MICROSTRUCTURE NOISE AND PRICE DISCOVERY IN OPTION MARKETS:
THEORY AND EMPIRICAL EVIDENCE
Michal Czerwonko*
Nabil Khoury**
Stylianos Perrakis***
Marko Savor****
JEL CODE: G14, G15
KEYWORDS: contingent claims, market microstructure, volatility inversion, price
discovery, tick size, informed trading
This version: May 2012

*Postdoctoral Fellow, McGill University, Concordia University and University of Quebec in Montreal; **
Professor of Finance, University of Quebec in Montreal; ***RBC Distinguished Professor of Financial Derivatives,
Concordia University; ****Associate Professor of Finance, University of Quebec in Montreal. Financial support
from the Social Sciences and Humanities Research Council of Canada, the Autorité des marchés financiers, the
Desjardins Chair in the Management of Derivative Products, and the RBC Distinguished Professorship in Financial
Derivatives, is gratefully acknowledged. The authors are also grateful to participants at the 2011 Montreal
Mathematical Finance Days, the 2011 European Financial Management Conference, and the 24th Australasian
Banking and Finance Conference 2011, for their constructive remarks.
1
MICROSTRUCTURE NOISE AND PRICE DISCOVERY IN OPTION MARKETS:
THEORY AND EMPIRICAL EVIDENCE
Abstract
We document theoretically a major dependence of the Information Shares (IS) price discovery metrics of a financial
market on its microstructure noise, as well as on parameter estimation errors. We then study the impact of the tick
size reduction in the option market on the discovery process in that market by introducing a new inversion
methodology. In contrast with previous studies, we find that equity options showed negligible price discovery before
the tick reduction but exhibit major increases in IS thereafter. Last, we present event-type empirical evidence of
informed trading in the equity option market after the tick change.
2
MICROSTRUCTURE NOISE AND PRICE DISCOVERY IN OPTION MARKETS:
THEORY AND EMPIRICAL EVIDENCE
I. Introduction
Price discovery refers to the process by which new information is embodied in trading activity. For
optioned stocks such discovery can take place either in the option market or in the market for the
underlying asset. The study of price discovery (or informed trading) through the option market goes
back to at least thirty years, starting with Manaster and Rendleman (1982), with important
contributions by Vijh (1990), Stephan and Whaley (1990), Easley, O’Hara and Srinivas (EOS, 1998),
and Chakravarty, Gulen and Mayhew (CGM, 2004). These studies followed a wide variety of
methodological approaches and reached radically different conclusions: the early and the later studies
concluded that significant price discovery was taking place in the option market, while the two 1990
studies alleged that all such discovery was taking place in the market for the underlying asset. The
methodological approaches also shifted over time, from lead-lag analysis between the two markets to
the Information Shares (IS) method presented in the influential study of Hasbrouk (1995), who argued
persuasively that the earlier method was misspecified. The IS approach forms currently the standard in
price discovery studies between different markets, and almost all its applications in the pair of
underlying and option markets conclude that a significant amount of discovery occurs through the
option market.1
In this paper we focus on the applications of the IS methodology in assessing the price discovery
process in the option and underlying asset markets. The main contributions of the study are as follows.
First we show theoretically under plausible structural models of the microstructure errors, and with
asset dynamics consistent with the available empirical techniques, that the size (as measured by the
1
See CGM (2004), Holowczak et al (2006), Anand and Chakravarty (2007), Hsieh et al (2008), and Chen and Gau (2009).
3
variance) of existing microstructure noise in the option market has a major negative impact on the IS
metric in that market. We conclude that an accurate assessment of price discovery in the option market
should strive to preserve as far as possible the existing microstructure noise without distorting it by the
empirical design. We identify two major sources of such distortion in earlier studies, arising
respectively out of the implied price estimation through volatility inversion, and from the method of
estimating the IS metric. We develop alternative volatility inversion methods and a statistical
framework to a priori appraise such methods under plausible and relatively weak assumptions, be it in
simulated or in real data.2 We then apply the IS estimation in ways that avoid both distortions. We
show with our unbiased approach that the earlier studies gave a highly misleading picture of price
discovery in the option market, which was virtually nonexistent for equity options prior to 2007. As a
robustness check, we also validate our findings by using the alternative Component Shares (CS)
approach of Gonzalo and Granger (1995) that yielded qualitatively identical results.
We confirm our main theoretical result empirically by analyzing the effects of a new policy that
reduced the microstructure noise in the option market. On January 26, 2007 the Chicago Board of
Options Exchange (CBOE) introduced the first phase of its experimental Penny Pilot Program for 13
option classes that reduced the minimum tick size from $0.05 to $0.01 for option series trading below
$3 and from $0.1 to $0.05 for series trading above that level. 3 The tick size for all option series on the
QQQ and the DFX indexes was also similarly reduced. A second phase of the program was introduced
on September 28, 2007 for 24 additional option classes4 which are the subject of this study.5 Since such
2
Our paper is the first to consider the basic time series properties of the implied price both in simulated data when the
option pricing model is fully specified and in real data when such model is left unspecified. See sections 2.3, 2.5 and 4.1
later on.
3
CBOE Regulatory Circular RG07-09, Penny Pilot Program, January 18, 2007.
4
CBOE Information Circular IC07-150 SEC Approval of Penny Pilot Expansion, September 27, 2007.
5
The Penny Pilot Program was subsequently extended to 300 new option classes in 2009,and 2010, while delisted classes
were replaced by new ones. At present (May 2012) there are 370 option classes participating in the program. See CBOE
Information Circulars RG09121, RG10 004, RG10 16, RG10 51 and RG10 87.
4
a tick size reduction is expected to improve option market liquidity6 and reduce the implied price error,
it should obviously affect the price discovery process. 7 Indeed, we show that this tick size reduction
resulted in a major increase in the option market discovery metrics for all volatility inversion methods,
the ones proposed in this paper as well as the ones used in earlier studies. Last but not least, we present
an event-type study that confirms the fact that the tick size reduction resulted in a major increase in
informed trading through the equity option market, with extreme values of the option market’s price
discovery metrics leading abnormal returns in the underlying asset market by as much as seven days.8
Although, as already noted, several studies investigated various metrics of price discovery in markets
for options, there are relatively few theoretical studies of this discovery. Such studies have noted that
the lower liquidity of the option market is counterbalanced by its higher leverage in maximizing the
advantages that can be achieved by informed trading. The EOS (1998) study develops conditions under
which informed traders choose the option rather than the underlying asset market to exploit their
informational advantages. Nonetheless, their empirical results refer mainly to option trading volume as
a signal for informed trading, without any analysis of the simultaneous pricing in the two markets.
Other theoretical studies are by Biais and Hillion (1994) and Capelle-Blancard (2003).
The currently dominant IS approach to the price discovery process involving more than one market for
the price of a given financial asset was first presented in the pioneering work of Hasbrouck (1995). 9
Hasbrouck assumes that new information in each market is embodied in a common efficient or “true”
6
The proportional option bid-ask spreads are typically more than twenty times as large as the equivalent spreads for the
underlying assets, even for the most liquid index options; see also note 11.
7
Very few studies have dealt with this topic, and only Chen and Gau (2009) has studied option markets.
8
Similar tick size reductions have been introduced and studied earlier in other financial markets. In almost all published
studies the focus was on the effect that these changes had on market liquidity. These studies concluded that tick size
changes did have some impact on various liquidity measures such as bid-ask spread, trading volume and market depth, but
that its size and significance were unclear. See, for instance, Bacidore (1997), Bessembinder (2003), Bollen and Busse
(2006), Jones and Lipson (2001), and other references discussed later on in the text.
9
Earlier empirical studies had adopted the lead-lag analysis, which is a less robust methodology for this study. For an
example of such a study see Stephen and Whaley (1990).
5
asset price, together with independent error terms around that price when it is observed in the various
markets. He then focuses on a particular representation of this efficient price known as the Vector Error
Correction Model (VECM), originally developed by Engle and Granger (1987). The IS metrics are
defined as the contributions of each market to the variance of the efficient price, evaluated from the
VECM. An alternative metric, the Component Shares (CS), was developed by Gonzalo and Granger
(1995). Most studies have used primarily IS in the study of simultaneous pricing of cross-listed
securities in two different markets.10 In our case we focus on IS with CS used as a robustness check. It
turns out that both measures yield virtually identical conclusions in all cases.
There are several objective factors that make the application of the IS method difficult and imprecise.
First of all, option markets are generally characterized by lower liquidity and wider bid-ask
proportional spreads compared to their equity counterparts. While trading takes place within the quoted
bid-ask spread, the effective spreads that result from observed trades are also very wide, thus rendering
the concept of a unique implied stock price difficult to define.11 Second and most important, the
recovery of the underlying implied stock price from the observed option price is also fraught with
imprecision, as discussed extensively in the next section. Note that these difficulties in the implied
price estimation are peculiar to the option markets and do not affect, for instance, the futures markets,
for which the inversion of the cost-of-carry model is straightforward and does not involve unknown
parameters or large errors.12
10
See, for instance, Bacidore and Sofianos (2002), Eun and Sabherwal (2003) and Normandin (2004). For the relative
merits of the two metrics see below, note 29.
11
In the option sample used by Khoury, Perrakis and Savor (2011) the average option price was $1.33 for the period 20022004, while the quoted and effective average bid-ask spreads never fell below 0.16 and 0.13 respectively, yielding
respective proportional spreads of 12% and 10%. By contrast, in Fleming et al (1996) the average spread for stocks in their
sample was 0.58%.
12
For instance, Constantinides et al (2011) report that for 1990-2002 the S&P 500 index futures basis risk error was less
than 0.5% for 95% of the observations.
6
There are two issues that arise in the derivation of the option-implied underlying asset price, both of
which affect the option market microstructure error and through it the IS estimation. First, there are
several option contracts on the same underlying asset with differing strike prices and exercise dates
trading simultaneously, each one of which can define an implied price. In deriving the IS metric we
need to decide on the number of such implied prices to be aggregated in order to define the “average”
option market implied price. This is important, since any averaging obviously reduces the error of the
implied price and increases the IS metric as the number of averaged contracts increases, as we show in
this paper with both simulated and actual data. The practice varies in previous studies,13 and in this
paper we present the results for both a fixed number of contracts and all contracts that pass our minimal
liquidity filters.
The second issue concerns the method of implied price estimation, which starts with the option bid-ask
spread midpoint as an estimate of the option price. There are two alternative approaches to the
extraction of the implied underlying price estimation from this option price. CGM (2004) use the
Black-Scholes-Merton implied volatility as a simple translation device in order to estimate the option
implied stock price. Since using the contemporaneous stock and option prices to find the implied
volatility would result in a tautology, they invert the option price using a half hour lag to estimate the
implied volatility, thus assuming that this volatility remains constant during that lag. With the time
series of the observed stock prices and options’ implied prices, CGM estimated the Hasbrouck IS
metric and concluded that the option market’s contribution to price discovery was about 17.5% on
average. We shall call the CGM inversion method the Lagged Implied Volatility (LIV) method. 14
13
CGM (2004) do not mention the number of contracts, while Hsieh et al (2008) have used only the nearest at-the-money
contract, and Holowczak et al (2006) use the most actively traded option that may differ between observations.
14
The LIV method is also used for part of the empirical results of Anand and Chakravarty (2007), which, however, involve
the analysis of relative price discovery between implied price series of different options on the same underlying asset,
differentiated by volume. The underlying market does not enter the comparisons.
7
An alternative IS estimation is by the so-called “portfolio approach”, which relies on put-call parity
(PCP) and estimates the implied price as the difference between a call and a put option with equal
strike prices and times to estimation.15 This approach has its own problems, stemming from the fact
that PCP does not hold in the presence of transaction costs. Indeed, Muravyev et al (2012) document
many cases where there is absolutely no overlap between the spreads of the underlying and implied
prices. Further, all North American equity options are American, for which PCP holds only
approximately. Since our simulation experiments cannot accommodate American options, we do not
use the PCP approach in this paper and concentrate on the implied volatility.
Our study shows that that the IS estimates for the sample of firms used by CGM are highly misleading,
insofar as they are subject to two systematic biases that go in opposite directions but still yield results
that are far too high for the market conditions of the CGM data. By using the LIV inversion and a
statistical averaging technique followed by CGM that is described in detail in Hasbrouk (2002b, 2003
note 9) and Anand and Chakravarty (2007) we find almost the identical CGM result for the average
option market IS.16 When, however, this averaging technique is not adopted, the average option market
IS falls to between 0.9 and 2.2% for equities depending on the sample and the volatility conditions; our
proposed new inversion methods raise these numbers to 1.2-4.2%. For exchange traded funds (ETF)
these numbers become 9.1% and 10.8% under LIV and rise to 17% and 14.7% under the new inversion
methods. In all cases these numbers are a far cry from the CGM results. There was, thus, virtually no
price discovery taking place in the equity option market prior to the tick size change. 17 The misleading
results in CGM, and similar misleading results in several other earlier studies, are fully consistent with
our theoretical finding: since averaging techniques reduce the existing microstructure noise in the
15
See Holowczak et al (2006), Hiseh et al (2008), and Muravyev et al (2012).
This technique consists in reducing distant lags to moving average terms that subsequently enter the VECM model.
17
A recent study by Muravyev et al (2011) that appeared almost simultaneously with the early versions of this paper
reaches the identical conclusion with a totally different method, based on the portfolio approach of implied price
determination. We discuss this study extensively further on in this paper.
16
8
option market, which is much larger than the noise in the underlying market, they increase the IS
metric of the option market and falsely attribute price discovery to that market.
Since the averaging technique bias can easily be avoided, we concentrate on the volatility inversion
issue, for which we develop two new estimation and averaging methods. Given the importance of the
accuracy of such inversion methods, we develop a statistical framework to a priori appraise them. In
deriving this framework, we assume the stochastic volatility dynamics for the underlying price 18
without using any equilibrium arguments necessary for the derivation of the option price. Specifically,
we adopt a discrete time formulation of the underlying and option price dynamics that includes
microstructure errors and converges to the assumed continuous time dynamics in the absence of such
errors. In such a framework we define the volatility of the implied price and its correlation with the
option price in accordance with the observed option market dynamics but independently of the
inversion method. We develop benchmarks for these two parameters in terms of observable statistics in
simulated data, but also in the actual data. We then compare these two parameters to the derived
benchmarks under various alternative inversion methods.
We demonstrate that the LIV inversion method as applied by CGM produces estimates of the option
market’s contributions to IS that are as little as half their “true” measures. We provide clear evidence
that the source of this bias lies in the extraneous noise or the measurement error of the parameters’
estimation inherent in the this method. For this reason we apply smoothing to the parameters used in
the inversion methods applied in this study, which efficiently aggregates the microstructure noise
present in point-in-time estimates such as the LIV method. We test this smoothing with simulated data
and find that it eliminates the error arising out of the parameters necessary for a given inversion
18
We assume the stochastic volatility framework to be sufficient for the objectives of this work since jumps either in the
underlying security or in the volatility are rare events. Carr and Wu (2009) derive relatively low estimates for the error
resulting from neglecting jumps in deriving the variance swap rate from the option prices.
9
method, while preserving this microstructure noise of the option market which should play a role in the
price discovery. We also confirm that our smoothing satisfies with the real data the earlier developed
benchmarks.
The smoothing - or filtering we apply consists of using the median value for a set of the inversion
parameters estimated within a lagged moving window relative to each time point of our implied price
series.19,20 We apply this filter to the LIV inversion method and call the result the Median LIV (MLIV),
which yields the best results in simulated data. We also develop and apply a new inversion method
based on the assumption of the homogeneity of the option price with respect to the underlying and
strike prices which is preserved under most stochastic volatility and jump-diffusion models, termed the
Homogeneity Inversion (HI), whose parameters are also smoothed. Although the MLIV method is
slightly better in our simulation experiments, both methods yield remarkably similar estimates of the
implied price and remarkably similar results with respect to the contribution of the option market
according to the IS metric. As noted earlier, this contribution is more than twice the estimate from the
conventional LIV method. This is true both before and after the change in tick size.
Turning now to the tick size reduction effect, Seppi (1997), Anshuman and Kalay (1998) and Sandas
(2001) presented stylized equilibrium models of competitive market making in which tick size and
other trading restrictions play a role. Surprisingly, the tick size reduction does not always result in more
liquid markets in these models; e.g. as Seppi (p. 104) noted, the relation between the tick size and
liquidity is non-monotone and discontinuous.
19
Data smoothing by using a running median achieves two important goals for the problem at hand: first, it aggregates the
measurement errors on the inversion parameters; second, it deals with the expected non-normality of those parameters as
opposed to a running mean. See Bowman and Azzalini (1997).
20
The underlying plausible assumption is that over a short time interval the variation in parameters governing a given
stochastic process is relatively small compared to the variation in those parameters induced by the microstructure noise.
10
Although there are several empirical studies on the impact of tick size reduction on price discovery in
equity and futures markets,21 only one of them, by Chen and Gau (2009), examined the role of options
in this discovery. The authors use the CGM methodology and examine the IS metric before and after
the tick size reduction in the underlying Taiwan index tracking fund. They assess the impact of this
reduction on the corresponding markets for the index tracking fund itself as well as for the markets of
its options and futures and report that the tick size reduction in the tracking fund’s market significantly
increased its IS at the expense of both the futures and options. They attribute this finding to the lower
transaction costs in the tracking fund induced by the smaller tick size. We note that the adoption of the
CGM averaging technique by Chen and Gau (2009) may have also introduced difficult to assess errors
in relative magnitudes of the IS metric before and after the tick reduction.
To complete the analysis, we also explore one of the reasons for the large increase in IS in the option
market following the tick size change, by examining the link between our metrics of informed trading
in options with the future information in the underlying market. Following a large literature that relates
volatility to the flow of information and by using a powerful non-parametric statistical technique
introduced by Corrado (1989), we show that abnormal values of option market’s IS lead to significant
increases in the volatility of the underlying asset following the event, after but not before the tick size
change; this effect is confined to equity options. We conclude that the tick size caused a migration of
informed traders from the underlying to the option market, contributing to an increase in IS.
The remainder of the paper is organized as follows. In the next section we present the price discovery
model, including the IS and CS metrics, the impact of the microstructure noise, as well as our analysis
of the new option price inversion methods in the context of the stochastic volatility option pricing
model. Section III then presents the data and section IV the corresponding empirical analysis, which
21
See Section III for more details.
11
includes the comparison of the inversion methods with respect to the information shares of the option
market, the impact of the tick size reduction and various robustness checks of the results; this section
also includes a comparison with the results of previous studies. Section V presents the event study of
the increase in informed trading via the option market following the tick size change. Section VI
concludes the paper.
II. Methodology
A fundamental issue in price discovery involving simultaneous pricing in stock and option markets is
the inversion of the option price. In the CGM study the LIV inversion method for the option implied
price was largely motivated by ‘tautology avoidance’, i.e. by the fact that no model may be estimated
when the implied price is equal to the underlying price. In a variant of the Black-Scholes-Merton
(BSM) model a lagged implied volatility was used to invert the present price of an option into its
implied price. Since the BSM expression was used merely as a translation device, this approach is free
of the option model misspecification error.22 Nonetheless, there is a major concern regarding this
inversion method due to the empirical properties of its critical parameter, the lagged IV: if the volatility
is not constant but random or time- and/or state-dependent then the LIV estimate contains errors. As a
result, while the LIV method uses an unbiased IV estimate, the error variance of this quantity is
expected to add to the noise of the implied price estimates.
We begin this section with a presentation of the total error of the option implied price that includes
microstructure noise as well as errors in the derivation of that price. This is followed by a description of
the econometric methods used to estimate the IS and CS metrics and then by a theoretical analysis of
the impact of the total error on these metrics. Next we present a framework for gauging the accuracy of
22
See Berkowitz (2010) for an asymptotic justification of the practitioners’ BSM model, whose variant the LIV inversion
method explores.
12
the implied price derivation methods under the commonly assumed model of stochastic volatility asset
dynamics in option pricing. The section concludes with the presentation of new volatility inversion
methods and their comparison with LIV. One of these methods turns out to be superior, and is used for
most subsequent results.
2.1 The IS and CS price discovery metrics
Consider high-frequency time series of a derivative f and its underlying security S. Hasbrouck (1995)
and Gonzalo and Granger (1995) present econometric frameworks for the estimation of the
contribution of each trading venue to the price discovery process for a single security traded in multiple
venues. Since either framework requires that the prices in those venues be cointegrated, we can’t
directly estimate the price discovery from the observables. However, since the possibility of arbitrage
closely links option and underlying security prices, the inversion of the option prices yields time series
cointegrated with the underlying price with a known cointegrating vector. We denote this series by I t :
It  h( ft ,t ) ,
(1)
where h(.) is a function yielding the implied underlying price and θ is a set of parameters necessary for
the inversion, assumed known exactly. Since this is clearly not a valid assumption, the series I t is
evaluated under an estimate ˆt of  t . The effect of this estimation is a major new element in this
paper, with the impact of possible estimation errors on the price discovery process discussed at length
further on.
First we examine the estimation of the IS and CS metrics. For some well specified inverting function
h(.) we have the following order-one cointegrated price vector:
13
 St   Et   S ,t 
Pt     
.
 I t   Et   I ,t 
(2)
In (2) E denotes the (unobservable) common efficient price,23 which is assumed to follow a random
walk:
Et  Et 1  vt ,
(3)
where E (vt )  0 , E (vt2 )   v2 and E (vt , vs )  0 for
t  s . The term  S ,t represents the microstructure
noise of the underlying asset market, while  I ,t is the total error in the option implied price, namely the
microstructure noise plus possible errors in its derivation. Microstructure noise denotes transient
microstructure effects in the two markets in the form of random variables with zero mean whose k-th
order autocovariance matrices depend only on k. In this paper we focus on the two particular sources of
the option implied price total error which are aggregated together in the term  I ,t , namely option
market microstructure noise arising out of the tick size and possible errors from the implied price
estimation.
Under the Granger Representation Theorem we have the following Vector Error Correction Model
(VECM) specification:
l
Pt l   ' Pt 1   Ai Pt i  et ,
(4)
i 1
where
 and
 are [2x1] matrices respectively containing adjustment parameters and cointegrating
vectors, the [2x2] matrices Ai describe the short-term dynamics of the process, l is the optimal number
23
The order-one cointegration follows from the assumption that there is a single common trend, the efficient price. See
Stock and Watson (1988).
14
of lags. We also denote by  the covariance of the serially-uncorrelated error terms et . The term
 ' Pt 1 represents the equilibrium dynamics between the prices.
The IS metrics are derived by a decomposition of the random walk variance, which may be estimated
even though the common efficient price is not observable:
2
 RW
  ' ,
(5)
where  are the (identical) rows of a [2x2] matrix  given by24

      


1
 1 0  l  

   Ai       ,
 0 1  i1  
the [2x1] matrices   and   are non-trivial orthogonal complements to
(6)

and  in (4) and the Ai
matrices are as in (4).25 The IS metric for a given market is defined26 as the proportion of the random
walk variance that is attributable to the innovations in that market. If the covariance matrix  is
diagonal, then we have a clean decomposition into contributions of each market to the total variance of
the permanent component, with the contribution for the j-th market equal to
 2j  jj
. This diagonal
2
 RW
property will hold if the underlying random walk hypothesis is well satisfied by the data, i.e. if there is
little contemporaneous correlation between the residuals in (4). If the off-diagonal elements are nonzero, which prevails in empirical work, then we apply the Cholesky decomposition to the covariance
24
See Johansen (1990).
See Johansen (1995) for the estimation methods of these orthogonal complements. Note also that by using (6), we avoid
estimating the matrix  via an impulse-response function with a priori unknown convergence properties. In fact, for our
data those functions may require an excessive number of iterations to achieve any reasonable convergence.
26
See Hasbrouck (1995).
25
15
matrix  and derive the lower and upper bounds on IS since this quantity varies with each ordering of
variables in the Cholesky decomposition.
The Gonzalo and Granger (1995) decomposition into the permanent and transitory components starts
with the same VECM model (3). Then we have the following decomposition of the price vector:
Pt  C1 gt  C2 Zt ,
(7)
where g t and Z t respectively represent the permanent and transitory components, C1 and C2 are
loading matrices, and gt    Pt and Zt   Pt with   and  as before. Note that here the efficient
price and the integrated of order one permanent component g t need not be random walks. Of interest
to us is the [2x1] matrix   . Under the additional assumption that Z t does not Granger-cause27 g t
this matrix may be identified up to a non-singular multiplication matrix. The interpretation of the
permanent component g t is that it is a weighted average (linear combination) of the observed prices
with the component weights in   . Booth et al. (1999) and Harris et al. (1995) suggest measuring price
discovery by defining CS as normalized to 1 elements of   . The interpretation of the CS metric is
that the market which reacts the least to the innovations in the other market will display the highest
relative weight in the permanent component. An attractive feature of the Gonzalo-Granger (1995)
approach is that statistical hypotheses may be tested by a χ2(1) test, which we apply in our empirical
work.
27
See Granger (1980).
16
2.2
Microstructure noise and the IS metric
To assess the effect of microstructure noise on the option market IS metric we need to provide some
structure to the error terms  S ,t and  I ,t of the time series ( St , I t ) and their relationship to the random
walk. We use two alternative such structural models. In the first one there is no informed trading and
the error terms are mutually independent random variables and independent of the random walk term vt
, with respective variances  S2 and  I2 . The second one is a modified version of a model presented in
Hasbrouk (2007, Ch. 9), itself an adaptation of the Roll (1984) microstructure model, that recognizes
the presence of informed trading and possible momentum effects. Our main result holds under both
assumptions, but our detailed proof will be given under the simpler assumption that  S ,t and  I ,t are
time-independent and uncorrelated random error terms. In both cases the inversion error

ˆI ,t  h  ft ,t   h ft ,ˆt
 can be considered part of the term 
I ,t
, together with the tick size and other
unspecified microstructure noise.
Under the Hasbrouk-Roll model the randomness of the microstructure noise affects the efficient price
revisions vt , which becomes now the sum of a pure random walk term ut and a component driven by
the direction of trade. Following Roll (1984), this direction is represented by the variable qt , which
determines both the location within the bid-ask spread but also possibly reflects private information and
affects, therefore, the efficient price. The two components of the time series ( St , I t ) have each one its
own revision proportional to the direction of trade. Analytically, while (3) still holds we now have:
vt  ut   wt , qt  wt , St  Et  cS wt , It  Et  cI wt   It .
(8)
17
In (8) the variables ut and wt represent the underlying asset market noise and information trading
respectively, while the term  It , with E ( It )  0, Var ( It )   I2 , Cov( It vt )  0, Cov( It  It  j )  0 j  0 ,
is again the option implied price total error that includes ˆIt ,. For both models (2)-(3) and (2)-(8) we
prove in the appendix the following result.
Proposition 1: The higher the variance  I2 of the error term  It in models (2)-(3) and (2)-(8) the lower
the information shares attributed to I t . The width of the information shares’ bounds also increases with
 I2 .
As noted earlier, the parameters of the structural model are not observable, and the price discovery
effects have to be assessed through the IS and CS metrics evaluated from (4). To compare these
metrics, we note that IS measures the impact of innovations on the random walk variance, whereas CS
is concerned with the linear impact of the innovations. Although the IS metric is more popular in the
market microstructure literature and probably more relevant for price discovery, a recent study
recommended using both metrics because the IS metric is subject to microstructure noise from high
frequency data.28 In addition, CS does not rely on the random walk assumption, which is usually not
satisfied by the data.29 In our results we focus on IS, with CS presented as an extension and robustness
check.
To estimate system (4), which forms the basis for the estimation of our metrics, we use the Johansen
(1990) maximum likelihood approach, which was shown to have clearly superior properties among
28
See Yan and Zivot (2010).
There is an ongoing debate in the market microstructure and the related econometric literature about the relative merits of
the Hasbrouck (1995) and Gonzalo and Granger (1995) approaches. As De Jong (2002) points out, CS for a given market
ignores the variance of an innovation in this market while it measures the weight of this innovation in the increment of the
efficient price. On the other hand, IS measures the share in the total estimated variance of the efficient price contributed by a
given market.
29
18
alternative VECM estimation methods.30 We discuss in Section 4 the important issue of the differences
in the results induced by the VECM estimation when it is done by applying the polynomial distributed
lag (PDL) with moving average terms (MA), as suggested by Hasbrouk (2002b) and adopted by CGM,
Anand and Chakravarty (2007) and Chen and Gau (2009). Before proceeding with the estimates,
however, we must deal with the microstructure noise issues, which in view of Proposition 1 need to be
addressed. Since the error  I ,t contains both the inversion error and microstructure noise such as tick
size effects, the implied price estimation should strive to minimize the former, which is added by the
empirical design and contaminates the IS metric, while preserving the latter, which belongs to the
discovery process. The procedures used can only be tested in simulated data that tracks the two errors
separately. In the next subsection we simulate the option implied price under a plausible and popular
model of asset dynamics in the presence of microstructure noise only and without any inversion error,
and derive diagnostic relations that should be satisfied by the implied price moments. Subsequent
sections introduce the volatility inversion methods and verify with the simulated data whether the
errors they introduce are sufficiently “small” to preserve these diagnostic relations.
2.3 Tick Size and the Accuracy of the Inversion Methods
We evaluate the accuracy of the inversion methods applied in this paper under the assumption of
stochastic volatility asset dynamics, with the inclusion of microstructure noise. As we argued in the
introduction, we assume that it is a sufficient framework for the analysis of the inversion problem.
Although our results hold for any type of noise, we limit ourselves to tick size noise in our simulated
data, since it is the most clearly defined and the focus of our empirical work. The state variables, that
is, the underlying price St and its volatility  t vary according to the following bivariate diffusion:
30
See Gonzalo (1992).
19
dSt  t St dt   t St dw1
d t2   t dt   t dw2
(9)
cov  dw1 , dw2    dt
where dw1 , dw2 are elementary Wiener processes and t is a function at most of the stock price, while
the parameters  t and  t are generally functions of  t but not of St or of past stock prices.31 We
assume that the correlation coefficient  is negative, in line with most econometric results.32 We
apply Ito’s lemma to a derivative f t resulting from (9) by an unspecified risk neutral valuation
process. In the absence of microstructure noise f t is a known function f (St ,  t2 ) . We have, by Ito’s
Lemma:
dft  ft ( S ) St  t dw1  ft ( )t dw2  other terms ,
(10)
where we use only those terms that contribute to the quadratic and cross variations, the superscripts
denote partial derivatives, and the ‘other terms’ clearly do not contain either Wiener process. Those
‘other terms’ will vanish in all further derivations.
Now we state and prove four lemmas that establish statistical benchmarks by which we assess the
accuracy of the inversion methods applied in this paper. We assume for simplicity and without loss of
generality that there is no microstructure noise in the market for the underlying asset and that the
parameter  t2 is known, but that the tick size induces a microstructure noise component  TS , uncorrelated
with the error components of the diffusion, such that at time t  t the derivative price becomes
31
It is possible to state a more general stochastic volatility process with respect to the functional dependence of the
parameters of this process. The presented specification is consistent with the homogeneity assumption that we use further
on in Lemma 3 and in one of the new inversion methods.
32
See, for instance, Christoffersen et al (2010, p. 14).
20
fˆt t  f (St t  TS ,  t2t )  f ( It t ,  t2t ) .
(11)
Setting St  I t , we rewrite the discretized version of (10) as follows:



fˆt t  ft ( S )  St t t   tt t t  TS   ft ( )t t t  1   2t t



t  o(t ) , (12)
wheret t and t t are mutually independent and N(0,1) error terms, and
St (t t   tt t t )  TS  St t  TS  It t .
Let also  
(13)
ft ( ) t 
; this variable is negative for calls and positive for puts. We may now prove the
St f t ( S )
following results:
Lemma 1
Under the state variables’ dynamics (9) and in the absence of microstructure noise we have the
following restriction on the correlation coefficients  fS of the underlying and the derivative prices, and
 IS of the inverted and underlying prices:
 fS   IS  1
(14)
Proof : By taking the appropriate quadratic and cross variations of the state dynamics (9) and the
stochastic differential of the option price (10), we have:
 fS 
df , dS
df , df
dS , dS

a  b
df , df

a  b
a 2  2  ab  b2
,
(15)
21
where a  St ft ( S ) t and b  ft ( ) t . It immediately follows that  fS  1 since the denominator exceeds
(a  b)2 , QED.
We can rewrite the left hand side (LHS) of (15) as follows:
 fS
2
(1   2 )
a 2  2  ab   2b 2
2
.

 1
2
a 2  2  ab  b 2
2
 t  2 t  2

(16)
Note that in the simple univariate diffusion, in which the second random term in (10) disappears, both
sides of (14) are equal to 1. The importance of the result stems from the fact that under (9) the
cointegrated price series clearly deviate from the random walk implied by the simple diffusion process.
Hence, the size of the correlation  fS is an indicator of such deviations. We assess its importance in the
numerical results further on in this section.
The next result is more important in establishing criteria to judge the effects of the tick size-induced
microstructure noise.
Lemma 2
Let ˆ fS indicate the correlation of the derivative price given by (11) with the underlying price. Then,
under the state dynamics (9) and in the presence of tick size microstructure noise, we have the
following relation between the correlation coefficients and the variances of the error terms:
 IS 
1
1

2
TS
2 2
t t
 S t
,
ˆ fS

 fS
1
1
 TS2
,
(17)
 2
2  2


2


S t
 t
t
 2  t

22
2
where  TS
is the variance of  TS .
Proof : We use the discretized version of (9), set for simplicity t  1 , and let  2f denote the variance of
2
df and  I the variance of I t t from (13). We have
 IS 
Cov(St t , I t t )
( t St )2
( t St ) 2
,


2
 t St I
( t St )2   TS
 t St ( t St )2   TS2
(18)
from which the first part of (17) follows obviously. Similarly, if ˆ f denotes the volatility of fˆt t we
have from (12) for the second part
ˆ fS 
Cov(fˆt t , St t )
a  b
.

2
 t Stˆ f
 f  ( ft ( S ) )2  TS2
(19)
Dividing now (19) by (15) and rearranging, we get the second part of (17), QED.
Next we establish a benchmark for the volatility of the implied price, but only for put options and only
under an additional assumption which, however, holds for most option models. Specifically we assume
that under the risk neutral transformation of the stochastic volatility asset dynamics (9) the option price
is linear homogeneous in ( St , X ) , where X denotes the strike price. This implies that the following
relation holds:
ft  ft ( S ) St  ft ( X ) X
(20)
We can then prove the following:
Lemma 3
23
2
Let  I  ( t St )2   TS
denote the volatility of the implied price. Then, under the homogeneity
assumption of the option price the following relation holds for put options
 
2
I
ˆ 2f
( ft ( S ) )2

(ˆ f St )2
( ft  Xft ( X ) )2
(21)
Proof: Taking the variance on both sides of (12), we have
ˆ 2f  ( ft ( S ) )2  I2  ( ft ( ) )2  2  2 ft ( S ) ft ( ) t St
(22)
Solving and re-arranging, we get
 
2
I
ˆ 2f
 2
 2

 2  2 t  St
(S ) 2
( ft )  

(23)
(21) then follows immediately if we note that  is positive for put options, QED.
The last result also establishes a lower bound for the all-important correlation between the underlying
and implied prices, depending again on the value of  .
Lemma 4
For call options, if   2 t  0 33 then in the presence of microstructure noise we have  IS  ˆ fS .
Proof:
From the RHS of (18) it is clear that  IS  0 ; it suffices, therefore, to show that (  IS )2  ( ˆ fS )2 .
Rewriting this last relation taking into account (18) and (19), we find that it holds if and only if
33
This condition is equivalent to a positive sign for the covariance of the call option stochastic differential with the similar
differential for the underlying price.
24
(  2  1)b2
 0 . Since the numerator is obviously negative, the relation holds if the denominator is
a 2  ( a   b) 2
positive, which holds if   2 t  0 , QED.
The variable  plays an important role in establishing benchmarks for the volatility inversion methods
in our empirical work. We observe in (17) that the right-hand-side (RHS) of the second relation
exceeds the RHS of the first second one, with the difference depending on the size of  . Similarly, in
(16)  determines the difference between 1 and |  fS | . On the other hand, there are reasons to believe
that
 makes a small contribution to these relations. By approximating the ratio of partial derivatives
in  by the Black-Scholes (1973) model quantities derived at the values of implied volatility, maturity
and moneyeness in our data at which
 reaches its highest absolute value, we find this value to be
lower than |  t | . As for the volatility of the volatility κ, it is clearly model-dependent; however, its
magnitude is generally low because of the documented persistence in volatility. 34 Hence, the highest
2
value of the quantity  for all options is lower than  2 t2 , which is clearly of a lower order of
magnitude than the term  t2 St2 . We expect, therefore, that in our data the RHS of the two equations in
(17) will be approximately equal and that Lemma 4 will hold, while to a lesser extent the correlation in
(16) will be close to 1 and (21) may also hold approximately for call options as well as exactly for put
options.
Since the sample counterparts of the quantities in (17) and (21) and of the correlations in Lemma 4 may
be easily estimated from the data, Lemmas 2, 3 and 4 establish powerful statistical benchmarks to
assess the accuracy of the inversion methods in simulated data. Note also that any additional errors
34
See Christoffersen at al. (2010), who find values of this quantity in the range 0.5-12.5% for the S&P 500 index at
different specifications.
25
from volatility inversion can also be considered as an additional factor increasing the error variance
 TS2 , resulting possibly in violations of (21) for put options in the actual data. In the following
subsections, as well as in Section IV, we use Lemmas 2, 3 and 4 in numerical results demonstrating the
superiority of our proposed new inversion methods. Last, we note from (17) that the correlation  IS
increases when the underlying price and its volatility are high. This establishes St as a determinant of
IS or CS for the option market for use in a later section.
2.4 Volatility Inversion Methods
Most empirical studies such as CGM and others approximate I t by setting It  hˆ(.)  BSM S1  ft , IVt  ,
with BSM and IVt denoting the Black-Scholes-Merton price and implied volatility respectively,
observed by necessity at some earlier time t-τ; this corresponds to an estimate ˆt of  t . As noted in the
beginning of this section, the key observation leading to an improvement over the LIV inversion
method is the fact that the parameter set  as defined in equation (1) is subject to measurement error.
We also saw that the error in measuring  reduces the price discovery metrics for that market.
Unlike microstructure noise, the parameter measurement error arising out of the volatility inversion
does not belong to the discovery process and needs to be eliminated. The “optimal” parameter
measurement is by polynomial fitting of the cross section of options on the same underlying asset at
any given time with respect to the strike price and time to expiration;35 we term this method the
Regression Implied Volatility or RIV. Unfortunately, RIV cannot be applied uniformly to our data set,
since the number of available options differs between stocks; further, the computational burden is
35
This was first applied in empirical option research by Dumas, Fleming and Whaley (1998). See also Christoffersen,
Heston and Jacobs (2010) and Berkowitz (2010).
26
prohibitively heavy if we want to do data smoothing. For this reason we examine this method only as a
benchmark and discuss its performance further on in this section.
We deal with the measurement error problem by data smoothing, 36 which leads to canceling out or
aggregation of the errors on  . In practice, the data smoothing we apply consists in using a running
median for the inversion parameter set  rather than a single lagged observation as in the earlier
studies. The use of the median is motivated by its ability to deal with the non-normality and with
possible data errors (outliers) that may still be present even after the screening that we apply. We also
show that our inversion methods compared to LIV admit shorter estimation windows, thereby allowing
the incorporation of more variability in the unobservable state variables. The relative performance of
the smoothing technique in reducing the volatility inversion error while leaving the microstructure error
unaffected is examined in the next subsection. Our main results focus on the smoothed LIV method or
MLIV, with the HI method described in the appendix and used as an extension.
The choice of the length of the moving window to estimate the inversion parameters involves balancing
several opposing effects. For the main results of this paper we used a five minute window, with 300
past observations in one-second intervals to estimate the inversion parameters. Such a relatively short
time interval that allows sufficient aggregation of the microstructure noise while admitting a relatively
high variation in the "true" parameters of the stochastic process appears to be the optimal choice for the
problem at hand.37
36
See Bowman and Azzalini (1997). Our smoothing is very similar to the one termed pre-averaging and described in detail
in Jacod et al (2009).
37
The risk of inducing a serial correlation in the error of the efficient price in equation (2) is small since by construction the
time series of the inversion parameters is smooth. Further, the optimal number of lags in the VECM model (3) is always
significantly lower than the length of a moving window. Note also that the earlier studies admitted relatively low variation
in those "true" parameters by selecting significantly higher lags.
27
2.5
The Relative Performance of the Volatility Inversion Methods
As noted, a volatility inversion method must minimize the parameter measurement error while leaving
intact the microstructure error. In order to test whether the proposed inversion methods achieve this
objective we use simulations that mimic more or less the characteristics of our data. In these
simulations we first assume the parameters known and use an option pricing model yielding a closed
form solution. In that model we introduce microstructure errors arising out of the minimum tick size
and examine the correlations between the underlying and implied prices. Next, we assume that the
option price parameters are unknown and compute the implied price series according to one of the
volatility inversion methods under consideration. A “satisfactory” inversion method should produce
implied price series that satisfy Lemmas 1-4 and have similar correlations with the underlying price
series as the ones arising out of the microstructure noise when the parameters are known.
For the simulations we choose the Heston (1993) model, which is the most popular stochastic volatility
model in option pricing and has a closed form solution. In that model the stock price instantaneous
mean is constant, while in (9)  t is mean-reverting and  t is proportional to  t . The model is an
“incomplete markets” one, which admits a closed form solution only if we assume that the market price
of volatility risk is proportional to  t . We use the following parameters for the base case: initial
volatility equal to its long-term mean of 0.3, volatility of volatility 0.4, speed of mean reversion 1,
correlation of Wiener processes -0.6, riskless rate 4%, stock drift 8%, price of volatility risk -5% and
initial underlying price 100.
Simulations of 300 paths for the stock were generated in one-second intervals for one trading day,
which totals 31,000 seconds. For these paths 15 option prices valued according to the Heston (1993)
model ('true prices') were generated for maturities of 1, 2 and 3 months and with five equally spaced
28
strike prices from 0.9 to 1.1 of the initial underlying price. We only simulated put option prices since
from those prices we may derive the upper bound on the implied price volatility (21). In the second
step, the prices for tick rules of 0.05 (below $3) and 0.1 (above $3) were derived from the model prices.
For the tick-adjusted or quoted midpoint prices, this price at time t was set equal to the quoted midpoint
price at t - 1 unless the midpoint price was at least one tick away from the model price. In this latter
case the quoted midpoint price moved by a minimal number of ticks so as to at least exceed (be below)
the 'true' price for an upward (downward) movement in the model price.
In the above setup it is apparent that the true implied price is simply the underlying price, since there is
no model or parameter uncertainty and the sole factor that may cause those prices to diverge is the
presence of frictions in the option market; we assume that there are no frictions in the underlying
market. The prices were inverted for HI, LIV and MLIV model with the interval or lag of 300 seconds.
In addition, the RIV model in the spirit of Berkowitz (2010), and Christoffersen et al (2010) was
applied. In that model, for the same lag as for LIV, we fit a least squares regression of the implied
volatility (IV) against a constant, the time to maturity, the strike price, and the squares of the latter two
variables. Finally, the fitted value for IV at time t was used for the inversion. We also added the best
estimate for the implied price without model or parameter uncertainty ('true implied price') denoted by
I, a quantity estimated by inverting the Heston (1993) model at the true parameters but for the option
prices affected by the tick rules as described above. The results for the performance of the inversion
methods are presented in Tables 2.1 and 2.2 below.38
[Table 2.1 approximately here]
[Table 2.2 approximately here]
38
For comparability purposes, we used only nine contracts for the HI, I, LIV and MLIV estimates, while all fifteen contracts
were used in the regression for the RIV inversion, since the inversion for the HI method was not possible at the extremes of
moneyness for each maturity.
29
In Table 2.1 we first note that all values of the true implied price volatility  I satisfy the upper bound
(21) of Lemma 3,39 thus validating its use as an indicator of performance of the inversion methods with
the real data. As for the methods themselves, the MLIV performance is excellent, with all results very
close to  I at the contract level in all cases. The HI results are somewhat higher but still close to the
benchmark for most contracts, while the LIV and RIV results are much higher than the benchmark in
all cases. On the other hand, at the aggregate level which is a simple average over all contracts, the
results for MLIV and HI closely bracket the similar average for  I from above and below respectively,
while those of the other two methods are, as expected, much higher.
In Table 2.2 we first observe that in the absence of frictions the correlation  fS between the derivative
price and the underlying price is lower than 1, as predicted by Lemma 1. It is, however, very close to 1
in all cases, as in the BSM model, implying that the latter model, with its random walk assumption,
may approximate quite well the more complex Heston model asset dynamics from the price discovery
point of view.40 Second, the microstructure noise arising out of the minimum tick size lowers
dramatically the size of the correlation ˆ fS , thus illustrating the importance of microstructure noise in
the price discovery metrics.41 We also note that this quantity is almost exactly equal to the “true”
correlation  IS , a result that is not surprising and fully consistent with the fact that the correlation
39
The upper bound in (21) was adjusted to account for microstructure noise, since it was estimated by using the time t
values of f | f  f X | that contain that noise.
40
A similar set of simulations for the BSM model, available from the authors on request, shows that the correlations
between the underlying and its derivative were always very close to 1, as predicted by the theory.
41
As an aside, we note that this dramatic drop in the correlation of underlying and implied prices as a result of the simple
tick size noise and without any other microstructure effects raises serious questions about the widespread practice of
estimating the underlying price’s risk neutral distribution from a cross section of option prices. Our results imply that such
an estimation is subject to major noise, even though it is the basis of several well-known studies such as, for instance, Carr
and Wu (2009).
(X)
30
between the underlying price and the frictionless derivative price is close to 1. Indeed, Lemma 2 shows
that when the variable  is low the RHS of both relations in (17) are equal.
The acid test of a volatility inversion method is the consistency of the correlation of the underlying
price with the implied price that it produces with the “true” correlation  IS that preserves the
microstructure but eliminates the inversion noise. As Table 2.2 shows, the MLIV results are virtually
indistinguishable from those of  IS , while the HI results are very close to them. On the other hand, the
LIV and RIV values are much lower than  IS in all cases, showing the contamination of the implied
price by the inversion method noise. For robustness check purposes, we varied the initial conditions
and verified the results of Tables 2.1 and 2.2. With respect to the relative performance of the inversion
methods, these results remained qualitatively almost completely unchanged when the initial conditions
were varied within the Heston model. Setting the initial volatility at 0.6 rather than 0.3 shows again that
the MLIV method performs best and was very close to the benchmarks (17) and (21), while HI
performs slightly worse and the other two methods much worse. On the other hand, the increase in the
initial volatility state had a significant effect on the level of correlations of the implied prices with the
underlying price. For instance, for MLIV the aggregate implied price correlation increased from 0.12 to
0.19, as predicted by relation (17) in Lemma 2. The same relation also explains the decrease in
correlation from 0.12 to 0.04 when the initial stock price is set at 20 instead of 100. To conclude, we
investigated the impact of the change in the tick rule, to 0.05 (0.01) for options priced above (below)
$3. For MLIV the same aggregate implied price correlation increases as a result of the change, from
2 42
0.12 to 0.28, which is again consistent with relation (17) because of the decrease in the variance  TS
.
Finally, changing the initial conditions brought little or no improvement in the results achieved by the
42
The full results for changing the initial conditions are available in our online supplement.
31
polynomial fitting of the RIV method. Since this method involves a heavy computational burden for
data smoothing, we ignore it henceforth in our results.
Figure 1 provides a graphic real data example of the implied prices derived by the HI and MLIV
methods compared to the LIV method.
The figure also plots the midpoints of the option and
underlying prices. It is clear from the figure that the HI and MLIV implied prices change only because
of the changes in the option price as we may expect a priori, while it is apparent that the implied price
for the LIV method contains variability not justified by the option dynamics. For instance, we observe
a large variability in the implied price for the LIV inversion for the flat segments in the option price;
we also observe directional changes for this method when the changes in the call price conform to short
lived demand/supply imbalances resulting in the widening or narrowing of the bid-ask spread shown in
the graph as the call price bounces back and forth around the same level. These anomalies, which
clearly do not reflect the option dynamics, don’t appear in the implied prices derived by the HI or
MLIV methods.
From the simulated option prices we may now estimate the VECM model (4) and the ensuing IS or CS.
Since the conclusions are similar for both metrics, we report the results only for the former quantity. In
the base case with the use all 18 option contracts the simulated IS midpoint was 25.2, 0.9 and 13.6%
respectively for the HI, LIV and MLIV methods. When we increased the initial volatility state to 60%
(decreased the option tick size), the corresponding estimates were 33.6, 0.8 and 18.9% (48.6, 0.5 and
42.1%). When we decreased the number of option contracts to four near ATM calls for the two nearest
maturities with the other parameters as in the base case, the respective estimates were 7.9, 1 and 3.2%.
These simulated results show that for the HI and MLIV inversions the increases or decreases in IS are
as predicted from the earlier discussion, with the metric increasing with volatility and decreasing with
32
tick size, and also increasing in the number of contracts for reasons discussed in Section IV. For the
LIV method, on the other hand, the behavior is erratic and often contrary to theoretical predictions.
[Fig 1 approximately here]
III. Data
Data for this study cover a 13 month period spanning from January 2007 to March 2008 inclusive, and
includes tick by tick quotes on options and their underlying securities. Our main data set is comprised
of 20 option classes out of 24 included in the second CBOE pilot program and their corresponding
underlying securities. Two option classes, namely Dendreon Corp. (ticker DNDN) and Financial
Select Sector SPDR (ticker XLF), were dropped from the sample because a substantial number of days
were missing in the CBOE tape. In addition, the Dow Jones (ticker DJX) and the mini S&P 500 (ticker
XSP) option classes were also dropped because the underlying data is reported approximately every 10
to 15 seconds, which precludes applying the VECM model on a one-second interval. Moreover, the
assets underlying the last two option classes are non-tradable, whereas our study focuses only on
options with publicly tradable underlying assets.
We included in our sample options maturing between 8 and 120 days. The data was first screened for
standard arbitrage violations, excessive bid-ask spreads and excessive implied volatility.43 Also, we
screen out the contracts for which in a substantial proportion of quotes we observe cA  S A  X  cB or
pA  X  S B  pB , since the option prices whose bid and ask prices bracket the early exercise value are
not informative. Since in our final sample we included only close to ATM options, the above screening
took place only for a limited number of close to maturity contracts. Note also that the presence of
43
Our screens resulted in discarding less than 0.5% of the option quotes.
33
dividends has little effect in our data since the only optimal time to exercise a call option is at the end
of the last cum dividend day.
One-second time series were then constructed for options and their underlying securities. Gaps in the
data were filled out with the last available quote. Finally, implied prices were derived for the three
nearest to the money call or put options for each maturity within the above limits. The final restriction
limited the moneyness of the included contracts, defined as the daily mean of the ratio of the strike to
the underlying prices, to a range of 0.9 to 1.1. Altogether, the data includes 273 trading days or 13
calendar months. More specifically, for the period before (after) the policy change, the data includes
146 (127) trading days, 2940 (2520) company-days and 38673 (36470) option contracts.44 Table 3.1
presents the description of the underlying securities in our sample.
A comparison sample was also constructed using the same sampling approach.
This comparison
sample was composed of the same number of randomly selected option classes where all available
classes in the Option Metrics database were first filtered according to the following criteria: i. the total
volume of transactions of the option class is no higher than that of the most traded class in the main
sample and no lower than that of the least traded, ii. the class remains actively traded during the entire
period under study.
[Table 3.1 approximately here]
Table 3.2 presents the descriptive statistics of the main and control samples for the 7 months before and
the 6 months after the minimum tick size reduction was implemented. We first note that there were
large and systematic differences in volume between the main and control samples before and after the
tick size change for both equities and ETFs, implying that volume size may have been a factor for
44
To derive the IS or CS we reduced the number of contracts. See the following section.
34
inclusion of an option series in the pilot program. 45 There was also an increase in volume in the postchange period, which appears small but was statistically significant: 0.24 (t-stat 4.06) and 0.34 (t-stat
7.77) for the main and control samples respectively for equities, and 0.57 (t-stat 5.80) for the main
sample ETFs, with the control sample’s volumes being approximately equal before and after the
change.
From the point of view of liquidity, the table shows that both quoted and effective spreads declined in a
statistically significant manner after the minimum tick size reduction was implemented. It is important
to note that prior to the change the differences in spreads between the main and control samples were
not statistically significant either for quoted or for effective spreads. These last results provide further
evidence of the effect of the minimum tick size reduction on transaction costs in option markets.
Overall, these findings are consistent with those reported for equity markets (e.g. Henker and Martens
(2005)) and for futures markets by Kurov (2008).46 The reductions in spreads were accompanied by
corresponding reductions in the sizes of the quotes. More specifically, the table shows that both the bid
and ask quote sizes of the option classes included in the pilot project decreased significantly after the
minimum tick size reduction was implemented, while the corresponding sizes increased moderately
and non-significantly for the option classes in the control sample.
[Table 3.2 approximately here]
These introductory results indicate the importance of the tick size change on transaction costs and the
efficiency of trading. Note also in Table 3.2 that the pilot and comparison samples exhibit small but
45
Note that our control sample selection was based on the best volume match in the three months prior to the period of
study. The fact that that sample’s volume remained much lower than the main sample’s indicates that the options in the
latter sample were heavily traded during the study period.
46
On the other hand, Bollen and Busse (2006), Chakravarty et al (2004), Graham et al (2003), Jones and Lipson (2001) and
Harris (1994) suggest that the reduction in tick size can increase the cost for liquidity providers thus adversely affecting the
bid-ask spread. These suggestions are not confirmed by our results.
35
statistically significant differences in their implied volatility levels prior to the tick size change, while
these levels increase sharply, especially for the control sample, after the tick change; these levels
reflect the general market environment at the time of the pilot project’s implementation. A similar
effect is observed in the comparison between realized volatilities in the underlying markets, which
were approximately equal for both main and control samples prior to the change, while in the later
period the realized volatility of the control sample is much higher than that of the main sample for both
equities and ETFs. The widening differences in both implied and observed volatilities between main
and control samples after the tick size change merit further research that requires a much larger sample
and transcends the purpose of this paper.
IV. Empirical analysis: inversion methods, IS and CS before and after the tick size change
To derive our main results we use a five-minute moving window (or lag) of data for the estimation of
the parameters of the inversion methods. Results for other time-frames are presented as robustness
checks. To derive the option implied price which enters the VECM model (4), we aggregate the prices
implied by four ATM call options47 with the two nearest maturities that exceed the applied screen of
eight days.
In view of Proposition 1, the number of options used to derive the implied price plays a role in deriving
the IS. If the implied prices are perfectly correlated then their aggregation will not affect the variance
of the microstructure noise and, therefore, the size of IS. Unfortunately this is not the case in practice,
since the tick size, for instance, has a different impact on each option, implying that the aggregation
will have a diversification effect and increase the IS. There are, therefore, important reasons to limit the
number of options used to derive the implied price. First, a large number of option contracts will
47
Anand and Chakravarty (2007) argue that near-term ATM equity call options contain a large proportion of the
information of the option market.
36
introduce an upward bias in IS through diversification effects, even though additional option contracts
may or may not introduce any additional information.48 Second, the economic analysis of the
comparisons of IS across underlying securities in Section V is more reliable if it is done for an equal
number of similar option contracts for every security. We present the results for all option contracts
which passed our screens as a robustness check.
4.1 Accuracy of Inversion Methods
In this section we present the accuracy of the inversion methods relative to the two benchmarks,
expressed by relation (21) and the inequality of the correlations of Lemma 4. Apart from assessing the
merits of the two methods proposed in this paper relative to LIV, the empirical results are of some
interest since to our best knowledge our study is the first to assess the implied price a priori. Both
inequalities apply to the instantaneous measures of the indicated variables, which vary continuously.
We estimate the benchmarks, the RHS of (21) for put options49 and the correlation ˆ fS for call options,
from the equivalent sample estimates in the sample of one-second interval observations in each day.50
In Table 4.1 we present the summary statistics for the performance of the inversion methods applied in
this study for a five-minute moving window or lag used to derive the inversion parameters, with the
results aggregated for implied prices specific to each appropriate option contract; specifically, the
correlation statistics are derived for call options and the volatility statistics for put options, with four
48
This argument is corroborated by simulations results in section 2.4 that show increases in IS as the number of aggregated
implied prices increases, even though there is little incremental information from these additional prices.
49
As with the call options, we include four ATM contracts for the two nearest maturities.
50
For the RHS of (21) we use the sample volatility to estimate ˆ f and the daily median of the values of the factor
St /( ft  Xft
(X )
) , with ft
(X )
estimated from (A6). Although the accuracy of the benchmark estimate cannot be assessed, any
possible bias would not affect the comparative performances of the volatility inversion methods.
37
contracts for each option type per firm day.51,52 We present the results for both our main and control
samples separately for equities and for Exchange Traded Funds (ETFs) because of significant
differences between these two groups in the main quantities of interest, as discussed in the following
subsection. In addition, we split the results around the date for which the tick policy change occurred.
We also show in the table the proportion of cases for which relation (21) and Lemma 4 were satisfied
in our data. For descriptive purposes, we also present the proportion of cases for which two null
hypotheses, H 0 : ρIS ≥ ρfS and H 0 : σI ≤ [RHS of (21)] were not rejected with a p-value of at least
0.1.53
[Table 4.1 approximately here]
The results in Table 4.1 show clearly that both the HI and MLIV inversion methods outperform the
LIV method in all cases. Further, the performance of LIV inversion worsened after the decrease in the
tick size uniformly across equities and ETFs and for both main and control samples. Out of the two
proposed inversion methods, MLIV outperforms HI in all cases, especially with respect to the volatility
benchmark (21), while their performances regarding the correlation benchmark of Lemma 4 are closer.
The performance statistics for our main and control samples before and after the policy change indicate
that the two groups are quite similar, especially for the equity group.
[Table 4.2 approximately here]
[Table 4.3 approximately here]
51
To derive the results in table 4.1 we used 9344 (8128) options contracts of each type before (after) the policy change for
equities either for the main or the comparison sample. For ETFs the corresponding numbers were 2336 (2032). To derive
the IS or CS we used similar numbers of call option contracts.
52
Similar results available from authors upon request for moving windows or lags up to thirty minutes were not
qualitatively different and are not presented here.
53
To test the first null, we apply a z-test for the Fisher transformation of the two correlations; to test the second, we apply a
t-test based on the standard error of the RHS of (21) by assuming it is an estimate of the true volatility.
38
Tables 4.2 and 4.3 provide descriptive statistics for the implied prices aggregated across those prices
specific to each option contract, i.e. the statistics for the correlations between implied and underlying
prices and for the volatilities of implied prices for equities and ETFs respectively. In Table 4.2 we
observe a large increase in the correlation between the implied and underlying prices and a decrease in
the volatility of the implied price even though, on average the volatility of the underlying increased
from one period to the other. The correlations are significantly larger for the HI and MLIV methods
relative to the LIV method, while the converse holds true for the volatilities.54 Further, even though the
MLIV method clearly dominated the HI method with respect to the benchmarks in Table 4.1, the two
methods were very similar in terms of aggregate statistics for both correlations and volatilities. As for
the LIV method, even after the policy change its integrated implied price volatility vastly exceeds the
one of the underlying price, which is not plausible given the diversification effects resulting from
aggregating 4 prices. Note that in the control sample we also observe an increase in the correlations
after the policy change, albeit to lower levels than in the main one. 55 Similar effects are observed for
the ETFs in Table 4.3, with the correlations increasing in all cases in the main sample (but not always
in the control sample) as a result of the policy change; the increase, however, is smaller than the one
observed for equities in Table 4.2. The volatilities, however, generally increase for both samples and all
inversion methods after the policy change, given the overall increase in the volatility in the more recent
period.
In conclusion, the results in Tables 4.1 to 4.3 confirm with real data the simulated data results in
Section 2: they indicate that the LIV inversion does not satisfy the benchmarks of Lemmas 3 and 4 and
generally underperforms the two methods proposed in this paper in terms of the basic time-series
54
Note, however, that the correlation between implied and underlying prices continues to be low even after the tick size
reduction, implying that the reservations expressed in footnote 40 about the extraction of the underlying price’s risk neutral
distribution from a cross section of options continue to hold.
55
We attribute this general increase in the correlation to the overall increase in the volatility of the underlying. Recall that a
more volatile underlying asset price stipulates a larger co-movement of options regardless of the tick size.
39
properties of the implied price. It is expected, therefore, that according to Proposition 1 the relative
information shares of the option market will be higher with these new inversion methods than with
LIV, both before and after the tick size change.
4.2 Option markets’ information shares
In this section we present our main results for the IS metrics, which assess the price discovery in the
option market before and after the minimal tick policy change. We compare our main sample to the
comparison sample also in terms of time series since our metrics are highly variable over time. We also
report separately the results for 16 equities and 4 ETFs, since their discovery results were significantly
different.
Table 4.4 presents the summary results for the IS metric. It is clear that the average IS increased after
the tick size reduction irrespectively of the inversion method for both main and comparison samples, as
predicted in Proposition 1. In addition, we note that all IS metrics are much higher for the HI and
MLIV inversions compared to the LIV inversion. Further, the width of the bounds decreased in all
cases for the MLIV method and remained approximately constant for the HI method, while they
widened significantly after the tick size reduction for LIV, contrary to the predictions of Proposition 1.
For the IS midpoint the LIV inversion yields about half the IS values before, and only about 36% of the
IS values of the other two methods after the tick size reduction. We conclude that the LIV method
underestimates the role of the option markets in price discovery by failing to eliminate the unwanted
volatility inversion error. Finally, we observe highly consistent results between the HI and MLIV
inversions.
[Table 4.4 approximately here]
40
Figure 2 plots the daily averages across underlying securities for the midpoint IS metric for the main
and comparison samples under the MLIV inversion method, separately for equities and ETFs. 56 We
observe a significant increase in IS for the main sample after the tick size reduction (day 0 in the
figure) without a corresponding increase for the comparison sample. This figure also underscores the
importance of the comparison sample since we observe significant variability in IS over time. Note
that the CGM (2004) results indicated little variability in time series of this metric.
[Fig 2 approximately here]
Since we observed highly consistent results for the HI and MLIV inversions, while we also observed
that the former method is slightly inferior with respect to the performance benchmarks, in what follows
we drop the results for the HI inversion. Table 4.4 shows that the IS metrics also increased for the
control sample during the later period, which we attribute to the overall increase in volatility relative to
the earlier period. For this reason we assess the statistical significance of the differences between the
two samples by performing t-tests for the difference in the respective means of their midpoint IS. In all
cases we used a two-sided test before the policy change and a one-sided test after this change, assuming
unequal variances in either case. For all underlying securities in the period before the policy change
(2940 firm-days in each sample) we recorded a difference in means of 0.002 (p-value of 0.026). A
similar difference after the policy change (2520 firm-days in each sample) was 0.176 (p-value of 0).
We also performed the tests separately for equities and ETFs. For equities (2940 firm-days in each
sample) the difference in means was -0.003 (p-value of 0) before, and 0.153 (p-value of 0) after the
policy change. For ETFs these differences were 0.037 (p-value of 0) before the policy change (588
firm-days in each sample) and 0.268 (p-value of 0) after the change, with 504 firm-days in each
56
The HI inversion yielded similar results.
41
sample. As a robustness check, we also repeated the t-tests for each day in our sample57 and found that
while the difference in means was significantly positive at the 10% level for only 18 out of the 147
days before the policy change, there were 116 out of 126 days with positive differences at the 5% or
better level after the change. We conclude that the tick size change had a major and statistically
significant impact on the option market price discovery. Table 4.5 presents the results for the midpoint
of the IS metric, aggregated for each underlying separately for the periods before and after the policy
change for the MLIV and LIV inversion methods. The results in this table confirm previous findings
both with respect to the effect of the policy change and to the impact of the inversion method on this
metric. This table also shows that before the policy change the option market IS were much higher for
the four exchange-traded funds (ETF) in our sample than for equities, differing by as much as a factor
of ten; there was virtually no price discovery in equity options trading. The tick size reduction, on the
other hand, seems to have had a much heavier impact on equities than on ETFs, increasing their
information shares and making them in many cases comparable to those of ETFs.
Overall, a
comparison of the mean IS midpoints between ETFs and equities across all firm-days shows that the
differences are significant at 0.187 (p-value of 0) for the main sample, and at 0.112 (p-value of 0) for
the comparison sample. We conclude that the price discovery process in the option markets is
significantly different for equities and ETFs. We explore this issue further in the next section.
[Table 4.5 approximately here]
4.3 Polynomial distributed lags, moving average terms and option markets information shares58
As noted in the introduction, our IS results in Tables 4.3 and 4.6 are far apart from the results of the
earlier CGM (2004) study of the discovery process for 60 optioned firms’ equities over the five-year
57
Since the number of ETFs was too low for statistical inference, these tests were carried out for the sample of all
underlying securities.
58
Detailed tables for all results mentioned in this subsection are in an appendix available in our online supplement.
42
period 1988-1992. The average IS for the option market in that study was about 17.5% and there was
comparatively little dispersion among firms, with very tight bounds for IS. The comparable LIV
inversion method in Table 4.6 yields IS measures before the tick size change that are about one tenth of
the CGM numbers. For the two firms that are common in both studies, General Motors has IS bounds
16.2-16.4 % in CGM, and 0.8-1.8% in our results, while for Bristol-Myers Squibb the corresponding
numbers are 18.1-18.3% in CGM and 1.0-1.9% in our results. Since such a difference is hard to justify
on economic grounds, we consider possible differences in the VECM estimation methodology, which
was not spelled out clearly in the CGM study.
There are several approaches to the estimation of the VECM. Our main results were derived by
applying a relatively low number of lags indicated by the Akaike information criterion. Hasbrouk
(1995) used solely polynomial distributed lag (PDL) terms. In the robustness tests of the following
subsection we use PDL of various lengths for our VECM estimations and we find that our results are
robust to this methodology. Nonetheless, in subsequent work Hasbrouk (2002b, 2003) combines PDL
with moving average (MA) terms to reduce the number of estimated coefficients; note that MA terms
also reduce the off-diagonal correlations in the covariance matrix, thus tightening the bounds on IS.
This estimation method was adopted without change by Anand and Chakravarty (2007), Chen and Gau
(2009) and, most probably, by CGM. We apply the exact combination of PDL and MA to our data and
we find an average IS midpoint for the equities in our sample equal to 17.6%, virtually identical to the
CGM results, albeit with a bit wider bounds. We conclude that this VECM estimation by means of the
PDL-MA combination is the source of the differences in IS and we investigate its properties in
simulated data.
We use the Heston (1993) stochastic volatility model (9) to simulate the underlying, and the
corresponding option pricing model with and without tick size noise for the implied price. By
43
construction, there is no difference in information between trading venues, which implies that the only
factors allocating the information shares to each venue are the microstructure noise and the inversion
accuracy. To abstract from the volatility inversion problem, we also artificially created two trading
venues by augmenting the underlying price with two independent zero-mean Gaussian noises with the
standard deviation of 0.02. In this case, with both our method and with the PDL-MA VECM estimation
the Monte Carlo average of the midpoint IS was a perfect' 49.9%, with a similar value for CS. The
only effect of PDL-MA was the tightening of the spread around the midpoint IS by reducing it from
47.0 to 22%.
When we move to options, however, the IS and CS results become drastically different between the
VECM estimations with the optimal number of lags as in this paper, and the PDL-MA method as in
Hasbrouk (2002b). To begin with, for the optimal number of lags estimation the IS and CS metrics
depend on the volatility inversion method. For the HI and MLIV methods the metrics behave as
expected on theoretical grounds, increasing when the initial volatility increases and (especially) when
the tick size decreases; the LIV method is always very low and appears to be insensitive to tick size,
with the midpoint decreasing slightly when the tick size decreases. For the PDL-MA method, on the
other hand, all metrics for the option market become quite high and behave erratically, with most IS
midpoints above 80% for HI, above 90% for MLIV and above 50% for LIV, while they do not always
increase with the tick size reduction. When we repeated our simulations by including only PDL terms
the results remained qualitatively unchanged from the ones estimated with the optimal number of lags.
Our pre-tick size change IS results receive a resounding confirmation in a recent study, by Muravyev et
al (2012), that appeared almost concurrently with the earlier versions of this paper. In that study, which
used the PCP portfolio approach and pre-2007 data to estimate the implied price, the main method
consisted in observing the instances in which there was no overlap between the spreads of the
44
underlying and implied prices. It concluded that in such instances it was the option market that adjusted
to the stock market and not vice versa. Further, the IS estimates that it produced were strikingly similar
to our own, in spite of the differences in samples, time periods and research methodologies: for the
pooled sample its estimates (Table 9) varied between 2.6% and 1.6%, depending on the averaging
method, while the similar aggregate estimates in our Table 4.4 for the optimal MLIV method for our
main sample prior to the tick change and the control sample before and after the change vary from
1.2% to 4.2%. Even individual firms that are common to both papers are very close to each other:
AMZN is at 1.5% in the day-to-day and 0.6% in the pooled sample in the Muravyev et al (2012) paper,
and 0.8% in our Table 4.5.; BMY is 2.9% and 0.55% in their results and 1.5% in ours; etc. The
inescapable conclusion, common to both studies, is that there was no price discovery in the equity
option market for the period before the tick size change.
The PDL-MA method is clearly a “pre-averaging” approach, as in Jacod et al (2009) that reduces the
microstructure noise volatility. Since this approach is applied to both time series in the VECM
estimation, its effect on the two markets’ IS depends, by Proposition 1, on the respective reductions of
their microstructure noise; recall that this noise should not be affected by the empirical design in order
not to contaminate the IS. As the simulations show, when the noise variances are equal in the two
markets the PDL-MA method leaves the IS unchanged. In the case of the underlying asset-option
market pair, however, the microstructure noise is much higher for the option market, with the result
that pre-averaging has highly asymmetric effects on the two markets’ IS. We conclude that when it
comes to option markets’ information shares based on the VECM estimation with the use of MA terms
the earlier studies’ results are strongly biased upwards and cannot be considered reliable under most
circumstances. The exact form and size of this bias are beyond the scope of this paper.
45
4.4 Robustness checks
We start with comparing our main results for the IS metric with the ones for the CS metric. The results
in Table 4.6 for the latter metric confirm the findings in Table 4.4. In addition, the proportion of
component shares CS that are statistically significant at a 5% or lower level increases after the tick size
reduction, from 30-60% to 60-85%. We conclude that both IS and CS metrics for option markets
increased significantly after the reduction of the minimum tick size irrespective of the inversion
method, especially for the MLIV method.
[Table 4.6 approximately here]
In all subsequent robustness checks, we present aggregate results for the pooled samples of equities and
ETFs.59 Table 4.7 presents the results for 15- and 30-minute moving windows (or lags) for the
estimation of the inversion parameters, compared to the 5-minute window (or lag) in our main results.
To shorten the presentation we present the IS metrics’ results only for the midpoint. The table shows
that our results are robust to the length of the window (or lag).
In the light of our earlier argument, the number of aggregated option implied prices is an important
determinant of price discovery metrics. For all available option contracts60 used to derive the aggregate
implied price and for the period after the policy change, the midpoint IS for the MLIV inversion
increased from 23.9% for the four ATM near to maturity call options to 33.1%. The corresponding
differences for the period before this change and for the LIV inversion were significantly lower. These
differences are substantial, but it is not clear whether the availability of additional option contracts
facilitates price discovery through the option market or the increase in IS is simply an artefact of the
59
The full results for the separate samples, available from the authors on request, were not qualitatively different.
By our screens, this number was at most 18 contracts, calls and puts in equal proportion for three nearest maturities
exceeding two weeks.
60
46
statistical analysis. Given this last argument, we remain on the conservative side and present as our
main results the IS for the limited number of call options.
[Table 4.7 approximately here]
The fundamental work of Hasbrouck (1995) used a PDL model to estimate the VECM. The main role
of PDL is to reduce the number of the estimated coefficients in case a large number of lags is applied.
We did not use PDL to derive our main results since we applied a relatively low number of lags
indicated by the Akaike information criterion. As a robustness check we verify if the inclusion of a
large number of lags may affect our results. The outcome of this analysis is summarized in Table 4.8.
This table shows that the IS metrics are unaffected by the inclusion of a large number of lags for the
MLIV inversion method. For the LIV method, these metrics increase slightly when the included
number of lags increases in the period before the policy change. We conclude that our results are
robust to the inclusion of a large number of lags in the estimation of the VECM model.
[Table 4.8 approximately here]
V. Tick size change and informed trading in equities and ETFs
A decrease in the option market’s tick size is expected a priori to increase trading in options and, thus,
improve the liquidity of the option market, given the strong association of trading volume with liquidity
documented in several microstructure studies. Similarly, the results of the two previous price discovery
studies in option markets, CGM (2004) and Chen and Gau (2009), document an increase in the price
discovery metrics of the option market in response to improved liquidity of the market as measured by
variables such as spread and volume. These results are confirmed in our data: regression results in our
online appendix from the entire sample of the daily discovery metrics across equities and ETFs show
that the relative spread is a strongly significant determinant of option market discovery in the expected
47
direction, while the influence of volume is small and/or non-significant. Similarly, the implied
volatility has positive effects on the option market discovery metrics, which are statistically significant
only for equity options. There is also a significant negative effect of the underlying price for both
equities and ETFs. A variable intended to capture informed trading and using the Lee and Ready
(1991) algorithm turned out to be not significant. Last, the tick size change indicator is positive and
strongly significant in all cases, as expected.
Such tests, though, cannot explain the fundamental issue in price discovery, which is the use of the
option market for informed trading as in the theoretical model of Easley et al (1998). The change in
tick size resulted in a major increase in the discovery metrics, which were very low for equities prior to
the change, and the key question is whether this increase signals a migration of informed trading from
the underlying to the option market as a result of the improved liquidity of that market. We know from
the analysis of the proprietary data used in Pan and Poteshman (2006) that informed trading did, in fact,
take place in the option market prior to the tick change,61 but our regression analysis of the option
market’s discovery metrics did not reveal any strong effects of the conventional measures of informed
trading on these metrics both before and after the tick size change: the proportion of informed traders
had a small, negative and insignificant effect.
5.1 An event study of informed trading in the option market
A full study of informed trading in the option market is a large topic that requires more data and is
beyond the scope of this paper. Nonetheless, given the failure of our conventional regression analysis
to associate option market price discovery metrics with publicly available measures of informed
trading, we examine the following hypothesis in an event study formulation: do extreme values of the
61
Informed trading in the option market also appears in Aanand and Chakravarty (2007), but that study’s IS metrics are
flawed because of the problem mentioned in Section 4.3.
48
IS metrics signal a subsequent abnormal behavior of the returns in the underlying market? If this
hypothesis is rejected before the tick size change and accepted after then there is at the very least
evidence of a qualitative change in the importance of informed trading through the option market as a
result of the tick size reduction, confirming the suspected migration of informed traders to that market.
As a first step we identify the “high” IS events, by applying a filter of 1.8 standard deviations above the
mean of the daily IS,62 estimated separately for each one of the before- and after-tick size change
periods. Of these identified high IS days we select the first and the last one, resulting in 32 and 8
events respectively for the equities and ETFs in each period, for both the main and the comparison
samples. For each event that we select we distinguish the event period, consisting of 91 days of
underlying asset return data used for the estimations,63 and the event window, which is included in the
event period and has a length that varies from 3 to 7 days after the event day, during which the
potentially abnormal returns are measured.64 These returns form the data set for our event study.
Since the information implied by a high IS may signal positive or negative news for the underlying
asset, we verify whether there are unusual absolute returns during the event window. A nonparametric
method for such a test was presented by Corrado (1989) and used subsequently by Bhattacharya et al
(2000) in their study of informed trading in the Mexican stock market. The test involves the rank
ordering of the observations during the event period and the test statistic is equal to the average excess
rank of the observations during the event window divided by the standard deviation of the ranks over
the entire event period. The exact procedure is explained in the appendix. As observations we use three
62
The choice of the filter was motivated by the need to balance the identification of extreme IS values with the necessity to
find a sufficient number of such values to avoid overlap in the subsequent event windows. In addition, since IS or CS were
relatively thin-tailed in our data, applying a more stringent filter would result in fewer events.
63
Part of this data for the pre-tick change period extends beyond the sample used for the IS estimations.
64
Contrary to normal practice, we do not include the event day or any prior day in the event window. The “high IS” event
cannot be predicted before the end of the day, let alone beforehand, since it goes from open to close of trading. The returns,
on the other hand, are as usual from close to close.
49
different types of returns: the absolute value of the raw returns; the absolute value of the returns
adjusted by subtracting the sample mean; the absolute value of the residuals of a market model with the
S&P 500 index as a market index (for the equities sample only), and with the event period as an
estimation period.
Table 5.1 shows the values of the test statistic for the various samples and periods and the various
definitions of the observations. We concentrate on the four samples of equities, the main and control
samples before and after the tick size change. Recall that the statistic tests whether there were
“unusual” returns during the event window as compared to the returns during the entire event period. It
is immediately apparent from the table that there is little evidence of unusual returns in the event
windows prior to the tick size change for both main and control samples, and no evidence at all for the
control sample after the change. By contrast, the test statistic for the main sample returns after the tick
size change is highly significant for all event windows and all types of returns. In other words, a high
IS derived from the event day’s trading is a signal that the underlying asset returns will change
significantly during the event window, from 3 to 7 days later. This provides important evidence for
both migration of informed traders to the equity options market and a significant impact of informed
trading on the price discovery metrics in this market.
[Table 5.1 approximately here]
The results of the event test for ETFs provide no evidence of informed trading. In some cases, p-values
were in the 15%-20% range, possibly indicating a need for further analysis. The next subsection
provides alternative specifications that seem to provide higher p-values. The meaning of informed
trading in the case of ETFs is unclear since there are other signals (such as the VIX index) that were not
part of this study and that ETF traders may respond to.
50
5.2 Robustness checks
We report briefly on the results of several alternative specifications of the test variables, all available in
our online supplement. First of all, we repeated the tests with the high CS metric as the event indicator
and with the same filter as for the IS midpoint. The results were qualitatively the same as in Table 5.1,
a fact that is not surprising given the very similar results for both metrics in Tables 4.4 and 4.5.
Second, we repeated our experiment for event period lengths of 61 and 121 days, again without
qualitative differences in the results. Setting the beginning of the event window at day 0 instead of 1
while holding the number of days constant resulted in a non-significant test statistic in the main sample
of equities for a three-day event window for raw or mean-adjusted raw returns, while the results
strengthened for the market model. The other event windows were not qualitatively affected in the
main sample for equities, and the results for the comparison sample were unaffected.
Next we verified whether we arrived at 'significant' events by chance. We changed the events from the
first and the last in a given period passing our filter to the second and before last with little effect on the
results.
A reciprocal verification, whether we arrived at 'insignificant' events in the comparison
samples by chance, again confirmed our main findings. Last, we varied our threshold for the inclusion
in the event set to as low as one standard deviation above the mean of the midpoint IS. This low
threshold had little effect on the results for equities; however, it led to an increase in p-values for ETFs
well beyond the 15%-20% range reported before.
On balance, therefore, taking into account our main results and robustness checks, we conclude that
there is strong evidence for the migration of informed trading for equities to the option market after the
decrease in the tick size. There is little evidence for such migration in ETFs.
51
V. Conclusions
We present an alternative approach to the price discovery process in option markets, in which we stress
the importance of microstructure noise from the structure of trading in the option market as a
determinant of that market’s IS. We show theoretically that the price discovery IS metrics are highly
sensitive to the microstructure noise, as well as to the errors introduced by the option price inversion
method; the latter is due to the research method and does not belong to the price discovery process. We
develop theoretical benchmark measures in order to assess the inversion methods under plausible
underlying asset dynamics and we verify empirically with simulated data and a known option pricing
model the adequacy of our proposed methods in eliminating the inversion errors. We show that the
dominant lagged implied volatility inversion method, which was used almost exclusively in earlier
studies, tends to seriously underestimate the contributions of the option markets to price discovery. Last
but not least, we show that a popular pre-averaging statistical estimation method that was used in
several well-known earlier studies introduces major upward biases in the estimation of the option
market’s price discovery metrics. We apply these insights to estimate these price discovery effects after
a change in minimum tick size in the option market implemented recently in a pilot study of 24
optioned stocks, exchange traded funds (ETFs) or indices in the CBOE. We examine price discovery
by using this pilot study sample and a matched sample of optioned stocks in which the minimum tick
size was not changed. We apply our proposed new inversion methods and we eliminate the upward
bias in the estimation. After these corrections our price discovery results show that there was no
significant discovery in the equity option market prior to the tick size change. We also show that tick
size change had a major impact on the information shares of the option market, increasing the
importance of this market in the discovery process. Last, we find significant and systematic differences
52
in the price discovery metrics between equities and ETFs both before and after the tick size change,
with equities being more responsive than ETFs to the tick size change.
We verify empirically whether the major increase in the discovery metrics for equities was partly due
to informed trader migration to the option market after the tick size change. Using a novel event study
approach, we associate abnormally high values of the discovery metrics with subsequent abnormal
behavior of underlying asset returns. This association is always weak or nonexistent for both main and
control samples before the tick size change; it is however strong and statistically significant for the
main sample after the tick size change for equities but not for ETFs. These results have important
implications for future research. They point out to a potentially tradable anomaly, in which public
information from the option market can be used to extract superior returns from the underlying asset
market. As for the ETF results, they require a more in depth analysis with additional data, in view of
the availability of additional signals to traders for the major stock index funds and the evidence for
systematic mispricing of the index options similar to the one uncovered by Constantinides et al (2011).
We conclude that option markets play a very complex role in price discovery that depends on market
conditions and was masked by flaws in the statistical estimation and the volatility inversion methods.
We also surmise that this role is going to become even more important if it is incorporated into several
important results from the asset pricing literature like the aforementioned studies of Carr and Wu
(2009) and Constantinides et al (2011).
53
Table 2.1
Simulated Performance of the Inversion Methods for the Implied Price Volatility
The table displays the results for the simulated Heston (1993) model with the microstructure noise in the option price
(Monte Carlo averages for 300 simulations). Parameter values are as follows: initial volatility equal to its long-term mean of
0.3, volatility of volatility 0.4, speed of mean reversion 1, correlation of Wiener processes -0.6, riskless rate 4%, stock drift
8%, price of volatility risk -5%, initial underlying price 100 and minimal tick size 0.1 (.05) for option prices above (below)
$3.
T
K/S0
True
(σS)
Benchmark
(eq. 21)
σI
σHI
σLIV
σMLIV
σRIV
A: Contract level implied price
1/12
1.05
0.0112
0.0119
0.0114
0.0130
0.0192
0.0114
0.0164
1/12
1.00
0.0112
0.0122
0.0117
0.0132
0.0196
0.0118
0.0186
1/12
0.95
0.0112
0.0123
0.0121
0.0135
0.0199
0.0120
0.0212
2/12
1.05
0.0112
0.0123
0.0117
0.0124
0.0195
0.0117
0.0168
2/12
1.00
0.0112
0.0127
0.0120
0.0138
0.0199
0.0120
0.0164
2/12
0.95
0.0112
0.0131
0.0124
0.0133
0.0201
0.0122
0.0168
3/12
1.05
0.0112
0.0127
0.0120
0.0127
0.0198
0.0119
0.0174
3/12
1.00
0.0112
0.0131
0.0123
0.0134
0.0201
0.0123
0.0161
3/12
0.95
0.0112
0.0136
0.0127
0.0135
0.0205
0.0128
0.0167
N/A
N/A
0.0112
n/a
0.0051
0.0048
0.0149
B: Aggregate (1/N) implied price
0.0057
0.0125
Table 2.2
Simulated Performance of the Inversion Methods for the Correlation between Implied and
Underlying Prices
The table displays the results for the simulated Heston (1993) model with the microstructure noise in the option price
(Monte Carlo averages for 300 simulations). Parameter values are as follows: initial volatility equal to its long-term mean of
0.3, volatility of volatility 0.4, speed of mean reversion 1, correlation of Wiener processes -0.6, riskless rate 4%, stock drift
8%, price of volatility risk -5%, initial underlying price 100 and minimal tick size 0.1 (.05) for option prices above (below)
$3.
T
K/S0
f-S
w/o noise
f-S
with noise
I-S
IHI-S
ILIV-S
IMLIV-S
IRIV-S
A: Contract level implied price
1/12
1.05
-0.996
-0.063
0.063
0.059
0.037
0.062
0.042
1/12
1
-0.992
-0.043
0.043
0.040
0.025
0.042
0.027
1/12
0.95
-0.985
-0.047
0.047
0.044
0.028
0.047
0.025
2/12
1.05
-0.992
-0.057
0.057
0.052
0.034
0.057
0.039
2/12
1
-0.986
-0.043
0.042
0.037
0.025
0.042
0.030
2/12
0.95
-0.978
-0.057
0.055
0.052
0.035
0.056
0.039
3/12
1.05
-0.988
-0.054
0.053
0.049
0.032
0.054
0.036
3/12
1
-0.981
-0.041
0.041
0.038
0.025
0.041
0.031
3/12
0.95
-0.972
-0.030
0.030
0.026
0.018
0.029
0.022
0.120
0.047
B: Aggregate (1/N) implied price
N/A
N/A
N/A
N/A
0.113
0.104
0.046
54
Table 3.1
Underlying Securities in the Main Sample
The table provides the names and ticker symbols for the underlying securities included in the main sample that were a part
of the tick size reduction pilot project.
Ticker
Underlying Name
AAPL
Apple
AMGN
Amgen
AMZN
Amazon
BMY
Bristol-Myers Squibb
C
Citigroup
COP
ConocoPhillips
CSCO
Cisco
FCX
Freeport-McMoran
GM
General Motors
MO
Altria Group
MOT
Motorola
NYX
NYSE Euronext
QCOM
Qualcomm
RIMM
Research in Motion
T
AT&T
YHOO
Yahoo
DIA*
Diamonds Trust
OIH*
Oil Services HLDRS
SPY*
SPDR S&P 500
XLE*
Energy Sector SPDR
*
Exchange traded funds
55
Table 3.2
Descriptive statistics before and after the Tick Size Reduction
The table displays descriptive statistics for the options on securities included in the main and control samples. The means
are provided for the following variables: Price, the national best offer and ask midpoint quoted price, Implied volatility, the
Black-Scholes-Merton implied volatility, Log volume, the natural logarithm of daily volume, Effective spread, twice the
absolute value of the difference between transaction prices and the average between quoted bid and ask prices, Quoted
spread, the difference between quoted bid and ask prices, Bid size and Ask size, the number of contracts available at the
quoted bid and ask prices, Volatility, the realized daily volatility derived for five-minute intervals and annualized. The tstatistics are for the difference in means between the main and control samples for a two-sided test. The symbols * and ***
denote a statistically significant difference of means for a given variable between the main and control sample, respectively
at the 10 and 1% level.
Variable
Price
Implied volatility (%)
Equity log volume
ETF log volume
Effective spread
Quoted spread
Bid size (‘000s)
Ask size (‘000s)
N obs. (‘000s)
Equity log volume
ETF log volume
Equity volatility
ETF volatility
Before Policy Change
After Policy Change
Main
Control
Main
Control
t-stat
t-stat
Sample
Sample
Sample
Sample
A: Option Markets
3.86
4.43
-71.59***
4.63
4.97
-38.92***
***
32.55
32.37
12.32
42.52
45.80
-164.63***
11.19
9.62
20.80***
11.43
9.96
17.70***
***
12.07
9.50
16.84
12.64
9.58
13.95***
0.128
0.171
-1.53
0.056
0.205
-355.77***
0.119
0.138
-0.67
0.062
0.166
-37.54***
1.87
0.45
652.02***
0.45
0.50
-54.71***
***
1.71
0.40
663.56
0.44
0.48
-34.71***
3,567
814
N/A
5,306
910
N/A
B: Underlying Markets
16.48
15.5
44.75***
16.63
15.78
36.23***
***
17.00
15.25
32.55
17.08
15.51
24.78***
*
0.2403
0.2463
-1.73
0.3468
0.4141
-10.59***
***
0.1531
0.1727
-3.73
0.2157
0.2860
-8.02***
56
Table 4.1
Performance Statistics for Volatility Inversion Methods
The table displays the performance of inversion methods relative to the Lemma 4 benchmark for the implied and underlying
prices correlations (call options), and volatility upper bound (21) (put options) before (147 days) and after (126 days) the
tick size policy change. The statistics presented in the table were derived for implied prices corresponding to option
contracts. The sample sizes for the main or control sample before (after) policy change were as follows: equities 9,344

(8128) call or put options, ETFs 2336 (2032) call or put options. Prop. Pr( H 0 ) > 0.1 denotes the proportion for which the

null hypothesis H 0 : ρIS ≥ ρfS was not rejected with p-value of at least 0.1 under the Fisher transformation for correlations,


Prop. Pr( H 0 ) > 0.1 denotes the proportion for which the hypothesis H 0 : σI ≤ RHS of (21) was not rejected with the p-value
of at least 0.1, with the volatility standard error derived from the RHS of (21). The volatility σI is the integrated volatility of
the squared log-increments of the implied price taken in one-second intervals.
Inversion
Method
Prop.
Lemma 4
satisfied
Prop.

Pr( H 0 )
Prop.
(21)
satisfied
> 0.1
Main Sample
Prop.

Pr( H 0 )
> 0.1
Prop.
Lemma 4
satisfied
Prop.

Pr( H 0 )
Prop.
Prop.
(21)
satisfied
Pr( H 0 )
> 0.1
Control Sample

> 0.1
A: Equities before Policy Change
HI
0.421
0.984
0.170
0.260
0.416
0.992
0.182
0.276
LIV
0.177
0.666
0.000
0.000
0.175
0.657
0.001
0.001
MLIV
0.469
1
0.559
0.603
0.463
1
0.612
0.649
B: Equities after Policy Change
HI
0.245
0.993
0.180
0.311
0.318
0.998
0.151
0.221
LIV
0.004
0.029
0.001
0.001
0.019
0.221
0.002
0.002
MLIV
0.338
0.996
0.755
0.784
0.416
0.999
0.808
0.837
C: ETFs before Policy Change
HI
0.221
0.999
0.064
0.106
0.281
0.994
0.096
0.135
LIV
0.073
0.492
0.002
0.002
0.234
0.720
0.011
0.011
MLIV
0.197
1
0.976
0.983
0.272
0.992
0.650
0.665
HI
0.054
0.986
0.036
0.066
0.177
0.990
0.045
0.068
LIV
0.013
0.080
0.006
0.006
0.174
0.651
0.003
0.003
MLIV
0.089
0.999
0.992
0.994
0.218
0.996
0.762
0.779
D: ETFs after Policy Change
57
Table 4.2
Descriptive Statistics for Aggregate Implied Price: Equities
The table displays the descriptive statistics for the aggregate implied price before (147 days) and after (126 days) the tick
size policy change. 16 equities were included in the main and control samples, which resulted in 2352 and 2016 firm-days
in each period, respectively. Q3-Q1 denotes the inter-quartile range.
Inversion
Method
Mean
Median
St. Dev.
Q3-Q1
Mean
Main Sample
Median
St. Dev.
Q3-Q1
Control Sample
A: Before Policy Change
Correlation between implied and underlying prices
HI
0.047
0.038
0.042
0.051
0.048
0.042
0.036
0.046
LIV
0.030
0.025
0.027
0.033
0.030
0.027
0.024
0.031
MLIV
0.048
0.039
0.041
0.050
0.048
0.042
0.036
0.046
Integrated volatility of implied price
HI
0.023
0.018
0.036
0.008
0.021
0.017
0.054
0.011
LIV
0.057
0.029
0.169
0.012
0.084
0.029
0.233
0.017
MLIV
0.031
0.018
0.024
0.008
0.029
0.018
0.061
0.011
B: After Policy Change
Correlation between implied and underlying prices
HI
0.267
0.253
0.103
0.146
0.109
0.104
0.043
0.058
LIV
0.156
0.144
0.065
0.087
0.065
0.062
0.026
0.034
MLIV
0.267
0.253
0.103
0.148
0.109
0.104
0.043
0.058
Integrated volatility of implied price
HI
0.016
0.015
0.006
0.007
0.027
0.021
0.055
0.017
LIV
0.034
0.027
0.114
0.013
0.048
0.037
0.114
0.028
MLIV
0.017
0.016
0.006
0.008
0.028
0.022
0.050
0.018
58
Table 4.3
Descriptive Statistics for Aggregate Implied Price: ETFs
The table displays the descriptive statistics for the aggregate implied price before (147 days) and after (126 days) the tick
size policy change. 4 ETFs were included in the main and control samples, which resulted in 588 and 504 firm-days in each
period, respectively. Q3-Q1 denotes the inter-quartile range.
Inversion
Method
Mean
Median
St. Dev.
Q3-Q1
Mean
Main Sample
Median
St. Dev.
Q3-Q1
Control Sample
A: Before Policy Change
Correlation between implied and underlying prices
HI
0.073
0.068
0.043
0.054
0.037
0.031
0.028
0.032
LIV
0.046
0.043
0.028
0.037
0.021
0.018
0.018
0.021
MLIV
0.075
0.070
0.043
0.054
0.037
0.032
0.029
0.033
Integrated volatility of implied price
HI
0.011
0.010
0.007
0.006
0.011
0.009
0.005
0.006
LIV
0.020
0.018
0.012
0.011
0.038
0.018
0.134
0.012
MLIV
0.012
0.011
0.008
0.006
0.013
0.010
0.023
0.006
B: After Policy Change
Correlation between implied and underlying prices
HI
0.183
0.181
0.086
0.128
0.040
0.036
0.028
0.038
LIV
0.101
0.094
0.057
0.085
0.021
0.018
0.017
0.020
MLIV
0.186
0.184
0.085
0.126
0.042
0.037
0.029
0.039
Integrated volatility of implied price
HI
0.014
0.009
0.028
0.004
0.015
0.011
0.011
0.009
LIV
0.026
0.020
0.029
0.010
0.075
0.026
0.131
0.026
MLIV
0.014
0.010
0.020
0.004
0.016
0.012
0.011
0.010
59
Table 4.4
Average IS before and after the Tick Size Reduction
The table provides descriptive statistics for information shares, IS, for the options implied price before and after the policy
change reducing the minimum tick size. IS are described by their average minimum value (min), the average value of the
midpoint (mid) and the average maximum value (max).
Inversion
Method
HI
LIV
MLIV
HI
LIV
MLIV
HI
LIV
MLIV
HI
LIV
MLIV
IS for Main Sample
IS for Comparison Sample
mid
max
min
mid
max
A: Equities Before Policy Change
0.008
0.013
0.018
0.010
0.016
0.021
0.007
0.009
0.012
0.009
0.012
0.014
0.007
0.012
0.017
0.010
0.015
0.021
B: Equities After Policy Change
0.081
0.194
0.306
0.020
0.042
0.064
0.035
0.084
0.132
0.013
0.022
0.031
0.082
0.195
0.308
0.019
0.042
0.064
C: ETFs Before Policy Change
0.124
0.157
0.190
0.110
0.124
0.139
0.074
0.091
0.109
0.088
0.096
0.103
0.134
0.170
0.206
0.117
0.133
0.148
D: ETFs After Policy Change
0.281
0.383
0.484
0.126
0.140
0.155
0.137
0.191
0.245
0.101
0.108
0.114
0.308
0.415
0.522
0.131
0.147
0.162
min
60
Table 4.5
Average Information Shares per Underlying before and after Policy Change
The table displays information shares, IS, for the options implied price before and after the policy change reducing the
minimum tick size for the securities in the main sample. IS are described by their the average value of the midpoint for each
security.
Before Policy
Change
Ticker
LIV
MLIV
inversion inversion
AAPL
0.010
0.012
AMGN
0.007
0.009
AMZN
0.006
0.008
BMY
0.013
0.015
C
0.010
0.012
COP
0.013
0.019
CSCO
0.007
0.009
FCX
0.013
0.015
GM
0.010
0.013
MO
0.013
0.018
MOT
0.010
0.013
NYX
0.012
0.015
QCOM
0.007
0.010
RIMM
0.006
0.009
T
0.010
0.014
YHOO
0.006
0.005
DIA*
0.071
0.121
OIH*
0.136
0.272
*
SPY
0.042
0.080
XLE*
0.116
0.207
*
exchange traded funds
After Policy
Change
LIV
MLIV
inversion inversion
0.163
0.322
0.051
0.120
0.125
0.268
0.059
0.138
0.055
0.135
0.064
0.179
0.084
0.187
0.086
0.230
0.095
0.218
0.055
0.137
0.079
0.175
0.058
0.159
0.103
0.236
0.106
0.260
0.095
0.225
0.058
0.127
0.150
0.340
0.281
0.573
0.142
0.296
0.190
0.453
61
Table 4.6
Average CS before and after the Tick Size Reduction
The table displays component shares, CS, for the options implied price before and after the policy change reducing the
minimum tick size. CS are described by the average value (mean) and Pr(p<5%), the proportion of CS that are statistically
significant at the 5% or lower level by the χ2 test with one degree of freedom.
Inversion
Method
HI
LIV
MLIV
HI
LIV
MLIV
HI
LIV
MLIV
HI
LIV
MLIV
Main Sample
Control Sample
mean
Pr(p<5%)
mean
Pr(p<5%)
A: Equities Before Policy Change
0.055
0.265
0.067
0.341
0.032
0.131
0.038
0.256
0.053
0.263
0.065
0.337
B: Equities After Policy Change
0.276
0.525
0.117
0.591
0.112
0.238
0.055
0.409
0.271
0.532
0.112
0.582
C: ETFs Before Policy Change
0.265
0.918
0.278
0.840
0.213
0.845
0.153
0.776
0.256
0.934
0.279
0.842
D: ETFs After Policy Change
0.525
0.986
0.332
0.831
0.238
0.954
0.173
0.718
0.532
0.986
0.333
0.847
Table 4.7
Results for Varying Moving Window or Lag
The table displays the average midpoint information shares, IS, before and after the policy change reducing the minimum
tick size. The main results (5-minute moving window or lag) are compared to the results for 10- and 30-minute moving
window or lag. The results are provided for the main sample, securities included in the pilot project with the pooled results
for equities and ETFs.
Inversion
Method
LIV
MLIV
5 minutes
before
after
0.026
0.105
0.044
0.239
Moving window or lag
10 minutes
before
after
0.028
0.107
0.048
0.246
30 minutes
before
after
0.031
0.112
0.053
0.258
Table 4.8
Results for PDL Model
The table displays average midpoint information shares, IS, before and after the policy change reducing the minimum tick
size. In the main results the optimal number of lags was selected by the Akaike information criterion. In each of the
polynomial distributed lag model (PDL) models a separate second-degree polynomial was fit into each of the first three 10lag segments. For PDL 150 (300) three similar polynomials were fit into additional segments of the subsequent lags with
the respective lengths of 30, 30 and 60 (30, 60 and 180). The results are provided for the main sample, securities included
in the pilot project with the pooled results for equities and ETFs.
Inversion
Method
LIV
MLIV
Main results
before
after
0.026
0.105
0.044
0.239
PDL 150
before
after
0.026
0.103
0.041
0.234
PDL 300
before
after
0.032
0.113
0.043
0.237
62
Table 5.1
The event study results
The table displays the t-statistic computed by the Corrado (1989) test and described in the appendix for events defined as
unusually high midpoint IS’s for the option market. A potential events is defined as an incidence of an IS at least 1.8
standard deviations above its mean, estimated separately for each underlying and in each period before or after the policy
change. From all potential events for a given underlying, the first and the last one in each period were included in the
sample, which corresponds to 32 (8) events for the main or comparison sample of equities (ETFs) in each period. Residuals
are estimated in three ways: first, raw returns are used; second, returns are adjusted by the event-period mean; third,
,residuals from a market model estimated with the S&P 500 as the market index and length equal to the event period of 91
days are used. This last method was not applied to ETFs since one of the underlying securities was an S&P 500 index
tracking fund. The null hypothesis is that the absolute values of residuals are the same in the event period and in the event
window. The symbols *,** and *** respectively denote statistical significance at the 10, 5 and 1% level.
Main sample
Residuals estimation
3 days in
event
5 days in
event
Comparison sample
7 days in
event
3 days in
event
5 days in
event
7 days in
event
A: Equities before policy change (N = 32)
Raw returns
Sample mean adj.
Market model
0.48
0.52
0.49
0.74
0.64
1.93
**
-0.83
1.53*
1.03
1.08
-1.00
1.62
*
1.07
0.92
0.40
0.70
0.36
0.12
B: Equities after policy change (N = 32)
Raw returns
3.52
***
4.46***
5.00***
0.81
0.52
-0.05
Sample mean adj.
3.20***
4.43***
4.91***
0.77
0.37
-0.18
**
***
***
0.12
0.72
0.82
Market model
1.99
2.56
3.98
C: ETFs before policy change (N = 8)
Raw returns
4.32
4.00
4.58
4.22
Sample mean adj.
4.38
4.07
3.76
4.81
4.58
D: ETFs after policy change (N = 8)
4.20
Raw returns
4.93
4.64
4.86
3.51
3.68
3.58
Sample mean adj.
4.84
4.57
4.80
3.61
3.70
3.61
3.70
4.72
63
Figure 1
Example of the Implied Prices
The figure plots the implied prices for the LIV, HI and MLIV inversion methods as well as the prices of the underlying
security and the call option. The observed prices are for Apple stock and a February ATM call option (strike 135) observed
on January 25, 2008. Graphs have been shifted to facilitate presentation.
Underlying
HI
1
Price
MLIV
0.8
LIV
0.6
0.4
0.2
Call
0
0
50
100
150
200
Time (seconds)
250
300
Figure 2
Time Series of Average Midpoint IS
The figure plots the midpoint information shares (IS) aggregated for each day separately for the main and comparison
samples of equities and ETFs.
B: ETFs
0.9
0.8
0.8
Information Share (daily average of midpoint)
Information Share (daily average of midpoint)
A: Equities
0.9
0.7
Main Sample
0.6
0.5
0.4
Comparison Sample
0.3
0.2
0.1
0
-150
Main Sample
0.7
0.6
0.5
0.4
Comparison Sample
0.3
0.2
0.1
-100
-50
0
50
Days relative to policy change
100
150
0
-150
-100
-50
0
50
Days relative to policy change
100
150
64
Appendix
Proof of Proposition 1
Under the error independence assumption we write the structural model (2)-(3) under the form of a
matrix equation
vt 
vt 1 
vt   S ,t   S ,t 1  1 1 0  vt  0 -1 0  vt 1 
 Pt  
  


 


 


   vt   I ,t   I ,t 1   1 0 1   S ,t   0 0 -1   S ,t 1     S ,t     S ,t 1 
vt  

 


 1 0 0    0 0 0  
vt

 I ,t 
 I ,t 1 
 I ,t 
 I ,t 1 
(A1)
Define now
 S ,t  eSt 
 S ,t 
eSt 
0 0 1 
   
 

1 
1
1
  I ,t   eIt   t   I ,t    eIt    t , where   1 0 -1  .
v  evt 
v 
evt 
0 1 -1
 t 
 t 
(A2)
Then (A1) can be written under the following Vector Moving Average (VMA) lag polynomial
representation, with I denoting the identity matrix
-1 0 1 
 Pt 


   I t  0 -1 1  t 1  ( I   ( L))t .
vt 
0 0 0
(A3)
From (A1)-(A3) the covariance matrix  of the error vector t is equal to
 S2   v2  v2
 v2

  Vart   v2
 I2 + v2  v2
 2
 v2
 v2
 v


,


(A4)
65
From (A3) the identical rows of the matrix  in (5) and (6) are equal to   [0 0 1] . From (5) and
(A4) we get '   v2 , the variance of the random walk.
Since the variance ˆ 2 of ˆIt forms part of  I2 , its effect on the IS bounds for I t can be found by
exploiting the symmetry of the vector P and the matrix  with respect to the two components of the
vector ( St , I t ). Recall that the IS upper (lower) bound of a price is derived when that price occupies the
top (bottom) position in the VMA representation.65 Symmetry here implies that it suffices to find the
impact of the variances  S2 and  I2 on IS from the Cholesky factorization of (A4), given that  is not
diagonal. Setting   FF ' , where F is the lower triangular matrix, we note from the form of that it
suffices to consider the last row of F , denoted by  a31 a32 a33  . Hence, the effect of ˆ 2 on the upper
(lower) bound of IS has the sign of
2
a32
a31
 v4
2
a

(
);
since
the upper bound is clearly
31
 S2  I2
 S2   v2
decreasing in  S2 , QED. The two bounds are given by
IS 
2
 v2
a32
 S2 v2
1
2
,
IS


[

]
,
where


 I2   v2
 v2
( I2  ) v2
 S2   v2
(A5)
The lower bound is also clearly decreasing in  I2 and, hence, in ˆ 2 . Last, the relative distance of the
bounds is found by the ratio of the two bounds shown in (A5), which turns out to be an increasing
function of  I2 . This completes the proof for the case of the error independence assumption.
A similar proof also holds for the Hasbrouk-Roll structural model (2)-(8), except for the fact that it
involves heavier computations. (A1) now becomes
65
See Hasbrouk (2002).
66
ut  (  cS ) wt  cS wt 1

 Pt  


  ut  (  cI ) wt  cI wt 1   I ,t   I ,t 1 
 Et  

ut   wt

1   cS

 1   cI
1 
ut 
ut 1 
0  ut  0  cS 0  ut 1 

 


  

1   wt   0  cI -1   wt 1     wt     wt 1 
 I ,t 
 I ,t 1 
0   I ,t  0 0 0   I ,t 1 
 


.
(A1’)
Similarly, (A2), (A3) and (A4) now become, with Var (ut )   u2 , Var (wt )   w2 , Cov(ut , wt )  0
 
(  cS ) 
0


cS
cS 





u
e
u
e
 t 
 t 
St
St
1
1 

   
 
1 
  wt   eIt   t   wt    eIt    1t , where  1  
0


cS
cS 

eqt 
 It  eqt 
 It 
 
 
 cI
c c 
1  S I

cS 
 cS
-1 0
 Pt 


  I t  0 -1
 Et 
0 0
1
1 t 1  ( I   ( L))t
0
 u2  (  cS ) 2  w2
 u2  (  cS )(  cI ) w2

  Vart   u2  (  cS )(  cI ) w2  u2  (  cI ) 2  w2 + I2
 2   (  c ) 2
 u2   (  cI ) w2
S
w
 u
(A2’)
(A3’)
 u2   (  cS ) w2 
(  cI ) w2
 u2   2 w2




(A4’)
From (A3’) the rows of  are again   [0 0 1] , with  '   u2   2 w2 , the random walk variance,
as expected.
The main difference between the two structural models is that the vector P and the matrix  are no
longer symmetric with respect to the two components of the vector ( St , I t ). As a result, the Cholesky
factorization of (A4’) as it now stands can only demonstrate the effect of  I2 on the IS lower bound,
67
and we need to interchange the rows of the matrix B and repeat the estimation in order to find the effect
on the IS upper bound. Both factorizations and demonstrations are straightforward but tedious and are
left as an exercise.
The HI method of volatility inversion
At any time t we observe simultaneously a stock price St and a cross section of options with a
common maturity but differing with respect to their strike prices. For any option in a given cross
section (except for the two extreme strike prices), the two neighboring options may be used to estimate
the partial derivatives, ft ( X ) and ft ( S ) . Once we have these estimates for the partials from past data, we
may then easily invert (11) for the implied price I t . We have from (20) for any   t  M ,..., t 1 :
f( X )  12  f  X  X   f  X  X   X
and

f( S )  f  f( X ) X

S , or
1
f
(S )


S
f  f( X ) X
,
(A6)

From (24), we have the following formula for the implied price:
It 
ft  ft ( X ) X
 at ft  bt X ,
ft ( S )
(A7)
where for a given moving window length M , at and bt are the respective medians of

f( X )
as derived by (24),   t  M ,..., t 1 .
f( S )
1
f( S )
and
Note that the implied price under this inversion
approach has a meaningful interpretation as a portfolio composed of  t options and t X in the money
market account. In empirical applications, where we use the average implied price derived from
68
several options, this portfolio interpretation holds, in this case as a portfolio of options with unequal
weights and the money market account.
The Corrado (1989) test
The test will be described for the market model returns, with the others left as an exercise. For each
underlying i, we rank 91 days of absolute values of the model’s residuals. The rank of the absolute
value of the residual in date t is denoted as Kit. Day 0 indexes the event date. The point estimate  ( K )
for the event window (day 1 to d, d = 3, 5, 7) is estimated as
 (K ) 
1
N
d
 ( K
N
it
 45.5)
(A8)
i 1 t 1
and the standard deviation  ( K ) is estimated using the entire event period (day −90 + d to d)
 (K ) 
2
 1 N ( K  45.5)  .

 it

t 90 d 
91
 N i1

d
d
(A9)
The null hypothesis is that the absolute values of the residuals are the same in the event period and in
the event window. The test statistic t is equal to
t
 (K )
1 N d
for N  10; t 
 Kit for N  10
 (K )
92d i 1 t 1
(A10)
For N  10 , the test is asymptotically N(0,1); for N  10 it is distributed as an N-fold convolution of a
[0, 1] uniform with itself, which for N = 8 corresponds to the 10% or better significance level if t  5.05
.
69
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