# Implied Variance and Market Index Reversal

## Transcript Of Implied Variance and Market Index Reversal

Implied Variance and Market Index Reversal

Christopher S. Jones Marshall School of Business University of Southern California [email protected]

Sung June Pyun Marshall School of Business University of Southern California [email protected]

Tong Wang Pamplin College of Business

Virginia Tech University [email protected] March 2016

Abstract We ﬁnd that the S&P 500 Index, its futures contracts, and its most popular ETF display strong evidence of daily return reversal when implied variance is high. This negative serial covariance is mostly observed following negative market returns, while positive returns show a moderate tendency to continue when implied variances are high. Price changes around the close and the open show a somewhat stronger tendency to reverse, though the serial covariance we document generally takes longer than a day to resolve. Furthermore, reversal tend to be stronger when option open interest is high, and the presence of return reversal has a signiﬁcant eﬀect on the performance of option trading strategies. Our results have important implications for how market participants interpret implied variances, which due to serial covariance provide an extremely biased forecast of longer-horizon return variances.

JEL classiﬁcation: G12, G13, G14. Keywords: Implied volatility, return reversal, liquidity.

1

1 Introduction

Option markets have the potential to provide extremely useful information to investors, even those who are not interested in trading options. An obvious example is an asset’s implied volatility, which may be valuable to an investor making a risk/reward calculation involving that asset. A precondition is that the implied volatility is reasonably correlated with actual return volatility, and that any bias in implied volatility be stable and therefore correctable.

Starting with the studies of Day and Lewis (1992) and Lamoureux and Lastrapes (1993), the empirical options literature has shown that implied variances from equity index options are highly informative predictors of future realized variances.1 At the same time, they display a systematic upward bias, and furthermore display a degree of variation that appears to be greater than that of actual ex ante variance, results that may be interpreted as reﬂecting the existence of risk premia on jumps and/or volatility. Fortunately, under plausible assumptions this bias can be well captured by a simple linear adjustment, as Chernov (2007) demonstrates empirically.

Evaluation of implied volatility forecasts have, since the inception of this literature, generally evaluated those forecasts on the basis of their ability to predict variances computed from all daily returns realized over the remaining life of the option. This choice is natural. Since the seminal work of Merton (1980), it has been well known that higher frequency data can be used to construct a more accurate proxy of a latent volatility process. Thus, forecast evaluation using daily realized variances should provide far more power than, say, forecasting the squared monthly return. Forecast evaluation based on realized variances computed from intra-day returns, an approach used by Blair et al. (2001), in theory has even greater informativeness, though microstructure issues complicate implementation and interpretation.

Under the assumption that returns are serially uncorrelated, a variance forecast that is constructed be an unbiased predictor of realized variance from daily returns will oﬀer a similarly unbiased forecast of monthly return variance. This is important given that many uses of variance

1Somewhat contradictory evidence was presented by Canina and Figlewski (1993), who found that implied variances are essentially uncorrelated with future realized variances. The unrepresentative result of this paper is explained by Christensen and Prabhala (1998).

1

forecasts, for example in portfolio optimization, require that the variance forecast’s horizon correspond to the anticipated holding period. Traditionally, this assumption has been accepted without very much scrutiny.

There is a long list of studies, however, that challenge the assumption of zero autocorrelation. One set focuses on the ﬁnding of positive index autocorrelation, ﬁrst identiﬁed by Lo and MacKinlay (1988) and Conrad and Kaul (1988) in studies of the CRSP market index. Campbell et al. (1993) observed that the autocorrelation was decreasing in the level of trading volume. They further hypothesized that autocorrelation should decrease with volatility, which was conﬁrmed by LeBaron (1992).

The diﬃculty in interpreting these results is that they could be due to the eﬀects of asynchronous prices, whereby the closing prices of diﬀerent securities are indicative of values at diﬀerent points in time. This has been shown by Fisher (1966) to produce spurious autocorrelation at the index level even when individual security returns are serially independent, and it is consistent with the ﬁnding from these papers that autocorrelations are smaller in value weighted portfolios than in equally weighted portfolios, which put more weight on stocks whose closing prices are more likely to be stale. Furthermore, Ahn et al. (2002) show that the positive autocorrelations observed in the so-called ‘cash’ market indexes are largely absent from futures returns, again suggesting that serial correlation is likely a ﬁgment of microstructure eﬀects and not something that is in any way exploitable by traders.

More recently, however, a group of studies has documented new evidence of serial correlation in stock market returns, but in this case the autocorrelation is negative and therefore inconsistent with stale prices. Bates (2012), for example, reports that the value weighted CRSP portfolio displayed signiﬁcantly negative autocorrelations for several years during the U.S. Financial Crisis. Etula et al. (2015) ﬁnd strong evidence of short-term return reversals in U.S. and international stock market indexes around the end of the month, a pattern that they attribute to the concentration of institutional trading at that time. Chordia et al. (2002) demonstrate that aggregate order imbalances, which are positively correlated with contemporaneous returns, negatively predict nextday market returns. Ben-Rephael et al. (2011) show that daily ﬂows in mutual funds generate a

2

similar pattern of reversal in the Israeli stock market. The primary goal of our paper is to reassess the ability of implied variances of S&P 500 Index

options to forecast future realized variances at diﬀerent horizons. Speciﬁcally, we are interested in whether implied variances are as useful in forecasting longer-horizon (e.g. monthly) realized variance as they are in forecasting short-run (e.g. daily) variances, where any diﬀerence is attributable to the presence of serial correlation.

Our main result is that the VIX index, a measure of the implied variance of the S&P 500 Index, is a much more biased forecast of longer-horizon variance than it is of the short-horizon variances typically examined. Speciﬁcally, in regressions in which implied variance is the sole predictor, the slope coeﬃcient in a regression in which the dependent variable is the monthly squared return is less than half the coeﬃcient of a regression in which the dependent variable is the sum of squared daily returns. Because the diﬀerence between these two dependent variables is a measure of inter-day serial covariance, these results are equivalent to a ﬁnding that serial covariance is predictable by implied variance. The predictability of serial covariance is highly signiﬁcant.

This result is very robust. Similar results are obtained for the cash index, the S&P 500 futures contract, and the SPY “Spider” ETF, suggesting that reversal-based trading strategies may be implementable, depending on transactions costs. We ﬁnd similar and signiﬁcant results in both halves of our sample, meaning that a single episode, such as the Financial Crisis, is not responsible for our ﬁndings. Finally, we ﬁnd predictable reversal using diﬀerent measures of implied variances, based on the Black-Scholes model and on a ‘model-free’ approach.

It is worth emphasizing that the serial covariance we are documenting is negative rather than positive. In addition, our primary focus is on the S&P 500 Index futures market rather than a cash index. Thus, our ﬁndings are fundamentally diﬀerent from those of Lo and MacKinlay (1988) and LeBaron (1992), for example, who may primarily be documenting patterns in the staleness of prices that imply spuriously positive autocovariance as a result of the bias identiﬁed by Fisher (1966).

We further show that most autocovariance is the result of the reversal of negative rather than positive returns. That is, as implied variances increase, a negative return is much more likely to be reversed in the next day or more, while positive returns are not. This raises the possibility that

3

either market liquidity is asymmetric, able to absorb buy orders with less transitory price impact than sell orders, or that available liquidity is symmetric but more likely to be overwhelmed by selling pressure, for example from ﬁre sale-type trades.

These results are complementary to recent work showing that the VIX is related to market liquidity. Most notably, Nagel (2012) shows that the level of the VIX is strongly positively related to the proﬁtability of the short-run reversal strategies of Lehmann (1990) and Lo and MacKinlay (1990). Interestingly, while reversal trading has typically been most proﬁtable in small, illiquid stocks (see Avramov et al., 2006), Collin-Dufresne and Daniel (2015) ﬁnd signiﬁcant evidence of reversal in the largest U.S.-listed stocks, but they see no relation between large-cap reversal returns and the VIX index.

Our results are notable in that they provide an additional link between the VIX index and market liquidity and substantially strengthen earlier ﬁndings by showing strong evidence of recurrent negative autocorrelation at the index level. Our results are also somewhat unique in showing that return reversal remains a signiﬁcant force even in the last two decades, during which the returns to traditional cross-sectional reversal strategies have been steadily declining (see Khandani and Lo, 2007).

Our second set of results concerns the exact timing of reversal. We see evidence of reversal in daily close-to-close returns, but is that where the tendency to reverse is strongest? To answer this question, we examine daily returns formed based on prices observed at times of day other than the close. We ﬁnd that reversal based on closing prices is indeed stronger than reversal based on prices at most other times of the day. The exception is that reversal is particularly strong when daily returns are based on prices immediately after the open. Regardless of the time at which daily prices are recorded, autocovariances are more negative when implied variances are high. The eﬀect is not particularly strong for returns based on closing prices.

We also examine reversal over horizons shorter than a day. Speciﬁcally, we look at how returns over the last N minutes of one day are reversed in the time from that day’s close to N minutes after the next day’s open. We ﬁnd that reversal is strongest for N between about 20 and 60. Furthermore, this very short-term reversal is most sensitive to the level if implied variance is above

4

average. When implied variances are high, futures price movements in the last 30 minutes or so of the day have a very strong tendency to reverse during the overnight periods and the ﬁrst 30 minutes of trading the following morning.

In light of the ﬁnding that returns around the end of the day seem to exhibit greater reversal, particularly when implied variances are high, we examine a potential explanation. Speciﬁcally, we hypothesize that end-of-day hedging by options traders may cause temporary price dislocation in the market index. We therefore investigate the relationship between reversal and the level of open interest in S&P 500 Index options. We ﬁnd that the VIX forecasts negative future autocovariance more strongly when this open interest is high. This result is obtained both for raw open interest and for a detrended measure of open interest.

Our ﬁnal set of results concerns implications for option trading strategies. As shown by Lo and Wang (1995), the fact that asset prices are discounted martingales under the risk neutral distribution means that option prices are unaﬀected by serial correlation – under the risk neutral distribution, it does not exist. As a result, at least in theory, option prices should be more closely related to daily return volatility than to monthly return volatility, which is impacted much more heavily by serial correlation. At the same time, the expected payoﬀ of the option is determined by the actual distribution of returns, which does depend on serial correlation. Hence serial correlation has a potentially important role to play in determining expected option returns.

The option strategy we focus on is the at-the-money straddle on the S&P 500 Index. This combination of an at-the-money put and an at-the-money call is constructed to have zero delta, so that it represents a bet not on the direction of prices but rather on the absolute value of their change, i.e. it is a bet on volatility. Negative autocorrelation in the returns on the underlying asset decreases the volatility of the underlying price at longer horizons, and hence reduces the expected payoﬀ of the straddle. Since the prices of the call and put are not reduced by that autocorrelation, the result of negative autocorrelation should be lower average returns on the straddle.

For the straddle buyer, there is a natural way to avoid this trap, which is to rebalance the portfolio daily such that at the end of every day the trader again holds an at-the-money zero delta straddle. This works by making the trader, at the end of each trading day, indiﬀerent as to whether

5

the next day’s underlying price change is positive or negative, implying that they are protected from the eﬀects of return reversal.

We examine the returns on these two versions of the straddle trade, as well as the diﬀerence in those returns, for straddles of one, two, and three months until expiration. On average, we ﬁnd that the buy-and-hold strategy underperforms the daily rebalanced strategy, reﬂecting the fact that serial covariance is on average negative. However, these diﬀerences are not signiﬁcant. When we regress the diﬀerence between the buy-and-hold and rebalanced returns on implied variance, we ﬁnd a signiﬁcant relation. When implied variance rises, serial covariance drops, and the buy-and-hold strategy substantially underperforms the rebalanced strategy.

Taken together, the evidence presented in this paper shows that even the most liquid assets display serial correlation in high uncertainty environments. The eﬀect is not small. It leads to a striking reinterpretations of what a change in implied variance actually means for investors, and it has considerable implications for how equiy and option traders should behave in the midst of a volatile market.

2 Variance and covariance forecasts

2.1 Regression framework

Traditionally, the literature examining implied variances as forecasts of future realized variances

has focused on the following speciﬁcation:

Nt

ri2,t = αd + βdIVt−1 + d,t

(1)

i=1

In this regression, ri,t represents the logarithmic return of an asset on day i of month t minus the

contemporaneous riskless return. Nt is the number of trading days within that month, and IVt−1 is the implied variance of the asset at the end of the prior month.2

In this paper, we propose to also examine the regression based on squared monthly excess

2As a test of predictability in variances, the regression suﬀers from a slight misspeciﬁcation because returns are not demeaned. It has been understood at least since French et al. (1987) that demeaning has very little impact on squared returns. We will return to this issue later to show that it is not aﬀecting any of our results materially.

6

returns,

Nt

2

ri,t = αm + βmIVt−1 + m,t,

(2)

i=1

which follows from our use of continuously compounded excess returns, which at the monthly level

are sums of daily values. Because

Nt

ri,t

2 Nt

=

Nt

ri,trj,t,

i=1

i=1 j=1

we can decompose the monthly squared excess return as follows:

Nt

2 Nt

Nt−1 Nt

ri,t = ri2,t + 2

ri,trj,t.

(3)

i=1

i=1

i=1 j=i+1

Aside from expected return eﬀects, which will be small, the second term can be interpreted as a mea-

sure of serial covariance, while the ﬁrst term is simply the sum of squared daily returns considered

above. Thus, the wedge between daily and monthly realized variance is inter-day autocovariance.

We compile several diﬀerent series for each dependent and independent variable. Daily returns

are based on closing prices of the S&P 500 ‘cash’ Index, the front month S&P 500 futures contract,

or the SPY exchange traded fund, which is the oldest and largest ETF tracking the S&P 500 Index.

The riskless rate we use is from Kenneth French’s website.

Our independent variable is usually based on the VIX, which is the Volatility Index constructed

by the Chicago Board Options Exchange (CBOE). The VIX a model-free implied volatility con-

structed similarly to that proposed by Britten-Jones and Neuberger (2000), and its construction

involves interpolation such that the measure can be interpreted as corresponding to a one-month

contract. In some cases we use the VXO instead. This is a predecessor of the VIX index that is

based on the Black-Scholes model and is constructed from options on the S&P 100 Index.

We create a rescaled measure of implied variance as IVt = VIX2t , (4) 120,000

where VIX is replaced by VXO in some cases. The denominator reﬂects the conversion of percentage

to decimal and annual to monthly, so that the resulting series is comparable to our realized variances.

Table 1 contains summary statistics for the data that will underlie most of the analysis in this

section. All data in the table are monthly. The sum of squared daily returns is the dependent vari-

7

able of regression (1), while the monthly squared return is the dependent variable of (2). Following (3), we deﬁne the diﬀerence as the latter minus the former. The table also shows the end-of-month values of the squared and rescaled VIX index, computed as in (4).

Several patterns in the data are immediately apparent. First, the average sum of squared daily returns exceeds the average monthly squared return by a signiﬁcant margin. From (3), this implies that serial covariances are negative on average.3 A second observations is that all variance proxies are persistent, though the sum of squared daily returns is more persistent than the monthly squared return. It is possible that the lower persistence of monthly squared returns is the result of more noise in that measure, but if so this noise is not obvious from the standard deviations of the two proxies, since monthly squared returns display less variability. The higher volatility of the sum of squared daily returns could be related to the presence of signiﬁcantly greater kurtosis in daily returns. A ﬁnal observation is that implied variances on average exceed both measures of realized variance, a standard result that likely reveals the presence of a volatility risk premium.

2.2 Main results

We report the results of variance forecasting regressions (1) and (2) in Table 2. The left side of the table contains results for regression (1), where the dependent variable is the sum of squared daily returns, which is the the traditional speciﬁcation for evaluating the bias and eﬃciency of implied variance forecasts. Consistent with a large literature, the slope coeﬃcient of the regression is highly signiﬁcant in all speciﬁcations, with substantial R-squares. The slope coeﬃcients are also all slightly less than one, which is also a common ﬁnding, though we cannot reject the unit slope.

The right side of the table reports results for regression (2), in which the dependent variable is the squared monthly return. Results here are much diﬀerent. In almost every sample, the slope coeﬃcient drops by at least half, in many cases more, indicating that the variance of monthly returns is much less responsive to changes in implied variances. This pattern holds for the S&P

3As we discuss below, our variance measures use returns that are not demeaned, and hence there is a small diﬀerence between the sum of squared daily returns and the monthly squared return that is driven by expected returns. However, this component should increase the monthly squared return relative to the sum of squared daily returns. Hence it cannot explain the average diﬀerence shown in the table.

8

500 Index, index futures, and the SPY ETF. It is present in our main sample period, in each half

of our sample, and in an extended sample. For the extended sample, we base our implied variance

measure on the VXO index rather than the VIX, since the former is available for four additional

years. Although regressions using the VXO are somewhat haphazard given that the dependent

variable is based on S&P 500 returns, while the independent variable is from S&P 100 options, the

results are nevertheless consistent with others. Furthermore, they show that the lower slope for

regression (2) is obtained in a period that includes the crash of 1987 and that it is found whether

we use model-free or Black-Scholes implied variances.

In comparing the two sets of results, we also see that the R-squares are usually much lower

for the monthly squared return regressions. This is to be expected. Since the work of Merton

(1980) and Andersen and Bollerslev (1998), it has been well understood that higher frequency

data contain signiﬁcantly more information about the latent variance process and can be used to

construct much more powerful tests of the accuracy of variance forecasts. The monthly squared

return simply contains more noise, and as a result it cannot be predicted as reliably. The only

exception is for the extended sample, which includes the crash of 1987. This caused an enormous

outlier in daily squared returns, which was not as obvious in monthly returns.

For high frequency data to truly oﬀer better inference they must be a proxy of the same latent

variance process, in that the variance rate of high-frequency returns be the same as that of low-

frequency returns. This is equivalent to stating that returns are serially uncorrelated.

To test for the predictability serial covariance directly, we use the result from (3) that a serial

covariance measure,

Nt−1 Nt

2

ri,trj,t,

(5)

i=1 j=i+1

can be constructed from the diﬀerence between monthly and daily return variances. We refer to

this measure as ‘total autocovariance’ to emphasize that it is reﬂects all lead-lag relations between

daily returns within a given month.

Regressing total autocovariance on implied variance will yield a slope coeﬃcient that is exactly

equal to βm − βd, the diﬀerence between the slopes of (2) and (1). Table 3 reports the results of

this regression, mainly to verify that the diﬀerence in slopes is statistically signiﬁcant. In short,

9

Christopher S. Jones Marshall School of Business University of Southern California [email protected]

Sung June Pyun Marshall School of Business University of Southern California [email protected]

Tong Wang Pamplin College of Business

Virginia Tech University [email protected] March 2016

Abstract We ﬁnd that the S&P 500 Index, its futures contracts, and its most popular ETF display strong evidence of daily return reversal when implied variance is high. This negative serial covariance is mostly observed following negative market returns, while positive returns show a moderate tendency to continue when implied variances are high. Price changes around the close and the open show a somewhat stronger tendency to reverse, though the serial covariance we document generally takes longer than a day to resolve. Furthermore, reversal tend to be stronger when option open interest is high, and the presence of return reversal has a signiﬁcant eﬀect on the performance of option trading strategies. Our results have important implications for how market participants interpret implied variances, which due to serial covariance provide an extremely biased forecast of longer-horizon return variances.

JEL classiﬁcation: G12, G13, G14. Keywords: Implied volatility, return reversal, liquidity.

1

1 Introduction

Option markets have the potential to provide extremely useful information to investors, even those who are not interested in trading options. An obvious example is an asset’s implied volatility, which may be valuable to an investor making a risk/reward calculation involving that asset. A precondition is that the implied volatility is reasonably correlated with actual return volatility, and that any bias in implied volatility be stable and therefore correctable.

Starting with the studies of Day and Lewis (1992) and Lamoureux and Lastrapes (1993), the empirical options literature has shown that implied variances from equity index options are highly informative predictors of future realized variances.1 At the same time, they display a systematic upward bias, and furthermore display a degree of variation that appears to be greater than that of actual ex ante variance, results that may be interpreted as reﬂecting the existence of risk premia on jumps and/or volatility. Fortunately, under plausible assumptions this bias can be well captured by a simple linear adjustment, as Chernov (2007) demonstrates empirically.

Evaluation of implied volatility forecasts have, since the inception of this literature, generally evaluated those forecasts on the basis of their ability to predict variances computed from all daily returns realized over the remaining life of the option. This choice is natural. Since the seminal work of Merton (1980), it has been well known that higher frequency data can be used to construct a more accurate proxy of a latent volatility process. Thus, forecast evaluation using daily realized variances should provide far more power than, say, forecasting the squared monthly return. Forecast evaluation based on realized variances computed from intra-day returns, an approach used by Blair et al. (2001), in theory has even greater informativeness, though microstructure issues complicate implementation and interpretation.

Under the assumption that returns are serially uncorrelated, a variance forecast that is constructed be an unbiased predictor of realized variance from daily returns will oﬀer a similarly unbiased forecast of monthly return variance. This is important given that many uses of variance

1Somewhat contradictory evidence was presented by Canina and Figlewski (1993), who found that implied variances are essentially uncorrelated with future realized variances. The unrepresentative result of this paper is explained by Christensen and Prabhala (1998).

1

forecasts, for example in portfolio optimization, require that the variance forecast’s horizon correspond to the anticipated holding period. Traditionally, this assumption has been accepted without very much scrutiny.

There is a long list of studies, however, that challenge the assumption of zero autocorrelation. One set focuses on the ﬁnding of positive index autocorrelation, ﬁrst identiﬁed by Lo and MacKinlay (1988) and Conrad and Kaul (1988) in studies of the CRSP market index. Campbell et al. (1993) observed that the autocorrelation was decreasing in the level of trading volume. They further hypothesized that autocorrelation should decrease with volatility, which was conﬁrmed by LeBaron (1992).

The diﬃculty in interpreting these results is that they could be due to the eﬀects of asynchronous prices, whereby the closing prices of diﬀerent securities are indicative of values at diﬀerent points in time. This has been shown by Fisher (1966) to produce spurious autocorrelation at the index level even when individual security returns are serially independent, and it is consistent with the ﬁnding from these papers that autocorrelations are smaller in value weighted portfolios than in equally weighted portfolios, which put more weight on stocks whose closing prices are more likely to be stale. Furthermore, Ahn et al. (2002) show that the positive autocorrelations observed in the so-called ‘cash’ market indexes are largely absent from futures returns, again suggesting that serial correlation is likely a ﬁgment of microstructure eﬀects and not something that is in any way exploitable by traders.

More recently, however, a group of studies has documented new evidence of serial correlation in stock market returns, but in this case the autocorrelation is negative and therefore inconsistent with stale prices. Bates (2012), for example, reports that the value weighted CRSP portfolio displayed signiﬁcantly negative autocorrelations for several years during the U.S. Financial Crisis. Etula et al. (2015) ﬁnd strong evidence of short-term return reversals in U.S. and international stock market indexes around the end of the month, a pattern that they attribute to the concentration of institutional trading at that time. Chordia et al. (2002) demonstrate that aggregate order imbalances, which are positively correlated with contemporaneous returns, negatively predict nextday market returns. Ben-Rephael et al. (2011) show that daily ﬂows in mutual funds generate a

2

similar pattern of reversal in the Israeli stock market. The primary goal of our paper is to reassess the ability of implied variances of S&P 500 Index

options to forecast future realized variances at diﬀerent horizons. Speciﬁcally, we are interested in whether implied variances are as useful in forecasting longer-horizon (e.g. monthly) realized variance as they are in forecasting short-run (e.g. daily) variances, where any diﬀerence is attributable to the presence of serial correlation.

Our main result is that the VIX index, a measure of the implied variance of the S&P 500 Index, is a much more biased forecast of longer-horizon variance than it is of the short-horizon variances typically examined. Speciﬁcally, in regressions in which implied variance is the sole predictor, the slope coeﬃcient in a regression in which the dependent variable is the monthly squared return is less than half the coeﬃcient of a regression in which the dependent variable is the sum of squared daily returns. Because the diﬀerence between these two dependent variables is a measure of inter-day serial covariance, these results are equivalent to a ﬁnding that serial covariance is predictable by implied variance. The predictability of serial covariance is highly signiﬁcant.

This result is very robust. Similar results are obtained for the cash index, the S&P 500 futures contract, and the SPY “Spider” ETF, suggesting that reversal-based trading strategies may be implementable, depending on transactions costs. We ﬁnd similar and signiﬁcant results in both halves of our sample, meaning that a single episode, such as the Financial Crisis, is not responsible for our ﬁndings. Finally, we ﬁnd predictable reversal using diﬀerent measures of implied variances, based on the Black-Scholes model and on a ‘model-free’ approach.

It is worth emphasizing that the serial covariance we are documenting is negative rather than positive. In addition, our primary focus is on the S&P 500 Index futures market rather than a cash index. Thus, our ﬁndings are fundamentally diﬀerent from those of Lo and MacKinlay (1988) and LeBaron (1992), for example, who may primarily be documenting patterns in the staleness of prices that imply spuriously positive autocovariance as a result of the bias identiﬁed by Fisher (1966).

We further show that most autocovariance is the result of the reversal of negative rather than positive returns. That is, as implied variances increase, a negative return is much more likely to be reversed in the next day or more, while positive returns are not. This raises the possibility that

3

either market liquidity is asymmetric, able to absorb buy orders with less transitory price impact than sell orders, or that available liquidity is symmetric but more likely to be overwhelmed by selling pressure, for example from ﬁre sale-type trades.

These results are complementary to recent work showing that the VIX is related to market liquidity. Most notably, Nagel (2012) shows that the level of the VIX is strongly positively related to the proﬁtability of the short-run reversal strategies of Lehmann (1990) and Lo and MacKinlay (1990). Interestingly, while reversal trading has typically been most proﬁtable in small, illiquid stocks (see Avramov et al., 2006), Collin-Dufresne and Daniel (2015) ﬁnd signiﬁcant evidence of reversal in the largest U.S.-listed stocks, but they see no relation between large-cap reversal returns and the VIX index.

Our results are notable in that they provide an additional link between the VIX index and market liquidity and substantially strengthen earlier ﬁndings by showing strong evidence of recurrent negative autocorrelation at the index level. Our results are also somewhat unique in showing that return reversal remains a signiﬁcant force even in the last two decades, during which the returns to traditional cross-sectional reversal strategies have been steadily declining (see Khandani and Lo, 2007).

Our second set of results concerns the exact timing of reversal. We see evidence of reversal in daily close-to-close returns, but is that where the tendency to reverse is strongest? To answer this question, we examine daily returns formed based on prices observed at times of day other than the close. We ﬁnd that reversal based on closing prices is indeed stronger than reversal based on prices at most other times of the day. The exception is that reversal is particularly strong when daily returns are based on prices immediately after the open. Regardless of the time at which daily prices are recorded, autocovariances are more negative when implied variances are high. The eﬀect is not particularly strong for returns based on closing prices.

We also examine reversal over horizons shorter than a day. Speciﬁcally, we look at how returns over the last N minutes of one day are reversed in the time from that day’s close to N minutes after the next day’s open. We ﬁnd that reversal is strongest for N between about 20 and 60. Furthermore, this very short-term reversal is most sensitive to the level if implied variance is above

4

average. When implied variances are high, futures price movements in the last 30 minutes or so of the day have a very strong tendency to reverse during the overnight periods and the ﬁrst 30 minutes of trading the following morning.

In light of the ﬁnding that returns around the end of the day seem to exhibit greater reversal, particularly when implied variances are high, we examine a potential explanation. Speciﬁcally, we hypothesize that end-of-day hedging by options traders may cause temporary price dislocation in the market index. We therefore investigate the relationship between reversal and the level of open interest in S&P 500 Index options. We ﬁnd that the VIX forecasts negative future autocovariance more strongly when this open interest is high. This result is obtained both for raw open interest and for a detrended measure of open interest.

Our ﬁnal set of results concerns implications for option trading strategies. As shown by Lo and Wang (1995), the fact that asset prices are discounted martingales under the risk neutral distribution means that option prices are unaﬀected by serial correlation – under the risk neutral distribution, it does not exist. As a result, at least in theory, option prices should be more closely related to daily return volatility than to monthly return volatility, which is impacted much more heavily by serial correlation. At the same time, the expected payoﬀ of the option is determined by the actual distribution of returns, which does depend on serial correlation. Hence serial correlation has a potentially important role to play in determining expected option returns.

The option strategy we focus on is the at-the-money straddle on the S&P 500 Index. This combination of an at-the-money put and an at-the-money call is constructed to have zero delta, so that it represents a bet not on the direction of prices but rather on the absolute value of their change, i.e. it is a bet on volatility. Negative autocorrelation in the returns on the underlying asset decreases the volatility of the underlying price at longer horizons, and hence reduces the expected payoﬀ of the straddle. Since the prices of the call and put are not reduced by that autocorrelation, the result of negative autocorrelation should be lower average returns on the straddle.

For the straddle buyer, there is a natural way to avoid this trap, which is to rebalance the portfolio daily such that at the end of every day the trader again holds an at-the-money zero delta straddle. This works by making the trader, at the end of each trading day, indiﬀerent as to whether

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the next day’s underlying price change is positive or negative, implying that they are protected from the eﬀects of return reversal.

We examine the returns on these two versions of the straddle trade, as well as the diﬀerence in those returns, for straddles of one, two, and three months until expiration. On average, we ﬁnd that the buy-and-hold strategy underperforms the daily rebalanced strategy, reﬂecting the fact that serial covariance is on average negative. However, these diﬀerences are not signiﬁcant. When we regress the diﬀerence between the buy-and-hold and rebalanced returns on implied variance, we ﬁnd a signiﬁcant relation. When implied variance rises, serial covariance drops, and the buy-and-hold strategy substantially underperforms the rebalanced strategy.

Taken together, the evidence presented in this paper shows that even the most liquid assets display serial correlation in high uncertainty environments. The eﬀect is not small. It leads to a striking reinterpretations of what a change in implied variance actually means for investors, and it has considerable implications for how equiy and option traders should behave in the midst of a volatile market.

2 Variance and covariance forecasts

2.1 Regression framework

Traditionally, the literature examining implied variances as forecasts of future realized variances

has focused on the following speciﬁcation:

Nt

ri2,t = αd + βdIVt−1 + d,t

(1)

i=1

In this regression, ri,t represents the logarithmic return of an asset on day i of month t minus the

contemporaneous riskless return. Nt is the number of trading days within that month, and IVt−1 is the implied variance of the asset at the end of the prior month.2

In this paper, we propose to also examine the regression based on squared monthly excess

2As a test of predictability in variances, the regression suﬀers from a slight misspeciﬁcation because returns are not demeaned. It has been understood at least since French et al. (1987) that demeaning has very little impact on squared returns. We will return to this issue later to show that it is not aﬀecting any of our results materially.

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returns,

Nt

2

ri,t = αm + βmIVt−1 + m,t,

(2)

i=1

which follows from our use of continuously compounded excess returns, which at the monthly level

are sums of daily values. Because

Nt

ri,t

2 Nt

=

Nt

ri,trj,t,

i=1

i=1 j=1

we can decompose the monthly squared excess return as follows:

Nt

2 Nt

Nt−1 Nt

ri,t = ri2,t + 2

ri,trj,t.

(3)

i=1

i=1

i=1 j=i+1

Aside from expected return eﬀects, which will be small, the second term can be interpreted as a mea-

sure of serial covariance, while the ﬁrst term is simply the sum of squared daily returns considered

above. Thus, the wedge between daily and monthly realized variance is inter-day autocovariance.

We compile several diﬀerent series for each dependent and independent variable. Daily returns

are based on closing prices of the S&P 500 ‘cash’ Index, the front month S&P 500 futures contract,

or the SPY exchange traded fund, which is the oldest and largest ETF tracking the S&P 500 Index.

The riskless rate we use is from Kenneth French’s website.

Our independent variable is usually based on the VIX, which is the Volatility Index constructed

by the Chicago Board Options Exchange (CBOE). The VIX a model-free implied volatility con-

structed similarly to that proposed by Britten-Jones and Neuberger (2000), and its construction

involves interpolation such that the measure can be interpreted as corresponding to a one-month

contract. In some cases we use the VXO instead. This is a predecessor of the VIX index that is

based on the Black-Scholes model and is constructed from options on the S&P 100 Index.

We create a rescaled measure of implied variance as IVt = VIX2t , (4) 120,000

where VIX is replaced by VXO in some cases. The denominator reﬂects the conversion of percentage

to decimal and annual to monthly, so that the resulting series is comparable to our realized variances.

Table 1 contains summary statistics for the data that will underlie most of the analysis in this

section. All data in the table are monthly. The sum of squared daily returns is the dependent vari-

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able of regression (1), while the monthly squared return is the dependent variable of (2). Following (3), we deﬁne the diﬀerence as the latter minus the former. The table also shows the end-of-month values of the squared and rescaled VIX index, computed as in (4).

Several patterns in the data are immediately apparent. First, the average sum of squared daily returns exceeds the average monthly squared return by a signiﬁcant margin. From (3), this implies that serial covariances are negative on average.3 A second observations is that all variance proxies are persistent, though the sum of squared daily returns is more persistent than the monthly squared return. It is possible that the lower persistence of monthly squared returns is the result of more noise in that measure, but if so this noise is not obvious from the standard deviations of the two proxies, since monthly squared returns display less variability. The higher volatility of the sum of squared daily returns could be related to the presence of signiﬁcantly greater kurtosis in daily returns. A ﬁnal observation is that implied variances on average exceed both measures of realized variance, a standard result that likely reveals the presence of a volatility risk premium.

2.2 Main results

We report the results of variance forecasting regressions (1) and (2) in Table 2. The left side of the table contains results for regression (1), where the dependent variable is the sum of squared daily returns, which is the the traditional speciﬁcation for evaluating the bias and eﬃciency of implied variance forecasts. Consistent with a large literature, the slope coeﬃcient of the regression is highly signiﬁcant in all speciﬁcations, with substantial R-squares. The slope coeﬃcients are also all slightly less than one, which is also a common ﬁnding, though we cannot reject the unit slope.

The right side of the table reports results for regression (2), in which the dependent variable is the squared monthly return. Results here are much diﬀerent. In almost every sample, the slope coeﬃcient drops by at least half, in many cases more, indicating that the variance of monthly returns is much less responsive to changes in implied variances. This pattern holds for the S&P

3As we discuss below, our variance measures use returns that are not demeaned, and hence there is a small diﬀerence between the sum of squared daily returns and the monthly squared return that is driven by expected returns. However, this component should increase the monthly squared return relative to the sum of squared daily returns. Hence it cannot explain the average diﬀerence shown in the table.

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500 Index, index futures, and the SPY ETF. It is present in our main sample period, in each half

of our sample, and in an extended sample. For the extended sample, we base our implied variance

measure on the VXO index rather than the VIX, since the former is available for four additional

years. Although regressions using the VXO are somewhat haphazard given that the dependent

variable is based on S&P 500 returns, while the independent variable is from S&P 100 options, the

results are nevertheless consistent with others. Furthermore, they show that the lower slope for

regression (2) is obtained in a period that includes the crash of 1987 and that it is found whether

we use model-free or Black-Scholes implied variances.

In comparing the two sets of results, we also see that the R-squares are usually much lower

for the monthly squared return regressions. This is to be expected. Since the work of Merton

(1980) and Andersen and Bollerslev (1998), it has been well understood that higher frequency

data contain signiﬁcantly more information about the latent variance process and can be used to

construct much more powerful tests of the accuracy of variance forecasts. The monthly squared

return simply contains more noise, and as a result it cannot be predicted as reliably. The only

exception is for the extended sample, which includes the crash of 1987. This caused an enormous

outlier in daily squared returns, which was not as obvious in monthly returns.

For high frequency data to truly oﬀer better inference they must be a proxy of the same latent

variance process, in that the variance rate of high-frequency returns be the same as that of low-

frequency returns. This is equivalent to stating that returns are serially uncorrelated.

To test for the predictability serial covariance directly, we use the result from (3) that a serial

covariance measure,

Nt−1 Nt

2

ri,trj,t,

(5)

i=1 j=i+1

can be constructed from the diﬀerence between monthly and daily return variances. We refer to

this measure as ‘total autocovariance’ to emphasize that it is reﬂects all lead-lag relations between

daily returns within a given month.

Regressing total autocovariance on implied variance will yield a slope coeﬃcient that is exactly

equal to βm − βd, the diﬀerence between the slopes of (2) and (1). Table 3 reports the results of

this regression, mainly to verify that the diﬀerence in slopes is statistically signiﬁcant. In short,

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