A Corridor Fix for VIX:Developing a Coherent Model-FreeOption-Implied Volatility Measure∗

Torben G. Andersen† Oleg Bondarenko‡ Maria T. Gonzalez-Perez§

July 2010; Revised: January 2011

Abstract

The VIX index is computed as a weighted average of SPX option prices over a rangeof strikes according to specific rules regarding market liquidity. Using tick-by-tickobservations on the underlying options, we document that this strike range varies sub-stantially in how much coverage it provides of the distribution of future S&P 500 indexprices, producing significant biases, or distortions, in the time series of VIX measures.We propose a novel high-frequency Corridor Implied Volatility index (CX) computedfrom a strike range covering an “economically invariant” proportion of the future S&P500 index values and using only reliable option quotes. Comparing the time seriesproperties of these alternative volatility indices from June 2008 through June 2010,we find that our CX measure is superior in terms of filtering out noise and avoidinglarge artificial jumps. Consequently, the properties of the two series at both the dailyand the intraday level are dramatically different in important dimensions of relevancefor asset pricing, risk management and real-time trading strategies.

∗We are indebted to the Zell Center for Risk at the Kellogg School of Management, Northwestern Univer-sity, for financial support. We thank seminar participants at the Federal Reserve Board of Governors, Wash-ington, D.C., the SoFiE-Creates Conference in Aarhus, October 2010, the University of Illinois at ChicagoCollege of Business Administration, and Pamplin College of Business, Virginia Tech for comments.

†Kellogg School of Management; Northwestern University; 2001 Sheridan Road, Evanston, IL 60208,NBER, and CREATES; e-mail: t-andersen@kellogg.northwestern.edu

‡Department of Finance (MC 168), University of Illinois at Chicago, 601 S. Morgan St., Chicago, IL60607; e-mail: olegb@uic.edu

§Kellogg School of Management; Northwestern University; 2001 Sheridan Road, Evanston, IL 60208;e-mail: m-gonzalezperez@kellogg.northwestern.edu

1 IntroductionThe VIX volatility index disseminated by the Chicago Board Options Exchange (CBOE)has attracted much attention in recent years. This index is computed from concurrent pricequotes on a wide range of out-of-the-money European style put and call options writtenon the S&P 500 equity index, and it is constructed to serve as a model-free option-impliedreturn volatility measure for the S&P 500 index over the coming 30 days, expressed inannualized percentage terms. One striking and widely noted feature is that the changes inthe VIX index displays a very strong negative correlation with the corresponding returnson the underlying S&P 500 index. This negative relationship with market returns suggeststhat holding long VIX positions provides portfolio diversification. On the other hand, theexistence of a large volatility risk premium means that the average return on long VIXfutures positions are negative, implying that speculators typically may desire short VIXpositions. Various trading strategies exploiting the latter feature have been launched underthe moniker of “volatility arbitrage” and associated performance measures are widely fol-lowed. As a result, the trading activity in VIX related derivatives has grown rapidly1, andthe relevance of VIX index as well.

The pronounced negative correlation with the overall market returns has also capturedthe attention of both the popular press and the more specialized financial media which oftenlabel the VIX a “fear gauge” due to the rapid spikes it displays during periods of marketstress. Naturally, the VIX garnered even more attention as the index soared to dramaticheights during the financial crisis of 2008-2009, reaching an intraday high of 89.5% onOctober 24, 2008, compared with an average value of about 19% in the period from 1990until October 2008. In short, the VIX is routinely cited by the main media when describingcurrent market conditions or “investor sentiment”.

On the other hand, the VIX is increasingly used in financial economics. In this case, themain attraction is that, in theory, VIX provides a model-free measure of the expected valueof the S&P 500 return variation under the risk-neutral (Q probability) measure. Hence, theVIX embodies a model-free2 market volatility forecast. The literature has exploited thisfeature going in three main directions. One set of studies explores the forecast power ofvariation in the VIX for future realized return volatility. In this line, Jiang & Tian (2005)deem the information content of the VIX volatility forecast superior to alternative impliedvolatility measures as well as forecasts based on historical volatility. A second group ofstudies emphasize that the VIX measure is constructed directly from observed option pricesso it will necessarily incorporate any pricing of variance risk that may be embedded in themarket prices. Thus, VIX does not represent a pure estimator of future return volatilityfor the underlying asset because it includes compensation for variance risk as well. In-stead, the wedge between regular time series forecasts for volatility, developed under the

1In fact, the CBOE considers the VIX options, introduced in February of 2006, the most successful newproduct launch in their history. Moreover, both the VIX futures and options have received “most innovativeindex product” awards at industry conferences.

2This is in marked contrast to earlier notions of option-implied volatility that derive implied volatilityfrom option prices plus specific modeling assumptions, e.g., the Black-Scholes model or more sophisticatedstochastic volatility models incorporating, potentially, multiple volatility factors and jumps.

1

actual or objective (P probability) measure, and the VIX, representing the risk-neutral re-turn volatility forecast, may be interpreted as a market volatility risk premium, see, e.g.,Bondarenko (2007), Bollerslev, Tauchen & Zhou (2009) and Carr & Wu (2009). Given thisresult, the empirical evidence favors a large, negative and strongly time-varying variancerisk premium that is also negatively correlated with market returns. Interestingly, this vari-ance risk premium has been found to carry predictability for future equity returns and to bepriced in the cross-section of equity returns: stocks that exhibit a positive covariation withthe premium have lower expected returns than stocks that are negatively related to the pre-mium3. Further studies document that jumps in prices and/or volatilities are critical for thedynamics of the variance risk premium and suggest that the large premium is tied to time-varying compensation for tail risks, see Todorov (2010) and Bollerslev & Todorov (2010).A third group of studies seeks to draw inference regarding the stochastic properties of theunderlying market return volatility process directly from the high-frequency behavior of theVIX. For example, while a jump in the return volatility of the S&P 500 index may be hardto identify, even from high-frequency observations on the stock index itself, the expectedfuture return volatility would, under weak regularity conditions, jump at the same time.Consequently, such events should be more readily identifiable from the VIX index becauseit directly reflects the expected future volatility under an equivalent probability measure.This observation motivates Todorov & Tauchen (2010) to explore the relative importanceas well as the finer probabilistic structure of the jumps in high-frequency VIX series.

Given all these trends, the high-frequency characteristics of model-free volatility se-ries, as exemplified by the VIX index, will inevitably become important areas of study intheir own right over the coming years. For example, how often do we see large discretemoves, or jumps, in these volatility measures and what economic circumstances triggersuch reactions? In exploring such questions it is critical to account for the microstructurefeatures of the index induced by data discreteness, market structure and other potentialsources of high-frequency noise. At first blush, VIX may appear less susceptible to suchproblems than regular assets because the index is constructed from individual asset priceswithin a large portfolio, typically relying on quote mid-points from more than 100 dis-tinct equity-index options, so individual pricing errors will tend to cancel. However, aswe detail below, the computation of model-free volatility confronts unique challenges thatrender precise real-time measurement of the index difficult. A variety of related issues havepreviously been noted by Jiang & Tian (2007) who discuss how to alleviate some specificproblems. Our concerns, however, run deeper.

This article (i) characterizes an important deficiency in the CBOE VIX methodologythat induces artificial jumps in the volatility index, (ii) proposes the estimation of alterna-tive model-free indices to alleviate the mentioned deficiency, and finally (iii) studies howthe CBOE methodology to compute the VIX affects the relationship between the CBOEVIX changes and S&P500 returns. Following this agenda we first systematically explorethe high-frequency properties of the official VIX index and compares it with a number of

3Intuitively, large variance risk premiums arise during periods of market stress when the marginal valueof wealth is particularly high, so assets that covary positively with the premium serve as hedges and requirelower risk premiums. Hence, the evidence suggests that variance risk is priced both in the time series andcross-sectional dimensions.

2

alternative measures that we constructe using the same underlying CBOE SPX options onthe S&P 500 index. These measures are grouped in three sets of implied volatility indices.Initially, to establish a proper benchmark for assessing the potential measures, we seek toreplicate the official VIX data using the exact rules for calculation laid out by the CBOE andusing option quotes from their official data vendor. Overall, we find a reasonable coherencebetween our constructed index and the official VIX. However, there are occasional dramaticdeviations that have a pronounced impact on the properties of the high-frequency volatilityseries. Hence, we make incremental modifications to our index computation that alleviatessome of the observed shortcomings. This further improves the coherence between our con-structed index and the VIX but important discrepancies remain on certain days. Inspectingthe options quotes carefully, we detect a substantial, occasionally abrupt, time variationin the range of options exploited in the VIX computations. Such changes in the range ofcovered strike prices induce substantial shifts in the implied volatility measures, not onlyfor our VIX replicating indices, constructed from tick-by-tick option quotes, but also forthe officially released VIX series. Since the coverage of strike prices should be absolute,such shifts, caused by idiosyncratic variation in strike coverage, induce artificial breaks or“jumps” in the associated implied volatility indices. We conclude that the high-frequencyVIX data released in real-time from the CBOE are not suitable for analyzing critical as-pects of the volatility dynamics due to inconsistencies and noise in the time series inducedby various market microstructure features. Moreover, any independent computation of theideal model-free volatility series from the underlying option quotes must confront seriousobstacles in order to construct coherent index measures over time. Inevitably, the pro-nounced variability in the available strike coverage necessitates ad hoc corrections that willinfuse the series with non-trivial measurement errors.

Next, we propose the construction of alternative indices to alleviate the critical problemwith the VIX computations, namely the need for model-free implied volatility to be com-puted from options across the full strike range, literally from zero to infinity. In practice,only strike prices around and moderately in- or out-of-the-money are actively quoted andtraded, so the tails of the risk-neutral return distribution are generally not well covered4.This concern is partially neutralized by the fact that the prices of far out-of-the-money op-tions rapidly converge to zero and become negligible beyond a certain point. Hence, thecritical issue is to determine a suitable cut-off point for the computation. The CBOE stip-ulates that, moving from at-the-money and towards the tails, the inclusion of additionalquotes is terminated once we encounter two consecutive out-of-the-money options with azero bid quote, thus implicitly deeming the remaining mid quotes unreliable and/or neg-ligible. As noted above, the problem with this strategy is two-fold. First, the eliminatedoption quotes often constitute a non-trivial part of the overall volatility measure. Hence,there is a systematic downward bias in the official measure. Second, the range of strikeprices covered by this cut-off criterion displays pronounced time variation5. Consequently,the downward bias displays a degree of (intraday) persistence, interspersed with dramatic

4Moreover, the relative precision of option prices declines rapidly in the tails as the percentage bid-askspread inflates and liquidity dies off. Hence, the option quotes become increasingly noisy as we move furtherinto the tails.

5It will usually remain relatively stable for periods within a given trading day, only to then change abruptly.

3

breaks that induce jump-like behavior in the index. Instead, we explicitly acknowledgethat inherent data problems render it impractical, if not infeasible, to consistently computethe full model-free implied volatility. Hence, we focus on a slightly more limited strikerange that, measured by a specific option pricing metric, covers an invariant portion ofthe underlying risk-neutral density at all times. Consequently, we introduce an alternativecut-off criterion, determined endogenously from the available option prices, that allows themeasure to reflect the pricing of volatility across an economically equivalent fraction of thestrike range over time, thus ensuring that the volatility measure is internally coherent in thetime series dimension. Any such measure is termed a model-free Corridor Implied Volatil-ity Index (CX), and is closely related to the corresponding theoretical concept discussed byCarr & Madan (1998). Obviously, different corridor measures may be obtained dependingon the width and positioning of the strike range used in the computations. We vary the cor-ridors moderately to understand the benefits and drawbacks of each choice. Our empiricalresults indicate that all our CX measures dominate the VIX measures in terms of providingaccurate and coherent volatility indicators over time. In particular, the CX measures avoidmany “false” jumps and display stronger time series persistence than the VIX counterparts,reflecting the improved intertemporal consistency and the lower level of idiosyncratic noiseof the CX measures.

Third, our analysis goes on to document the implications of the distortions inducedby the current CBOE methodology for computing model-free option-implied volatility. Inparticular, we find that the frequency of jumps in the implied volatility series is inflatedand the high-frequency correlation between equity-index volatility and market returns isseverely downward biased. In contrast, carefully constructed high-frequency CX mea-sures produce much more robust measures of the time series properties of market volatility.Consequently, the CX measures facilitate model-free identification of important underly-ing features of the volatility series that will be useful in discriminating between alternativemodels and explanations for observed high-frequency asset price dynamics. We also notethat the development of robust CX measures is not limited to monthly equity volatility in-dices. They may be readily computed across alternative asset classes and maturities as longas the requisite high-frequency options data are available. As such, these measures shouldhave broad applicability in future research. This result is of high relevance to improve riskmeasures based on VIX, and the use of VIX in risk managing exercises.

Finally, we explore whether the problems plaguing the intraday VIX measures carryover to the end-of-day closing values. Although ultra-high-frequency VIX is becomingmore studied in the academia, and more used in trading strategies, the end-of-day VIXcontinues being the frequency most used by the academia and media to study the VIXdynamic. This article also reports that the end-of-day values are linked directly to the cor-responding distortions in the intraday measure towards the end of the trading day. Hence,the daily VIX values are also deficient. Moreover, this distortion is found to have a signifi-cant impact on the time series properties of the daily VIX measure, raising the concern thatthe idiosyncratic errors in the index can be detrimental to empirical studies exploiting theseries. Such issues are again greatly alleviated by our CX measures.

4

2 The Methodology behind the VIX Computation

2.1 Brief Sketch of the Historical BackgroundOver-the-counter trading of volatility contracts emerged in the 1990s with liquidity build-ing primarily in the so-called variance swaps. Despite the label, these swaps are forwardcontracts involving no initial or intermediary cash flows. At expiry, the long party paysa fixed premium, determined at contract initiation, and in exchange receives a paymentreflecting the (random) realized variance of the underlying index. Typically, the realizedvariance would be measured by the cumulative sum of daily squared index returns from ini-tiation to expiry. A critical backdrop for these developments was breakthroughs in theorydemonstrating the feasibility of replicating the realized return variation of an asset over agiven future horizon. This path-dependent payoff will, under general conditions, equal thepayoff from a strategy involving only a static options portfolio combined with delta hedg-ing in the underlying asset.6 These results provided practical pricing and hedging toolsnecessary for sustaining market liquidity and depth. In particular, they imply that the “fair”value of a claim promising to pay the future index return variation is given by market priceof the replicating options portfolio. As such, the premium on a variance swap directly val-ues the future return variation. Moreover, since this reasoning is model-independent, thecorresponding notion is referred to as the model-free implied variance, and its square-rootas the model-free implied volatility, or simply MFIV.

In order to facilitate the development of an exchange traded market for volatility deriva-tives, the CBOE, on September 22, 2003, aligned their methodology for computing thevolatility index, VIX, with that of the trading community. Hence, while the “old” VIX,now denoted VXO, was based on “at-the-forward” Black-Scholes implied volatility, the“new” VIX is designed to closely approximate the MFIV. In this manner, market partici-pants may rely on existing procedures for pricing and hedging of volatility contracts whendealing with exchange traded VIX products.

2.2 From Theory to ImplementationIn theory, the computation of the model-free implied variance for a given maturity

requires availability of market prices for a continuum of European-style options with strikeprices spanning the support of the possible future index prices, from zero to infinity, for thegiven maturity. A convenient representation of the valuation formula takes the form,

σ2T =

2erT T

T

[ ∫ F

0

P(T,K)K2 dK +

∫∞

F

C(T,K)K2 dK

]=

2erT T

T

[ ∫∞

0

Q(T,K)K2 dK

], (1)

6Initial work leading to these insights is represented by Neuberger (1994), Dupire (1993) and Dupire(1996), with the latter published in Dupire (2004). Building on these insights, Carr & Madan (1998) con-structed an explicit and robust replication strategy for the pay-off on the variance swap, while Britten-Jones& Neuberger (2000) provide additional results exploiting these concepts.

5

where rT is the (annualized) risk-free interest rate for the period [0,T ] as represented bythe corresponding U.S. Treasury bill rate, T is time-to-maturity measured in units of a year,F denotes the forward price for transaction at maturity T , P(T,K) and C(T,K) are themid-quotes for European put and call options with strike K and time to maturity T , andQ(T,K) = min{C(T,K),P(T,K)} denotes the out-of-the money option at strike K. Hence,for K < F, Q(t,K) denotes the price of an out-of-the-money put option while, for K > F,it refers to the price of an out-of-the-money call option. Correspondingly, we label anoption with strike K = F (and C(T,K) = P(T,K)) an “at-the-money” option.

Of course, in practice the number of available strikes is limited so the CBOE approxi-mates σ2

T by the following expression,

σ̂2T =

2erT T

T

n

∑i=1

∆Ki

K2i

Q(Ki)︸ ︷︷ ︸Discrete Approximation

− 1T

[FK f−1]2

︸ ︷︷ ︸Correction Term

(2)

where 0 < K1 < · · ·< K f ≤ F < K f +1 < · · ·< Kn refer to the strikes included in the com-putation, K f denotes the first strike price available below the forward rate, F , so f is apositive integer, 2 < f < n−1. Moreover, the increments in the strike ranges are calculatedas ∆K1 = K2−K1, ∆Kn = Kn−Kn−1 , and for 1 < i < n, ∆Ki = (Ki+1−Ki−1)/2, whilethe prices of the OTM options, Q(Ki), are given by the midpoint of the option bid and askquotes. Finally, the second term in (2) reflects a correction for the discrepancy between K fand the forward price.7

Another practical issue is that only a few options maturity dates are available on anygiven day. The V IX is defined for a fixed calendar maturity of TM = 30

365 , or thirty days.Hence, the CBOE obtains the V IX measure via a linear combination of the MFIV forthe two expiration dates closest to thirty calendar days, but excluding options with lessthan seven calendar days to expiry. The interpolated quantity is annualized and quoted involatility units, using the following procedure,8

V IX = 100 ×

√[w1(

σ̂21 T1

)+ w2

(σ̂2

2 T2)]× 365

30, (3)

where w1 = T2−TMT2−T1

and w2 = TM−T1T2−T1

, so that, obviously, w1 + w2 = 1.

There are different sources of measurement error in VIX. Jiang & Tian (2005) classifythe errors in σ̂2 as arising from: (i) truncation errors, due to the minimum and maximumstrike prices being far removed from zero and infinity in (2); (ii) discretization errors, dueto the lack of numerical integration in (2); (iii) “expansion” errors, due to the Taylor seriesapproximation of the log function used by the CBOE in defining the correction term in (2);and (iv) interpolation errors, caused by the interpolation of maturities in (3).

7In the ideal MFIV formula (1), K f = F , so the adjustment seeks to correct for the bias in the volatilitymeasure arising from K f not being equal to the exact at-the-money strike.

8Detailed information is available at the URL: http://www.cboe.com/micro/vix/vixwhite.pdf

6

In this article we focus on an additional critical source of measurement error in the VIX,namely the CBOE methodology to determine the minimum and maximum strikes in thecomputation of σ̂2. This induces a time-varying effective strike range that can result inspurious breaks in the VIX time series, entirely unrelated with contemporaneous develop-ments in the underlying return series (artificial jumps). We argue that this problem can bealleviated by implementing the Corridor Implied Volatility index (CX) described in somedetail in Andersen & Bondarenko (2007). The corridor methodology directly controls therange (K1,Kn) in a time-consistent manner and thus prescribes which strike prices to includein the model-free volatility computation at any given time.

3 DataWe compute model-free volatility indices from S&P 500 option quotes obtained from theofficial vendor of CBOE data, Market Data Express, or MDX. The data set is based onMDR (Market Data Retrieval) data captured by CBOE’s internal system. The underlyingtick-by-tick option quotes cover the period June 2, 2008 - June 30, 2010. We constructfifteen second series for each individual option by retaining the last available quotes priorto the end of each 15 second interval throughout the active trading day on the CBOE from8:30:15 AM to 3:15:00 PM CT. If no new quote arrives in a given 15 second interval, the lastavailable quote is retained. Hence, in the case of inactivity, the quotes may remain constantfor some time although we impose a strict limit on the allowable duration of inactivity,as described below. In addition, as a benchmark, we will also on occasion exploit thecorresponding intraday quotes for S&P 500 futures on the Chicago Mercantile ExchangeGroup (CME).

A few filters are applied to guard against data errors: (i) to avoid excessive staleness,we deem an option quote missing if the discrepancy between the quote time-stamp in theMDR-SPX data base and the time it appears in our data set exceeds five minutes; (ii) ifan entire block of adjacent OTM option prices have not been updated for five minutes, wedeem the index itself “not available” (n.a.) and eliminate the corresponding time periodfrom our analysis; (iii) exploiting internal CBOE regulation, we impose a maximum bid-ask spread that every option quote pair must conform to in order to be deemed valid; (iv)we impose a maximum threshold for the degree of no-arbitrage violations implied by thereported option mid-quotes and declare the volatility index n.a. over the correspondingperiod whenever the threshold is exceeded. We lose close to 5% of the 15 second observa-tions due to this filtering procedure, with the vast majority of the missing values resultingfrom violations of the “lack of staleness” requirement (ii). This condition is essential foravoiding artificial jumps in the volatility index when an entire set of stale option quotes areupdated simultaneously after some temporary malfunction of the dissemination system. Amore detailed account of our filtering procedure is provided in the appendix.

The fifteen second series for the official VIX, covering June 2008 - June 2010 is ob-tained from different sources although they all ultimately stem from the CBOE. The firstdata set, from June 2008 to September 2008, includes the official VIX every fifteen secondsprovided directly by the exchange. The second data set is labeled MDR-VIX and contains

7

tick-by-tick quotes of VIX options, covering October 2008 to April 2009. This source alsoprovides the latest available observation on the underlying VIX measure at the time the op-tion quote is recorded, thus ensuring a high-frequency recording of the VIX values. Finally,the May 2009 - June 2010 period is again covered by the VIX series released directly bythe CBOE, but this time secured from the official data vendor.

We apply a simple filter to the intraday VIX series: we remove a few observationsthat fall outside, and thus are inconsistent with, the high-low range for VIX over the dayreported by the CBOE. We denote this lightly filtered VIX series VIX∗. Figure 1 showsthe evolution of the daily VIX closing value over recent years, with the thick green linedenoting the period analyzed in this article.

4 Practical Issues with the Computation of the VIX

4.1 VIX Replication Indices, RX1 and RX2We first seek to replicate the official VIX index from the underlying SPX options datausing the exact CBOE methodology. The first step involves the computation of σ̂2 foreach relevant maturity. However, the forward price, F , and the strike price just below theforward price, K f , in equation (2) are not directly observable. The CBOE determines thesebasic variables from the available option quotes in three separate steps. First, at a givenpoint in time, and for each relevant maturity separately, they identify the strike price K∗

for which the distance between the quoted midpoints of the call and put prices is minimal.This strike is then used to compute the “implied” forward price according to put-call parity,F0 = K∗+ erT T × [C(K∗,T )−P(K∗,T )]. Finally, K f0 is determined as the first availablestrike price below F0. We refer to the VIX index constructed from (2) and (3), using thesevalues for F and K f , as our first “Replication indeX”, or RX1. This index constitutes ourdirect equivalent of the CBOE VIX, computed according to official rules, from the SPXoption quotes in the MDR data set.

Our subsequent analysis reveals that this feature of the CBOE methodology entails acertain lack of robustness which, in some instances, has dramatic consequences. It arisesfrom the use of a single (K∗) put-call option price pair from which to infer F0 and K f0 .Occasional problems, like a temporary gap in the updating for a subset of the option pricesor outright recording errors, can cause the minimum distance between the call and putprices to, erroneously, appear at a point far away from the true at-the-forward strike. In turn,this can generate a large bias in the VIX and may even render its computation infeasible.9

This problem is exemplified in Figure 2 which depicts the prices of SPX options on June 9,2008, at around 9:00, for the maturity date July 19, 2008. The minimization of the absoluteoption price differential leads to an implied forward price of F0 = 1390, while the shape ofthe put and call price schedules suggests a forward price substantially below that value asthese curves should intersect at a strike close to the implied forward.

9Since we obtain an overall good coherence with the official VIX but also, at times, observe very signif-icant discrepancies, it is apparent that our option data set is not always identical to the one used in real-timeby the CBOE. We provide further comments on this issue later on.

8

Figure 1: Daily Closing Value for VIX, January 2004 - August 2010. Our sample period,June 2, 2008 - June 30, 2010, is indicated in green. Source: CBOE

Official Daily CBOE closing VIX

2004 2005 2006 2007 2008 2009 2010

10

20

30

40

50

60

70

80

90

100

9

Figure 2: The CBOE and our Robust Implied Forward Prices: The minimum distancebetween the mid-quote call (C) and put (P) on 09-June-2008 is marked with a red star inthe top-right panel and the associated K∗ is indicated on the horizontal axis. This valueis used for computing F0. In contrast, the robust forward price is obtained as the medianof all implied forward prices extracted from strike prices for which |C−P| is less than$25. These price differentials are marked by the red stars in the bottom-right panel. Thecorresponding put and call price curves are depicted in the left side panels. On this day:F0 = 1390 and F = 1367.5

1300 1350 1400 14500

20

40

60

80

100

120

Strike Price

Pric

e

F0

CP

1300 1350 1400 14500

20

40

60

80

100

120

Strike Price

|Call - Put|

1300 1350 1400 14500

20

40

60

80

100

120

Strike Price

Pric

e

F

1300 1350 1400 14500

20

40

60

80

100

120

Strike Price

|Call - Put|

10

Motivated by this finding, we introduce a slight modification that ensures a more robustprocedure for computing the implied forward. In a first step, we determine the full rangeof strike prices for which the absolute put-call price discrepancy is below $25. Secondly,for each of these put-call option price pairs, we compute the implied forward price. Third,we designate the “robust implied forward” price to be the median of this range of impliedforward prices. Using a broad set of option prices helps prune the forward price series ofmajor outliers. On the other hand, this wider set includes some less liquid out-of-the-moneyoptions for which the relative pricing errors are larger. Hence, the robust implied forwardmay, in most cases, be more affected by noise than the one determined from a truly near-the-money strike. Consequently, as our final step, we retain the original F0 value rather than therobust forward, except when the two approaches produce substantially different answers.Following some diagnostic checks, we settled on a threshold value of 0.5%.10 Hence, ifthe two implied forward prices are in reasonable agreement, we stick with F0. However, ifthey differ by more than 0.5%, we instead adopt the robust implied forward price. Thus,this procedure generates a new implied forward price, F , which typically coincides with F0,but deviates whenever the robust value indicates that F0 is subject to a sizeable error. Thenew VIX replication index, computed from F , is denoted RX2. Importantly, it retains thefeature that, apart from the risk-free interest rate, it is computed exclusively from CBOEoption prices, as is the case for the official VIX.

Figure 2 illustrates a real example in our sample where the difference between F0 andF is significant. The bottom row in the figure shows that the robust procedure generates animplied forward price of F = 1367.5 at 9:00, June 9, 2008, implying a substantial reductionrelative to F0 and driving a sizeable gap between RX2 and RX1 at this point. This isreadily apparent from Figure 3 which displays the RX1, RX2 and VIX∗ series from 8:30to just after 9:00. The jump in RX1 at 9:00 represents an isolated outlier. Both before andjust after 9:00, RX1 coincides with RX2 and they qualitatively match the evolution of thereported VIX. The fact that VIX∗ does not spike at 9am provides further corroboration ofour conjecture that this jump in RX1 reflects a faulty implied forward value induced by anerrant option quote in our data base.

Another noteworthy (market-microstructure) feature of the upper panel of Figure 3 isthe obvious delay in the VIX∗ series relative to our RX indices. We find that the RX serieslead the real-time reporting of VIX by about 15 to 30 seconds. This is likely due to capacityconstraints for the CBOE server which processes quote and trade data for a large set ofderivatives contracts simultaneously.11. Since this reporting delay fluctuates over time inways that, given the nature of the current dissemination system, cannot be observed directlyor even reconstructed ex post, it is literally impossible to match these VIX values perfectly.If anything, this renders it even more critical to understand whether we can reproduce allmain qualitative features of the high-frequency VIX series, or whether random delays andother potential sources of noise in the VIX index may distort our inference regarding theproperties of the high-frequency implied volatility equity-index series.

10For a level of the S&P 500 around 1000, the threshold is about 5 which is the same order of magnitudeas the gap between the strike prices.

11This explanation arose from conversations with John Hiatt of the CBOE

11

Figure 3: CBOE Forward vs Robust Forward in the replication of VIX. The first paneldepicts the RX1, RX2 and VIX∗ indices on June 9, 2008. The second and third panels plotthe implied forward price over the trading day for the first and second maturity, accordingto the CBOE methodology (F0) and the more robust procedure (F), respectively.

Prc

te

Volatility Indexes: RX1 vs RX2. Day: 09-Jun-2008

8:30 9:00

22.9

23

23.1

23.2

23.3

23.4

23.5

23.6

RX2 RX1 VIX*

F0 (CBOE Rule): Two maturities used for RX

8:30 9:001360

1380

1400

F: Two maturities used for RX

8:30 9:001360

1365

1370

12

Back to June 9, 2008, note that the issue of an erroneous implied forward value appearsto impact only the single observation discussed above. But significant differences in thetwo forward prices can also hold during a time interval. This is reported on June 3, 2008.Figure 5 reveals a major distortion on this day in the implied forward price, F0, for thelonger of the two option maturities over an extended period between 10:30 and 12:30. Thisrenders it infeasible to compute the model-free implied volatility and, by implication, RX1cannot be derived. In contrast, the F values are well behaved and RX2 is readily compiledfor the entire day. Since official VIX values are also reported throughout this entire periodit is, again, evident that the CBOE exploits a real time data set that is different from ours.

As these examples illustrate, the RX2 index succeeds in removing some artificial jumpsand eliminating some gaps from the RX1 index series. However, it is also readily apparentfrom Figures 4 and 5 that major discrepancies between the official VIX and our RX2 seriesremain. Clearly, the implied forward computation is not the only, nor the most important,issue in reconciling the series.

4.2 A Broader VIX Replication Index, RX3The preceding sections describe our practical procedures ensuring that the option and im-plied forward prices employed in the computations are reliable. The only remaining inputsinto the VIX recipe in equations (2) and (3) are the lowest and highest strike prices to in-clude in the calculations. In principle, all (available) options should be exploited. However,far out-of-the money options are near worthless and are traded infrequently, if at all. Asa result, the relative bid-ask spread rises dramatically and even the midpoint quotes maybecome poor indicators of the underlying value. In fact, CBOE discards OTM options ac-cording to a specific rule: as we move from the forward price and into the out-of-the-moneyregion and first encounters two consecutive strikes with zero bid option prices, we neglectall further out-of-the-money options, irrespective of whether they have a positive bid priceor not. Strikes with bid price equal to zero are not considered to compute the VIX, so theextreme strike is the last one with positive bid before the two consecutive strikes with zerobid. Hence, market liquidity and pricing jointly determines the cut-off levels. This rule hasintuitively appealing features. For example, if volatility rises substantially and quoted op-tion (bid) prices increase correspondingly, then additional, and newly valuable, options willbe included in the VIX calculation. As a result, the computation generally should includeall options with non-trivial (bid) valuations while avoiding the potential noise and distor-tions associated with the quotes on far OTM options that supposedly have only a minorimpact on the overall measure.

If we ignore this CBOE stopping rule but continue to exclude all option quotes involv-ing a zero bid price, we obtain a broader strike range which may be interpreted as themaximum coverage offered by the prevailing liquidity in the options market. This proce-dure allows us to construct a broader volatility benchmark index, exploiting the mid-quotesof all OTM options with positive bid prices. We denote this index RX3. As for RX2, weexploit the robust forward price, F, in the calculation of RX3. This alternative index shouldprovide a natural upper bound on the VIX values reported by the CBOE. Moreover, when-ever there are no positive bid prices for OTM options beyond the CBOE truncation point,

13

Figure 4: CBOE Forward vs Robust Forward in the replication of VIX. The first paneldepicts the RX1, RX2 and VIX∗ indices on June 9, 2008. The second and third panels plotthe implied forward price over the trading day for the first and second maturity, accordingto the CBOE methodology (F0) and the more robust procedure (F), respectively.

Prc

te

Volatility Indexes: RX1 vs RX2. Day: 09-Jun-2008

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:0019

20

21

22

23

24

25

RX2 RX1 VIX*

F0 (CBOE Rule): Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00

1360

1380

1400

F: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:001350

1360

1370

1380

14

Figure 5: CBOE Forward vs Robust Forward in the replication of VIX. The first paneldepicts the RX1, RX2 and VIX∗ indices on June 3, 2008. The second and third panels plotthe implied forward price over the trading day for the first and second maturity, accordingto the CBOE methodology (F1) and the more robust procedure (F2), respectively.

Prc

te

Volatility Indexes: RX1 vs RX2. Day: 03-Jun-2008

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:0018.5

19

19.5

20

20.5

21

RX2 RX1 VIX*

F0 (CBOE Rule): Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00800

1000

1200

1400

F: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:001370

1380

1390

1400

15

the two measures will coincide. Finally, since far OTM option prices converge to zerorapidly, RX3 should constitute a sensible benchmark for the official V IX∗ in the majorityof cases.

In this article we exploit the concept of an effective range, or ER, to gauge the coverageof options embedded within the VIX computation. It is defined formally in equation (4). Itexploits an at-the-money implied Black-Scholes volatility measure, σ̂BS, which is obtainedfrom a linear interpolation of four near-the-money Black-Scholes implied volatilities, ex-tracted from the options with strike prices just above and below the forward price for eachof the two maturities closest to 30 calendar days. Strikes K1 and Kn represent the minimumand maximum strikes included in the computation of the model-free volatility index, and Tis the time to maturity. The VIX is computed using two different maturities, so we get aneffective strike range for each of the two maturities.

ER =[

ln(K1/F )σ̂BS√

T,

ln(Kn/F )σ̂BS√

T

]. (4)

Assuming the Black-Scholes measure, σ̂BS, captures the overall variation in volatilityacross time appropriately, this standardized strike range indicates the degree of strike cov-erage provided by the quoted equity index options at any given moment. As such, it offersa tool for assessing the effectiveness and coherence of the CBOE rule that terminates thestrike coverage upon encountering two consecutive zero bid prices.

Figure 6 again plots the official VIX and RX2 series for June 9, 2008, but we nowalso include the RX3 index on the graph, and the corresponding effective range of optionsavailable for the volatility index calculation in the panel below. This figure intends tohighlight the relationship between changes in the relevant strike coverage and the associatedRX2 values for this trading day. The figure shows that between 9:00 and 10:00 the effectivestrike range for RX2 displays abrupt shifts, leading to a dramatic drop in the index values.Next, just prior to 10:00, the range widens again and is compatible with the level that wasin effect during the early part of the trading day from 8:30-9:00, and the RX2 measurejumps back up to a higher plateau where it appears to remain for the rest of the tradingday. At the same time, the thick drawn lines in the bottom panel indicate that additionalnon-zero bid price OTM options were available, but were explicitly excluded from theRX2 computation. Hence, the large shifts in the effective range for RX2 stem from ourapplication of the CBOE stopping rule, governed by the occurrence of two consecutivestrikes with zero bid option prices, to the options available in our data set. Furthermore,it is apparent that the official computation of the VIX measure must be exploiting someof these additional option quotes, thus confirming that there are discrepancies between ourdata set and the one used by the CBOE in real time.

In summary, at least for this trading day, the RX3 measure is successful in alleviatingsome of the abrupt drops observed for RX2 due to sudden occurrences or disappearancesof two consecutive zero bid prices for some moderately out-of-the-money options and itproduces a vastly superior fit to the official VIX, even if there are some notable deviationsbetween RX3 and V IX∗ in the early parts of the day. Hence, in this case it seems preferable

16

Figure 6: Effective Strike Range, RX2 and RX3. The top panel depicts the RX2, RX3 andVIX∗ volatility indices. The bottom panel shows the corresponding effective strike rangeavailable from the quoted options at a given point in time for each of the two maturities.The thin lines correspond to the range covered by the options included in the computationof RX2, while the thicker lines indicate the strike range available from all valid optionquotes for the given maturity. The strike ranges for the nearby maturities are indicated bydashed lines while the longer maturity ranges are displayed in continuous lines.

Volatility Indexes: RX2 and RX3 vs VIX*. Day: 09-Jun-2008

Prc

te

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00

19

20

21

22

23

24

25

RX3 RX2 VIX*

Effective Strike Range: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00

-6

-4

-2

0

2

4

17

to utilize all available strikes in the construction of model-free volatility. Of course, thisspecific illustration does not prove the case for RX3 in general. We now turn to a moregeneral assessment of the coherence between our RX replication series and the official(filtered) V IX∗.

4.3 Replicating the Official VIX∗ Series by the RX IndicesWe have studied the behavior of two separate model-free replication volatility series, RX2and RX3, in some detail. It is evident that both do, on occasion, display striking deviationsfrom VIX∗, but are these deviations significant or frequent? Are we very far to replicate theofficial V IX∗? In this subsection we intend to answer these questions. Figure 7 depicts theaverage Mean Absolute Percentage Error and Median Absolute Percentage Error for eachmonth (bars) and for the full sample (horizontal lines), and for each RX index. The lossfunctions are computed comparing the indexes at fifteen seconds frequency. The figureshows that we nonetheless match the real-time data for the VIX released by the CBOEreasonably well on average. The level of both RX indices is clearly highly correlated withVIX, as the average discrepancies are less than 0.5%, while the median deviation for RX2is less than 0.2%, and even generally less than 0.1% over the last ten months of the sample.

In order to gain some insight into the sources of discrepancy between the VIX∗ and RXmeasures, we first determine the proportion of time the indices coincide versus differ, usinga 0.5% relative deviation as the criterion for satisfactory replication. Within the tradingday, the VIX∗ is matched by the RX3 about 83% of the time, while the correspondingnumber for the end-of-day indices is 81%. Hence, more than 80% of the time, the VIX∗

values are compatible with having been computed on the basis of all the quoted optionsacross both maturities. The scenario where VIX∗ is below RX3, but matches RX2 accountsfor, respectively, 9% and 12% of the valid observations at the intraday and end-of-daylevel. Thus, about an additional 10% of the time the VIX∗ is evidently not based on allavailable option quotes, but the reported values are consistent with following the CBOE“stopping rule.” This leaves roughly 7%-8% of the observations where we are not able torationalize the reported VIX values. We have 1620 high-frequency VIX observations perday (from 8:30:15 to 15:15:00) which amounts to an average of around 1335 times whenall options seemingly are utilized, but leaves roughly 160 instances where the VIX∗ is closeto RX2 but below RX3, and about 125 cases where we cannot readily classify the sourceof deviation between the VIX and RX indices.12 As a robustness check, we also triedusing a stricter definition for acceptable replication of 0.25%. This produces qualitativelysimilar results, as we now rationalize the observed VIX∗ value with RX2 and/or RX3 for

12As discussed in connection with Figure 3, there is a time-varying delay, averaging somewhere between15 and 30 seconds, in the official intraday VIX series, likely due to various real-time computational andreporting issues. Since our RX series is constructed using the time stamps for the options quotes in theMDR data, they do not suffer any such delay. Hence, a change in the volatility level will induce an artificialdeviation between the VIX and our RX series. Consequently, we also explored the match between the VIX∗

and the 15 or 30 second lagged RX series. While the matched observations increase for both lagged series, theeffect is truly marginal. Thus, given the likely presence of reporting delays for the official VIX, we concludethat our count of unexplained deviations may be mildly inflated relative to the “true” discrepancies.

18

Figure 7: Approximation Error. Mean and Median Absolute Percentage Error (%) forthe RX2 and RX3 indices. MAPE for each month and for the whole sample (straightlines). The MAPE values are computed as indicated below, while the corresponding medianabsolute percentage error uses all valid observations within each month to form the medianpercentage statistic.

MAPE = 100× 1T

T

∑t=1

∣∣∣∣V IX∗t −RXt

V IX∗t

∣∣∣∣, RX = RX2,RX3

Prc

tge

Mean Absolute Percentage Error (June 2008 - June 2010)

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

0.2

0.4

0.6

0.8

1RX2RX3

Prc

tge

Median Absolute Percentage Error (June 2008 - June 2010)

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

0.2

0.4

0.6

0.8

1RX2RX3

19

the intraday measure about 75.5% of the time and for the end-of-day measure about 76%of the time. However, the cases where RX3 coincides with VIX∗ declines significantly andnow constitutes only roughly half of the total number of replications.

The observed pattern suggests that the VIX series oscillates frequently between the twobenchmark cases of RX2 and RX3. In fact, Figures 8 and 9 highlight this phenomenon.First, in Figure 8 the VIX is clearly below RX3, but fairly closely approximated by RX2from 8:30-11:00, and then for the remainder of the trading day well described by both RX2and RX3. Hence, in spite of some visible discrepancies between VIX and RX2 over theearlier parts of the day, we still obtain a sense of how the VIX values were obtained. At theend of the day, all indices (RX2, RX3, and V IX∗) coincide and it is thus quite transparenthow the closing value is set. In the second example, Figure 9, the picture is qualitativelysimilar although the pattern is reversed, with VIX matched by both indices during most ofthe early trading hours, but then dropping abruptly along with RX2 below the level of RX3.Thus, at the end of trading VIX is significantly below RX3 due to the CBOE stopping ruledetermining the effective strike range. For these two trading days, it is quite evident thatthe “jumps” in the VIX level up to or down and away from RX3 are driven solely by shiftsin the effective range and not by any major movement in the overall volatility level, asindicated by the lack of any perceptible change in RX3 which, during this period, is basedon a largely invariant strike range.

In summary, we are able to replicate the VIX index well in most instances. However,we identify some economically troublesome features in the process. The VIX displayserratic movements corresponding to abrupt shifts in the range of option strikes utilized inthe computation. In many cases “jumps” do not appear related to any discernible changein market volatility. In contrast, the RX3 measure seems less susceptible to such errantmovements and may constitute a more reliable basis for gauging volatility. In other words,neither RX2 nor VIX∗ look compelling as candidate high-frequency model-free volatilitymeasures, while the RX3 index may be a serious candidate. Of course, the fact that the RX3index uses all available strikes also implies that the measure may be excessively sensitive tothe relatively large bid-ask spread of the far out-of-the-money options and that the availableeffective strike range for RX3 may vary substantially over time as option market liquidityin the tails of the distribution fluctuates. One way to gauge the severity of this problem is tocontrast the properties of RX3 with an alternative index which exploits an “economicallyinvariant” strike range in the computation of a MFIV style index.

5 Corridor Implied Volatility, or CX, Indices

5.1 Defining the CX IndicesIn order to establish a benchmark from which to assess how much variation in the effectivestrike range impacts practical model-free volatility measurement, we turn to the concept ofcorridor implied volatility indices, or CX, introduced by Carr & Madan (1998) and studiedempirically in Andersen & Bondarenko (2007). The point is to, a priori, fix the range ofstrikes used for the volatility index computation at a level that provides broad coverage but

20

Figure 8: RX2, RX3 and VIX∗. The top panel depicts the RX2, RX3 and VIX∗ volatilityindices for February 16, 2010. The middle panel shows the corresponding effective strikerange available from the quoted options at a given point in time for each of the two matu-rities. Finally, the bottom panel indicates the evolution of implied forward rate across thetrading day.

Volatility Indexes. Day: 16-Feb-2010

Prc

tge

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00

22

22.5

23

23.5

24

24.5

RX3 RX2 VIX*

Effective Strike Range: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00-10

-5

0

5

F

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:001080

1085

1090

1095

21

Figure 9: RX2, RX3 and VIX∗. The top panel depicts the RX2, RX3 and VIX∗ volatilityindices for March 2, 2010. The middle panel shows the corresponding effective strike rangeavailable from the quoted options at a given point in time for each of the two maturities.Finally, the bottom panel indicates the evolution of implied forward rate across the tradingday.

Volatility Indexes. Day: 02-Mar-2010

Prc

tge

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:0016.5

17

17.5

18

18.5

19

19.5

20RX3 RX2 CX2 VIX*

Effective Strike Range: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00-10

-5

0

5

F

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:001116

1118

1120

1122

22

avoids reliance on excessive extrapolation of noisy or non-existing option quotes for farout-of-the-money options. Clearly, by restricting the range at a more narrow level than isavailable on the typical trading day, the Corridor Index, or CX, measures will be downwardbiased relative to the RX3 measure that exploits all available options. On the other hand,the adoption of an invariant coverage across time ensures that the measure is coherent inthe time series dimension and not subject to random variation due to exogenous shifts inthe strike range. Hence, the relative stability of the two indices should speak to the impactof extraneous factors in the computation of the model-free volatility index.

The main question is how to define an economically invariant portion of the strike rangewhich ensures that the associated implied volatility measures are compatible over time. Weemphasize a few desirable features. First, the forward price constitutes a natural focal pointas it defines the ATM position and thus determines the set of OTM call and put optionsthat enter the implied volatility computation. Moreover, the near ATM options are themost valuable and contribute most to the volatility measure. It is therefore desirable todefine the corridor through a metric that naturally centers on the forward rate and capturesthe degree of moneyness of the OTM put and call options relative to the forward rate.Second, the strike range should expand as volatility, and option prices, increase as thissimply reflects a more dispersed risk-neutral distribution. Third, the corridor should beas closely linked to observed option prices as possible in order to ensure transparencyregarding the construction of the measure. The latter is critical if the measure is to achieveacceptance among market participants as a viable alternative to the existing VIX measure.

One common approach to meet approximate some of the desirata stressed above isto measure moneyness relative to the forward price via the Black-Scholes ATM impliedvolatility. However, this is unsatisfactory in relying on a model-dependent, and clearlymisspecified, assumption of constant volatility and log-normality. In particular, the Black-Scholes volatility is only sensitive to the ATM option price and thus fails to adapt to varia-tion in the shape of the risk-neutral distribution (RND) caused by, e.g., shifts in the skew-ness or the fatness of the tails. A different and popular approach is to rely directly on thepercentiles of the RND. Unfortunately, in the current context, percentile corridors have thesignificant drawback that they are not centered on the forward price. Due to the pronouncedskew in the risk neutral equity-index distribution, the forward price is located anywhere be-tween the 38’th and 47’th percentiles of the distribution with a mean around the 43’thpercentile over our sample period. This reflects the fact that the percentiles do not readilytranslate into a stable relative price for the OTM options associated with the correspond-ing strikes. Since the option prices are critical determinants of the volatility measures, theinstability of the percentiles relative to the ATM strike is a serious concern. In addition, es-timation of the RND requires certain auxiliary assumptions and computational conventionsthat render the approach less than fully transparent to market participants.13

In response to the shortcomings identified above, we develop an alternative approachwhich possesses a number of distinct advantages. It builds on the notion of tail moments.Specifically, we define the left and right tail moments for any distribution of a strictly

13The convention of defining corridors via the percentiles of the RND was adopted by Andersen & Bon-darenko (2007), so the alternative approach developed below is novel.

23

positive random variable, x, with strictly positive density f (x) for all x > 0 by,

LT (K) =∫ K

0(K− x) f (x)dx and RT (K) =

∫∞

K(x−K ) f (x)dx . (5)

We then introduce the ratio statistic, R(K), as an indicator of how far in the tail a givenpoint, K, is located within the support of x,

R(K) =LT (K)

LT (K)+RT (K). (6)

First, we note that the R(K) attains the properties of a cumulative density function, orCDF , as it is strictly increasing on (0, ∞) with R(0) = 0 and R(∞) = 1. Moreover, it iscentered at the mean value of x as R(K) = 1

2 , where K = E[x] =∫

∞

0 x f (x) dx . Second, fora given point, or percentile, q, in the range of the R(K) function, we define the quotient ofthe ratio function, Kq, as

Kq = R−1(q) for any q ∈ [0,1] . (7)

Now, letting f (x) denote the risk-neutral density for the S&P 500 forward price for thematurity date T = 1

12 , or one month ahead, and K be the strike price of European styleout-of-the-money put or call options, we obtain the following expression,

R(K) =P(K)

P(K)+C(K). (8)

It follows that the ratio statistic is directly obtainable from the market prices of European-style put and call options. Hence, as long as we remain within the range where option pricesmay be reliably extracted from the existing quotes in the market, the value of R(K) is trivialto determine, and it is independent on any estimation procedure for the underlying RND.Moreover, the median, or 50’th percentile, of R(K) corresponds to the expected value ofthe underlying S&P 500 index at T , because the forward rate is given as the mean of therisk-neutral distribution for the index level,

K = F0 and K0.50 = R−1(0.50) = F0 . (9)

This implies that the forward price, F , that separates the strike range into the out-of-the-money put and call regions is identified by the median of the R(K) function. Consequently,the percentiles of the ratio function automatically centers the strike range appropriatelyrelative to the computation of model-free implied volatility. This is particularly convenientwhen considering a set of nested corridor implied volatility measures as we do below.

It is now natural to define the truncation strike points for the implied volatility compu-tation via symmetric percentiles of the R(K) function. For example, we may use the leftand right truncation strikes obtained as,

K0.01 = R−1(0.01) and K0.99 = R−1(0.99) . (10)

24

As illustrated in Figure 10, the determination of the option prices for a wide range ofstrikes is quite straightforward from existing option quotes. Only towards the tails of therisk-neutral distribution are the option prices, and thus the distribution itself, hard to as-certain with precision. This issue is circumvented by the R function which only dependson the observable option prices out until the truncation point. Hence, the truncation levelsare determined directly from observed prices and largely independent of any extrapolationtechnique or other exogenous approximation rules. Of course, the intention is also that thisprovides a stable basis for the computation of a comparable volatility measure over timeas the support of the strike range is kept invariant in a specific economic metric. Specifi-cally, we compute the CX1 and CX2 series by exploiting the [0.01,0.99] and [0.03,0.97]percentile quotients for the R function.

While it is impossible to provide a universal rule for the optimal variation in the rangeover time, it is instructive to compare the effective ranges implied by the various measuresin terms of the corresponding Black-Scholes implied volatilities. Figure 11 displays thecoverage of the strike range for the end-of-day volatlity measures in this fashion. It isstriking how the RX measures display dramatic variation in their strike coverage across thedays in our sample, while the CX measures provide a relatively stable coverage, expressedin terms of the Black-Scholes volatilities. Inspecting the RX coverage more closely, itis further evident that the RX2 series have outliers that largely match those of the RX3series, but the RX2 displays rather extreme inlier observations as well. Since a narrowingof the strike coverage typically has a more significant impact on the associated variancemeasure than widening coverage, simply because the options closer-to-the-money carry arelatively heavier price tag, the RX3 series may well be the more reliable one in termsof capturing the intertemporal fluctuations in market volatility. Nonetheless, the strikecoverage of the RX3 series remains extraordinarily variable, reaching a low on the out-of-the-money put side of about−3 and a high of around−13 at the nearby maturity and withinthe range [−21

2 ,−12] at the longer maturity. In comparison, the variation in the coverageprovided by the CX1 and CX2 series is small, with values fluctuating within the intervals[−11

2 ,−212 ] and [−2,−31

2 ], respectively. While this does not prove that the CX measureswill be superior in applications, it does raise serious questions regarding the impact of theabrupt shifts in coverage provided by the more traditional RX style of VIX measures.

5.2 Comparing the VIX, RX, and CX IndicesWe now explore whether the observed differences in the strike coverage between the VIX,RX and CX indices impact the statistical properties of the series. We focus on criticalfeatures of the stock index return volatility such as the persistence of the series, the preva-lence of abrupt moves, or jumps, and the extent of the (negative) correlation of changesin volatility with the underlying stock returns, often labeled the leverage effect. However,we first exemplify the impact that variation in the effective strike range can have on RX2,RX3 and VIX∗ relative to the CX measures. Since the qualitative behavior of the two CXmeasures near indistinguishable, we include only the more narrow corridor measure, CX2,in the illustration below.

25

Figure 10: Determination of the CX Truncation. The top left panel displays the normal-ized out-of-the-money option prices at the end of trading on June 16, 2010 for the thirtyday maturity. Moneyness is defined as k = K

F , while the normalized option prices are givenas M(k) = Q(K)

F and Q(K) = min(C(K),P(K)). The top right panel depicts the correspond-ing Black-Scholes implied volatilities. The bottom right panel provides an estimate of thecorresponding risk-neutral density, whereas the left bottom panel shows the extracted R(k)function. The vertical dashed lines indicate the 1,10,50,90, and 99 percentile quotients ofR(K).

0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

Moneyness k

Implied Volatility

0.8 0.9 1 1.10

2

4

6

8

Moneyness k

Risk Neutral Density

0.8 0.9 1 1.10

0.005

0.01

0.015

0.02

0.025

Moneyness k

Normalized Option Price M(k)

0.8 0.9 1 1.10

0.2

0.4

0.6

0.8

1

Moneyness k

R(k)

26

Figure 11: Effective Strike Range for End-of-Day Implied Volatility Measures. Thetop panel displays the effective strike range, expressed in Black-Scholes volatilities, for thevarious implied volatility measures at the end-of-the-day across the entire sample periodfor the first maturity exploited in the index calculation. The bottom panel provides thecorresponding ranges for the second maturity.

27

5.2.1 A Pair of Illustrative Trading Days

We present two Figures to illustrate the sensitivity of the model-free volatility indexesto changes in the effective strike range used in their computation. Figure 12 depicts theevolution of the indices during a highly volatile day in the equity markets, October 14,2008, where the intraday range of the VIX spans values below 50 and above 60. First,we notice the apparent problems experienced in the real-time computation of the officialVIX figures. At first, from 8:30-8:40, V IX∗ is frozen at an inexplicably high level of about55%, compared to the contemporaneously observed values for the RX2 and RX3 indices,only to then drop to a similarly incomprehensibly low set of values around 46% and 47%.After some additional wild fluctuations, the V IX∗ values start roughly mimicking the RX2index in the time period 9:40-12:55, and then after an apparent jump at 12:55 coincidingreasonably well with both RX2 and RX3 until about 14:30. Finally, from this point onwardsthe RX2 drops below RX3 while V IX∗ remains fairly close to the RX3 level until the marketclose. The reason for the “jumps” in RX2 at 12:55 and after 14:30 is readily identified fromthe second panel as the strike range for the nearby maturity widens in the first case and thennarrows again later, in exact concert with the observed shifts in the volatility index level.

Even disregarding the obvious problems during the early parts of the trading day, theVIX series appears “indecisive” as it first attains a level consistent with RX2 over 9:40-12:55 and then roughly follows the RX3 index for the remainder of the day. Such randomoscillation between two distinct volatility levels induces artificial breaks in the V IX∗ seriesthat are unrelated to the underlying equity index, and probably related to a new zero bidprice quoted. Moreover, the identical reasoning applies to the RX2 series itself, as it issubject to the same type of jump-like behavior due to unpredictable changes in its strikerange. Finally, the RX3 index indicates a strongly elevated volatility level over the pe-riod 10:05-10:25 which may also be attributed directly to an expansion of the associatedstrike range available for the nearby maturity over that time interval. This seemingly arti-ficial shift increases the volatility measure by close to 4%, while none of the other indicesdisplays any major discontinuities over this period. In short, the RX3 measure is also vul-nerable to idiosyncratic shifts in the underlying strike range. In contrast, the CX2 measureevolves continuously, albeit also somewhat erratically, throughout the trading day. More-over, the qualitative pattern nicely mirrors the movements in the RX indices apart fromthe absence of glaring jumps. In terms of capturing the evolution of volatility across thiseventful trading day, the CX index appears to reflect all relevant variation in the RX indiceswhile annihilating their more suspect breaks.

Figure 13 depicts a more typical trading day with a distinctly lower volatility level,namely February 9, 2010. In this case, the RX2 index replicates the official VIX series verywell throughout most of the trading day, although there are a number of striking outliers inboth the RX2 and VIX∗ series between 11:00 and 11:30, and a fairly abrupt drop in RX2around 3:00. In contrast, the RX3 index is more erratic between 10:50 and 11:45 and thenagain after 15:00. Not surprisingly, over these time spans we also observe a widening inthe strike range for RX3. As before, CX2 appears to capture the qualitative features ofthe volatility evolution satisfactorily while avoiding any abnormal jumps. The only sharpmovement in the CX2 series, just after 11:40, is associated with a contemporaneous drop

28

Figure 12: VIX∗, RX2, RX3 and CX. The top panel depicts the VIX∗, RX2, RX3 andCX volatility indices for October 14, 2008. The middle panel shows the correspondingeffective strike range available from the quoted options at a given point in time for eachof the two maturities. Finally, the bottom panel indicates the evolution of implied forwardrate across the trading day.

Volatility Indexes. Day: 14-Oct-2008

Prc

tge

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00

46

48

50

52

54

56

58

60

62

RX3 RX2 CX2 VIX*

Effective Strike Range: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00-6-4-202

F

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00980

1000102010401060

29

in the S&P 500 forward price, as indicated in the bottom panel, and thus likely representsa genuine change in volatility.

For both of these trading days, the evidence is consistent with the view that the CXmeasure captures an economically invariant portion of the risk-neutral density, so that thefairly smoothly shifting CX2 values may be interpreted as actual changes in volatility.Of course, these examples are merely indicative and do not speak to the robustness ofthe findings throughout the sample. Hence, we now turn towards a more comprehensiveinvestigation of the properties of the alternative volatility indices.

5.2.2 General Features of Alternative Model-Free Volatility Indices

We first explore the extent and relative size of outliers in the volatility indices, beforecomparing their general features, and finally the relation between the implied volatility in-dices and the underlying equity returns. Our motives for focusing on these features aretwofold. First, appropriate modeling of volatility dynamics is of major interest in appliedwork within asset pricing, portfolio choice, risk management and derivatives pricing. Oneimportant dimension of this question is the prevalence of large changes, or “jumps”, in thevolatility process and their interaction, or correlation, with shifts in the underlying equityindex, especially since long volatility positions serve as a hedge for broader exposures toequity if there is a robust negative correlation between the two. Second, the possibilitythat random fluctuations in the strike range, used by a given index, induce artificial breaksin the corresponding volatility measure suggests that stark discrepancies in outlier activityacross alternative indices may reflect differences in robustness to the quoting patterns andvariation of liquidity of the options market.

i. Volatility Index Jumps

We proceed nonparametrically by defining large moves relative to a robust measure ofreturn volatility for each trading day. In order to alleviate the distortions arising from noisein the 15-second series, we focus on observations at a slightly lower frequency. Specifically,we first compute the series of one-minute changes, or returns, for the volatility index. Next,we obtain an initial estimate of the index volatility by computing the daily standard devi-ation from the one-minute returns. Then, we delete all observations whose absolute valueexceeds six standard deviations and, finally, we recompute the daily index volatility usingonly the remaining observations. On the basis of this robust measure of the daily standarddeviation, we identify outliers for each volatility index at the one-minute frequency acrossthe full sample. Table 1 tabulates the findings.The table reveals a startling discrepancy in the number of large moves across the alterna-

tive indices. Irrespective of the threshold adopted for defining the large changes, or returns,the conclusions are identical. For example, including moves beyond six standard devia-tions on either the upside or downside, we find around 1600 outliers in the VIX indicesand close to 1500 for RX1 and RX2. In contrast, the numbers are approximately 950 forRX3, and below 625 and 560, respectively, for the CX1 and CX2 measures. Given that thesample covers 25 months, all indices display, on average, more than one large move per

30

Figure 13: VIX∗, RX2, RX3 and CX. The top panel depicts the VIX∗, RX2, RX3 and CXvolatility indices for February 9, 2010. The middle panel shows the corresponding effectivestrike range available from the quoted options at a given point in time for each of the twomaturities. Finally, the bottom panel indicates the evolution of implied forward rate acrossthe trading day.

Volatility Indexes. Day: 09-Feb-2010

Prc

tge

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00

22

23

24

25

26

27

RX3 RX2 CX2 VIX*

Effective Strike Range: Two maturities used for RX

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00-10

-5

0

5

F

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:001060

1070

1080

31

Table 1: Distribution of Extreme Returns (“Jumps”), 1-min Frequency

RX1 RX2 RX3 CX1 CX2 VIX∗ VIX

(−∞,−30) 46 47 7 1 0 24 31(−30,−15) 99 97 29 6 7 91 93(−15,−9) 193 195 91 45 34 189 191(−9,−6) 384 383 333 266 251 512 512(−6,−4) 1178 1179 1125 1017 987 1323 1322

(4,6) 1140 1141 1098 1000 949 1333 1334(6,9) 401 401 352 249 221 444 445

(9,15) 218 218 101 47 35 204 204(15,30) 106 105 33 7 9 102 102(30,∞) 47 48 7 2 0 21 28

Note: Range is measured in multiples of the (robust) standard deviation.

trading day while the VIX measures top the count with close to three per day. Even moretelling is the difference in the number of extremely large moves of more than 15 standarddeviations. Here, the CX measures have 16 such moves, while the RX3 index has almostsix times more and the other indices all sport more than 15 times as many. This is obvi-ously anomalous: any genuinely large shift in volatility should manifest itself in significantelevation of option prices across a broad range of strikes and should thus be reflected inall broadly based model-free volatility indices. This suggests that various sources of mea-surement error, including the documented idiosyncratic variation in the strike range, aresufficiently commonplace that they severely inflate the outlier count for some indices.

A second interesting feature is the near symmetry of positive and negative “jumps”within each size category. Part of the explanation may be that misclassified jumps oftenreverse themselves, as an unusual widening of the strike range, say, is followed by a returnto the usual range. However, even for the measures that are less prone to this type of er-ror, we observe a near symmetric distribution of positive and negative jumps. This suggeststhat high-frequency volatility undergoes frequent and abrupt moves in both the positive andnegative direction. The evidence of numerous large negative volatility jumps is inconsis-tent with many popular parametric specifications of asset price dynamics that allow onlyfor positive jumps in volatility.

ii. General Features of the Volatility Indices

The pronounced variation in jump intensity and size across indices is likely to leave itsmark on their general distributional characteristics. Table 2 provides an overview of thesummary statistics for the alternative volatility measures. The most noteworthy aspect ofTable 2 is the huge variation in the kurtosis statistic for the volatility changes, or returns, inPanel B. While the sample kurtosis for the CX series are sizeable, falling in the range of 24

32

Table 2: Summary Statistics, 1-Minute Frequency

Panel A: Volatility Index Levels

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

Mean 31.82 31.82 31.90 30.51 29.26 31.74 31.74Std Dev 13.38 13.38 13.43 12.89 12.41 13.43 13.43

Skewness 1.31 1.31 1.30 1.30 1.30 1.31 1.31Kurtosis 4.15 4.15 4.12 4.15 4.14 4.14 4.13

Panel B: Volatility Index Returns

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

Mean ×104 -0.08 -0.08 -0.10 -0.12 -0.13 -0.15 -0.15Std Dev ×104 39.78 40.18 20.16 20.88 21.96 25.33 31.62

Skewness 14.52 14.05 0.57 0.31 0.23 -0.46 -0.71Kurtosis 6886.11 6660.04 44.62 27.83 24.49 754.64 3783.17ρ1×102 1.80 1.79 4.88 4.90 4.04 3.78 3.45

ρ2,5×102 1.97 1.97 2.66 2.66 2.28 2.92 2.80ρ6,21×102 1.07 1.06 1.29 1.27 1.13 1.05 1.03

ρ21,45×102 0.08 0.08 0.09 0.13 0.08 -0.02 -0.02

Panel A: Percent of missing is 5.28% for RX1-CX2, 0.03% for V IX∗, and 0% for V IX . Panel B:Percent of missing is 5.86% for RX1-CX2, 0.03% for V IX∗, and 0% for V IX .

33

to 28, the value for RX3 is considerably larger at about 45, while they attain rather imposingvalues of about 755 and 3783for the VIX series, and, finally, truly outsized values of around6600 for the RX1 and RX2 indices. Notice also that mild filtering reduces the kurtosis ofVIX∗ sharply relative to VIX. This serves as further indirect evidence that erroneous dataare an important source of the inflated kurtosis statistics. Overall, the table is consistentwith the view that all series, bar perhaps the CX indices, suffer from excessive and artificialoutliers.In spite of these major discrepancies in the properties of the high-frequency changes of thevarious indices, they are basically identical in their portrayal of the general volatility level.This is evident from the summary statistics in Panel A of Table 2 which refer to the volatilitylevels rather than their first differences. Apart from the fact that the CX indices, as a directconsequence of the deliberate truncation, represent a slightly down-scaled version of themodel-free implied volatility, it is striking how similar the statistics are across the table,e.g., all the kurtosis measures cluster around 4.14. Hence, apart from a slight reduction inthe mean level and standard deviation, the CX measures capture the identical features ofthe volatility process as the VIX and replication (RX) indices. This further translates intoan extraordinarily high degree of correlation between the index levels, as may be confirmedfrom Panel A of Table 3.

Of course, Tables 1 and 2 suggest that the correlations are much lower for index changes.Panel B of Table 3 verifies this conjecture. In fact, the indices now fall into three distinctcategories with similar high-frequency characteristics, namely the RX1 and RX2 measuresin one group, the VIX and VIX∗ in another, and the CX1 and CX2 in a third, with thelatter also having RX3 loosely associated with it. Most noteworthy is the poor coherencebetween VIX and RX2 as the latter explicitly is designed to mimic the VIX. Instead, VIXchanges correlate better, albeit still very imperfectly, with RX3 and the CX indices. Over-all, the evidence confirms that VIX falls somewhere between RX2 and RX3 in terms ofits high-frequency behavior. An obvious concern is that random oscillation of the VIX be-tween these two index levels may render it excessively noisy as an indicator of short termmovements in model-free implied volatility. Finally, we observe that the implied volatilityreturns display only weak serial correlation. This is not surprising as the behavior of theimplied volatility returns should be akin to those of the VIX futures which are traded onthe CBOE. Hence, although the VIX is not directly a traded asset, the VIX returns should,approximately, behave as such. Consequently, the VIX series can only display a modestdegree of predictability at high frequencies. Otherwise it would violate the semimartingaleproperty which is necessary to rule out arbitrage opportunities.

It is worth reflecting on the striking discrepancies between Panels A and B of Table3. It highlights the point that even exceptionally high levels of correlation do not ensurethat the high-frequency dynamic behavior of the series is similar. Instead, the question ofwhether a given series provides a suitable representation of volatility hinges critically onthe application. For example, it is evident that the degree of low frequency correlation ofmodel-free implied volatility with other economic variables may be addressed equally wellusing any of the series in the table. At the same time, the indices have radically different

34

Table 3: Correlations, 1-Minute Frequency

Panel A: Volatility Index Levels

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

RX1 1.0000RX2 1.0000 1.0000RX3 0.9997 0.9997 1.0000CX1 0.9994 0.9994 0.9996 1.0000CX2 0.9990 0.9990 0.9992 0.9999 1.0000V IX∗ 0.9997 0.9997 0.9997 0.9995 0.9991 1.0000V IX 0.9997 0.9997 0.9997 0.9995 0.9991 1.0000 1.0000

Panel B: Volatility Index Returns

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

RX1 1.000RX2 0.996 1.000RX3 0.467 0.462 1.000CX1 0.475 0.470 0.908 1.000CX2 0.460 0.456 0.896 0.957 1.000V IX∗ 0.318 0.314 0.528 0.545 0.539 1.000V IX 0.318 0.314 0.528 0.545 0.539 1.000 1.000

35

implications for the nature of the volatility generating process at the high-frequency level.14

iii. Implied Volatility - Equity Return Correlations

We now explore the dynamic relation between the returns (changes) of the volatilityindices and the underlying stock index returns. The pronounced negative correlation be-tween daily changes in the VIX and the daily S&P 500 returns have long been noted andserves as an argument for diversifying long equity positions with an exposure to the volatil-ity index. The CBOE web-site provides annual estimates for the sample correlation of thetwo series in the range of -0.75 to -0.85 at the daily level for 2004-2009, with the -0.75estimate referring to the year 2009 which constitutes about half of our sample. The originof such large negative correlations has been much debated in the literature. Among otherthings, it is not clear whether it arises from a corresponding correlation at the very highestreturn frequencies (often labeled a “leverage effect”) or whether it is generated through afeedback mechanism where negative jumps, say, lead to subsequent elevation of volatilitywhich then increases the required rate of return on the equity index and a drop in stockprices. Although the latter effects may play out following the negative jumps, the aggre-gation of the high-frequency returns and volatility into daily measures could make themappear contemporaneous.

Table 4, Panel A, provides one-minute correlations between VIX returns and returnson the S&P 500 index. This panel exploits the implied forward price obtained from theS&P 500 index options to construct the high-frequency equity return proxy. Since thisforward price is also an input into the computation of each of the contemporaneous volatil-ity indices, any noise or error in the forward price will simultaneously impact the impliedvolatility measures, thus potentially inducing artificial correlation between the measures.Thus, as a robustness check, we also report, in Panel B, the corresponding correlationsusing the S&P 500 futures returns obtained from the Chicago Mercantile Exchange (CMEGroup). In this case there might potentially be a slight mismatch in the timing of thevolatility and equity index returns which could induce a downward bias in the estimatedcorrelations.

Panel A of Table 4 conveys a clear message. For the smaller returns, the correlationswith the equity returns are similar across most of the implied volatility indices and stronglynegative, coming in at around -0.70, except for the VIX indices that display correlationsaround -0.50. For the “small” jumps, in the (6,9) category, most of the correlations dropnoticeably, and the CX indices are now clearly the most negatively correlated with the stockreturns. Finally, for the “larger” volatility jumps, all indices except the CX measures fall offdramatically. Hence, the CX indices are the only ones to display consistent equity returncorrelations across all size categories for the volatility returns. This is also manifest in the

14Informally, it may be instructive to think of the indices as fractionally co-integrated so that, even ifthey all are highly persistent, they move in unison over time. While pronounced deviations from the typicalrelationship between the series may be observed, for example due to measurement errors, they tend to beshort-lived as the errors are corrected relatively quickly. This “error-correcting” feature restores “equilibrium”to the system. Although formal modeling along these lines may provide interesting insights it will lead usastray relative to our objectives of the present paper.

36

Table 4: Correlations of Volatility Indexes with S&P 500 Return, 1-Minute Frequency

Panel A: S&P 500 Implied Forward

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

(0,6) -0.69 -0.69 -0.70 -0.72 -0.73 -0.49 -0.49(6,9) -0.57 -0.58 -0.56 -0.61 -0.62 -0.48 -0.48(9,∞) -0.13 -0.13 -0.36 -0.61 -0.64 -0.24 -0.13(0,∞) -0.34 -0.34 -0.66 -0.70 -0.72 -0.40 -0.32

Panel B: S&P 500 Futures

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

(0,6) -0.65 -0.65 -0.66 -0.67 -0.68 -0.51 -0.51(6,9) -0.59 -0.59 -0.63 -0.71 -0.70 -0.53 -0.54(9,∞) -0.09 -0.09 -0.42 -0.69 -0.76 -0.22 -0.16(0,∞) -0.32 -0.31 -0.62 -0.67 -0.68 -0.40 -0.32

Notes: Range is measured in multiples of standard deviation.

overall correlations, reported in the last row of the table, which are much higher for the CXmeasures than for the rest. Moving on to Panel B, we obtain qualitatively identical resultsusing the equity futures returns.

The findings are compatible with the hypothesis that only the large moves in the CXindices are credible, while many of the outliers in the alternative volatility measures arethe result of idiosyncratic errors or random variation in their effective strike range. Sinceartificial outliers, by definition, have no systematic relation to the contemporaneous eq-uity returns, the dramatic reduction in the correlation measures is a direct implication ofthis hypothesis. Assuming the CX measures provide a reasonably accurate account of thevolatility dynamics, we conclude that the high-frequency intraday correlations between im-plied volatility and equity index returns largely mirror the values observed at the daily leveland the relation is quite stable across different return size categories. This has interestingimplications for practical risk management and derivatives pricing as well as the more gen-eral specification of asset price models in continuous time.15

iv. Regression-Based Evidence15Moreover, our results, based on model-free option implied volatility measures, nicely complement the

high-frequency S&P 500 futures study by Bollerslev, Litvinova & Tauchen (2006) who also conclude thatthe evidence favors the leverage type effect but rely on absolute five-minute equity returns as their volatilityproxy.

37

Finally, we investigate the asymmetric return-volatility relation through a complimen-tary approach based on a robust regression framework. This facilitates formal testing ofthe significance of various features within a uniform setting. Denoting a generic impliedvolatility index V X , we define the i’th one-minute volatility and equity-index returns asvi = log(V Xi /V Xi−1 ) and ri = log(S&P500i /S&P500i−1), respectively. The strengthof the contemporaneous return-volatility effect may then be assessed through the simpleregression,

vi = α + β ri + γ |ri|1I(ri < 0) + εi , i ∈ Ig . (11)

This regression includes two coefficients β and γ that capture the general negativevolatility-return relation (or leverage effect) and the asymmetries in the effect for positiveversus negative returns, respectively. Moreover, by restricting the collection of one-minutereturns to those falling within a specific set Ig, we may investigate whether the relation isstable across different scenarios of interest. However, we first concentrate on the full sam-ple evidence. Table 5 reports the results for the regression (11) based on all observations.

The table provides a consistent picture: the β coefficients indicate the presence of alarge and highly significant leverage effect for all indices, and the γ coefficients are statis-tically significant and negative in most cases, even if the magnitude is dramatically smallerthan for the direct leverage effect. In other words, the basic leverage effect is pronouncedand slightly asymmetric so that negative returns of a given magnitude are associated withan increase in volatility that exceeds the reduction in volatility associated with positive re-turns of the same magnitude. Moreover, the constant term, α , is generally negative but oftrivial economic size. Finally, the R2 statistics differ widely, with the group of CX1, CX2and RX3 providing a much higher degree of explanatory power than the VIX measureswhich again dominate the direct replication measures, RX1 and RX2. This ranking is alsoconsistent with the relative significance of the leverage coefficient, β , across the volatilityindex regressions. Again, this supports the interpretation that the CX indices succeed inmaintaining the intertemporal coherence in the volatility series which, in turn, allows theleverage effect to stand out more prominently in the regressions. In particular, if the basicRX and VIX series are contaminated by erroneous jumps, then a number of large volatilityreturns will be unrelated to the underlying index returns and the associated β coefficientswill be downward biased, while the R2 deteriorate correspondingly.

In order to illustrate the potential sources of discrepancy in the findings across volatil-ity indices, we split the observations into three sets, corresponding to whether the size ofthe volatility return falls into the categories (0,6) (NJ, or “No Jump”), (6,9) (SJ, or “SmallJump”) or (9,∞)) (LJ, or “Large Jump”), exactly as done in Table 4 as well. Since theterm corresponding to γ generally becomes insignificant within each of these groupingsand overall has an immaterial impact on the slope estimates, we exclude the correspondingasymmetric term in the leverage effect from the regressions, thus in effect “zeroing out” theγ coefficients. Figure 14 now depicts the resulting regression lines arising from regressionsof the S&P 500 returns on the volatility index returns. Visually, they fully corroborate ourprior conclusions. The regressions involving RX2 and VIX∗ have low slopes for the volatil-

38

Table 5: OLS regression onto S&P 500 Return, 1-Minute Frequency

Panel A: S&P 500 Implied Forward

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

α×104 -0.42 -0.42 -0.47 -0.45 -0.47 -0.27 -0.21-4.58 -4.57 -4.42 -4.09 -4.16 -3.89 -2.58

β -1.62 -1.63 -1.58 -1.76 -1.91 -1.19 -1.20-58.96 -58.97 -44.48 -46.35 -47.93 -45.68 -42.02

γ -0.14 -0.14 -0.15 -0.13 -0.14 -0.07 -0.04-3.11 -3.10 -3.07 -2.63 -2.63 -2.19 -1.19

R2 0.12 0.11 0.43 0.49 0.52 0.16 0.10

Panel B: S&P 500 Futures

RX1 RX2 RX3 CX1 CX2 V IX∗ V IX

α×104 -0.29 -0.30 -0.39 -0.39 -0.39 -0.28 -0.23-3.04 -3.07 -5.22 -5.18 -5.28 -3.69 -2.71

β -1.41 -1.41 -1.38 -1.54 -1.66 -1.13 -1.15-67.48 -67.44 -83.50 -84.01 -88.52 -49.63 -43.48

γ -0.07 -0.08 -0.10 -0.09 -0.09 -0.05 -0.03-2.37 -2.39 -3.31 -3.03 -2.97 -1.64 -0.83

R2 0.10 0.10 0.39 0.44 0.46 0.16 0.10

Notes: For each coefficient, the second row reports corresponding t-statisics, which are HAC-corrected with 20 lags.

39

ity jump returns due to the numerous large volatility index moves without any counterpartin unusual equity index returns. In contrast, the regressions involving the RX3 “jump” re-turns dramatically reduce the scattering of observations along the x-axis and the regressionslopes become much more pronounced. Finally, the in the case of the regressions involvingCX2, almost all large volatility return observations are accompanied by large equity-indexreturns in the opposite direction. As a result, all the associated scatter plots are clusteredaround steeply, negatively sloped lines.

In summary, the CX series contain much fewer abrupt movements than the other can-didate implied volatility indices, but when the CX indices do “jump” these changes, orvolatility returns, are very strongly, and negatively, related to the returns in the stock mar-ket. On the other hand, those jumps identified from RX1 and RX2 or the VIX indices,that do not also manifest themselves in the CX indices, are generally unrelated to the un-derlying S&P 500 returns. This suggests they are induced by deficiencies in the real-timedata collection, processing and dissemination procedures rather than from true jumps in op-tion implied volatility. In particular, these events are not associated with any broad-basedincrease in option prices across the strike range - as this also would invoke significant reac-tions in the CX and RX3 series. Hence, they are likely attributable to idiosyncratic errorsor unwarranted variation in the effective strike range underlying their computation.

6 ConclusionsWe have pursued two goals. First, we seek to replicate the official real-time fifteen sec-

ond VIX series from tick-by-tick quotes on S&P500 option prices obtained from the officialCBOE vendor. The purpose is to gauge the reliability of high-frequency movements in theVIX index and understand which features of the real-time VIX series are robust to noiseand measurement error issues. This first step has provided us with valuable insights intopotential data problems plaguing the VIX series. We identify a particularly important defi-ciency in the CBOE methodology: the rule for determining the effective strike range usedto compute the volatility index is not robust to fairly common data errors or random shiftsin market liquidity. This induces spurious jumps in VIX as well as any alternative indexconstructed on the basis of similar rules for the determination of the effective strike range.These “artificial” jumps have a profound impact on certain high-frequency characteristicsof the VIX return series. Importantly, the identical issue affects the daily closing value ofthe VIX reported by the CBOE and cited widely in the financial press. Day-to-day changesin the index are not reliable indicators of the shifts in implied volatility. On the other hand,the level of the official daily VIX index is sufficiently precise for many applications. Inother words, all the implied volatility measures considered convey almost identical mes-sages concerning the evolution in the overall level of volatility. The critical discrepanciesarise only when focusing on the changes in the indices at a daily or high-frequency intradaylevel.

Our second goal is to demonstrate that one may construct corridor implied volatilitystyle indices, CX, that retain the theoretical advantages of the model-free volatility conceptwhile avoiding the pitfalls associated with the official VIX series. These CX indices may

40

Figure 14: Volatility Index Return vs. S&P500 Return Regressions. This figure depictsthe regressions between one-minute returns on the volatility indices and the underlying eq-uity futures index where we decompose the (absolute) volatility returns into: Small Movesor No Jump, NJ (returns fall within the (0,6) group), Small Jump Returns, SJ (fall withinthe (6,9) group) and Large Jump Returns, LJ (the remainder).

Note: Small Move (No Jump) [Blue, NJ] Small Jump [Black, SJ] Large Jump [Red, LJ]

41

be computed directly from the real-time option quotes via a simple and transparent com-putational rule. The main change from the VIX is simply the determination of a differenttruncation point for when to exclude options far out of the money in the computation of theindex. By ensuring intertemporal consistency in this dimension, we obtain CX indices thatcover economically equivalent parts of the strike range. Consequently, they are internallyconsistent over time and the index values may more meaningfully be compared across time,independently of the underlying liquidity in the options market. As a result, we find that theamount of spurious jumps in the volatility index are dramatically reduced, if not entirelyeliminated, and the jump returns now consistently display a pronounced negative relationto the underlying equity index returns.In short, the CX and VIX return series display strik-ingly different time series characteristics along important dimensions that are of relevancefor active real-time financial decision-making and risk management. The same issues af-flict the widely used daily VIX series which may be obtained from the CBOE website. Thedaily changes, or volatility returns, implied by this official VIX series are inherently noisyand appropriate caution should be exercised if analysis is based on those daily shifts in thevolatility index. For many purposes, our CX measures represents a superior alternative andthe CX indices are straightforward to calculate from the underlying SPX option quotes.

Our findings point towards a number of avenues for future research into the construc-tion of implied volatility measures. One important application of such measures is theidentification and modeling of the equity variance risk premium. This premium is givenby the wedge between the (expected) realized variance of the equity index returns and theoption implied variance measure over the corresponding horizon. In other words, it rep-resents the discrepancy between the actual return variation and the pricing of this returnvariation embodied in the (squared) option implied volatility measure. In the literature,this is most often approximated by the difference between an (expected) realized variancemeasure, constructed from the cumulative sum of squared high-frequency equity-index re-turns, and the (squared) implied volatility index, represented by the CBOE VIX measure.Hence, to the extent VIX is a noisy, downward-biased, and excessively volatile measureof implied volatility, the quality of the risk premium measures are similarly degraded. Inthis case, however, the CX indices do not represent a natural alternative. In theory, theimplied variance measure should captures the entire theoretical strike range for the equityreturns, from zero to infinity. Hence, using (squared) CX measures in this capacity wouldresult in a downward biased estimate of the theoretical implied variance measure, as the CXmeasures effectively, and consistently, exclude far OTM options from the computation. Incomparison, the VIX measure typically provides a broader, but also inconsistent, coverageof the tails, depending on the truncation point determined by the configuration of non-zerobid options quotes at any point in time. Hence, on average, the VIX provides a less biasedimplied volatility measure but the extent of the downward bias is strongly time-varying.There are two obvious approaches to remedy the situation. First, one may directly seek toestimate reliable prices for far out-of-the-money options so that the effective strike range isexpanded to ensure that all non-trivial contributions to the measure may be included. Sinceobserved quotes for far OTM options are unreliable in this regard, such procedures hingeon specific assumptions about the shape and fatness and the risk-neutral distribution. Oneinteresting approach towards this problem is provided by Bollerslev and Todorov (2010a),

42

and it is also the subject of ongoing work in Andersen and Bondarenko (2011). Second, onemay resort to exploring the variance risk premium only within ranges, or corridors, of thestrike range for which we can reliably measure both the model-free implied variance andthe corresponding realized return variation of the underlying equity-index returns. This ap-proach provides a decomposition of the variance risk premium into segments of the returnspace, thus enabling direct identification of the size of the variance risk premium acrossdifferent equity return states. For example, one may explore how much of the variancerisk premium stems from high premiums on the return variation in states where the equityprices have declined over the given horizon, or one may investigate whether the variancerisk premium is positive also for states where equity prices have been rising. This approachis explored in detail in Andersen and Bondarenko (2010).

A second important avenue for future research is to enhance our understanding of thenature of the random variation in the quality and strike coverage of the SPX option quotesthat constitute the most critical input to the computation of the implied volatility measures.The SPX options market is primarily an floor based open outcry system. However, theelectronic data feed providing the official SPX quotes do not stem directly from this marketbut are instead constructed from real-time quotes by two lead market maker firms at theCBOE as well as customer and broker orders listed on the same electronic platform. Itis evident that the electronically disseminated quotes generally have significantly widerspreads than the effective bid-ask spreads prevailing in the SPX trading pit. By studyingthe interaction of the quoting patterns from the lead market makers, customers and brokers,we may obtain a deeper understanding of the features in the data that induce noise as wellas spurious jumps into the real-time released VIX measures. As a consequence, improvedprocedures for extracting the relevant information from the real-time option quotes maybe within reach. Gonzalez-Perez (2011) is currently exploring the relevant microstructurefeatures of the SPX option market and the associated electronic quoting mechanism in orderfacilitate such direct inquiry into the origins of the apparent distortions in the VIX index.

43

ReferencesAndersen, T.G. & O. Bondarenko. 2007. Construction and Interpretation of Model-Free

Implied Volatility. Volatility as an Asset Class, (London: Risk Books). Ed. by I.Nelken.

Bollerslev, T., G. Tauchen & H. Zhou. 2009. “Expected Stock Returns and Variance RiskPremia.” The Review of Financial Studies 22:4463–4492.

Bollerslev, T., J. Litvinova & G. Tauchen. 2006. “Leverage and Volatility Feedback Effectsin High-Frequency Data.” Journal of Financial Econometrics 4:353–384.

Bollerslev, T. & V. Todorov. 2010. “Jumps and Betas: A New Theoretical Framework forDisentangling and Estimating Systematic Risks.” Journal of Econometrics; 157:220–235.

Bondarenko, O. 2007. “Variance Trading and Market Price of Variance Risk.” WorkingPaper, University of Illinois, Chicago .

Britten-Jones, M. & A. Neuberger. 2000. “Option Prices, Implied Price Processes, andStochastic Volatility.” Journal of Finance 55(2):839–866.

Carr, P. & D.B. Madan. 1998. “Option Valuation Using the Fast Fourier Transform.” Jour-nal of Computational Finance 2:61–73.

Carr, P. & L. Wu. 2009. “Variance Risk Premiums.” The Review of Financial Studies22:1311–1341.

Dupire, B. 1993. “Model Art.” Risk 6(9):118–124.

Dupire, B. 1996. “A Unified Theory of Volatility.” Working Paper, BNP Paribus .

Dupire, B. 2004. “A Unified Theory of Volatility”. Derivatives Pricing: The Classic Col-lection. (London: Risk Books). Ed. by P. Carr.

Jiang, G.J. & Y.S. Tian. 2005. “The Model-Free Implied Volatility and Its InformationContent.” The Review of Financial Studies 18 (4):1305–1342.

Jiang, G.J. & Y.S. Tian. 2007. “Extracting Model-Free Volatility from Option Prices: AnExamination of the VIX Index.” Journal of Derivatives Spring:1–26.

Neuberger, A.J. 1994. “The Log Contract.” Journal of Portfolio Management 20:74–80.

Todorov, V. 2010. “Variance Risk Premium Dynamics: The Role of Jumps.” The Reviewof Financial Studies 23:345–383.

Todorov, V. & G. Tauchen. 2010. “Activity Signature Functions for High-Frequency DataAnalysis.” Journal of Econometrics 154:125–138.

44

A Data Filtering

SPX option classes

We acquired the MDR (Market Data Retrieval) data for S&P 500 options from the CBOEsubsidiary, Market Data Express (http://www.marketdataexpress.com/). The MDRdata set includes tick-by-tick quotes and transactions throughout the trading day for alloption classes issued by the CBOE on the S&P 500. Each option class is characterizedby a three or four letter code. For example, the JXA, JXB, JXC, JXD and JXE classesare weekly options, while LEAPS may be denoted QSE, QSZ, QZQ, SAQ, SKQ, SLQ,SQG, SQP, SZQ or SZU. We only consider the options that the CBOE actually uses intheir computation of the VIX over our sample period, namely those belonging to theSPB, SPQ, SPT, SPV, SPX, SPZ, SVP, SXB, SXM, SXY, SXZ, SYG, SYU, SYV andSZP categories. The latter are generally known as SPX equity options and they ma-ture on the Saturday immediately following the third Friday of the expiration month (seehttp://www.cboe.com/Products/EquityOptionSpecs.aspx). Henceforth, we referto this entire set of options classes by the label “SPX options.”

Maximum quote delay between MDR time-stamp and price record

We begin by constructing an equally-spaced fifteen second option quote record across allavailable strikes for two different maturities for each trading day, starting at 8:30:15 CSTand ending at 15:15:00 CST. This produces up to 1620 bid-ask quote pairs per day for agiven option. Since more than one hundred distinct strikes routinely are quoted for eachmaturity, the amount of quote data stored is quite substantial. The raw tick data are at timesrather noisy and include both typos and errors. Hence, we rely on a variety of filters toensure that the analysis is not excessively impacted by faulty or flawed observations.

For each fifteen second mark on the time grid, we record the last available SPX quotepair for all call and put options at the two “closest-to-maturity” expiration dates.16. How-ever, the time difference between the MDR option quote time-stamp and our time grid pointmust be less than five minutes to be included in our quote series. If this timeliness condi-tion is violated we tag the SPX quote as missing. Figure 15 plots the percentage of missingdata after applying this filter. On average, we remove around 20% of the initial data dueto this specific form of staleness. As is evident from the figure, put option quotes are moreaffected by this filter, in particular during the second part of the sample. We also found,unsurprising, that missing data are more prevalent for far-away-from-the-money options.17

16In line with the CBOE methodology for the VIX computation, we replace the first (second) maturityoptions by the second (third) maturity options when the time to maturity on the closest-to-maturity option isless than seven calendar days

17The peaks in Figure 15 for July 3, November 28, and December 24 reflect partial Holidays where tradingwas terminated prior to the usual close, so they do not represent any anomalous market conditions.

45

Figure 15: The percentage of put and call quotes removed due to the five-minute asyn-chrony filter

First maturity options

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

10

20

30

40

50

60

70

80CallPut

Second maturity options

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

10

20

30

40

50

60

70

80

46

Finally, we note that part of our analysis is based on a one-minute, rather than fifteensecond, sampling frequency. However, the 405 potential one-minute daily quotes for eachoption are obtained from the underlying higher frequency series by retaining the data ateach one minute mark. As such, these lower frequency quote observations are subject tothe same filters and inherit the associated properties from the fifteen second series.

Abnormal bid-ask spreads

The preceding filters do not guard directly against data flaws or errors so we now turntowards that task. Given our focus on quotes, a natural starting point is a scheme designedto detect erratic quoting behavior.

To motivate our specific implementation, we note that the CBOE imposes an upperbound on the spread of option quotes. These maximum allowable spreads are specifiedas a step function which, to a first approximation, constitutes a concave increasing func-tion in the bid quote. Hence, as we move further into-the-money and the options becomemore valuable, the maximum spread generally increases in absolute terms but declines inrelative, or percentage, terms. Moreover, during our sample period, this maximum quoteschedule reaches its highest value for bid quotes in excess of $20 and remains fixed there-after. Finally, in the vast majority of cases, the actual quoted spreads for deep-in-the-moneyoptions coincide with this upper bound. This feature is particularly useful when this max-imum spread size is not known, as we then effectively can infer it from the actual quotedspreads on the deep in-the-money options.18

Inspired by the features of the general quoting schedule, we construct a conservativefilter for unreliable quote spreads. First, we classify our SPX options into four categoriesaccording to their expiration date (first versus second maturity) and option type (put versuscall). We then implemented our spread filter through a sequence of steps:

• Define Bi,t(K,T ) and Ai,t(K,T ) as the bid and ask prices quoted at the grid point i onday t for an SPX option with strike K and maturity time T . The associated spreadsare, Si,t(K,T ) = Ai,t(K,T )−Bi,t(K,T ).

• At each fifteen second time grid point, define the mode, si,t , of the quoted spreadswith bid quotes above $20 within each of the four option categories. Likewise, de-fine the mode, St , of all quoted spreads with bid quotes above $20 for each optioncategory across the full trading day. Next, we define the maximum of these spreadmodes across the four option categories for each grid point, s∗i,t , and for the dailyspread modes, S∗t . Finally, the spread bound indicator, SBi,t , is obtained at each fif-teen second grid point as the maximum of s∗i,t , and S∗t . Hence, at a given point intime, the spread bound reflects the highest typical spreads observed across the optioncategories at that moment as well as the maximum representative spreads observedacross the entire trading day.

18During the turbulent market conditions in late 2008 and early 2009, the market makers were repeatedlyoffered “spread relief” by the exchange so that they could scale up all quoted spreads by a certain multiple.We do not have access to a precise account of these temporary changes in the maximum quote schedule.

47

• Using SBi,t as a benchmark, we now define a step function for the maximum spread,SMi,t , across the different bid quote ranges employed by the CBOE.19 As indicatedin equation (12), we add $0.25 to the benchmark spread to allow for relatively minorviolations before we eliminate any quote pair. All quotes with spreads exceedingthose indicated below are then deemed erratic and removed from our data set. Finally,we also eliminate quotes with zero or negative spreads.

(SMi,t) =

1 · (SBi,t +1/4) if Bi,t(K,T ) > 204/5 · (SBi,t +1/4) if Bi,t(K,T ) ∈ (10,20]3/5 · (SBi,t +1/4) if Bi,t(K,T ) ∈ (5,10]2/5 · (SBi,t +1/4) if Bi,t(K,T ) ∈ (2,5]1/4 · (SBi,t +1/4) if Bi,t(K,T ) ∈ (0,2]

(12)

Figure 16 displays the auxiliary variables s∗i,t and S∗t used to construct SBi,t . The local-ization of the bound, associated with using the spread mode computed at a specific pointin time, s∗i,t , is critical in retaining a sufficient number of option quotes across all tradingperiods when volatility shifts abruptly within the day. On such occasions, the spreads mayshift to a higher plateau during a small part of the day, and spread relief may even havebeen offered at some unknown point in time. Our implementation adapts to such featuresby letting the prevailing spread modes determine the bounds.

This spread filter completes the description of our cleaning of the individual optionquotes. Figure 17 depicts the percentage of missing option quotes after the individualquote filters described in this section have been applied.

One implication of removing quotes associated with erratic spreads is that we may in-troduce some additional asynchrony into our data set. The grid point associated with anerratic quote is designated as “missing” only if there are no existing quotes for this optionavailable over the prior five minutes. Otherwise, the removed quote will, in accordancewith our maximum delay rule, be replaced by the last valid quote. Such a quote could thusrepresent a lag of five minutes relative to the time stamp for the erratic quote it replaces.However, since the procedure is applied sequentially, after the initial construction of thefifteen second grid where there also is a potential time gap of up to five minutes, this sub-stitution for erratic quotes may induce a time delay of up to ten minutes. This staleness inquotes may impact the quality of our volatility measures. We control for these effects aspart of our additional screening procedures described in Appendix B.

B Quality Control for Implied Volatility MeasuresAppendix A explains how we construct our SPX option data set covering quotes across

the available strikes for the two nearly maturities. This section describes how we ascertainwhether the option quotes, available at a given fifteen second grid point, are of sufficient

19This is in the spirit of CBOE Rule 8.7(b)(iv).

48

Figure 16: The sequence of measures used to compute the Spread Bound schedule.

Intraday and Daily Auxiliary Variables: s*i,t and S*

i,t

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

5

10

15

20

25

30

35

40s*

i,t

S*i,t

Intraday Threshold for the Bid-Ask Spread

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

5

10

15

20

25

30

35

40

49

Figure 17: Percentage of missing quotes in the original data versus after filtering.

Call and First Maturity Options

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

10

20OriginalAfter Filters

Put and First Maturity Options

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

10

20

Call and Second Maturity Options

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

10

20

Put and Second Maturity Options

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

20

40

50

quality to ensure reliable computations of implied model-free volatility measures. If thequotes are found lacking for a particular point in time, we deem the corresponding volatil-ity measures to be “not available.”

Our quality criteria focus on two aspects, namely timeliness, as captured by the overalldegree of quote staleness, and pricing coherence, as captured by the extent to which thebasic no-arbitrage conditions are violated.

Systemic Staleness in quotes

By far, the most influential options for the computation of model-free implied volatilitymeasures are the out-of-the-money (OTM) put and call options with strikes closest to theat-the-money (ATM) strike which is given by the forward price. Hence, we focus on theset of twenty OTM put and twenty OTM call options closest to ATM, as well as the ATMcall and put option price. We label this group of options the “pivotal” ones.

Our systemic staleness filter turns out to be fairly restrictive. It flags instances wherea period of five minutes or more transpires without any quote update among the pivotaloptions in the bid or the ask price at either of the two maturities. This dummy variableequals one at such times and zero otherwise. We provide a depiction of the percentage ofthe trading day impacted by such staleness in Figure 18. When we encounter such lackof activity for all options within one of our four pivotal option categories, we classify theentire inactive period of five minutes or more as unreliable for computing the volatility in-dices.

Coherence of option quotes

No-arbitrage conditions imply that the option prices, for a given maturity date, must beconvex in the strike price. We obtain the option prices as the average of the bid and askquotes. Next, we construct a “non-convexity” index to indicate the degree of violationof the no-arbitrage condition for a given time grid point. The index is constructed byfirst computing the (discrete) second derivative of each option price with respect to thestrike price using standard discrete approximation techniques. Second, we identify all the(absolute) violations as given by second derivatives that are negative and cumulate theminto an overall measure. Finally, the statistic is averaged across all the available strikes.Hence, an elevated value of the non-convexity indicator signals poor pricing coherence assuch violations should induce arbitrageurs to enter the market and profit from the relativeinconsistency in option valuations. Consequently, we treat such scenarios as indicatinginferior quote quality and classify the associated volatility measures as “not available.”Figure 19 displays the non-convexity indicator across the sample period. The vertical linerepresents our chosen threshold so all time grids for which the indicator value reachesbeyond this line are excluded from our model-free implied volatility computations.

51

Figure 18: Percentage of the trading day with staleness within one of the pivotal optioncategories

52

Figure 19: Non-convexity indicator recorded every 15 seconds. June 2008 - June 2010

First Maturity (SPX Options)

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Second Maturity (SPX Options)

Jun08 Sep08 Dec08 Mar09 Jun09 Sep09 Dec09 Mar10 Jun100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

53