A Corridor Fix for VIX: Developing a Coherent Model-Free Option-Implied Volatility Measure

A Corridor Fix for VIX: Developing a Coherent Model-Free Option-Implied Volatility Measure Torben G. Andersen Oleg Bondarenko Maria T. Gonzalez-Perez July 21; Revised: January 211 Abstract The VIX index is computed as a weighted average of SPX option prices over a range of strikes according to specific rules regarding market liquidity. Using tick-by-tick observations on the underlying options, we document that this strike range varies substantially in how much coverage it provides of the distribution of future S&P 5 index prices, producing significant biases, or distortions, in the time series of VIX measures. We propose a novel high-frequency Corridor Implied Volatility index (CX) computed from a strike range covering an economically invariant proportion of the future S&P 5 index values and using only reliable option quotes. Comparing the time series properties of these alternative volatility indices from June 28 through June 21, we find that our CX measure is superior in terms of filtering out noise and avoiding large artificial jumps. Consequently, the properties of the two series at both the daily and the intraday level are dramatically different in important dimensions of relevance for 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 University, for financial support. We thank seminar participants at the Federal Reserve Board of Governors, Washington, D.C., the SoFiE-Creates Conference in Aarhus, October 21, the University of Illinois at Chicago College of Business Administration, and Pamplin College of Business, Virginia Tech for comments. Kellogg School of Management; Northwestern University; 21 Sheridan Road, Evanston, IL 628, NBER, and CREATES; e-mail: t-andersen@kellogg.northwestern.edu Department of Finance (MC 168), University of Illinois at Chicago, 61 S. Morgan St., Chicago, IL 667; e-mail: olegb@uic.edu Kellogg School of Management; Northwestern University; 21 Sheridan Road, Evanston, IL 628; e-mail: m-gonzalezperez@kellogg.northwestern.edu

1 Introduction The VIX volatility index disseminated by the Chicago Board Options Exchange (CBOE) has attracted much attention in recent years. This index is computed from concurrent price quotes on a wide range of out-of-the-money European style put and call options written on the S&P 5 equity index, and it is constructed to serve as a model-free option-implied return volatility measure for the S&P 5 index over the coming 3 days, expressed in annualized percentage terms. One striking and widely noted feature is that the changes in the VIX index displays a very strong negative correlation with the corresponding returns on the underlying S&P 5 index. This negative relationship with market returns suggests that holding long VIX positions provides portfolio diversification. On the other hand, the existence of a large volatility risk premium means that the average return on long VIX futures positions are negative, implying that speculators typically may desire short VIX positions. Various trading strategies exploiting the latter feature have been launched under the moniker of volatility arbitrage and associated performance measures are widely followed. As a result, the trading activity in VIX related derivatives has grown rapidly 1, and the relevance of VIX index as well. The pronounced negative correlation with the overall market returns has also captured the attention of both the popular press and the more specialized financial media which often label the VIX a fear gauge due to the rapid spikes it displays during periods of market stress. Naturally, the VIX garnered even more attention as the index soared to dramatic heights during the financial crisis of 28-29, reaching an intraday high of 89.5% on October 24, 28, compared with an average value of about 19% in the period from 199 until October 28. In short, the VIX is routinely cited by the main media when describing current market conditions or investor sentiment. On the other hand, the VIX is increasingly used in financial economics. In this case, the main attraction is that, in theory, VIX provides a model-free measure of the expected value of the S&P 5 return variation under the risk-neutral (Q probability) measure. Hence, the VIX embodies a model-free 2 market volatility forecast. The literature has exploited this feature going in three main directions. One set of studies explores the forecast power of variation in the VIX for future realized return volatility. In this line, Jiang & Tian (25) deem the information content of the VIX volatility forecast superior to alternative implied volatility measures as well as forecasts based on historical volatility. A second group of studies emphasize that the VIX measure is constructed directly from observed option prices so it will necessarily incorporate any pricing of variance risk that may be embedded in the market prices. Thus, VIX does not represent a pure estimator of future return volatility for the underlying asset because it includes compensation for variance risk as well. Instead, the wedge between regular time series forecasts for volatility, developed under the 1 In fact, the CBOE considers the VIX options, introduced in February of 26, the most successful new product launch in their history. Moreover, both the VIX futures and options have received most innovative index product awards at industry conferences. 2 This is in marked contrast to earlier notions of option-implied volatility that derive implied volatility from option prices plus specific modeling assumptions, e.g., the Black-Scholes model or more sophisticated stochastic volatility models incorporating, potentially, multiple volatility factors and jumps. 1

actual or objective (P probability) measure, and the VIX, representing the risk-neutral return volatility forecast, may be interpreted as a market volatility risk premium, see, e.g., Bondarenko (27), Bollerslev, Tauchen & Zhou (29) and Carr & Wu (29). Given this result, the empirical evidence favors a large, negative and strongly time-varying variance risk premium that is also negatively correlated with market returns. Interestingly, this variance risk premium has been found to carry predictability for future equity returns and to be priced in the cross-section of equity returns: stocks that exhibit a positive covariation with the premium have lower expected returns than stocks that are negatively related to the premium 3. Further studies document that jumps in prices and/or volatilities are critical for the dynamics of the variance risk premium and suggest that the large premium is tied to timevarying compensation for tail risks, see Todorov (21) and Bollerslev & Todorov (21). A third group of studies seeks to draw inference regarding the stochastic properties of the underlying market return volatility process directly from the high-frequency behavior of the VIX. For example, while a jump in the return volatility of the S&P 5 index may be hard to identify, even from high-frequency observations on the stock index itself, the expected future return volatility would, under weak regularity conditions, jump at the same time. Consequently, such events should be more readily identifiable from the VIX index because it directly reflects the expected future volatility under an equivalent probability measure. This observation motivates Todorov & Tauchen (21) to explore the relative importance as 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 series, as exemplified by the VIX index, will inevitably become important areas of study in their own right over the coming years. For example, how often do we see large discrete moves, or jumps, in these volatility measures and what economic circumstances trigger such reactions? In exploring such questions it is critical to account for the microstructure features of the index induced by data discreteness, market structure and other potential sources of high-frequency noise. At first blush, VIX may appear less susceptible to such problems than regular assets because the index is constructed from individual asset prices within a large portfolio, typically relying on quote mid-points from more than 1 distinct equity-index options, so individual pricing errors will tend to cancel. However, as we detail below, the computation of model-free volatility confronts unique challenges that render precise real-time measurement of the index difficult. A variety of related issues have previously been noted by Jiang & Tian (27) who discuss how to alleviate some specific problems. Our concerns, however, run deeper. This article (i) characterizes an important deficiency in the CBOE VIX methodology that induces artificial jumps in the volatility index, (ii) proposes the estimation of alternative model-free indices to alleviate the mentioned deficiency, and finally (iii) studies how the CBOE methodology to compute the VIX affects the relationship between the CBOE VIX changes and S&P5 returns. Following this agenda we first systematically explore the high-frequency properties of the official VIX index and compares it with a number of 3 Intuitively, large variance risk premiums arise during periods of market stress when the marginal value of wealth is particularly high, so assets that covary positively with the premium serve as hedges and require lower risk premiums. Hence, the evidence suggests that variance risk is priced both in the time series and cross-sectional dimensions. 2

alternative measures that we constructe using the same underlying CBOE SPX options on the S&P 5 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 to replicate the official VIX data using the exact rules for calculation laid out by the CBOE and using option quotes from their official data vendor. Overall, we find a reasonable coherence between our constructed index and the official VIX. However, there are occasional dramatic deviations that have a pronounced impact on the properties of the high-frequency volatility series. Hence, we make incremental modifications to our index computation that alleviates some of the observed shortcomings. This further improves the coherence between our constructed index and the VIX but important discrepancies remain on certain days. Inspecting the options quotes carefully, we detect a substantial, occasionally abrupt, time variation in the range of options exploited in the VIX computations. Such changes in the range of covered strike prices induce substantial shifts in the implied volatility measures, not only for our VIX replicating indices, constructed from tick-by-tick option quotes, but also for the 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-frequency VIX data released in real-time from the CBOE are not suitable for analyzing critical aspects of the volatility dynamics due to inconsistencies and noise in the time series induced by various market microstructure features. Moreover, any independent computation of the ideal model-free volatility series from the underlying option quotes must confront serious obstacles in order to construct coherent index measures over time. Inevitably, the pronounced variability in the available strike coverage necessitates ad hoc corrections that will infuse the series with non-trivial measurement errors. Next, we propose the construction of alternative indices to alleviate the critical problem with the VIX computations, namely the need for model-free implied volatility to be computed 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 and traded, so the tails of the risk-neutral return distribution are generally not well covered 4. This concern is partially neutralized by the fact that the prices of far out-of-the-money options rapidly converge to zero and become negligible beyond a certain point. Hence, the critical issue is to determine a suitable cut-off point for the computation. The CBOE stipulates that, moving from at-the-money and towards the tails, the inclusion of additional quotes is terminated once we encounter two consecutive out-of-the-money options with a zero bid quote, thus implicitly deeming the remaining mid quotes unreliable and/or negligible. As noted above, the problem with this strategy is two-fold. First, the eliminated option 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 strike prices covered by this cut-off criterion displays pronounced time variation 5. Consequently, the downward bias displays a degree of (intraday) persistence, interspersed with dramatic 4 Moreover, the relative precision of option prices declines rapidly in the tails as the percentage bid-ask spread inflates and liquidity dies off. Hence, the option quotes become increasingly noisy as we move further into the tails. 5 It 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 acknowledge that inherent data problems render it impractical, if not infeasible, to consistently compute the full model-free implied volatility. Hence, we focus on a slightly more limited strike range that, measured by a specific option pricing metric, covers an invariant portion of the underlying risk-neutral density at all times. Consequently, we introduce an alternative cut-off criterion, determined endogenously from the available option prices, that allows the measure to reflect the pricing of volatility across an economically equivalent fraction of the strike range over time, thus ensuring that the volatility measure is internally coherent in the time series dimension. Any such measure is termed a model-free Corridor Implied Volatility Index (CX), and is closely related to the corresponding theoretical concept discussed by Carr & Madan (1998). Obviously, different corridor measures may be obtained depending on the width and positioning of the strike range used in the computations. We vary the corridors moderately to understand the benefits and drawbacks of each choice. Our empirical results indicate that all our CX measures dominate the VIX measures in terms of providing accurate and coherent volatility indicators over time. In particular, the CX measures avoid many false jumps and display stronger time series persistence than the VIX counterparts, reflecting the improved intertemporal consistency and the lower level of idiosyncratic noise of the CX measures. Third, our analysis goes on to document the implications of the distortions induced by the current CBOE methodology for computing model-free option-implied volatility. In particular, we find that the frequency of jumps in the implied volatility series is inflated and the high-frequency correlation between equity-index volatility and market returns is severely downward biased. In contrast, carefully constructed high-frequency CX measures produce much more robust measures of the time series properties of market volatility. Consequently, the CX measures facilitate model-free identification of important underlying features of the volatility series that will be useful in discriminating between alternative models and explanations for observed high-frequency asset price dynamics. We also note that the development of robust CX measures is not limited to monthly equity volatility indices. They may be readily computed across alternative asset classes and maturities as long as the requisite high-frequency options data are available. As such, these measures should have broad applicability in future research. This result is of high relevance to improve risk measures based on VIX, and the use of VIX in risk managing exercises. Finally, we explore whether the problems plaguing the intraday VIX measures carry over to the end-of-day closing values. Although ultra-high-frequency VIX is becoming more studied in the academia, and more used in trading strategies, the end-of-day VIX continues being the frequency most used by the academia and media to study the VIX dynamic. This article also reports that the end-of-day values are linked directly to the corresponding 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 significant impact on the time series properties of the daily VIX measure, raising the concern that the idiosyncratic errors in the index can be detrimental to empirical studies exploiting the series. 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 Background Over-the-counter trading of volatility contracts emerged in the 199s with liquidity building primarily in the so-called variance swaps. Despite the label, these swaps are forward contracts involving no initial or intermediary cash flows. At expiry, the long party pays a fixed premium, determined at contract initiation, and in exchange receives a payment reflecting the (random) realized variance of the underlying index. Typically, the realized variance would be measured by the cumulative sum of daily squared index returns from initiation to expiry. A critical backdrop for these developments was breakthroughs in theory demonstrating the feasibility of replicating the realized return variation of an asset over a given future horizon. This path-dependent payoff will, under general conditions, equal the payoff from a strategy involving only a static options portfolio combined with delta hedging in the underlying asset. 6 These results provided practical pricing and hedging tools necessary 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 price of the replicating options portfolio. As such, the premium on a variance swap directly values the future return variation. Moreover, since this reasoning is model-independent, the corresponding notion is referred to as the model-free implied variance, and its square-root as the model-free implied volatility, or simply MFIV. In order to facilitate the development of an exchange traded market for volatility derivatives, the CBOE, on September 22, 23, aligned their methodology for computing the volatility 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 participants may rely on existing procedures for pricing and hedging of volatility contracts when dealing with exchange traded VIX products. 2.2 From Theory to Implementation In 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 strike prices spanning the support of the possible future index prices, from zero to infinity, for the given maturity. A convenient representation of the valuation formula takes the form, σ 2 T = 2er T T T [ F P(T, K) ] C(T, K) K 2 dk + F K 2 dk = 2er T T T [ Q(T, K) K 2 ] dk, (1) 6 Initial work leading to these insights is represented by Neuberger (1994), Dupire (1993) and Dupire (1996), with the latter published in Dupire (24). Building on these insights, Carr & Madan (1998) constructed an explicit and robust replication strategy for the pay-off on the variance swap, while Britten-Jones & Neuberger (2) provide additional results exploiting these concepts. 5

where r T is the (annualized) risk-free interest rate for the period [,T ] as represented by the 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 the mid-quotes for European put and call options with strike K and time to maturity T, and Q(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 an option 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 approximates σt 2 by the following expression, ˆσ T 2 = 2er T T n [ ] K i F 2 Q(K i ) (2) T i=1 Ki 2 } {{ } Discrete Approximation 1 T 1 K f } {{ } Correction Term where < K 1 < < K f F < K f +1 < < K n refer to the strikes included in the computation, K f denotes the first strike price available below the forward rate, F, so f is a positive integer, 2 < f < n 1. Moreover, the increments in the strike ranges are calculated as K 1 = K 2 K 1, K n = K n K n 1, and for 1 < i < n, K i = (K i+1 Ki 1)/2, while the prices of the OTM options, Q(K i ), are given by the midpoint of the option bid and ask quotes. Finally, the second term in (2) reflects a correction for the discrepancy between K f and the forward price. 7 Another practical issue is that only a few options maturity dates are available on any given day. The V IX is defined for a fixed calendar maturity of T M = 365 3, or thirty days. Hence, the CBOE obtains the V IX measure via a linear combination of the MFIV for the two expiration dates closest to thirty calendar days, but excluding options with less than seven calendar days to expiry. The interpolated quantity is annualized and quoted in volatility units, using the following procedure, 8 [ ( V IX = 1 w 1 ˆσ 2 1 T ) ( ) ] 1 + w2 ˆσ 2 2 T 2 365 3, (3) where w 1 = T 2 T M T 2 T 1 and w 2 = T M T 1 T 2 T 1, so that, obviously, w 1 + w 2 = 1. There are different sources of measurement error in VIX. Jiang & Tian (25) classify the errors in ˆσ 2 as arising from: (i) truncation errors, due to the minimum and maximum strike prices being far removed from zero and infinity in (2); (ii) discretization errors, due to the lack of numerical integration in (2); (iii) expansion errors, due to the Taylor series approximation 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). 7 In the ideal MFIV formula (1), K f = F, so the adjustment seeks to correct for the bias in the volatility measure arising from K f not being equal to the exact at-the-money strike. 8 Detailed 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 the computation of ˆσ 2. This induces a time-varying effective strike range that can result in spurious breaks in the VIX time series, entirely unrelated with contemporaneous developments in the underlying return series (artificial jumps). We argue that this problem can be alleviated by implementing the Corridor Implied Volatility index (CX) described in some detail in Andersen & Bondarenko (27). The corridor methodology directly controls the range (K 1,K n ) in a time-consistent manner and thus prescribes which strike prices to include in the model-free volatility computation at any given time. 3 Data We compute model-free volatility indices from S&P 5 option quotes obtained from the official vendor of CBOE data, Market Data Express, or MDX. The data set is based on MDR (Market Data Retrieval) data captured by CBOE s internal system. The underlying tick-by-tick option quotes cover the period June 2, 28 - June 3, 21. We construct fifteen second series for each individual option by retaining the last available quotes prior to the end of each 15 second interval throughout the active trading day on the CBOE from 8:3:15 AM to 3:15: PM CT. If no new quote arrives in a given 15 second interval, the last available quote is retained. Hence, in the case of inactivity, the quotes may remain constant for 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 the corresponding intraday quotes for S&P 5 futures on the Chicago Mercantile Exchange Group (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 the MDR-SPX data base and the time it appears in our data set exceeds five minutes; (ii) if an entire block of adjacent OTM option prices have not been updated for five minutes, we deem the index itself not available (n.a.) and eliminate the corresponding time period from our analysis; (iii) exploiting internal CBOE regulation, we impose a maximum bidask 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 the reported option mid-quotes and declare the volatility index n.a. over the corresponding period whenever the threshold is exceeded. We lose close to 5% of the 15 second observations due to this filtering procedure, with the vast majority of the missing values resulting from violations of the lack of staleness requirement (ii). This condition is essential for avoiding artificial jumps in the volatility index when an entire set of stale option quotes are updated simultaneously after some temporary malfunction of the dissemination system. A more detailed account of our filtering procedure is provided in the appendix. The fifteen second series for the official VIX, covering June 28 - June 21 is obtained from different sources although they all ultimately stem from the CBOE. The first data set, from June 28 to September 28, includes the official VIX every fifteen seconds provided directly by the exchange. The second data set is labeled MDR-VIX and contains 7

tick-by-tick quotes of VIX options, covering October 28 to April 29. This source also provides the latest available observation on the underlying VIX measure at the time the option quote is recorded, thus ensuring a high-frequency recording of the VIX values. Finally, the May 29 - June 21 period is again covered by the VIX series released directly by the CBOE, but this time secured from the official data vendor. We apply a simple filter to the intraday VIX series: we remove a few observations that fall outside, and thus are inconsistent with, the high-low range for VIX over the day reported by the CBOE. We denote this lightly filtered VIX series VIX. Figure 1 shows the evolution of the daily VIX closing value over recent years, with the thick green line denoting the period analyzed in this article. 4 Practical Issues with the Computation of the VIX 4.1 VIX Replication Indices, RX1 and RX2 We first seek to replicate the official VIX index from the underlying SPX options data using the exact CBOE methodology. The first step involves the computation of ˆσ 2 for each relevant maturity. However, the forward price, F, and the strike price just below the forward price, K f, in equation (2) are not directly observable. The CBOE determines these basic variables from the available option quotes in three separate steps. First, at a given point 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, F = K + e r T T [C(K,T ) P(K,T )]. Finally, K f is determined as the first available strike price below F. We refer to the VIX index constructed from (2) and (3), using these values for F and K f, as our first Replication index, or RX1. This index constitutes our direct equivalent of the CBOE VIX, computed according to official rules, from the SPX option quotes in the MDR data set. Our subsequent analysis reveals that this feature of the CBOE methodology entails a certain lack of robustness which, in some instances, has dramatic consequences. It arises from the use of a single (K ) put-call option price pair from which to infer F and K f. Occasional problems, like a temporary gap in the updating for a subset of the option prices or outright recording errors, can cause the minimum distance between the call and put prices 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, 28, at around 9:, for the maturity date July 19, 28. The minimization of the absolute option price differential leads to an implied forward price of F = 139, while the shape of the put and call price schedules suggests a forward price substantially below that value as these curves should intersect at a strike close to the implied forward. 9 Since we obtain an overall good coherence with the official VIX but also, at times, observe very significant discrepancies, it is apparent that our option data set is not always identical to the one used in real-time by the CBOE. We provide further comments on this issue later on. 8

Figure 1: Daily Closing Value for VIX, January 24 - August 21. Our sample period, June 2, 28 - June 3, 21, is indicated in green. Source: CBOE 1 Official Daily CBOE closing VIX 9 8 7 6 5 4 3 2 1 24 25 26 27 28 29 21 9

Figure 2: The CBOE and our Robust Implied Forward Prices: The minimum distance between the mid-quote call (C) and put (P) on 9-June-28 is marked with a red star in the top-right panel and the associated K is indicated on the horizontal axis. This value is used for computing F. In contrast, the robust forward price is obtained as the median of 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. The corresponding put and call price curves are depicted in the left side panels. On this day: F = 139 and F = 1367.5 F Call - Put 12 1 C P 12 1 8 8 Price 6 4 2 6 4 2 13 135 14 145 Strike Price 13 135 14 145 Strike Price F Call - Put 12 12 1 1 8 8 Price 6 4 2 6 4 2 13 135 14 145 Strike Price 13 135 14 145 Strike Price 1

Motivated by this finding, we introduce a slight modification that ensures a more robust procedure for computing the implied forward. In a first step, we determine the full range of 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 implied forward prices. Using a broad set of option prices helps prune the forward price series of major outliers. On the other hand, this wider set includes some less liquid out-of-the-money options for which the relative pricing errors are larger. Hence, the robust implied forward may, in most cases, be more affected by noise than the one determined from a truly near-themoney strike. Consequently, as our final step, we retain the original F value rather than the robust forward, except when the two approaches produce substantially different answers. Following some diagnostic checks, we settled on a threshold value of.5%. 1 Hence, if the two implied forward prices are in reasonable agreement, we stick with F. However, if they differ by more than.5%, we instead adopt the robust implied forward price. Thus, this procedure generates a new implied forward price, F, which typically coincides with F, but deviates whenever the robust value indicates that F is subject to a sizeable error. The new VIX replication index, computed from F, is denoted RX2. Importantly, it retains the feature that, apart from the risk-free interest rate, it is computed exclusively from CBOE option prices, as is the case for the official VIX. Figure 2 illustrates a real example in our sample where the difference between F and F is significant. The bottom row in the figure shows that the robust procedure generates an implied forward price of F = 1367.5 at 9:, June 9, 28, implying a substantial reduction relative to F and driving a sizeable gap between RX2 and RX1 at this point. This is readily apparent from Figure 3 which displays the RX1, RX2 and VIX series from 8:3 to just after 9:. The jump in RX1 at 9: represents an isolated outlier. Both before and just after 9:, RX1 coincides with RX2 and they qualitatively match the evolution of the reported VIX. The fact that VIX does not spike at 9am provides further corroboration of our conjecture that this jump in RX1 reflects a faulty implied forward value induced by an errant option quote in our data base. Another noteworthy (market-microstructure) feature of the upper panel of Figure 3 is the obvious delay in the VIX series relative to our RX indices. We find that the RX series lead the real-time reporting of VIX by about 15 to 3 seconds. This is likely due to capacity constraints for the CBOE server which processes quote and trade data for a large set of derivatives contracts simultaneously. 11. Since this reporting delay fluctuates over time in ways that, given the nature of the current dissemination system, cannot be observed directly or 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 all main qualitative features of the high-frequency VIX series, or whether random delays and other potential sources of noise in the VIX index may distort our inference regarding the properties of the high-frequency implied volatility equity-index series. 1 For a level of the S&P 5 around 1, the threshold is about 5 which is the same order of magnitude as the gap between the strike prices. 11 This 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 panel depicts the RX1, RX2 and VIX indices on June 9, 28. The second and third panels plot the implied forward price over the trading day for the first and second maturity, according to the CBOE methodology (F ) and the more robust procedure (F), respectively. Volatility Indexes: RX1 vs RX2. Day: 9-Jun-28 23.6 23.5 23.4 Prcte 23.3 23.2 23.1 23 RX2 RX1 VIX * 22.9 8:3 9: F (CBOE Rule): Two maturities used for RX 14 138 136 8:3 9: 137 F: Two maturities used for RX 1365 136 8:3 9: 12

Back to June 9, 28, note that the issue of an erroneous implied forward value appears to impact only the single observation discussed above. But significant differences in the two forward prices can also hold during a time interval. This is reported on June 3, 28. Figure 5 reveals a major distortion on this day in the implied forward price, F, for the longer of the two option maturities over an extended period between 1:3 and 12:3. This renders it infeasible to compute the model-free implied volatility and, by implication, RX1 cannot be derived. In contrast, the F values are well behaved and RX2 is readily compiled for the entire day. Since official VIX values are also reported throughout this entire period it 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 jumps and eliminating some gaps from the RX1 index series. However, it is also readily apparent from Figures 4 and 5 that major discrepancies between the official VIX and our RX2 series remain. 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, RX3 The preceding sections describe our practical procedures ensuring that the option and implied forward prices employed in the computations are reliable. The only remaining inputs into the VIX recipe in equations (2) and (3) are the lowest and highest strike prices to include 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. As a result, the relative bid-ask spread rises dramatically and even the midpoint quotes may become poor indicators of the underlying value. In fact, CBOE discards OTM options according to a specific rule: as we move from the forward price and into the out-of-the-money region and first encounters two consecutive strikes with zero bid option prices, we neglect all further out-of-the-money options, irrespective of whether they have a positive bid price or not. Strikes with bid price equal to zero are not considered to compute the VIX, so the extreme strike is the last one with positive bid before the two consecutive strikes with zero bid. Hence, market liquidity and pricing jointly determines the cut-off levels. This rule has intuitively appealing features. For example, if volatility rises substantially and quoted option (bid) prices increase correspondingly, then additional, and newly valuable, options will be included in the VIX calculation. As a result, the computation generally should include all options with non-trivial (bid) valuations while avoiding the potential noise and distortions associated with the quotes on far OTM options that supposedly have only a minor impact on the overall measure. If we ignore this CBOE stopping rule but continue to exclude all option quotes involving a zero bid price, we obtain a broader strike range which may be interpreted as the maximum coverage offered by the prevailing liquidity in the options market. This procedure allows us to construct a broader volatility benchmark index, exploiting the mid-quotes of all OTM options with positive bid prices. We denote this index RX3. As for RX2, we exploit the robust forward price, F, in the calculation of RX3. This alternative index should provide a natural upper bound on the VIX values reported by the CBOE. Moreover, whenever 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 panel depicts the RX1, RX2 and VIX indices on June 9, 28. The second and third panels plot the implied forward price over the trading day for the first and second maturity, according to the CBOE methodology (F ) and the more robust procedure (F), respectively. 25 Volatility Indexes: RX1 vs RX2. Day: 9-Jun-28 24 23 Prcte 22 21 2 RX2 RX1 VIX * 19 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: F (CBOE Rule): Two maturities used for RX 14 138 136 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: F: Two maturities used for RX 138 137 136 135 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 14

Figure 5: CBOE Forward vs Robust Forward in the replication of VIX. The first panel depicts the RX1, RX2 and VIX indices on June 3, 28. The second and third panels plot the implied forward price over the trading day for the first and second maturity, according to the CBOE methodology (F1) and the more robust procedure (F2), respectively. 21 Volatility Indexes: RX1 vs RX2. Day: 3-Jun-28 2.5 Prcte 2 19.5 19 RX2 RX1 VIX * 18.5 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 14 12 1 F (CBOE Rule): Two maturities used for RX 8 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 14 139 138 137 F: Two maturities used for RX 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 15

the two measures will coincide. Finally, since far OTM option prices converge to zero rapidly, RX3 should constitute a sensible benchmark for the official V IX in the majority of cases. In this article we exploit the concept of an effective range, or ER, to gauge the coverage of options embedded within the VIX computation. It is defined formally in equation (4). It exploits an at-the-money implied Black-Scholes volatility measure, ˆσ BS, which is obtained from a linear interpolation of four near-the-money Black-Scholes implied volatilities, extracted from the options with strike prices just above and below the forward price for each of the two maturities closest to 3 calendar days. Strikes K 1 and K n represent the minimum and maximum strikes included in the computation of the model-free volatility index, and T is the time to maturity. The VIX is computed using two different maturities, so we get an effective strike range for each of the two maturities. ER = [ ln(k1 /F ), ln(k ] n/f ). (4) ˆσ BS T ˆσ BS T Assuming the Black-Scholes measure, ˆσ BS, captures the overall variation in volatility across time appropriately, this standardized strike range indicates the degree of strike coverage provided by the quoted equity index options at any given moment. As such, it offers a tool for assessing the effectiveness and coherence of the CBOE rule that terminates the strike coverage upon encountering two consecutive zero bid prices. Figure 6 again plots the official VIX and RX2 series for June 9, 28, but we now also include the RX3 index on the graph, and the corresponding effective range of options available for the volatility index calculation in the panel below. This figure intends to highlight the relationship between changes in the relevant strike coverage and the associated RX2 values for this trading day. The figure shows that between 9: and 1: the effective strike range for RX2 displays abrupt shifts, leading to a dramatic drop in the index values. Next, just prior to 1:, the range widens again and is compatible with the level that was in effect during the early part of the trading day from 8:3-9:, and the RX2 measure jumps back up to a higher plateau where it appears to remain for the rest of the trading day. At the same time, the thick drawn lines in the bottom panel indicate that additional non-zero bid price OTM options were available, but were explicitly excluded from the RX2 computation. Hence, the large shifts in the effective range for RX2 stem from our application of the CBOE stopping rule, governed by the occurrence of two consecutive strikes 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 some of these additional option quotes, thus confirming that there are discrepancies between our data 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 alleviating some of the abrupt drops observed for RX2 due to sudden occurrences or disappearances of two consecutive zero bid prices for some moderately out-of-the-money options and it produces a vastly superior fit to the official VIX, even if there are some notable deviations between 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 and VIX volatility indices. The bottom panel shows the corresponding effective strike range available 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 computation of RX2, while the thicker lines indicate the strike range available from all valid option quotes for the given maturity. The strike ranges for the nearby maturities are indicated by dashed lines while the longer maturity ranges are displayed in continuous lines. 25 Volatility Indexes: RX2 and RX3 vs VIX *. Day: 9-Jun-28 24 23 Prcte 22 21 2 RX3 RX2 VIX * 19 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 4 Effective Strike Range: Two maturities used for RX 2-2 -4-6 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 17

to utilize all available strikes in the construction of model-free volatility. Of course, this specific illustration does not prove the case for RX3 in general. We now turn to a more general 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 Indices We have studied the behavior of two separate model-free replication volatility series, RX2 and RX3, in some detail. It is evident that both do, on occasion, display striking deviations from VIX, but are these deviations significant or frequent? Are we very far to replicate the official V IX? In this subsection we intend to answer these questions. Figure 7 depicts the average Mean Absolute Percentage Error and Median Absolute Percentage Error for each month (bars) and for the full sample (horizontal lines), and for each RX index. The loss functions are computed comparing the indexes at fifteen seconds frequency. The figure shows that we nonetheless match the real-time data for the VIX released by the CBOE reasonably well on average. The level of both RX indices is clearly highly correlated with VIX, as the average discrepancies are less than.5%, while the median deviation for RX2 is less than.2%, and even generally less than.1% over the last ten months of the sample. In order to gain some insight into the sources of discrepancy between the VIX and RX measures, we first determine the proportion of time the indices coincide versus differ, using a.5% relative deviation as the criterion for satisfactory replication. Within the trading day, the VIX is matched by the RX3 about 83% of the time, while the corresponding number for the end-of-day indices is 81%. Hence, more than 8% of the time, the VIX values are compatible with having been computed on the basis of all the quoted options across both maturities. The scenario where VIX is below RX3, but matches RX2 accounts for, respectively, 9% and 12% of the valid observations at the intraday and end-of-day level. Thus, about an additional 1% of the time the VIX is evidently not based on all available 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 to rationalize the reported VIX values. We have 162 high-frequency VIX observations per day (from 8:3:15 to 15:15:) which amounts to an average of around 1335 times when all options seemingly are utilized, but leaves roughly 16 instances where the VIX is close to RX2 but below RX3, and about 125 cases where we cannot readily classify the source of deviation between the VIX and RX indices. 12 As a robustness check, we also tried using a stricter definition for acceptable replication of.25%. This produces qualitatively similar results, as we now rationalize the observed VIX value with RX2 and/or RX3 for 12 As discussed in connection with Figure 3, there is a time-varying delay, averaging somewhere between 15 and 3 seconds, in the official intraday VIX series, likely due to various real-time computational and reporting issues. Since our RX series is constructed using the time stamps for the options quotes in the MDR data, they do not suffer any such delay. Hence, a change in the volatility level will induce an artificial deviation between the VIX and our RX series. Consequently, we also explored the match between the VIX and the 15 or 3 second lagged RX series. While the matched observations increase for both lagged series, the effect is truly marginal. Thus, given the likely presence of reporting delays for the official VIX, we conclude that 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 (%) for the RX2 and RX3 indices. MAPE for each month and for the whole sample (straight lines). The MAPE values are computed as indicated below, while the corresponding median absolute percentage error uses all valid observations within each month to form the median percentage statistic. MAPE = 1 1 T T t=1 V IX t RX t V IXt, RX = RX2,RX3 1.8 Mean Absolute Percentage Error (June 28 - June 21) RX2 RX3 Prctge.6.4.2 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 1.8 Median Absolute Percentage Error (June 28 - June 21) RX2 RX3 Prctge.6.4.2 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 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 and now constitutes only roughly half of the total number of replications. The observed pattern suggests that the VIX series oscillates frequently between the two benchmark 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 RX2 from 8:3-11:, and then for the remainder of the trading day well described by both RX2 and RX3. Hence, in spite of some visible discrepancies between VIX and RX2 over the earlier parts of the day, we still obtain a sense of how the VIX values were obtained. At the end of the day, all indices (RX2, RX3, and V IX ) coincide and it is thus quite transparent how the closing value is set. In the second example, Figure 9, the picture is qualitatively similar although the pattern is reversed, with VIX matched by both indices during most of the 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 rule determining the effective strike range. For these two trading days, it is quite evident that the jumps in the VIX level up to or down and away from RX3 are driven solely by shifts in the effective range and not by any major movement in the overall volatility level, as indicated by the lack of any perceptible change in RX3 which, during this period, is based on 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 displays erratic movements corresponding to abrupt shifts in the range of option strikes utilized in the computation. In many cases jumps do not appear related to any discernible change in market volatility. In contrast, the RX3 measure seems less susceptible to such errant movements 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 volatility measures, while the RX3 index may be a serious candidate. Of course, the fact that the RX3 index uses all available strikes also implies that the measure may be excessively sensitive to the relatively large bid-ask spread of the far out-of-the-money options and that the available effective strike range for RX3 may vary substantially over time as option market liquidity in the tails of the distribution fluctuates. One way to gauge the severity of this problem is to contrast the properties of RX3 with an alternative index which exploits an economically invariant strike range in the computation of a MFIV style index. 5 Corridor Implied Volatility, or CX, Indices 5.1 Defining the CX Indices In order to establish a benchmark from which to assess how much variation in the effective strike range impacts practical model-free volatility measurement, we turn to the concept of corridor implied volatility indices, or CX, introduced by Carr & Madan (1998) and studied empirically in Andersen & Bondarenko (27). The point is to, a priori, fix the range of strikes used for the volatility index computation at a level that provides broad coverage but 2

Figure 8: RX2, RX3 and VIX. The top panel depicts the RX2, RX3 and VIX volatility indices for February 16, 21. The middle panel shows the corresponding effective strike range available 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 trading day. 24.5 Volatility Indexes. Day: 16-Feb-21 24 RX3 RX2 VIX * 23.5 Prctge 23 22.5 22 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 5-5 Effective Strike Range: Two maturities used for RX -1 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 195 19 185 18 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: F 21

Figure 9: RX2, RX3 and VIX. The top panel depicts the RX2, RX3 and VIX volatility indices for March 2, 21. The middle panel shows the corresponding effective strike range available 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 trading day. 2 19.5 Volatility Indexes. Day: 2-Mar-21 RX3 RX2 CX2 VIX * 19 Prctge 18.5 18 17.5 17 16.5 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 5-5 Effective Strike Range: Two maturities used for RX -1 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 1122 112 1118 1116 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: F 22

avoids reliance on excessive extrapolation of noisy or non-existing option quotes for far out-of-the-money options. Clearly, by restricting the range at a more narrow level than is available on the typical trading day, the Corridor Index, or CX, measures will be downward biased 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 in the time series dimension and not subject to random variation due to exogenous shifts in the strike range. Hence, the relative stability of the two indices should speak to the impact of 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 range which ensures that the associated implied volatility measures are compatible over time. We emphasize a few desirable features. First, the forward price constitutes a natural focal point as it defines the ATM position and thus determines the set of OTM call and put options that enter the implied volatility computation. Moreover, the near ATM options are the most valuable and contribute most to the volatility measure. It is therefore desirable to define the corridor through a metric that naturally centers on the forward rate and captures the 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 this simply reflects a more dispersed risk-neutral distribution. Third, the corridor should be as closely linked to observed option prices as possible in order to ensure transparency regarding the construction of the measure. The latter is critical if the measure is to achieve acceptance among market participants as a viable alternative to the existing VIX measure. One common approach to meet approximate some of the desirata stressed above is to measure moneyness relative to the forward price via the Black-Scholes ATM implied volatility. However, this is unsatisfactory in relying on a model-dependent, and clearly misspecified, 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 variation in the shape of the risk-neutral distribution (RND) caused by, e.g., shifts in the skewness or the fatness of the tails. A different and popular approach is to rely directly on the percentiles of the RND. Unfortunately, in the current context, percentile corridors have the significant drawback that they are not centered on the forward price. Due to the pronounced skew in the risk neutral equity-index distribution, the forward price is located anywhere between the 38 th and 47 th percentiles of the distribution with a mean around the 43 th percentile over our sample period. This reflects the fact that the percentiles do not readily translate into a stable relative price for the OTM options associated with the corresponding strikes. Since the option prices are critical determinants of the volatility measures, the instability of the percentiles relative to the ATM strike is a serious concern. In addition, estimation of the RND requires certain auxiliary assumptions and computational conventions that render the approach less than fully transparent to market participants. 13 In response to the shortcomings identified above, we develop an alternative approach which 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 13 The convention of defining corridors via the percentiles of the RND was adopted by Andersen & Bondarenko (27), so the alternative approach developed below is novel. 23

positive random variable, x, with strictly positive density f (x) for all x > by, LT (K) = K (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 given point, 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, or CDF, as it is strictly increasing on (, ) with R() = and R( ) = 1. Moreover, it is centered at the mean value of x as R(K) = 1 2, where K = E[x] = x f (x) dx. Second, for a given point, or percentile, q, in the range of the R(K) function, we define the quotient of the ratio function, K q, as K q = R 1 (q) for any q [,1]. (7) Now, letting f (x) denote the risk-neutral density for the S&P 5 forward price for the maturity date T = 12 1, or one month ahead, and K be the strike price of European style out-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 Europeanstyle put and call options. Hence, as long as we remain within the range where option prices may be reliably extracted from the existing quotes in the market, the value of R(K) is trivial to determine, and it is independent on any estimation procedure for the underlying RND. Moreover, the median, or 5 th percentile, of R(K) corresponds to the expected value of the underlying S&P 5 index at T, because the forward rate is given as the mean of the risk-neutral distribution for the index level, K = F and K.5 = R 1 (.5) = F. (9) This implies that the forward price, F, that separates the strike range into the out-of-themoney 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 appropriately relative to the computation of model-free implied volatility. This is particularly convenient when 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 computation via symmetric percentiles of the R(K) function. For example, we may use the left and right truncation strikes obtained as, K.1 = R 1 (.1) and K.99 = R 1 (.99). (1) 24

As illustrated in Figure 1, the determination of the option prices for a wide range of strikes is quite straightforward from existing option quotes. Only towards the tails of the risk-neutral distribution are the option prices, and thus the distribution itself, hard to ascertain with precision. This issue is circumvented by the R function which only depends on the observable option prices out until the truncation point. Hence, the truncation levels are determined directly from observed prices and largely independent of any extrapolation technique or other exogenous approximation rules. Of course, the intention is also that this provides a stable basis for the computation of a comparable volatility measure over time as the support of the strike range is kept invariant in a specific economic metric. Specifically, we compute the CX1 and CX2 series by exploiting the [.1,.99] and [.3,.97] percentile quotients for the R function. While it is impossible to provide a universal rule for the optimal variation in the range over time, it is instructive to compare the effective ranges implied by the various measures in terms of the corresponding Black-Scholes implied volatilities. Figure 11 displays the coverage of the strike range for the end-of-day volatlity measures in this fashion. It is striking how the RX measures display dramatic variation in their strike coverage across the days in our sample, while the CX measures provide a relatively stable coverage, expressed in terms of the Black-Scholes volatilities. Inspecting the RX coverage more closely, it is further evident that the RX2 series have outliers that largely match those of the RX3 series, but the RX2 displays rather extreme inlier observations as well. Since a narrowing of the strike coverage typically has a more significant impact on the associated variance measure than widening coverage, simply because the options closer-to-the-money carry a relatively heavier price tag, the RX3 series may well be the more reliable one in terms of capturing the intertemporal fluctuations in market volatility. Nonetheless, the strike coverage of the RX3 series remains extraordinarily variable, reaching a low on the out-ofthe-money put side of about 3 and a high of around 13 at the nearby maturity and within the range [ 2 1 2, 12] at the longer maturity. In comparison, the variation in the coverage provided by the CX1 and CX2 series is small, with values fluctuating within the intervals [ 1 1 2, 2 1 2 ] and [ 2, 3 1 2 ], respectively. While this does not prove that the CX measures will be superior in applications, it does raise serious questions regarding the impact of the abrupt shifts in coverage provided by the more traditional RX style of VIX measures. 5.2 Comparing the VIX, RX, and CX Indices We 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 critical features of the stock index return volatility such as the persistence of the series, the prevalence of abrupt moves, or jumps, and the extent of the (negative) correlation of changes in 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 CX measures near indistinguishable, we include only the more narrow corridor measure, CX2, in the illustration below. 25

Figure 1: Determination of the CX Truncation. The top left panel displays the normalized out-of-the-money option prices at the end of trading on June 16, 21 for the thirty day maturity. Moneyness is defined as k = K F, while the normalized option prices are given as M(k) = Q(K) F and Q(K) = min(c(k), P(K)). The top right panel depicts the corresponding Black-Scholes implied volatilities. The bottom right panel provides an estimate of the corresponding risk-neutral density, whereas the left bottom panel shows the extracted R(k) function. The vertical dashed lines indicate the 1,1,5,9, and 99 percentile quotients of R(K). Normalized Option Price M(k) Implied Volatility.25.6.2.5.15.1.4.3.2.5.1.8.9 1 1.1 Moneyness k.8.9 1 1.1 Moneyness k 1 R(k) 8 Risk Neutral Density.8.6 6.4 4.2 2.8.9 1 1.1 Moneyness k.8.9 1 1.1 Moneyness k 26

Figure 11: Effective Strike Range for End-of-Day Implied Volatility Measures. The top panel displays the effective strike range, expressed in Black-Scholes volatilities, for the various implied volatility measures at the end-of-the-day across the entire sample period for the first maturity exploited in the index calculation. The bottom panel provides the corresponding 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 indexes to changes in the effective strike range used in their computation. Figure 12 depicts the evolution of the indices during a highly volatile day in the equity markets, October 14, 28, where the intraday range of the VIX spans values below 5 and above 6. First, we notice the apparent problems experienced in the real-time computation of the official VIX figures. At first, from 8:3-8:4, V IX is frozen at an inexplicably high level of about 55%, 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 RX2 index in the time period 9:4-12:55, and then after an apparent jump at 12:55 coinciding reasonably well with both RX2 and RX3 until about 14:3. Finally, from this point onwards the RX2 drops below RX3 while V IX remains fairly close to the RX3 level until the market close. The reason for the jumps in RX2 at 12:55 and after 14:3 is readily identified from the second panel as the strike range for the nearby maturity widens in the first case and then narrows 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, the VIX series appears indecisive as it first attains a level consistent with RX2 over 9:4-12:55 and then roughly follows the RX3 index for the remainder of the day. Such random oscillation between two distinct volatility levels induces artificial breaks in the V IX series that are unrelated to the underlying equity index, and probably related to a new zero bid price quoted. Moreover, the identical reasoning applies to the RX2 series itself, as it is subject to the same type of jump-like behavior due to unpredictable changes in its strike range. Finally, the RX3 index indicates a strongly elevated volatility level over the period 1:5-1:25 which may also be attributed directly to an expansion of the associated strike range available for the nearby maturity over that time interval. This seemingly artificial shift increases the volatility measure by close to 4%, while none of the other indices displays any major discontinuities over this period. In short, the RX3 measure is also vulnerable to idiosyncratic shifts in the underlying strike range. In contrast, the CX2 measure evolves continuously, albeit also somewhat erratically, throughout the trading day. Moreover, the qualitative pattern nicely mirrors the movements in the RX indices apart from the absence of glaring jumps. In terms of capturing the evolution of volatility across this eventful trading day, the CX index appears to reflect all relevant variation in the RX indices while annihilating their more suspect breaks. Figure 13 depicts a more typical trading day with a distinctly lower volatility level, namely February 9, 21. In this case, the RX2 index replicates the official VIX series very well throughout most of the trading day, although there are a number of striking outliers in both the RX2 and VIX series between 11: and 11:3, and a fairly abrupt drop in RX2 around 3:. In contrast, the RX3 index is more erratic between 1:5 and 11:45 and then again after 15:. Not surprisingly, over these time spans we also observe a widening in the strike range for RX3. As before, CX2 appears to capture the qualitative features of the volatility evolution satisfactorily while avoiding any abnormal jumps. The only sharp movement in the CX2 series, just after 11:4, is associated with a contemporaneous drop 28

Figure 12: VIX, RX2, RX3 and CX. The top panel depicts the VIX, RX2, RX3 and CX volatility indices for October 14, 28. The middle panel shows the corresponding effective strike range available 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 trading day. 62 Volatility Indexes. Day: 14-Oct-28 6 58 56 Prctge 54 52 5 48 46 RX3 RX2 CX2 VIX * 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: Effective Strike Range: Two maturities used for RX 2-2 -4-6 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 16 14 12 1 98 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: F 29

in the S&P 5 forward price, as indicated in the bottom panel, and thus likely represents a genuine change in volatility. For both of these trading days, the evidence is consistent with the view that the CX measure captures an economically invariant portion of the risk-neutral density, so that the fairly 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 of the findings throughout the sample. Hence, we now turn towards a more comprehensive investigation 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, before comparing their general features, and finally the relation between the implied volatility indices and the underlying equity returns. Our motives for focusing on these features are twofold. First, appropriate modeling of volatility dynamics is of major interest in applied work within asset pricing, portfolio choice, risk management and derivatives pricing. One important dimension of this question is the prevalence of large changes, or jumps, in the volatility process and their interaction, or correlation, with shifts in the underlying equity index, especially since long volatility positions serve as a hedge for broader exposures to equity if there is a robust negative correlation between the two. Second, the possibility that random fluctuations in the strike range, used by a given index, induce artificial breaks in the corresponding volatility measure suggests that stark discrepancies in outlier activity across alternative indices may reflect differences in robustness to the quoting patterns and variation of liquidity of the options market. i. Volatility Index Jumps We proceed nonparametrically by defining large moves relative to a robust measure of return volatility for each trading day. In order to alleviate the distortions arising from noise in 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 deviation from the one-minute returns. Then, we delete all observations whose absolute value exceeds six standard deviations and, finally, we recompute the daily index volatility using only the remaining observations. On the basis of this robust measure of the daily standard deviation, we identify outliers for each volatility index at the one-minute frequency across the full sample. Table 1 tabulates the findings. The table reveals a startling discrepancy in the number of large moves across the alternative indices. Irrespective of the threshold adopted for defining the large changes, or returns, the conclusions are identical. For example, including moves beyond six standard deviations on either the upside or downside, we find around 16 outliers in the VIX indices and close to 15 for RX1 and RX2. In contrast, the numbers are approximately 95 for RX3, and below 625 and 56, respectively, for the CX1 and CX2 measures. Given that the sample covers 25 months, all indices display, on average, more than one large move per 3

Figure 13: VIX, RX2, RX3 and CX. The top panel depicts the VIX, RX2, RX3 and CX volatility indices for February 9, 21. The middle panel shows the corresponding effective strike range available 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 trading day. 27 26 Volatility Indexes. Day: 9-Feb-21 RX3 RX2 CX2 VIX * Prctge 25 24 23 22 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 5-5 Effective Strike Range: Two maturities used for RX -1 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 18 F 17 16 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: 31

Table 1: Distribution of Extreme Returns ( Jumps ), 1-min Frequency RX1 RX2 RX3 CX1 CX2 VIX VIX (, 3) 46 47 7 1 24 31 ( 3, 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 117 987 1323 1322 (4, 6) 114 1141 198 1 949 1333 1334 (6, 9) 41 41 352 249 221 444 445 (9, 15) 218 218 11 47 35 24 24 (15, 3) 16 15 33 7 9 12 12 (3, ) 47 48 7 2 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 more telling is the difference in the number of extremely large moves of more than 15 standard deviations. Here, the CX measures have 16 such moves, while the RX3 index has almost six times more and the other indices all sport more than 15 times as many. This is obviously anomalous: any genuinely large shift in volatility should manifest itself in significant elevation of option prices across a broad range of strikes and should thus be reflected in all broadly based model-free volatility indices. This suggests that various sources of measurement error, including the documented idiosyncratic variation in the strike range, are sufficiently 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 often reverse themselves, as an unusual widening of the strike range, say, is followed by a return to the usual range. However, even for the measures that are less prone to this type of error, we observe a near symmetric distribution of positive and negative jumps. This suggests that high-frequency volatility undergoes frequent and abrupt moves in both the positive and negative direction. The evidence of numerous large negative volatility jumps is inconsistent with many popular parametric specifications of asset price dynamics that allow only for 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 its mark on their general distributional characteristics. Table 2 provides an overview of the summary statistics for the alternative volatility measures. The most noteworthy aspect of Table 2 is the huge variation in the kurtosis statistic for the volatility changes, or returns, in Panel 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 31.9 3.51 29.26 31.74 31.74 Std Dev 13.38 13.38 13.43 12.89 12.41 13.43 13.43 Skewness 1.31 1.31 1.3 1.3 1.3 1.31 1.31 Kurtosis 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 1 4 -.8 -.8 -.1 -.12 -.13 -.15 -.15 Std Dev 1 4 39.78 4.18 2.16 2.88 21.96 25.33 31.62 Skewness 14.52 14.5.57.31.23 -.46 -.71 Kurtosis 6886.11 666.4 44.62 27.83 24.49 754.64 3783.17 ρ 1 1 2 1.8 1.79 4.88 4.9 4.4 3.78 3.45 ρ 2,5 1 2 1.97 1.97 2.66 2.66 2.28 2.92 2.8 ρ 6,21 1 2 1.7 1.6 1.29 1.27 1.13 1.5 1.3 ρ 21,45 1 2.8.8.9.13.8 -.2 -.2 Panel A: Percent of missing is 5.28% for RX1-CX2,.3% for V IX, and % for V IX. Panel B: Percent of missing is 5.86% for RX1-CX2,.3% for V IX, and % for V IX. 33

to 28, the value for RX3 is considerably larger at about 45, while they attain rather imposing values of about 755 and 3783for the VIX series, and, finally, truly outsized values of around 66 for the RX1 and RX2 indices. Notice also that mild filtering reduces the kurtosis of VIX sharply relative to VIX. This serves as further indirect evidence that erroneous data are an important source of the inflated kurtosis statistics. Overall, the table is consistent with the view that all series, bar perhaps the CX indices, suffer from excessive and artificial outliers. In spite of these major discrepancies in the properties of the high-frequency changes of the various 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 volatility levels rather than their first differences. Apart from the fact that the CX indices, as a direct consequence of the deliberate truncation, represent a slightly down-scaled version of the model-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 in the mean level and standard deviation, the CX measures capture the identical features of the volatility process as the VIX and replication (RX) indices. This further translates into an extraordinarily high degree of correlation between the index levels, as may be confirmed from 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 distinct categories with similar high-frequency characteristics, namely the RX1 and RX2 measures in one group, the VIX and VIX in another, and the CX1 and CX2 in a third, with the latter also having RX3 loosely associated with it. Most noteworthy is the poor coherence between VIX and RX2 as the latter explicitly is designed to mimic the VIX. Instead, VIX changes correlate better, albeit still very imperfectly, with RX3 and the CX indices. Overall, the evidence confirms that VIX falls somewhere between RX2 and RX3 in terms of its high-frequency behavior. An obvious concern is that random oscillation of the VIX between these two index levels may render it excessively noisy as an indicator of short term movements in model-free implied volatility. Finally, we observe that the implied volatility returns display only weak serial correlation. This is not surprising as the behavior of the implied volatility returns should be akin to those of the VIX futures which are traded on the 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 modest degree of predictability at high frequencies. Otherwise it would violate the semimartingale property which is necessary to rule out arbitrage opportunities. It is worth reflecting on the striking discrepancies between Panels A and B of Table 3. It highlights the point that even exceptionally high levels of correlation do not ensure that the high-frequency dynamic behavior of the series is similar. Instead, the question of whether a given series provides a suitable representation of volatility hinges critically on the application. For example, it is evident that the degree of low frequency correlation of model-free implied volatility with other economic variables may be addressed equally well using 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. RX2 1. 1. RX3.9997.9997 1. CX1.9994.9994.9996 1. CX2.999.999.9992.9999 1. V IX.9997.9997.9997.9995.9991 1. V IX.9997.9997.9997.9995.9991 1. 1. Panel B: Volatility Index Returns RX1 RX2 RX3 CX1 CX2 V IX V IX RX1 1. RX2.996 1. RX3.467.462 1. CX1.475.47.98 1. CX2.46.456.896.957 1. V IX.318.314.528.545.539 1. V IX.318.314.528.545.539 1. 1. 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 volatility indices and the underlying stock index returns. The pronounced negative correlation between daily changes in the VIX and the daily S&P 5 returns have long been noted and serves as an argument for diversifying long equity positions with an exposure to the volatility index. The CBOE web-site provides annual estimates for the sample correlation of the two series in the range of -.75 to -.85 at the daily level for 24-29, with the -.75 estimate referring to the year 29 which constitutes about half of our sample. The origin of such large negative correlations has been much debated in the literature. Among other things, it is not clear whether it arises from a corresponding correlation at the very highest return frequencies (often labeled a leverage effect ) or whether it is generated through a feedback mechanism where negative jumps, say, lead to subsequent elevation of volatility which then increases the required rate of return on the equity index and a drop in stock prices. Although the latter effects may play out following the negative jumps, the aggregation of the high-frequency returns and volatility into daily measures could make them appear contemporaneous. Table 4, Panel A, provides one-minute correlations between VIX returns and returns on the S&P 5 index. This panel exploits the implied forward price obtained from the S&P 5 index options to construct the high-frequency equity return proxy. Since this forward price is also an input into the computation of each of the contemporaneous volatility indices, any noise or error in the forward price will simultaneously impact the implied volatility measures, thus potentially inducing artificial correlation between the measures. Thus, as a robustness check, we also report, in Panel B, the corresponding correlations using the S&P 5 futures returns obtained from the Chicago Mercantile Exchange (CME Group). In this case there might potentially be a slight mismatch in the timing of the volatility and equity index returns which could induce a downward bias in the estimated correlations. Panel A of Table 4 conveys a clear message. For the smaller returns, the correlations with the equity returns are similar across most of the implied volatility indices and strongly negative, coming in at around -.7, except for the VIX indices that display correlations around -.5. For the small jumps, in the (6,9) category, most of the correlations drop noticeably, and the CX indices are now clearly the most negatively correlated with the stock returns. Finally, for the larger volatility jumps, all indices except the CX measures fall off dramatically. Hence, the CX indices are the only ones to display consistent equity return correlations across all size categories for the volatility returns. This is also manifest in the 14 Informally, it may be instructive to think of the indices as fractionally co-integrated so that, even if they all are highly persistent, they move in unison over time. While pronounced deviations from the typical relationship between the series may be observed, for example due to measurement errors, they tend to be short-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 us astray relative to our objectives of the present paper. 36

Table 4: Correlations of Volatility Indexes with S&P 5 Return, 1-Minute Frequency Panel A: S&P 5 Implied Forward RX1 RX2 RX3 CX1 CX2 V IX V IX (, 6) -.69 -.69 -.7 -.72 -.73 -.49 -.49 (6, 9) -.57 -.58 -.56 -.61 -.62 -.48 -.48 (9, ) -.13 -.13 -.36 -.61 -.64 -.24 -.13 (, ) -.34 -.34 -.66 -.7 -.72 -.4 -.32 Panel B: S&P 5 Futures RX1 RX2 RX3 CX1 CX2 V IX V IX (, 6) -.65 -.65 -.66 -.67 -.68 -.51 -.51 (6, 9) -.59 -.59 -.63 -.71 -.7 -.53 -.54 (9, ) -.9 -.9 -.42 -.69 -.76 -.22 -.16 (, ) -.32 -.31 -.62 -.67 -.68 -.4 -.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 CX measures than for the rest. Moving on to Panel B, we obtain qualitatively identical results using the equity futures returns. The findings are compatible with the hypothesis that only the large moves in the CX indices are credible, while many of the outliers in the alternative volatility measures are the result of idiosyncratic errors or random variation in their effective strike range. Since artificial outliers, by definition, have no systematic relation to the contemporaneous equity returns, the dramatic reduction in the correlation measures is a direct implication of this hypothesis. Assuming the CX measures provide a reasonably accurate account of the volatility dynamics, we conclude that the high-frequency intraday correlations between implied volatility and equity index returns largely mirror the values observed at the daily level and the relation is quite stable across different return size categories. This has interesting implications for practical risk management and derivatives pricing as well as the more general specification of asset price models in continuous time. 15 iv. Regression-Based Evidence 15 Moreover, our results, based on model-free option implied volatility measures, nicely complement the high-frequency S&P 5 futures study by Bollerslev, Litvinova & Tauchen (26) who also conclude that the evidence favors the leverage type effect but rely on absolute five-minute equity returns as their volatility proxy. 37

Finally, we investigate the asymmetric return-volatility relation through a complimentary approach based on a robust regression framework. This facilitates formal testing of the significance of various features within a uniform setting. Denoting a generic implied volatility index V X, we define the i th one-minute volatility and equity-index returns as v i = log(v X i /V X i 1 ) and r i = log(s&p5 i /S&P5 i 1 ), respectively. The strength of the contemporaneous return-volatility effect may then be assessed through the simple regression, v i = α + β r i + γ r i 1I(r i < ) + ε i, i I g. (11) This regression includes two coefficients β and γ that capture the general negative volatility-return relation (or leverage effect) and the asymmetries in the effect for positive versus negative returns, respectively. Moreover, by restricting the collection of one-minute returns to those falling within a specific set I g, we may investigate whether the relation is stable across different scenarios of interest. However, we first concentrate on the full sample 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 a large and highly significant leverage effect for all indices, and the γ coefficients are statistically significant and negative in most cases, even if the magnitude is dramatically smaller than for the direct leverage effect. In other words, the basic leverage effect is pronounced and slightly asymmetric so that negative returns of a given magnitude are associated with an increase in volatility that exceeds the reduction in volatility associated with positive returns of the same magnitude. Moreover, the constant term, α, is generally negative but of trivial economic size. Finally, the R 2 statistics differ widely, with the group of CX1, CX2 and RX3 providing a much higher degree of explanatory power than the VIX measures which again dominate the direct replication measures, RX1 and RX2. This ranking is also consistent with the relative significance of the leverage coefficient, β, across the volatility index regressions. Again, this supports the interpretation that the CX indices succeed in maintaining the intertemporal coherence in the volatility series which, in turn, allows the leverage effect to stand out more prominently in the regressions. In particular, if the basic RX and VIX series are contaminated by erroneous jumps, then a number of large volatility returns will be unrelated to the underlying index returns and the associated β coefficients will be downward biased, while the R 2 deteriorate correspondingly. In order to illustrate the potential sources of discrepancy in the findings across volatility indices, we split the observations into three sets, corresponding to whether the size of the volatility return falls into the categories (,6) (NJ, or No Jump ), (6,9) (SJ, or Small Jump ) or (9, )) (LJ, or Large Jump ), exactly as done in Table 4 as well. Since the term corresponding to γ generally becomes insignificant within each of these groupings and overall has an immaterial impact on the slope estimates, we exclude the corresponding asymmetric term in the leverage effect from the regressions, thus in effect zeroing out the γ coefficients. Figure 14 now depicts the resulting regression lines arising from regressions of the S&P 5 returns on the volatility index returns. Visually, they fully corroborate our prior conclusions. The regressions involving RX2 and VIX have low slopes for the volatil- 38

Table 5: OLS regression onto S&P 5 Return, 1-Minute Frequency Panel A: S&P 5 Implied Forward RX1 RX2 RX3 CX1 CX2 V IX V IX α 1 4 -.42 -.42 -.47 -.45 -.47 -.27 -.21-4.58-4.57-4.42-4.9-4.16-3.89-2.58 β -1.62-1.63-1.58-1.76-1.91-1.19-1.2-58.96-58.97-44.48-46.35-47.93-45.68-42.2 γ -.14 -.14 -.15 -.13 -.14 -.7 -.4-3.11-3.1-3.7-2.63-2.63-2.19-1.19 R 2.12.11.43.49.52.16.1 Panel B: S&P 5 Futures RX1 RX2 RX3 CX1 CX2 V IX V IX α 1 4 -.29 -.3 -.39 -.39 -.39 -.28 -.23-3.4-3.7-5.22-5.18-5.28-3.69-2.71 β -1.41-1.41-1.38-1.54-1.66-1.13-1.15-67.48-67.44-83.5-84.1-88.52-49.63-43.48 γ -.7 -.8 -.1 -.9 -.9 -.5 -.3-2.37-2.39-3.31-3.3-2.97-1.64 -.83 R 2.1.1.39.44.46.16.1 Notes: For each coefficient, the second row reports corresponding t-statisics, which are HACcorrected with 2 lags. 39

ity jump returns due to the numerous large volatility index moves without any counterpart in unusual equity index returns. In contrast, the regressions involving the RX3 jump returns dramatically reduce the scattering of observations along the x-axis and the regression slopes become much more pronounced. Finally, the in the case of the regressions involving CX2, almost all large volatility return observations are accompanied by large equity-index returns in the opposite direction. As a result, all the associated scatter plots are clustered around steeply, negatively sloped lines. In summary, the CX series contain much fewer abrupt movements than the other candidate implied volatility indices, but when the CX indices do jump these changes, or volatility returns, are very strongly, and negatively, related to the returns in the stock market. 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 underlying S&P 5 returns. This suggests they are induced by deficiencies in the real-time data collection, processing and dissemination procedures rather than from true jumps in option implied volatility. In particular, these events are not associated with any broad-based increase in option prices across the strike range - as this also would invoke significant reactions in the CX and RX3 series. Hence, they are likely attributable to idiosyncratic errors or unwarranted variation in the effective strike range underlying their computation. 6 Conclusions We have pursued two goals. First, we seek to replicate the official real-time fifteen second VIX series from tick-by-tick quotes on S&P5 option prices obtained from the official CBOE vendor. The purpose is to gauge the reliability of high-frequency movements in the VIX index and understand which features of the real-time VIX series are robust to noise and measurement error issues. This first step has provided us with valuable insights into potential data problems plaguing the VIX series. We identify a particularly important deficiency in the CBOE methodology: the rule for determining the effective strike range used to compute the volatility index is not robust to fairly common data errors or random shifts in market liquidity. This induces spurious jumps in VIX as well as any alternative index constructed 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 characteristics of the VIX return series. Importantly, the identical issue affects the daily closing value of the VIX reported by the CBOE and cited widely in the financial press. Day-to-day changes in 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. In other words, all the implied volatility measures considered convey almost identical messages concerning the evolution in the overall level of volatility. The critical discrepancies arise only when focusing on the changes in the indices at a daily or high-frequency intraday level. Our second goal is to demonstrate that one may construct corridor implied volatility style indices, CX, that retain the theoretical advantages of the model-free volatility concept while avoiding the pitfalls associated with the official VIX series. These CX indices may 4

Figure 14: Volatility Index Return vs. S&P5 Return Regressions. This figure depicts the regressions between one-minute returns on the volatility indices and the underlying equity futures index where we decompose the (absolute) volatility returns into: Small Moves or No Jump, NJ (returns fall within the (,6) group), Small Jump Returns, SJ (fall within the (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 computational rule. The main change from the VIX is simply the determination of a different truncation point for when to exclude options far out of the money in the computation of the index. By ensuring intertemporal consistency in this dimension, we obtain CX indices that cover economically equivalent parts of the strike range. Consequently, they are internally consistent 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 the amount of spurious jumps in the volatility index are dramatically reduced, if not entirely eliminated, and the jump returns now consistently display a pronounced negative relation to the underlying equity index returns.in short, the CX and VIX return series display strikingly different time series characteristics along important dimensions that are of relevance for active real-time financial decision-making and risk management. The same issues afflict the widely used daily VIX series which may be obtained from the CBOE website. The daily changes, or volatility returns, implied by this official VIX series are inherently noisy and appropriate caution should be exercised if analysis is based on those daily shifts in the volatility index. For many purposes, our CX measures represents a superior alternative and the 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 construction of implied volatility measures. One important application of such measures is the identification and modeling of the equity variance risk premium. This premium is given by the wedge between the (expected) realized variance of the equity index returns and the option implied variance measure over the corresponding horizon. In other words, it represents the discrepancy between the actual return variation and the pricing of this return variation embodied in the (squared) option implied volatility measure. In the literature, this is most often approximated by the difference between an (expected) realized variance measure, constructed from the cumulative sum of squared high-frequency equity-index returns, 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 measure of implied volatility, the quality of the risk premium measures are similarly degraded. In this case, however, the CX indices do not represent a natural alternative. In theory, the implied variance measure should captures the entire theoretical strike range for the equity returns, from zero to infinity. Hence, using (squared) CX measures in this capacity would result in a downward biased estimate of the theoretical implied variance measure, as the CX measures effectively, and consistently, exclude far OTM options from the computation. In comparison, the VIX measure typically provides a broader, but also inconsistent, coverage of the tails, depending on the truncation point determined by the configuration of non-zero bid options quotes at any point in time. Hence, on average, the VIX provides a less biased implied 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 to estimate reliable prices for far out-of-the-money options so that the effective strike range is expanded to ensure that all non-trivial contributions to the measure may be included. Since observed quotes for far OTM options are unreliable in this regard, such procedures hinge on specific assumptions about the shape and fatness and the risk-neutral distribution. One interesting approach towards this problem is provided by Bollerslev and Todorov (21a), 42

and it is also the subject of ongoing work in Andersen and Bondarenko (211). Second, one may resort to exploring the variance risk premium only within ranges, or corridors, of the strike range for which we can reliably measure both the model-free implied variance and the corresponding realized return variation of the underlying equity-index returns. This approach provides a decomposition of the variance risk premium into segments of the return space, thus enabling direct identification of the size of the variance risk premium across different equity return states. For example, one may explore how much of the variance risk premium stems from high premiums on the return variation in states where the equity prices have declined over the given horizon, or one may investigate whether the variance risk premium is positive also for states where equity prices have been rising. This approach is explored in detail in Andersen and Bondarenko (21). A second important avenue for future research is to enhance our understanding of the nature of the random variation in the quality and strike coverage of the SPX option quotes that 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, the electronic data feed providing the official SPX quotes do not stem directly from this market but are instead constructed from real-time quotes by two lead market maker firms at the CBOE as well as customer and broker orders listed on the same electronic platform. It is evident that the electronically disseminated quotes generally have significantly wider spreads than the effective bid-ask spreads prevailing in the SPX trading pit. By studying the 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 well as spurious jumps into the real-time released VIX measures. As a consequence, improved procedures for extracting the relevant information from the real-time option quotes may be within reach. Gonzalez-Perez (211) is currently exploring the relevant microstructure features of the SPX option market and the associated electronic quoting mechanism in order facilitate such direct inquiry into the origins of the apparent distortions in the VIX index. 43

References Andersen, T.G. & O. Bondarenko. 27. 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. 29. Expected Stock Returns and Variance Risk Premia. The Review of Financial Studies 22:4463 4492. Bollerslev, T., J. Litvinova & G. Tauchen. 26. Leverage and Volatility Feedback Effects in High-Frequency Data. Journal of Financial Econometrics 4:353 384. Bollerslev, T. & V. Todorov. 21. Jumps and Betas: A New Theoretical Framework for Disentangling and Estimating Systematic Risks. Journal of Econometrics; 157:22 235. Bondarenko, O. 27. Variance Trading and Market Price of Variance Risk. Working Paper, University of Illinois, Chicago. Britten-Jones, M. & A. Neuberger. 2. Option Prices, Implied Price Processes, and Stochastic Volatility. Journal of Finance 55(2):839 866. Carr, P. & D.B. Madan. 1998. Option Valuation Using the Fast Fourier Transform. Journal of Computational Finance 2:61 73. Carr, P. & L. Wu. 29. 22:1311 1341. Variance Risk Premiums. The Review of Financial Studies Dupire, B. 1993. Model Art. Risk 6(9):118 124. Dupire, B. 1996. A Unified Theory of Volatility. Working Paper, BNP Paribus. Dupire, B. 24. A Unified Theory of Volatility. Derivatives Pricing: The Classic Collection. (London: Risk Books). Ed. by P. Carr. Jiang, G.J. & Y.S. Tian. 25. The Model-Free Implied Volatility and Its Information Content. The Review of Financial Studies 18 (4):135 1342. Jiang, G.J. & Y.S. Tian. 27. Extracting Model-Free Volatility from Option Prices: An Examination of the VIX Index. Journal of Derivatives Spring:1 26. Neuberger, A.J. 1994. The Log Contract. Journal of Portfolio Management 2:74 8. Todorov, V. 21. Variance Risk Premium Dynamics: The Role of Jumps. The Review of Financial Studies 23:345 383. Todorov, V. & G. Tauchen. 21. Activity Signature Functions for High-Frequency Data Analysis. Journal of Econometrics 154:125 138. 44

A Data Filtering SPX option classes We acquired the MDR (Market Data Retrieval) data for S&P 5 options from the CBOE subsidiary, Market Data Express (http://www.marketdataexpress.com/). The MDR data set includes tick-by-tick quotes and transactions throughout the trading day for all option classes issued by the CBOE on the S&P 5. Each option class is characterized by a three or four letter code. For example, the JXA, JXB, JXC, JXD and JXE classes are 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 in their computation of the VIX over our sample period, namely those belonging to the SPB, SPQ, SPT, SPV, SPX, SPZ, SVP, SXB, SXM, SXY, SXZ, SYG, SYU, SYV and SZP categories. The latter are generally known as SPX equity options and they mature on the Saturday immediately following the third Friday of the expiration month (see http://www.cboe.com/products/equityoptionspecs.aspx). Henceforth, we refer to 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 all available strikes for two different maturities for each trading day, starting at 8:3:15 CST and ending at 15:15: CST. This produces up to 162 bid-ask quote pairs per day for a given option. Since more than one hundred distinct strikes routinely are quoted for each maturity, the amount of quote data stored is quite substantial. The raw tick data are at times rather noisy and include both typos and errors. Hence, we rely on a variety of filters to ensure 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 quote pair for all call and put options at the two closest-to-maturity expiration dates. 16. However, the time difference between the MDR option quote time-stamp and our time grid point must be less than five minutes to be included in our quote series. If this timeliness condition is violated we tag the SPX quote as missing. Figure 15 plots the percentage of missing data after applying this filter. On average, we remove around 2% of the initial data due to this specific form of staleness. As is evident from the figure, put option quotes are more affected 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 16 In line with the CBOE methodology for the VIX computation, we replace the first (second) maturity options by the second (third) maturity options when the time to maturity on the closest-to-maturity option is less than seven calendar days 17 The peaks in Figure 15 for July 3, November 28, and December 24 reflect partial Holidays where trading was 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 asynchrony filter 8 7 6 5 4 3 2 1 First maturity options Call Put Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 8 Second maturity options 7 6 5 4 3 2 1 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 46

Finally, we note that part of our analysis is based on a one-minute, rather than fifteen second, sampling frequency. However, the 45 potential one-minute daily quotes for each option are obtained from the underlying higher frequency series by retaining the data at each one minute mark. As such, these lower frequency quote observations are subject to the 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 turn towards that task. Given our focus on quotes, a natural starting point is a scheme designed to detect erratic quoting behavior. To motivate our specific implementation, we note that the CBOE imposes an upper bound on the spread of option quotes. These maximum allowable spreads are specified as a step function which, to a first approximation, constitutes a concave increasing function in the bid quote. Hence, as we move further into-the-money and the options become more valuable, the maximum spread generally increases in absolute terms but declines in relative, or percentage, terms. Moreover, during our sample period, this maximum quote schedule reaches its highest value for bid quotes in excess of $2 and remains fixed thereafter. Finally, in the vast majority of cases, the actual quoted spreads for deep-in-the-money options coincide with this upper bound. This feature is particularly useful when this maximum spread size is not known, as we then effectively can infer it from the actual quoted spreads on the deep in-the-money options. 18 Inspired by the features of the general quoting schedule, we construct a conservative filter for unreliable quote spreads. First, we classify our SPX options into four categories according to their expiration date (first versus second maturity) and option type (put versus call). We then implemented our spread filter through a sequence of steps: Define B i,t (K,T ) and A i,t (K,T ) as the bid and ask prices quoted at the grid point i on day t for an SPX option with strike K and maturity time T. The associated spreads are, S i,t (K,T ) = A i,t (K,T ) B i,t (K,T ). At each fifteen second time grid point, define the mode, s i,t, of the quoted spreads with bid quotes above $2 within each of the four option categories. Likewise, define the mode, S t, of all quoted spreads with bid quotes above $2 for each option category across the full trading day. Next, we define the maximum of these spread modes across the four option categories for each grid point, s i,t, and for the daily spread modes, St. Finally, the spread bound indicator, SB i,t, is obtained at each fifteen second grid point as the maximum of s i,t, and S t. Hence, at a given point in time, the spread bound reflects the highest typical spreads observed across the option categories at that moment as well as the maximum representative spreads observed across the entire trading day. 18 During the turbulent market conditions in late 28 and early 29, the market makers were repeatedly offered 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 SB i,t as a benchmark, we now define a step function for the maximum spread, SM i,t, across the different bid quote ranges employed by the CBOE. 19 As indicated in equation (12), we add $.25 to the benchmark spread to allow for relatively minor violations before we eliminate any quote pair. All quotes with spreads exceeding those indicated below are then deemed erratic and removed from our data set. Finally, we also eliminate quotes with zero or negative spreads. (SM i,t ) = 1 (SB i,t + 1/4) if B i,t (K,T ) > 2 4/5 (SB i,t + 1/4) if B i,t (K,T ) (1,2] 3/5 (SB i,t + 1/4) if B i,t (K,T ) (5,1] 2/5 (SB i,t + 1/4) if B i,t (K,T ) (2,5] 1/4 (SB i,t + 1/4) if B i,t (K,T ) (,2] Figure 16 displays the auxiliary variables s i,t and S t used to construct SB i,t. The localization of the bound, associated with using the spread mode computed at a specific point in time, s i,t, is critical in retaining a sufficient number of option quotes across all trading periods when volatility shifts abruptly within the day. On such occasions, the spreads may shift to a higher plateau during a small part of the day, and spread relief may even have been offered at some unknown point in time. Our implementation adapts to such features by letting the prevailing spread modes determine the bounds. This spread filter completes the description of our cleaning of the individual option quotes. Figure 17 depicts the percentage of missing option quotes after the individual quote filters described in this section have been applied. One implication of removing quotes associated with erratic spreads is that we may introduce some additional asynchrony into our data set. The grid point associated with an erratic quote is designated as missing only if there are no existing quotes for this option available over the prior five minutes. Otherwise, the removed quote will, in accordance with our maximum delay rule, be replaced by the last valid quote. Such a quote could thus represent 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 the fifteen second grid where there also is a potential time gap of up to five minutes, this substitution for erratic quotes may induce a time delay of up to ten minutes. This staleness in quotes may impact the quality of our volatility measures. We control for these effects as part of our additional screening procedures described in Appendix B. (12) B Quality Control for Implied Volatility Measures Appendix 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 ascertain whether the option quotes, available at a given fifteen second grid point, are of sufficient 19 This 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. 4 Intraday and Daily Auxiliary Variables: s * i,t and S* i,t 35 3 s * i,t S * i,t 25 2 15 1 5 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 4 Intraday Threshold for the Bid-Ask Spread 35 3 25 2 15 1 5 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 49

Figure 17: Percentage of missing quotes in the original data versus after filtering. 2 1 Call and First Maturity Options Original After Filters Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 2 Put and First Maturity Options 1 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 2 Call and Second Maturity Options 1 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 4 Put and Second Maturity Options 2 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 5

quality to ensure reliable computations of implied model-free volatility measures. If the quotes are found lacking for a particular point in time, we deem the corresponding volatility measures to be not available. Our quality criteria focus on two aspects, namely timeliness, as captured by the overall degree of quote staleness, and pricing coherence, as captured by the extent to which the basic no-arbitrage conditions are violated. Systemic Staleness in quotes By far, the most influential options for the computation of model-free implied volatility measures are the out-of-the-money (OTM) put and call options with strikes closest to the at-the-money (ATM) strike which is given by the forward price. Hence, we focus on the set of twenty OTM put and twenty OTM call options closest to ATM, as well as the ATM call 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 where a period of five minutes or more transpires without any quote update among the pivotal options in the bid or the ask price at either of the two maturities. This dummy variable equals one at such times and zero otherwise. We provide a depiction of the percentage of the trading day impacted by such staleness in Figure 18. When we encounter such lack of activity for all options within one of our four pivotal option categories, we classify the entire inactive period of five minutes or more as unreliable for computing the volatility indices. Coherence of option quotes No-arbitrage conditions imply that the option prices, for a given maturity date, must be convex in the strike price. We obtain the option prices as the average of the bid and ask quotes. Next, we construct a non-convexity index to indicate the degree of violation of the no-arbitrage condition for a given time grid point. The index is constructed by first computing the (discrete) second derivative of each option price with respect to the strike price using standard discrete approximation techniques. Second, we identify all the (absolute) violations as given by second derivatives that are negative and cumulate them into 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 as such violations should induce arbitrageurs to enter the market and profit from the relative inconsistency in option valuations. Consequently, we treat such scenarios as indicating inferior quote quality and classify the associated volatility measures as not available. Figure 19 displays the non-convexity indicator across the sample period. The vertical line represents our chosen threshold so all time grids for which the indicator value reaches beyond 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 option categories 52

Figure 19: Non-convexity indicator recorded every 15 seconds. June 28 - June 21.4 First Maturity (SPX Options).35.3.25.2.15.1.5 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1.4 Second Maturity (SPX Options).35.3.25.2.15.1.5 Jun8 Sep8 Dec8 Mar9 Jun9 Sep9 Dec9 Mar1 Jun1 53