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 preva-lence 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.

Figure 10: Determination of the CX Truncation. The top left panel displays the normal-ized out-of-the-money option prices at the end of trading on June 16, 2010 for the 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 correspond-ing 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, 10, 50, 90, and 99 percentile quotients of R(K).

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.

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, 2008, where the intraday range of the VIX spans values below 50 and above 60. First, we notice the apparent problems experienced in the real-time computation of the official VIX figures. At first, from 8:30-8:40, 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 9:40-12:55, and then after an apparent jump at 12:55 coinciding reasonably well with both RX2 and RX3 until about 14:30. 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 14:30 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 9:40-12:55 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 pe-riod 10:05-10:25 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 arti-ficial 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 vul-nerable to idiosyncratic shifts in the underlying strike range. In contrast, the CX2 measure evolves continuously, albeit also somewhat erratically, throughout the trading day. More-over, 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, 2010. 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 11:00 and 11:30, and a fairly abrupt drop in RX2 around 3:00. In contrast, the RX3 index is more erratic between 10:50 and 11:45 and then again after 15:00. 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 11:40, is associated with a contemporaneous drop

Figure 12: VIX^∗, RX2, RX3 and CX. The top panel depicts the VIX^∗, RX2, RX3 and CX volatility indices for October 14, 2008. 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.

Volatility Indexes. Day: 14-Oct-2008

Prctge

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 46

48 50 52 54 56 58 60 62

RX3 RX2 CX2 VIX^*

Effective Strike Range: Two maturities used for RX

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

-4 -2 0 2

F

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 980

1000 1020 1040 1060

in the S&P 500 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 in-dices 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 devi-ation from the one-minute returns. Then, we delete all observdevi-ations 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 alterna-tive indices. Irrespecalterna-tive of the threshold adopted for defining the large changes, or returns, the conclusions are identical. For example, including moves beyond six standard devia-tions on either the upside or downside, we find around 1600 outliers in the VIX indices and close to 1500 for RX1 and RX2. In contrast, the numbers are approximately 950 for RX3, and below 625 and 560, 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

Figure 13: VIX^∗, RX2, RX3 and CX. The top panel depicts the VIX^∗, RX2, RX3 and CX volatility indices for February 9, 2010. 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.

Volatility Indexes. Day: 09-Feb-2010

Prctge

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 22

23 24 25 26 27

RX3 RX2 CX2 VIX^*

Effective Strike Range: Two maturities used for RX

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

-5 0 5

F

8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 1060

1070 1080

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

RX1 RX2 RX3 CX1 CX2 VIX^∗ VIX

(−∞, −30) 46 47 7 1 0 24 31

(−30, −15) 99 97 29 6 7 91 93

(−15, −9) 193 195 91 45 34 189 191

(−9, −6) 384 383 333 266 251 512 512

(−6, −4) 1178 1179 1125 1017 987 1323 1322 (4, 6) 1140 1141 1098 1000 949 1333 1334

(6, 9) 401 401 352 249 221 444 445

(9, 15) 218 218 101 47 35 204 204

(15, 30) 106 105 33 7 9 102 102

(30, ∞) 47 48 7 2 0 21 28

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

trading day while the VIX measures top the count with close to three per day. Even 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 obvi-ously 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 mea-surement 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 er-ror, 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 inconsis-tent 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

Table 2: Summary Statistics, 1-Minute Frequency Panel A: Volatility Index Levels

RX1 RX2 RX3 CX1 CX2 V IX^∗ V IX Mean 31.82 31.82 31.90 30.51 29.26 31.74 31.74 Std Dev 13.38 13.38 13.43 12.89 12.41 13.43 13.43 Skewness 1.31 1.31 1.30 1.30 1.30 1.31 1.31 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 ×10⁴ -0.08 -0.08 -0.10 -0.12 -0.13 -0.15 -0.15 Std Dev ×10⁴ 39.78 40.18 20.16 20.88 21.96 25.33 31.62

Skewness 14.52 14.05 0.57 0.31 0.23 -0.46 -0.71

Kurtosis 6886.11 6660.04 44.62 27.83 24.49 754.64 3783.17

ρ₁× 10² 1.80 1.79 4.88 4.90 4.04 3.78 3.45

ρ_2,5× 10² 1.97 1.97 2.66 2.66 2.28 2.92 2.80

ρ_6,21× 10² 1.07 1.06 1.29 1.27 1.13 1.05 1.03

ρ_21,45× 10² 0.08 0.08 0.09 0.13 0.08 -0.02 -0.02

Panel A: Percent of missing is 5.28% for RX 1-CX 2, 0.03% for V IX^∗, and 0% for V IX . Panel B:

Percent of missing is 5.86% for RX 1-CX 2, 0.03% for V IX^∗, and 0% for V IX .

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 6600 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. Over-all, 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 be-tween 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

Table 3: Correlations, 1-Minute Frequency Panel A: Volatility Index Levels

RX1 RX2 RX3 CX1 CX2 V IX^∗ V IX

RX1 1.0000

RX2 1.0000 1.0000

RX3 0.9997 0.9997 1.0000

CX1 0.9994 0.9994 0.9996 1.0000

CX2 0.9990 0.9990 0.9992 0.9999 1.0000

V IX^∗ 0.9997 0.9997 0.9997 0.9995 0.9991 1.0000

V IX 0.9997 0.9997 0.9997 0.9995 0.9991 1.0000 1.0000 Panel B: Volatility Index Returns

RX1 RX2 RX3 CX1 CX2 V IX^∗ V IX RX1 1.000

RX2 0.996 1.000

RX3 0.467 0.462 1.000

CX1 0.475 0.470 0.908 1.000

CX2 0.460 0.456 0.896 0.957 1.000

V IX^∗ 0.318 0.314 0.528 0.545 0.539 1.000

V IX 0.318 0.314 0.528 0.545 0.539 1.000 1.000

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 be-tween daily changes in the VIX and the daily S&P 500 returns have long been noted and serves as an argument for diversifying long equity positions with an exposure to the volatil-ity index. The CBOE web-site provides annual estimates for the sample correlation of the two series in the range of -0.75 to -0.85 at the daily level for 2004-2009, with the -0.75 estimate referring to the year 2009 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 aggre-gation 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 500 index. This panel exploits the implied forward price obtained from the S&P 500 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 volatil-ity 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 500 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 -0.70, except for the VIX indices that display correlations around -0.50. 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

14Informally, it may be instructive to think of the indices as fractionally co-integrated so that, even if

14Informally, it may be instructive to think of the indices as fractionally co-integrated so that, even if