 0.066*** 0.284 0.066*** 0.286 0.067*** 0.286 0.070***
AQS (×102) 0.617 0.042***
AES (×102) 0.489 0.043***
OVOL (×10–6
) 1.362 0.276***
DVOL (×10–6
) 0.531 0.095***
OAQS –0.040 0.012***
OI (× 10–8
) 8.822 3.051***
SYS –0.059 0.015*** –0.083 0.015*** –0.107 0.015*** –0.106 0.015*** –0.108 0.015*** –0.109 0.015***
SKEW 0.035 0.005*** 0.041 0.005*** 0.045 0.005*** 0.045 0.005*** 0.046 0.006*** 0.046 0.005***
KURT 0.021 0.003*** 0.022 0.003*** 0.023 0.003*** 0.023 0.003*** 0.023 0.003*** 0.023 0.003***
SPMFIV 0.841 0.024*** 0.897 0.029*** 1.003 0.028*** 0.998 0.028*** 0.992 0.028*** 1.016 0.028***
SIZE (×10–12
) –0.966 0.330*** –1.337 0.337*** –1.595 0.342*** –1.591 0.340*** –1.565 0.349*** –1.509 0.350***
LEV –0.473 0.119*** –0.572 0.122*** –0.651 0.125*** –0.593 0.126*** –0.601 0.126*** –0.631 0.126***
Liquidity Variable
t-value > 1.96 (%) 100.00 93.33 76.67 70.00 00.00 53.33
Liquidity Variable
t-value < –1.96 (%) 0.00 0.00 0.00 0.00 80.00 16.67
Adj. R2
(%) 89.40 88.30 87.50 87.60 87.10 87.20
38
Table 5 Regression results of model-free implied volatility on selected liquidity proxies, 2001-2004 This table presents the OLS regression results for the 30 component stocks of the Dow Jones Industrial Average Stock Index over the period from 1 January 2001 to 31
December 2004 and four yearly sub-samples. The variable definitions are detailed in Table 1. The time-series regression is run separately for each stock with the regression
coefficients then being averaged across the stocks. S.E. refers to the averaged standard errors, and Adj. R2 refers to the cross-sectional means of the
R2 for the time-series
regressions. Liquidity variable t-value >1.96 (< –1.96) refers to the percentage of the regression coefficients of the liquidity variables with t-statistics of greater (less) than
1.96 (–1.96). *** indicates significance at the 1% level; ** indicates significance at the 5% level; and * indicates significance at the 10% level.
Variables
Year
2001-2004 2001 2002 2003 2004
Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E.
Intercept 0.183 0.063 *** 0.431 0.246 0.673 0.460 0.137 0.322 0.205 0.226
VOL (×10–9
)
0.191 0.722 –0.963 1.840 2.723 1.854 1.383 1.346 –0.380 1.018
ATS (×10–6
) 5.372 3.909 1.993 7.071 –4.685 9.132 –5.580 7.956 3.272 6.023
AQS (×102) 0.533 0.044 *** 0.273 0.060 *** 0.254 0.140 * 0.197 0.161 0.548 0.267 **
DVOL (×10–6
) 0.297 0.095 *** 0.337 0.258 0.744 0.357 ** 0.139 0.161 0.178 0.125
OAQS –0.032 0.010 *** –0.034 0.020 * –0.034 0.023 –0.019 0.014 –0.011 0.008
OI (×10–8
) 7.766 2.773 *** 34.400 8.780 *** 4.100 7.358 –0.500 4.810 0.900 2.805
Control Variables Yes Yes Yes Yes Yes
VOL t-value > 1.96 (%) 40.00 13.33 50.00 33.33 16.67
VOL t-value < –1.96 (%) 30.00 16.67 6.67 6.67 20.00
ATS t-value > 1.96 (%) 43.33 23.33 16.67 13.33 33.33
ATS t-value < –1.96 (%) 26.67 6.67 23.33 20.00 13.33
AQS t-value > 1.96 (%) 100.00 76.67 56.67 36.67 46.67
AQS t-value < –1.96 (%) 0.00 3.33 3.33 3.33 3.33
DVOL t-value > 1.96 (%) 40.00 13.33 43.33 10.00 20.00
DVOL t-value < –1.96 (%) 0.00 0.00 3.33 0.00 0.00
OAQS t-value > 1.96 (%) 0.00 0.00 0.00 0.00 0.00
OAQS t-value < –1.96 (%) 70.00 33.33 33.33 26.67 26.67
OI t-value > 1.96 (%) 53.33 60.00 30.00 30.00 36.67
OI t-value < –1.96 (%) 10.00 13.33 23.33 33.33 33.33
Adj. R2
(%) 90.4 86.8 91.5 90.2 73.7
39
Table 6 Regression results of model-free implied volatility on selected liquidity proxies for various maturity periods This table presents the OLS regression results for the 30 component stocks of the Dow Jones Industrial Average Stock Index over the period from 1 January 2001 to 31
December 2004. The variable definitions are detailed in Table 1. The time-series regression for each stock is run separately for each maturity category with the regression
coefficients then being averaged across the stocks. S.E. refers to the averaged standard errors, and Adj. R2 refers to the cross-sectional means of the
R2 for the time-series
regressions. Liquidity variable t-value >1.96 (< –1.96) refers to the percentage of the regression coefficients of the liquidity variables with t-statistics of greater (less) than 1.96
(–1.96). *** indicates significance at the 1% level; ** indicates significance at the 5% level; and * indicates significance at the 10% level.
Variables
Maturity
30-days 60-days 91-days 182-days
Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E.
Intercept 0.183 0.063 *** 0.297 0.065 *** 0.453 0.060 *** 0.775 0.065 ***
VOL (×10–9
)
0.191 0.722 –0.206 0.660 –0.872 0.590 –1.027 0.504 **
ATS (×10–6
) 5.372 3.909 4.569 3.607 5.159 3.233 6.201 2.770 **
AQS (×102) 0.533 0.044 *** 0.562 0.041 *** 0.493 0.036 *** 0.385 0.031 ***
DVOL (×10–6
) 0.297 0.095 *** 0.188 0.105 * 0.084 0.088 0.101 0.145
OAQS –0.032 0.010 *** –0.032 0.011 ** –0.033 0.011 *** –0.032 0.014 **
OI (×10–8
) 7.766 2.773 *** 13.400 2.325 *** 12.000 2.616 *** –5.214 2.467 **
Control Variables Yes Yes Yes Yes
VOL t-value > 1.96 (%) 40.00 40.00 33.33 13.33
VOL t-value < –1.96 (%) 30.00 36.67 50.00 46.67
ATS t-value > 1.96 (%) 43.33 46.67 46.67 43.33
ATS t-value < –1.96 (%) 26.67 26.67 26.67 23.33
AQS t-value > 1.96 (%) 100.00 100.00 96.67 93.33
AQS t-value < –1.96 (%) 0.00 0.00 0.00 0.00
DVOL t-value > 1.96 (%) 40.00 16.67 16.67 6.67
DVOL t-value < –1.96 (%) 0.00 3.33 3.33 6.67
OAQS t-value > 1.96 (%) 0.00 0.00 3.33 0.00
OAQS t-value < –1.96 (%) 70.00 66.67 83.33 56.67
OI t-value > 1.96 (%) 53.33 76.67 50.00 23.33
OI t-value < –1.96 (%) 10.00 3.33 13.33 40.00
Adj. R2
(%) 90.4 90.1 90.9 91.9
40
Table 7 Regression results of the level effect tests This table presents the two-step regression results of the level effect tests. In the first step, for each stock,
j, we regress the difference in moneyness between implied volatility and historical volatility for
non-overlapping periods of one month: 0 1IV hisjk j j j jk j jka a y y , thereby obtaining the monthly
time-series of the intercept α0 j and the slope coefficient α1 j for all stocks included in the Dow Jones
Industrial average stock index. The moneyness variable is adjusted by the sample mean for the month so
that the intercept α0 j is the average of the difference between implied and historical volatility. In the
second step, we cross-sectionally regress the intercept on the systematic risk proportion, the risk-neutral
skewness, the risk-neutral kurtosis, spot liquidity (AQS), and option liquidity (OAQS). The regressions,
which are performed separately for four different maturity categories, are undertaken for every month in
the following two different forms:
0 0 1 2 3j j j j ja SYS Skew Kurt e and
0 0 1 2 3j j j ja SYS Skew Kurt 4 5j j jAQS OAQS e .
We average the monthly regression coefficients and then calculate the corresponding t-values as
( ) 48 / . .( )i imean S D . To conserve space, we omit the regression intercept and its t-value. The entries
under γ1 > 0, γ4 > 0 and γ5 < 0 refer to the percentages of the 48 monthly coefficients which satisfy inequality.
The reported R2 is the average R
2 obtained from the monthly cross-sectional regressions. The risk-neutral
skewness and kurtosis are estimated using the Bakshi et al. (2003) methodology. *** indicates significance
at the 1% level.
Variables Maturity
30-days 60-days 91-days 182-days
Panel A: Systematic risk proportion, risk-neutral skewness and kurtosis
γ1
Avg. 0.100 0.117 0.083 0.056
t-value 5.660 *** 6.080 *** 4.640 *** 4.530 ***
γ1 > 0 (%) 95.80 87.50 79.20 77.10
γ2 Avg. 0.149 –0.003 –0.080 –0.119
t-value 25.060 *** –0.100 –2.400 *** –7.320 ***
γ3
Avg. 0.039 –0.164 –0.281 –0.442
t-value 4.970 *** –3.900 *** –6.430 *** –15.520 ***
R2
(%) 59.90 56.70 50.30 52.80
Panel B: Systematic risk proportion, risk-neutral skewness, kurtosis, spot and option liquidity
γ1
Avg. 0.077 0.102 0.070 0.048
t-value 5.700 *** 6.500 *** 4.980 *** 4.060 ***
γ1 > 0 (%) 93.80 89.60 77.10 64.60
γ2 Avg. 0.110 –0.045 –0.133 –0.169
t-value 14.180 *** –1.490 –4.040 *** –10.710 ***
γ3 Avg. 0.033 –0.145 –0.261 –0.393
t-value 4.170 *** –3.980 *** –6.400 *** –14.700 ***
γ4 (x 102
)
Avg. 0.960 0.908 1.111 1.050
t-value 8.810 *** 14.200 *** 11.800 *** 9.090 ***
γ4 > 0 (%) 89.60 97.90 95.80 93.80
γ5
Avg. –0.227 –0.183 –0.173 –0.174
t-value –7.670 *** –3.800 *** –3.270 *** –2.900 ***
γ5 < 0 (%) 85.40 83.30 68.80 68.80
R2
(%) 70.20 66.40 64.60 66.40
41
Table 8 Regression results of the slope effect tests
This table presents the two-step regression results of the slope effect tests. In the first pass, for each
stock, j, we regress the difference in moneyness between implied volatility and historical volatility for
non-overlapping periods of one month: 0 1IV hisjk j j j jk j jka a y y , thereby obtaining the monthly
time-series of the intercept α0 j and the slope coefficient α1 j for all stocks included in the Dow Jones
Industrial average stock index. The moneyness variable is adjusted by the sample mean for the month so
that the intercept α0 j is the average of the difference between implied and historical volatility. In the
second step, we cross-sectionally regress the slope coefficient on the systematic risk proportion, the
risk-neutral skewness, the risk-neutral kurtosis, spot liquidity (AQS), and option liquidity (OAQS). The
regressions, which are performed separately for four different maturity categories, are undertaken for
every month in the following two different forms:
1 0 1 2 3j j j j ja SYS Skew Kurt e and
1 0 1 2 3j j j ja SYS Skew Kurt 4 5j j jAQS OAQS e .
We average the monthly regression coefficients and then calculate the corresponding t-values as
( ) 48 / . .( )i imean S D . To conserve space, we omit the regression intercept and its t-value. The entries under
γ′1 < 0, γ′4 < 0 and γ′5 > 0 refer to the percentages of the 48 monthly coefficients which satisfy inequality. The
reported R2 is the average R
2 obtained from the monthly cross-sectional regressions. The risk-neutral
skewness and kurtosis are estimated using the Bakshi et al. (2003) methodology. *** indicates significance at
the 1% level.
Variables Maturity
30-days 60-days 91-days 182-days
Panel A: Systematic risk proportion, risk-neutral skewness and kurtosis
γ′1
Avg. –0.206 –0.084 –0.034 0.002
t-value –8.300 *** –8.220 *** –4.010 *** 0.410
γ′1 < 0 (%) 89.60 93.80 77.10 45.80
γ′2 Avg. 0.419 0.414 0.441 0.411
t-value 17.440 *** 10.560 *** 13.760 *** 27.150 ***
γ′3
Avg. –0.190 –0.072 –0.005 0.060
t-value –10.04 *** –2.99 *** –0.25 5.52 ***
R2
(%) 41.00 54.60 65.90 76.10
Panel B: Systematic risk proportion, risk-neutral skewness, kurtosis, spot and option liquidity
γ′1
Avg. –0.169 –0.083 –0.034 0.003
t-value –6.18 *** –6.75 *** –3.83 *** 0.49
γ′1 < 0 (%) 79.20 83.30 75.00 50.00
γ′2 Avg. 0.417 0.395 0.432 0.405
t-value 17.49 *** 10.75 *** 13.66 *** 25.51 ***
γ′3 Avg. –0.195 –0.087 –0.004 0.057
t-value –9.36 *** –3.68 *** –0.19 4.57 ***
γ′4 (x 102
)
Avg. –0.445 0.318 0.269 –0.003
t-value –1.25 1.50 2.80 *** –0.04
γ′4 < 0 (%) 54.17 31.30 27.10 33.30
γ′5
Avg. 0.566 0.363 0.146 0.119
t-value 6.26 *** 4.69 *** 2.85 *** 2.61 ***
γ′5 > 0 (%) 83.3 83.3 64.6 60.4
R2
(%) 46.7 61.2 69.6 78.3