Trading the VIX Futures Roll and Volatility Premiums with VIX Options

David P. Simon Dept of Finance Bentley University Waltham, MA 02452 [email protected] (781) 891 2489

November 27, 2014

This study examines the efficiency of VIX option trading strategies that exploit the effects of the slope of the VIX futures curve on VIX futures returns and the substantial volatility premiums in VIX futures prices over the sample period from January 2007 through March 2014. The study first assesses the related issue of whether VIX options are overpriced by examining long VIX option delta hedged returns and demonstrates that average losses on front contract calls and puts over 5-business day horizons are either not statistically significant or are economically small. The study then examines whether these results vary when ex-ante VIX option volatility premiums are above or below their top quartile cutoffs. The results demonstrate that P&Ls are significantly negative for the former, but more importantly are not significantly negative for the roughly 75% of the sample period when ex-ante volatility premiums are below their top quartile cutoffs. In light of this evidence that VIX option buyers on average do not overpay substantially for the limited risk associated with VIX options, the study then turns to whether long VIX option positions can be used profitably to exploit the tendencies of VIX futures prices to fall and rise when the VIX futures curve is in contango and in backwardation, respectively, as well as the tendency of VIX futures prices to build in large ex-ante volatility premiums. The results demonstrate that buying VIX puts when the VIX futures curve is in contango offers attractive risk reward strategies, but less so for buying VIX calls when the VIX futures curve is in backwardation. The evidence also demonstrates that buying VIX put options when ex-ante VIX futures volatility premiums are high leads to favorable risk-reward tradeoffs. The results also demonstrate mixed effects from not entering these trades when VIX option ex-ante volatility premiums are in their highest quartile. Overall, the results indicate that VIX options typically are not substantially overpriced, which makes long VIX option positions attractive vehicles for exploiting the roll and the often large ex-ante VIX futures volatility premiums.

The author thanks Bentley University for a summer research grant. Trading the VIX Futures Roll and Volatility Premiums with VIX Options

Volatility has become an important and widely traded asset class since the introduction of

VIX futures contracts in 2004. This popularity stems from the hedging properties of implied volatility, owing to its strong and reliable negative correlation with equity index returns.1

Nevertheless, the historical performance of VIX exchange traded products that roll over long

VIX futures contracts has been very poor owing to the negligible predictive power of the VIX futures curve for subsequent (spot) VIX changes and because the VIX futures curve has been upward sloping or in contango roughly 3/4 of the time.2 Because the VIX does not tend to rise when VIX futures prices are higher, VIX futures prices tend to roll down the VIX futures curve to converge with a lower VIX at settlement. Likewise, when the VIX futures curve is in backwardation and VIX futures prices are below the VIX, the VIX does not tend to fall and VIX future prices roll up the curve to converge with a higher VIX at settlement. Studies, such as

Simon and Campasano (2014), demonstrate that shorting VIX futures contracts when the curve is sufficiently in contango and hedging the risk of a volatility spike with short S&P futures contracts and buying VIX futures when the curve is sufficiently in backwardation and hedging with long S&P futures contracts are highly profitable trading strategies from 2007 through 2012.

Issues related to the overpricing of S&P 500 index options have been studied extensively by Coval and Shumway (2001), Bakshi and Kapadia (2003), Eraker (2007) and Brodie,

1 See Szado (2009) for an analysis of the hedging properties of VIX futures and VIX options during the financial panic beginning in 2008. 2 This study uses the term VIX to refer to the spot VIX as opposed to VIX futures contracts or prices. The iPath S&P 500 VIX ST Futures ETN (ticker symbol VXX), which is the most heavily traded ETF that rolls over front VIX futures contracts has lost more than 99% of its value since its inception in February 2009. However, much of this loss owes to declines in the VIX, which fell by 70% over the same period. More compelling evidence of the importance of the roll when the VIX futures curve is in contango is that from the beginning of 2014 through mid- August 2014 when the curve was in contango much of the time, the VIX fell only about 5%, while the VXX plummeted by 34%. During the January 2007 through March 2014 sample period of the present study, the VIX futures curve is in contango 74% of the time, with contango defined as the price of the front VIX futures contract with at least 10 business days to settlement being above the VIX . See Alexander and Korovilas (2011) and Whaley (2013) for more evidence on the poor performance of VIX-related exchange traded funds. 2

Chervnov and Johannes (2009). These studies demonstrate that S&P 500 index options price in substantial ex-ante volatility premiums with S&P 500 index option implied volatilities typically hovering well above realized volatility. This phenomenon causes S&P 500 index option selling strategies to be extremely attractive on a risk-adjusted basis.3 Because VIX futures contracts are based on model-free implied volatilities of S&P 500 index options and because the VIX futures curve typically is in contango, VIX futures contracts necessarily build in even more substantial ex-ante volatility premiums than S&P index options or the VIX.

By contrast, only a few studies have focused on the potential overpricing of VIX options, which were introduced in April 2006.4 Song (2012) examines zero-delta long VIX option straddles and delta hedged long VIX option portfolios and finds that these positions have significantly negative returns. Barnea and Hogan (2012) examine the existence of VIX option volatility premiums by creating synthetic variance swaps on VIX futures contracts. They find that the realized variance of VIX futures contracts is on average 3.26% below variance swap rates implicit in VIX options and conclude that their findings indicate that selling VIX options on a delta hedged basis should be profitable. However, they do not test this hypothesis.

The present study first examines whether VIX options tend to be overpriced over a sample period that extends 3 years beyond those of the previously mentioned studies by examining unconditional delta hedged VIX option returns over 5-business day horizons.5 The findings demonstrate that long delta hedged VIX call and put average returns across several strikes of the front contract are either not statistically negative or average losses are economically

3 Eraker (2009) shows that the VIX averages 19% between 1990 and 2007, while the annualized standard deviation of S&P 500 index returns is 15.7%. Coval and Shumway (2001) demonstrate that zero beta 1-month S&P 500 index straddles lose 3.2% of their values per week from January 1986 through October 1995. Bakshi and Kapadia (2003) show that delta hedged long S&P 500 index call options with 31-60 days to expiration and moneyness ranging from 0 to 2.5% out of the money lose on average 4.44% of their values from 1988 through 1995. Brodie, Chernov and Johannes (2009) show that for 1 month at the money S&P 500 index puts, CAPM adjusted returns are a highly statistically significant -22.5% and Sharpe ratios are -.23 from August 1987 through June 2005. 4 Much of the literature on VIX options, such as Wang and Daigler (2011), Grunbichler and Longstaff (1996) and Carr and Lee (2007), focuses on the relative explanatory powers of different option pricing models rather than on overpricing. 5 Song (2012) examines VIX option returns through April 2011, while Barnea and Hogan’s sample period also ends in April 2011. 3 modest. The study then examines delta hedged 5-business day returns when ex-ante VIX option volatility premiums--defined as the spread between VIX option implied volatility and lagged 10- day VIX futures volatility--are above and below their 75 percent fractile cutoffs. The results indicate that that while average losses become greater and are always statistically significant for all degrees of moneyness in the former case; they are negligible and not statistically significant for any degrees of moneyness over the other 3/4 of the sample.

The evidence that VIX options are not overpriced over much of the sample period suggests that defined risk strategies of buying either VIX calls or puts to exploit the systematic tendencies of VIX futures may be attractive. The present study examines trading strategies based on the roll, such as buying VIX puts when the VIX futures curve is in contango and buying VIX calls when the VIX futures curve is in backwardation. The study also examines the profitability of buying VIX puts when ex-ante VIX futures volatility premiums are unusually high. The results demonstrate that buying puts when the VIX futures curve is in contango as well as when the VIX futures volatility premium is large are both highly statistically and economically significantly profitable, whereas buying calls when the VIX futures curve is in backwardation is not significantly profitable. The study then examines whether these strategies are improved by avoiding purchasing VIX options when VIX option ex-ante volatility premiums are in their highest quartiles over the sample period and finds mixed results. Overall, the findings indicate that, unlike S&P 500 index options, VIX options are not substantially overpriced much of the time and can be used effectively to exploit systematic tendencies of VIX futures prices.

The paper proceeds as follows. The first section presents background information on VIX options, describes the methodology used to calculate implied volatility and the greeks and then provides a summary of the data. The second section examines unconditional 5-business day

P&Ls of delta hedged long VIX option positions to determine whether VIX options are overpriced on average. The third section re-examines 5-business day delta hedged long VIX

4 option positions when VIX option ex-ante volatility premiums are above and below their upper quartile cutoffs. The fourth section examines the profitability of outright VIX option purchases that attempt to exploit the VIX futures curve’s lack of predictive power for subsequent VIX levels as well as the presence of large volatility premiums in VIX futures prices. The final section discusses the implications of the findings.

I. Background on VIX Options

VIX options were introduced in April 2006 and are European options that are cash-settled at the same time as corresponding VIX futures contracts. Consequently, VIX options are priced off the corresponding VIX futures contracts rather than the VIX. VIX futures and VIX option contracts expire on Wednesdays that are 30 days before S&P 500 index options expire and settle based on opening Wednesday quotes of S&P 500 index options that expire in 30 days.6 VIX futures prices are risk neutral measures of expected future 30-day implied volatilities of S&P 500 index options and VIX option implied volatilities are risk neutral measures of expected future volatilities of VIX futures returns.

This study uses Black’s (1976) futures option pricing model to calculate the implied volatility and the greeks of VIX options. This choice is supported by Wang and Daigler (2011), who find that the Black model used by Whaley (1993) outperforms more sophisticated stochastic volatility models in pricing VIX options, such as those developed by Grunbichler and Longstaff

(1996) and Carr and Lee (2007). The present study specifies the Black (1976) model in business

6 Both VIX futures and VIX options contracts stop trading at the close on the last business day before expiration day. For example, the September 2014 VIX futures contract and September 2014 VIX options settled on Wednesday September 18, 2014 based on opening S&P 500 index option prices that day, but stopped trading at the close on Tuesday September 17. Pavlova and Daigler (2008) demonstrate convergence problems sometimes occur at the final settlement of VIX futures contracts. This is not an issue here because we do not examine options that are close to settlement. 5 days rather than in calendar days because VIX futures return volatilities from Friday to Monday closes are in line with those from consecutive days during the business week.7

The VIX options data are from ivolatility.com and are matched with corresponding VIX futures settlement prices from Bloomberg to calculate implied volatilities and the greeks. Short term interest rates are proxied with interpolated 1- and 3-month Eurodollar deposit rates from the

Federal Reserve Board. The analysis examines the 5 closest to the money VIX calls and puts of the closest VIX futures contract that has at least 10 business days to expiration. The data are filtered for errors by calculating deviations from put-call parity, taking quoted option bid-ask spreads into account. Observations with deviations greater than 5 cents are excluded from the dataset, which implicitly allows for a 10 cent closing bid-ask spread on VIX futures contracts.8

Violations from put-call parity are considered present and observations are excluded if

(-rt) / Cbid – Pask - (F-X) e > .05 or (1 )

(-rt) // Cask – Pbid - (F-X) e < -.05. (1 )

These filters cause less than 1 percent of quotes for all levels of moneyness for the front contract to be thrown out.

Exhibit 1 shows mid-point closing VIX option quotes, bid-ask spreads, implied volatilities, lagged 10-day historical volatilities of corresponding VIX futures returns and ex-ante

7 From January 2007 through March 2014, the annualized volatility of front VIX futures log returns from Friday to Monday closes is 69.6 percent compared to 68.4 percent for close to close returns during consecutive days during the business week. The major benefit of using business days rather than calendar days for calculating implied volatility and the greeks is a more accurate measure of time decay or theta. 8 For example, we test the possibility that call prices are too high relative to put prices in (1/) by examining potential arbitrage profits from getting short VIX futures synthetically by selling calls at the bid price and buying puts at the ask price and buying VIX futures. We test the opposite by examining potential arbitrage profits from getting long synthetically by buying calls at the ask price and selling puts at the bid price and shorting VIX futures. 6 volatility premiums, defined as the difference between VIX option implied volatilities and lagged 10-day historical VIX futures return volatilities. The exhibit shows that average call prices range from 1.32 for out of the money calls to 3.87 for in the money calls, and average put prices range from .77 for out of the money puts to 4.49 for in the money puts. Average closing bid-ask spreads are wide and range from .12 for out of the money calls to .25 for in the money calls and from .10 for out of the money puts to .26 for in the money puts. Traders who pay these average quoted bid-ask spreads on round trip transactions would lose roughly 5 to 10 percent of the value of options, which would severely undermine the profitability of any short-term VIX option trading strategy including those examined in this study. However, the data also show that closing bid-ask spreads can be as low as a much more manageable .05. In the trading simulations conducted here, we abstract from the cost of bid-ask spreads, but it should be clear that most of the strategies examined in this study would not be successful unless the bid-ask spread is finessed. However, the many of the strategies examined in this study generally are robust to paying .05 to .10 bid-ask spreads.9 The implications of wide quoted VIX option bid- ask spreads are discussed later in the study.

VIX option implied volatilities exhibit a pronounced volatility skew. Low strike implied volatilities average 74 percent, while high strike implied volatilities average 96 percent. The volatility skew is consistent with strong buying pressure in out of the money calls to hedge equity portfolios and strong selling pressure in out of the money puts to defray part of the cost of buying out of the money calls.10 These average VIX option implied volatilities exceed 10-day

VIX futures actual lagged volatilities, which average 56 percent over the sample period. The pronounced volatility skew results in a wide range of ex-ante volatility premiums across strikes,

9 Another possibility is that trading strategies similar to those examined in this study could be executed with the iPath S&P 500 VIX ST Futures ETN (VXX) options, where perhaps surprisingly bid-ask spreads are extremely tight and frequently only 1-2 cents. 10 If two strike out of the money calls are priced with the average 85 percent implied volatility of at the money calls rather than their actual average implied volatility of 95.6 percent, the average price of these calls falls from $1.32 to $1.07. 7 which average from 17 percentage points for low strike options to 39 percentage points for high strike options. These large ex-ante volatility premiums also exhibit a great deal of variation, as the top decile cutoff for at the money calls and puts is around 50 percentage points, while the bottom decile cutoff is around 4-1/2 percentage points, with similarly wide ranges for the other options. Whether these seemingly large ex-ante volatility premiums translate into statistically significant and economically meaningful losses on delta hedged long VIX call and put positions is examined in the next section.11

II. Unconditional delta hedged returns

This section provides evidence on whether VIX options are overpriced by examining 5- business day delta hedged long VIX option returns. If the average returns from buying VIX options and delta hedging with VIX futures contracts are statistically significantly negative and economically meaningful, they would be consistent with VIX options building in ex-post volatility premiums. This finding would suggest that VIX option buyers overpay for the limited risk offered by VIX options, which would diminish the effectiveness of buying VIX options to exploit the well-known tendencies of VIX futures contracts. The options examined are the closest to expiration calls and puts that have at least 10 business days to expiration for strikes that are closest to the money (ATM), one and two strikes below the ATM strike (ATM-1 and

ATM-2) and one and two strikes above the ATM strike (ATM+1 and ATM+2). The results reported include P&Ls as well as returns, defined as P&Ls divided by beginning mid-price option quotes. We show P&Ls because for low option prices, seemingly large percentage

11 We refer to the spread between VIX option implied volatility and the lagged 10 day volatility of the corresponding VIX futures contract as the ex-ante volatility premium to emphasize that even large spreads between implied volatility and recent historical volatility does not necessarily translate into losses on delta hedged long VIX option positions, as is seen in the next section. 8 returns could be highly misleading.12 Because VIX option contracts are specified with one point worth $100 and because VIX futures contracts are specified with one point worth $1,000, the analysis assumes that 10 option contracts are purchased.13 The deltas used to set up and rebalance hedges daily as well as the other greeks are calculated from Black’s (1976) futures option pricing model, adjusted to a business day basis. The greeks are calculated every day and the impacts of the greeks on profitability are aggregated over each day that trades are open.

Exhibit 2 shows that the results are mixed for 5-business day delta hedged long VIX call and put positions with daily delta rebalancing. For calls, average losses are statistically significant at the 5% level for 3 of the 5 strikes. However, these average losses are relatively modest and range from $30 to $40 on call option positions that average roughly $3,000. Hence, average losses when statistically significant, typically amount to only about 1-1/4 percent of option values. The percentage of winning trades is centered around 40% and the relation between the average gains on winning trades and average losses on losing trades is mixed across the strikes. However, the top and bottom decile P&L cutoffs occur at smaller magnitude gains than losses. The presence of the statistically significant average losses owes to losses from time decay

(theta) outpacing gains from being long gamma, and gains from positive vega offsetting only part of this differential. 14

Average losses on delta hedged long put positions are significantly negative at the 5% level for only 1 of the strikes examined, with average losses ranging from $16 to $35. These losses corresponds to statistically insignificant (at the 5% level) negative returns ranging from

12 For example, a 50% return from .20 to .30 cents could exaggerate the magnitude of dollar gains and thus it is important to get a sense of dollar gains as well. Because the risk of delta hedged long option positions is loosely limited to the value of options purchased, it makes sense to express returns as gains divided by starting option values rather than dividing by the absolute net outlays that include VIX futures notional values. 13 For example, a trader who buys ten 50 delta VIX calls for 2.5 points would pay $2,500 and would delta hedge by shorting 1/2 VIX futures contract. Because position sizes can be grossed up, we allow for fractional VIX futures positions. 14 The results presented here are not exactly comparable to Song (2012) because he examines one week returns on long straddle positions that are delta hedged at the outset but not on a daily basis and he examines long VIX option portfolios that are held to expiration. Nevertheless, if we end our sample period when Song’s sample ends we find larger and more significant average losses on long call and long put positions that are delta hedged over 5-business day holding periods. 9

-1/2 % to -1-3/8 percent of initial put values. Overall, losses on the puts from time decay exceed gains from favorable gamma and the trades benefit from being long vega, as implied volatility rises on balance over the sample period. Unreported results demonstrate for both calls and puts that adjusting delta hedges for the volatility skew and for the tendency for VIX option implied volatilities to increase when VIX futures prices rise along the lines of Crépey (2004), Vähämaa

(2004) and Simon (2012) has little effect on the results.15 Consistent with this evidence, conditioning on whether VIX futures prices rise or fall does not lead to markedly different outcomes for P&Ls.

Overall, the results indicate that average ex-post VIX option volatility premiums are fairly modest, which suggests that on average VIX option buyers typically do not greatly overpay for the limited risk associated with being long VIX options.16 More importantly, the results enhance the possibility that long VIX option strategies could be used effectively to exploit the well-known regularities of VIX futures contracts. However, the results do not rule out the possibility that VIX options are substantially overpriced at times and that avoiding buying calls or puts during these times would be beneficial. The next section examines this issue.

III. Conditional VIX Option Volatility Premiums

The previous section demonstrates that VIX option ex-post volatility premiums are fairly small over the sample period. This section examines whether losses on 5-day delta hedged long

15 Because the volatility skew causes VIX option implied volatilities to be greater at higher strike prices, the adjusted deltas used for delta hedging are ∆ADJ = ∆BL + ν (dIV/dVIX), where the adjustment to the Black delta is equal to vega times the change in VIX option implied volatility for a given change in VIX futures prices. Because the latter term is positive, the adjustment involves selling more VIX futures contracts against long VIX call positions and buying fewer VIX futures contracts to hedge long put positions. The intuition for calls is that their prices rise more than unadjusted deltas suggest when VIX futures rise and their prices fall more than unadjusted deltas suggest when VIX futures fall and the adjustment is to short more VIX futures contracts. The intuition for puts is that their prices rise less than unadjusted deltas would suggest when VIX futures fall and their prices fall less than unadjusted deltas suggest when VIX futures rise and the adjustment is to buy fewer VIX futures contracts. 16 While claims that these losses are not economically meaningful may appear overly subjective, they are made in light of the losses being inadequate to spur successful trading strategies and also in the context of the much greater profitability of trading strategies shown later that take advantage of the systematic tendencies of VIX futures prices. 10

VIX option positions occur across the entire sample period or whether they vary according to whether ex-ante VIX option volatility premiums are above or below their top quartile cutoffs.

Given the pronounced implied volatility skew of VIX options, the thresholds for entering trades are that for each moneyness category differentials between implied volatilities and 10-day lagged historical VIX futures volatilities are above or below their highest quartile cutoffs over the sample period.17 These thresholds range from a low of 30 percentage points for calls and puts that are two strikes below at the money strikes to a high of 52 percentage points for calls and puts that are two strikes above at the money strikes18 Long VIX option positions again are delta hedged at the outset and deltas are rebalanced daily. While it would be more natural for high ex- ante volatility premiums to trigger short rather than long delta hedged VIX option positions, we nonetheless assume that traders are long VIX options to conform to our previous analyses. As a result, the findings provide information about whether the average losses on long VIX option positions over the sample period can be attributed solely to when ex-ante volatility premiums are in their highest quartiles and whether average losses occur over the other 3/4 of the sample period when ex-ante volatility premiums are less extreme.

Exhibit 3 shows the results for call options. When ex-ante call volatility premiums are in their highest quartiles, 5-business day P&Ls and delta hedged long call returns are negative and highly statistically significant across all moneyness categories. Average losses range from $64 to

$97, which correspond to average negative returns of -2.6 to -4 percent. These losses can be attributed to the roughly 3:1 ratio of losses versus gains as the average gains of winning trades tend to be about in line with the average losses of losing trades. As was the case with

17 This study uses lagged 10-day historical volatility as a simple proxy for near-term future volatility despite the fact that more sophisticated forecasts could be devised. Nevertheless, the important issue is whether this gauge is useful for determining whether VIX options will prove to be overpriced in an ex-post sense and whether the losses shown in exhibit 2 are general or are accounted for by a relatively small part of the data. 18 The cutoffs for the highest quartile VIX option volatility premiums are roughly 30, 35, 41, 47 and 52 percentage points for the ATM-2, ATM-1, ATM, ATM+1 and ATM+2 strikes, respectively. 11 unconditionally entered long VIX call 5-day trades, losses from theta outpace gains from gamma but gains from vega now offset less of this differential.

When ex-ante volatility premiums are below their top quartile cutoffs the results indicate that average P&Ls and returns on delta hedged long call or long put positions held for 5-business days are nowhere near statistically significant. Average P&Ls range from $12 gains to $24 losses, which correspond to returns ranging between plus .6 percent to losses minus .5 %. These results owe to the somewhat unfavorable frequency of winning to losing trades being offset by larger average gains on winning trades relative to losses on losing trades. Thus, ex-post volatility premiums are absent for VIX call options for roughly ¾ of the sample period.

Exhibit 4 shows the 5-business day results for delta-hedged long VIX put options when ex-ante volatility premiums are above and below their top quartile cutoffs. The results are fairly similar to those of calls. When ex-ante volatility premiums are in their highest quartiles, delta- hedged long put positions across all of the strikes have significantly negative average P&Ls that range from losses of $65 to $96, and correspond to significantly negative 5-business day returns of 2.5 to 4.4 %. Once more the frequency of winning to losing trades is unfavorable by a roughly

3:1 margin and the size of average gains on winning trades to losses on losing trades does not provide an offset. By contrast, when ex-ante VIX put volatility premiums are not in their highest quartiles, average P&Ls and returns for delta hedged long put positions again are nowhere near significantly negative across any of the strikes. Average P&Ls range from roughly $3 gains to losses of $17-1//2, which correspond to average returns ranging between plus and minus 3/10 percent. The absence of significant ex-post volatility premiums can be attributed to the slightly unfavorable ratio of winning to losing trades being offset by greater magnitude average gains on winning trades relative to average losses on losing trades.

The results for both calls and puts suggests that our measure of ex-ante VIX option volatility premiums--VIX option implied volatilities minus 10-day lagged VIX futures historical

12 volatilities--is a rough but effective measure for determining whether VIX options are overpriced and consequently ex-post delta hedged long VIX positions are likely to be negative. 19 The statistically significant and larger losses when VIX option ex-ante volatility premiums are in their highest quartiles suggests that strategies involving purchasing VIX options to exploit VIX futures tendencies might benefit from avoiding such positions when ex-ante VIX option volatility premiums are in their highest quartiles and VIX options are more expensive than usual.

At the same time, the relatively small average losses on long VIX option delta hedged positions suggests that the relatively modest average ex-post VIX option volatility premiums may not severely undermine the profitability of using long VIX option positions to try to exploit VIX futures tendencies.

IV. Trading the roll with VIX options

The strategies examined in this section use options to take advantage of the findings that the VIX futures curve does not have forecasting power for subsequent VIX changes. This lack of forecasting power results in a tendency for VIX futures prices to fall when the VIX futures curve is in contango and to rise when the curve is in backwardation. Simon and Campasano (2014) demonstrate the profitability of strategies that sell VIX futures contracts hedged by short mini-

S&P futures positions when the VIX futures curve is in contango and buy VIX futures contracts hedged by long mini-S&P futures positions when the VIX futures curve is in backwardation.

While hedging with short mini-S&P futures positions greatly reduces the tail risk of shorting

VIX futures contracts, these strategies do not enjoy limited risk. By contrast, buying VIX puts when the VIX futures curve is in contango and buying VIX calls when the VIX futures curve is in backwardation are limited risk strategies.

19 Thus, the volatility of S&P index options is far more overpriced than the volatility of S&P index volatility. 13

Along the lines of Simon and Campasano (2014), we assume that the VIX futures curve triggers option trades when the daily roll--the spread between front VIX futures prices and the

VIX divided by the number of business days until the settlement of the VIX futures contract--is greater than .10 percentage points in the case of contango when puts are purchased and less than

-.10 percentage points in the case of backwardation when calls are purchased.20 A daily roll of

.10 VIX percentage points corresponds to $100 per business day as the value of one VIX futures point is $1,000. The profitability of these trades is examined assuming 5-business day holding periods to focus on the impact of entry rules on profitability. We later examine the impact of not entering these strategies when VIX option ex-ante volatility premiums are unusually high. As earlier, trades are entered and exited at the mid-point of closing bid-ask quotes. New trades in options with the specified moneyness are entered as soon as they are triggered, which could be on the same day that the previous trade is exited. It is important to note that unreported results show that rolling over both long call and long put positions every 5-business days without delta hedging generally is not significantly profitable over the sample period for any of the options examined in this study.21

Exhibit 5 shows the profitability of 5-business day trades involving buying closest to expiration VIX puts that have at least 10 business days to expiration across the five closest to the money strikes when the VIX futures curve is in contango and the daily roll is greater than .10 futures point ($100) per business day. The trades are all significantly profitable at better than the

1 percent significance level with mean P&Ls ranging from $167 for two strike out of the money puts to $410 for two strike in the money puts. These average profits correspond to roughly 5-1/2 to 13-1/2 percent average returns, defined again as gains scaled by initial values of purchased

20 In contrast to the previous sections where we examine whether VIX options are overpriced and consequently assumed that long VIX option positions are delta hedged, the strategies examined here are directional bets on VIX futures prices and hence are not delta hedged. 21 These unreported results demonstrate that P&Ls are significantly negative at the 5 percent level for one strike out of the money calls and for one strike in the money puts with both having average losses over 5-business days of roughly $126. The same trades involving other VIX options do not lead to statistically significant gains or losses. 14 puts. The findings also indicate that profitability stems from the average size of gains being considerably greater than the average size of losses, as only somewhat more than half of the trades result in gains. Large gains are also far more prevalent than large losses, as reflected by the top decile P&L cutoffs occurring at gains that typically are about twice the magnitude of losses at bottom decile cutoffs. This is not surprising given the limited risk associated with buying put options. The average gains typically are more than explained by the negative delta of long put positions with VIX futures prices falling as they roll down an upward sloped curve. The benefits of positive gamma and favorable vega owing to the tendency of VIX option implied volatility to rise under these circumstances neutralizes much of the losses from time decay.

The results also indicate that sizeable portions of the roll can be earned with highly favorable risk reward tradeoffs and that buying in the money puts rather than out of the money puts leads to more compelling results owing to the greater negative delta of in the money puts.

The average gains for 2 strike in the money puts are $410 versus average gains of $167 for 2 strike out of the money puts, which correspond to average returns of 13.5% versus 5.4%. These outcomes occur despite the strong volatility skew, which causes in the money VIX puts to trade at considerably higher implied volatilities than out of the money VIX puts. Because synthetic

VIX put positions can be created by shorting VIX futures and buying VIX calls, an interpretation of the results from this perspective indicates that shorting VIX futures and buying dollar cheap but higher implied volatility out of the money calls outperforms shorting VIX futures and buying dollar expensive but low implied volatility in the money calls. Unreported results and casual inspection of exhibit 5 indicate that buying bear put spreads is not an efficient way to exploit the roll over 5-business day horizons, as losses from selling lower strike puts offsets much of the gains from buying higher strike puts.

The bottom panel of exhibit 5 shows the results of the same long put strategies, modified by not entering these trades when VIX option volatility premiums are in their top quartiles over

15 the sample period. The results are not qualitatively different as the P&Ls for each degree of moneyness continue to be statistically significantly positive, as are average returns. However avoiding these trades when VIX option volatility premiums are high increases the average gains by $100 to $140 for the in the money puts, which corresponds to increasing average returns by 4 to 6%. For the low strike puts the improvements in results are much smaller.

Exhibit 6 shows the results of buying VIX call options and holding for 5-business days when the VIX futures curve is in backwardation and the daily roll up inverted futures curves is worth at least .10 VIX futures points ($100) per business day. The bet for these trades is that the

VIX does not fall to levels consistent with lower VIX futures prices, in which case VIX futures prices rise and converge to higher VIX levels. The results indicate that average profits are large and range from $460 to 1,032, but are not statistically significant. The same is true for average returns, which range from 17 to 38 percent of the initial value of purchased call options. The lack of statistical significance can be attributed both to the relatively low number of trades entered and the much higher P&L volatility owing to inverted VIX futures curves being associated with higher levels of the VIX and greater uncertainty.22 Nevertheless, the risk reward tradeoffs appear compelling with the percentage of winning trades across the various strikes hovering just below

50 percent and with average gains of winning trades being about twice the magnitude of average losses of losing trades. Again, large gains are more prevalent than large losses as the top decile

P&L cutoffs are at gains that are twice the losses at bottom decile cutoffs. The greeks indicate that the gains from delta more than account for the overall gains, while the gains from gamma about offset the losses from time decay, while long vega exposure results in small and mixed results. Overall, the results indicate that buying VIX calls to take advantage of the tendency of

22 While roughly 144 put trades per strike are triggered in the previous exhibit when the VIX futures curve is in contango, only roughly 46 trades per strike are triggered when the VIX futures curve is in backwardation and the average level of the VIX futures contract when put trades are entered is 23.4 percent compared to 32.6 percent when call trades are entered and the standard deviation of the P&L for at the money calls is over 3 times the P&L for at the money puts. 16

VIX futures prices to roll up inverted VIX futures curves offers favorable risk reward tradeoffs and limited risk, despite the lack of statistically significant profits.23

The bottom panel of exhibit 5 shows that the results are not qualitatively different when trades are not entered when VIX call option ex-ante volatility premiums are in their highest quartiles. Average P&Ls remain large and positive ranging from $375 to $1,272 but are not statistically significant as is the case for returns, which range from 13% to 38%. Overall, avoiding entering these trades when VI call option volatility premiums are high does not improve the outcomes. In the next section we examine the efficacy of using VIX options to take advantage of the volatility premium in VIX futures contracts.

V. Trading the VIX futures volatility premium with VIX options

A large literature has established that S&P 500 index options typically are overpriced, as reflected by the tendency of delta hedged short S&P 500 index option positions to be highly profitable.24 Because the VIX is a measure of S&P 500 index implied volatility and because

VIX futures contracts settle to the value of S&P 500 index option implied volatilities, VIX futures contracts necessarily share this overpricing. The overpricing is further increased by the

VIX futures curve being in contango on 74% of the days in the sample period. This section examines whether when VIX futures prices are unusually high relative to recent actual S&P 500 volatility, convergence back to more typical levels can be exploited by buying VIX put options.25

To determine whether VIX futures prices embed unusually large ex-ante volatility premiums, we

23 Long call trades are fairly well spread out throughout the sample period rather than heavily concentrated during the financial panic. For example, only 21 of the 46 trades for at the money calls are entered in 2008 and 2009. Also, unreported results demonstrate that entering bullish vertical call spreads under the same circumstances does not lead to significantly profitable trades because gains on calls purchased are largely offset by losses on calls sold over 5- business day horizons. 24 See the studies cited earlier in the paper. 25 Recall that it was mentioned earlier that unreported results show that rolling over long VIX puts for 5-business day period generally is not significantly profitable. 17 compare VIX futures prices with the lagged 10-day historical volatility of S&P 500 returns. We enter long put trades when this differential is greater than 10 percentage points, which represents a compromise between a threshold that triggers a reasonable number of trades and is high enough to be meaningful. While a 10 percentage point volatility premium may seem high, this threshold results in 5-business day trades being open during roughly 1/4 of the sample period.26

Exhibit 7 shows the results of these 5-business day trades again assuming that puts are bought and sold at the mid-point of the bid-ask spread. The trades for all 5 strikes examined have significantly positive average P&Ls and returns. Average P&Ls range from $144 for puts that are two strikes out of the money to $438 for puts that are two strikes in the money, which correspond to returns ranging from 5 percent to 16 percent. The average gains stem mostly from roughly 50 percent greater average gains on winning trades relative to average losses on losing trades, as the frequency of winning versus losing trades tends to be only somewhat higher than

50 percent. Consistent with the limited losses associated with buying put options, the cutoffs for the top versus the bottom return deciles occur at considerably greater gains than losses. The gains owe to the large declines in VIX futures prices, as average gains on the puts attributed to negative deltas about match total average gains. In addition, gains from being long gamma and long vega, with VIX implied volatility rising in these circumstances, neutralize losses from theta.

As was the case for trades involving buying VIX puts when the VIX futures curve is in contango, the results here show much greater average profitability and returns for 2 strike in the money puts ($438 mean profits and 16% mean returns) versus 2 strike out of the money puts ($144 and

5-1/4%). These results occur despite the presence of a strong volatility skew that causes out of the money puts to trade at much higher implied volatilities than out of the money puts.

26 Again, it is important to note that rolling over long put positions that are held for 5-business days does not result in average gains for any of the expirations or strike prices examined in this study and for some moneyness categories result in statistically significant losses. 18

The bottom panel of exhibit 7 demonstrates that the results are qualitatively similar when these trades are entered only when VIX put option ex-ante volatility premiums are not in their top quartiles. The results are improved for the in the money puts as average P&Ls increase from

$100 to $200, while average returns rise by 3 to 7 percent. For out of the money puts the results are slightly worsened.

VI. Conclusion

This study first examines whether substantial volatility premiums are built into VIX options and then examines whether buying VIX calls and puts are effective strategies for exploiting the systematic tendencies of VIX futures prices. The study finds that 5-business day delta hedged returns on front month long call and long put positions are relatively modest over the sample period from January 2007 through March 2014. Average losses for calls are statistically significant for 3 of the 5 moneyness categories and range from -$18 to -$40 and associated negative returns from -.7% to -1.6%. Average losses for puts are significant for only

1 of the 5 moneyness categories and average losses range from -$16 to -$35, which correspond to negative average returns ranging from -.1% to -1.3%. The results also demonstrate that the statistically significant losses on 5-business delta-hedged long VIX option positions owe to the roughly ¼ of the sample period when ex-ante VIX option volatility premiums are in their highest quartiles, as losses are not statistically significant for any of the options examined when only the other ¾ of the sample period is included.

The relatively modest 5-business day negative returns on delta hedged long VIX call and put positions are surprising given the very large ex-ante volatility premiums of VIX options.

Implied volatilities of VIX options exceed lagged 10-day volatilities of corresponding VIX futures contracts by an average of 17 percentage points for options that are 2 strikes below at the

19 money strikes and an average of 39 percentage points for options that are 2 strikes above at the money strikes. An important unresolved issue is why does the roughly 300 basis point difference between S&P 500 index option implied volatility and S&P 500 historical volatility reported by

Eraker (2009) translate into outsized negative returns on long delta hedged S&P 500 index option positions, while the much larger ex-ante VIX option volatility premiums translate into much more modest losses on delta hedged long VIX option positions. A partial explanation is that being long vega contributes gains of roughly $21 to $120 to 5-business day delta hedged long VIX option positions, as increases in the implied volatility of VIX options offsets some of the losses that occur from the greater negative effect of time decay relative to the positive effect of being long gamma. An unanswered question remains as to why VIX option implied volatility rose steadily for the options examined against a backdrop where the VIX increased on balance only from about 11.56% to 13.88% over the sample period. One possibility is that short term

VIX futures tend to be more volatile as they move toward settlement days, which causes VIX option implied volatility to rise, leading to profits from being long vega.

In any event, the evidence suggests that market participants do not egregiously overpay for the limited risk associated with being long VIX options, which suggests that long VIX option positions that enjoy limited risk could be effective strategies for exploiting the well-established tendencies of VIX futures. The study shows that 5-business day trades involving buying VIX puts to exploit the roll of VIX futures contracts down the VIX futures curve when in contango provide highly significant profits that range from $167 to $410 and associated returns of 5.4% to

13.5 percent of the value of options. By contrast, 5-business day trades involving buying VIX calls to take advantage of the roll of VIX futures up the VIX futures curve when in backwardation are not significantly profitable although the risk-reward tradeoffs appear attractive with average profits ranging from $460 to $1,032 and profitable outcomes occurring from 41% to 49% of the time and average gains on winning trades about double the average

20 losses on losing trades. The results also indicate that buying VIX put options when VIX futures build in substantial ex-ante volatility premiums relative to lagged 10-day volatility of S&P 500 index returns is also a highly significantly profitable strategy with average profits over 5- business days ranging from $144 to $438 and returns ranging from 5.2 % to 15.9%. The profitability of buying puts both to exploit the tendency of VIX futures prices to fall when the

VIX futures curve is in contango and to exploit unusually high ex-ante volatility premiums in

VIX futures prices is not qualitatively improved when these trades are not entered when VIX option ex-ante volatility premiums are in their highest quartiles over the sample period.

However, the profitability of these trades is improved by buying in the money puts rather than out of the money puts, despite the volatility skew in VIX options that causes high strike options to be priced with substantially higher implied volatilities. The greater profitability owes to the benefits of larger deltas from in the money puts offsetting the incremental cost of paying higher implied volatilities. Because synthetic and equivalent put exposures can be achieved by shorting

VIX futures contracts and buying VIX calls, the results demonstrate that maintaining large negative deltas by shorting VIX futures contracts and buying dollar cheap but high implied volatility out of the money VIX calls is a more profitable strategy.

Finally, the option strategies that have been shown to be highly profitable have assumed that options trades are entered and exited at the mid-point of bid-ask spreads, which raises issues about the profitability of the trades after transactions costs. Given the large average bid-ask spreads of VIX options reported in the paper, very few strategies examined in this study or any short-term strategy involving VIX options would be robust to incurring the full costs of these spreads. However, the profitability of strategies involving buying in the money puts would remain robust to paying more reasonable effective bid-ask spreads of .05 to .10 point, which would lower the average profits of buying puts by $100 to $200. Another strategy to avoid potentially large bid-ask spreads on trades involving buying in the money put options instead

21 would be to short VIX futures contracts and buy more liquid and lower priced out of the money

VIX calls, which likely would enjoy tighter bid-ask spreads. In any event, the success of the strategies examined in this study requires non-market makers to work VIX option bid-ask spreads.

`

22

Exhibit 1. VIX Option Data from January 2007-March 2014. The exhibit shows the closing mid-point daily closing quote, closing bid-ask spread and implied volatility for calls and puts that from two strikes below the at the money strike (ATM-2) to two strikes above the at the money strike (ATM+2) for the closest to expiration options with at least 10 business days to expiration. Implied volatility is calculated from the Black (1976) futures option pricing model on a business day basis. 10 business day historical volatilities are calculated from the corresponding VIX futures contracts. Volatility premiums are the spreads between implied volatility and lagged 10-business day historical volatility.

Calls Puts ATM-2 ATM-1 ATM ATM+1 ATM+2 ATM-2 ATM-1 ATM ATM+1 ATM+2

Mid-point Quote mean 3.87 2.99 2.26 1.72 1.32 .77 1.36 2.25 3.40 4.79 (std. dev) (1.86) (1.49) (1.16) (.92) (.74) (.67) (.90) (1.20) (1.67) (2.32) top decile 6.25 4.75 3.70 2.93 2.33 1.68 2.53 3.80 5.50 7.40 bottom decile 2.20 1.60 1.15 .83 .58 .13 .45 1.08 1.83 2.63

Bid-Ask Spread mean .25 .20 .15 .13 .12 .10 .12 .16 .21 .26 (std. dev) (.15) (.13) (.11) (.09) (.08) (.07) (.09) (.11) (.13) (.16) top decile .40 .30 .30 .20 .20 .15 .20 .30 .30 .40 bottom decile .10 .10 .05 .05 .05 .05 .05 .05 .10 .10

Implied Volatility mean 73.51 79.08 84.93 90.32 95.58 73.82 78.74 85.05 90.72 95.94 (std. dev) (19.04) (17.48) (16.57) (16.04) (15.96) (17.88) (17.34) (16.78) (16.24) (16.19) top decile 96.47 100.73 106.22 111.79 116.71 96.37 100.98 106.75 11.80 117.09 bottom decile 51.58 59.60 66.40 72.24 77.76 53.18 58.90 65.68 72.37 77.87

10 day VIX Hist Vol. mean 56.14 56.12 56.43 56.39 56.54 56.46 56.12 56.43 56.39 56.54 (std. dev) (25.86) (25.87) (26.12) (26.01) (26.07) (25.82) (25.87) (26.12) (26.01) (26.07) top decile 93.87 93.86 94.44 94.12 94.45 93.97 93.86 94.45 94.12 94.45 bottom decile 29.53 29.52 29.38 29.52 29.43 29.95 29.52 29.38 29.52 29.43

Volatility Premium mean 17.37 22.96 28.50 33.93 39.04 17.36 22.63 28.61 34.33 39.40 (std. dev) (22.20) (20.97) (20.70) (20.71) (20.94) (20.92) (20.04) (20.58) (20.83) (20.97) top decile 39.54 44.40 49.91 55.94 61.55 39.46 43.86 50.19 56.70 62.56 bottom decile -8.11 -0.69 4.43 9.23 13.77 -8.17 -0.65 4.43 9.76 14.49

NOBS 1750 1728 1747 1775 1774 1730 1728 1747 1775 1774

23

Exhibit 2. Five-business day delta hedged P&Ls on front month VIX call and put long positions. Option transactions in the front month contracts that have more than 10-business days until expiration are assumed to take place at the mid-point of the closing bid-ask spread. The greeks are calculated with the Black (1976) model adjusted to a business day basis. The closest to the money (ATM) strike and the two lower (ATM-1 and ATM-2) and two higher (ATM+1 and ATM+2) strikes are examined. The P&Ls from the greeks are cumulated over the holding period with the greeks recalculated each day. The sample period is from January 2007 through March 2014.

Front Month Calls

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) -17.51 -40.18 -30.38 -28.42 -35.72 (.14) (.023) (.05) (.07) (.018) Return (p-value) -.007 -.016 -.013 -.011 -.014 (.17) (.03) (.08) (.13) (.04) Top decile 216.22 229.96 219.44 225.33 228.08 Bot. Decile -265.81 -314.22 -292.04 -303.59 -315.41 Pct. Gains 41% 42% 39% 36% 37% Avg. Gain 166.36 196.17 211.80 230.31 207.70 Avg. Loss -144.51 -208.99 -185.92 -176.44 -178.34 Gamma P&L 154.75 199.97 203.51 190.14 169.30 Vega P &L 21.08 55.74 90.66 116.29 120.44 Theta P&L -194.55 -273.10 -313.63 -329.50 -319.87 NOBS 360 360 358 360 360

Front Month Puts

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) -16.16 -18.00 -29.42 -34.78 -27.57 (.18) (.23) (.06) (.03) (.08) Return (p-value) -.005 -.0076 -.011 -.013 -.001 (.29) (.34) (.123) (.06) (.14) Top decile 197.03 278.51 269.46 268.22 246.64 Bot. Decile -355.52 -296.45 -311.87 -318.57 -293.03 Pct. Gains 41% 45% 36% 36% 38% Avg. Gain 167.77 197.24 237.84 228.96 221.15 Avg. Loss -144.57 -192.14 -179.98 -185.65 -180.38 GAMMA P&L 157.61 199.68 203.23 189.12 168.74 Vega P &L 40.69 71.21 91.82 100.51 107.07 Theta P&L -197.40 -274.12 -314.27 -330.42 -319.15 NOBS 360 360 358 360 360

24

Exhibit 3. Five-business day delta hedged P&Ls on front month VIX call options when ex-ante volatility premiums of specific options are above and below their highest quartile cutoffs over the sample period. Ex-ante volatility premiums are defined as specific option implied volatilities minus 10-day lagged volatility of the corresponding VIX futures contract. Option transactions are assumed to take place at the mid-point of the closing bid- ask spread in the closest to expiration contract that has at least 10-business days until expiration. The greeks are calculated with the Black (1976) model adjusted to a business day basis. The closest to the money (ATM) strike and the two lower (ATM-1 and ATM-2) and two higher (ATM+1 and ATM+2) strikes are examined. The P&Ls from the greeks are cumulated over the holding period with the greeks recalculated each day. The sample period is from January 2007 through March 2014.

VIX call option ex-ante volatility premiums are above their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) -85.03 -96.67 -75.63 -84.34 -63.87 (.000) (.000) (.001) (.000) (.007) Return (p-value) -.031 -.040 -.035 -.038 -.026 (.000) (.000) (.001) (.001) (.032) Top decile 157.43 167.30 166.47 133.27 123.65 Bot. Decile -336.87 -344.54 -299.82 -372.96 -310.63 Pct. Gains 35% 32% 30% 30% 35% Avg. Gain 138.73 157.13 189.83 175.24 164.25 Avg. Loss -203.91 -214.01 -188.59 -196.73 -185.54 Gamma P&L 142.96 145.19 142.88 143.72 139.38 Vega P &L -23.01 29.41 59.99 67.49 80.84 Theta P&L -221.99 -271.04 -277.28 -296.12 -290.53 NOBS 147 136 134 139 138

VIX call option ex-ante volatility premiums are below their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 12.39 -5.62 -10.92 -15.20 -23.73 (.363) (.753) (.534) (.362) (.143) Return (p-value) .006 -.001 -.001 -.003 -.005 (.230) (.409) (.128) (.165) (.419) Top decile 254.89 308.97 342.31 271.35 275.96 Bot. Decile -227.49 -255.27 -280.85 -287.02 -270.47 Pct. Gains 47% 43% 42% 43% 41% Avg. Gain 183.07 227.10 238.40 215.81 199.54 Avg. Loss -141.40 -180.51 -192.40 -186.12 -181.41 Gamma P&L 163.61 211.09 218.34 196.54 186.16 Vega P &L -193.50 73.71 104.04 130.58 130.40 Theta P&L -193.49 -271.04 -327.43 -338.21 -334.65 NOBS 308 303 301 301 302

25

Exhibit 4. Five-business day delta hedged P&Ls on front month VIX put options when ex-ante volatility premiums of specific options are above and below their highest quartile cutoffs over the sample period. Ex-ante volatility premiums are defined as specific option implied volatilities minus 10-day lagged volatility of the corresponding VIX futures contract. Option transactions are assumed to take place at the mid-point of the closing bid- ask spread in the closest to expiration contract that has at least 10-business days until expiration. The greeks are calculated with the Black (1976) model adjusted to a business day basis. The closest to the money (ATM) strike and the two lower (ATM-1 and ATM-2) and two higher (ATM+1 and ATM+2) strikes are examined. The P&Ls from the greeks are cumulated over the holding period with the greeks recalculated each day. The sample period is from January 2007 through March 2014.

VIX put option ex-ante volatility premiums are above their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) -65.16 -80.27 -95.80 -88.47 -66.87 (.001) (.000) (.000) (.001) (.011) Return (p-value) -.025 -.034 -.044 -.040 -.030 (.002) (.000) (.000) (.001) (.024) Top decile 161.60 163.45 149.96 162.36 133.19 Bot. Decile -291.26 -341.80 -360.15 -347.74 -296.95 Pct. Gains 33% 33% 25% 26% 31% Avg. Gain 151.12 164.24 193.15 228.50 199.07 Avg. Loss -169.73 -199.80 -194.05 -200.34 -183.77 Gamma P&L 147.42 142.38 149.69 154.56 143.84 Vega P &L 4.07 39.35 35.91 54.31 69.88 Theta P&L -221.09 -268.92 -284.80 -303.17 -291.84 NOBS 135 134 134 138 131

VIX put option ex-ante volatility premiums are below their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 2.84 -11.31 -3.79 -13.47 -17.49 (.822) (.466) (.823) (.423) (.285) Return (p-value) .003 -.002 .003 -.002 -.003 (.470) (.223) (.704) (.82) (.677) Top decile 227.45 285.35 313.70 331.04 340.33 Bot. Decile -203.88 -277.30 -301.77 -282.01 -302.99 Pct. Gains 47% 46% 44% 40% 43% Avg. Gain 160.38 192.82 226.96 235.94 209.47 Avg. Loss -136.10 -182.04 -187.84 -181.13 -188.71 Gamma P&L 162.06 203.75 213.15 197.53 183.39 Vega P &L 48.97 78.61 117.42 123.32 122.00 Theta P&L -190.51 -275.43 -327.85 -337.20 -333.35 NOBS 303 303 302 301 300

26

Exhibit 5. Five-business day P&Ls from buying VIX put options when the VIX futures curve is in contango and the average daily roll is greater than .10 point or $100 per business day. The roll is defined as the spread between VIX futures prices and the VIX divided by the number of business days to settlement. Option transactions are assumed to take place at the mid-point of the closing bid-ask spread in the closest to expiration contract that has at least 10- business days until expiration. The greeks are calculated with the Black (1976) model adjusted to a business day basis. The closest to the money (ATM) strike and the two lower (ATM-1 and ATM-2) and two higher (ATM+1 and ATM+2) strikes are examined. The P&Ls from the greeks are cumulated over the holding period with the greeks recalculated each day. The results are shown for the whole sample period from January 2007 through March 2014 and when the trades are not entered if ex-ante volatility premiums are above their highest decile cutoffs.

The whole sample

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 166.67 236.27 286.98 344.48 410.34 (.001) (.001) (.001) (.001) (.001) Return (p-value) .054 .076 .095 .111 .135 (.002) (.002) (.002) (.003) (.002) Top decile 950.00 1,200 1,450.00 1,750.00 2,050 Bot. Decile -300.00 -500.00 -725.00 -950.00 -1,150 Pct. Gains 53% 53% 58% 59% 61% Avg. Gain 513.31 754.67 892.77 1,066.76 1,199.43 Avg. Loss -228.31 -344.03 -537.30 -678.75 -807.89 Delta P&L 156.05 283.59 309.98 384.22 435.25 Gamma P&L 117.68 132.77 140.06 129.66 114.96 Vega P &L 76.92 100.37 136.18 138.27 149.19 Theta P&L -194.32 -258.11 -303.38 -313.48 -305.47 NOBS 144 142 144 145 145

With trades not entered if ex-ante volatility premiums are above their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 136.19 229.09 358.70 450.43 547.01 (.018) (.007) (.001) (.001) (.000) Return (p-value) .048 .080 .124 .155 .192 (.019) (.008) (.001) (.001) (.000) Top decile 650.00 1,000.00 1,525.00 1,775.00 2,100 Bot. Decile -300.00 -450.00 -650.00 -925.00 -1,075.00 Pct. Gains 53% 56% 61% 65% 66% Avg. Gain 452.23 678.88 937.86 1,085.67 1,275.32 Avg. Loss -225.00 -338.04 --542.22 -740.63 -855.00 Delta P&L 115.21 253.62 372.96 475.98 542.09 Gamma P&L 112.60 127.63 143.33 132.85 119.24 Vega P &L 75.57 106.84 152.97 160.75 170.80 Theta P&L -184.11 -253.44 -307.46 -317.90 -306.80 NOBS 105 104 115 115 117

27

Exhibit 6. Five-business day P&Ls from buying VIX call options when the VIX futures curve is in backwardation and the average daily roll is greater than .10 point or $100 per business day. The roll is defined as the spread between VIX futures prices and the VIX divided by the number of business days to settlement. Option transactions are assumed to take place at the mid-point of the closing bid-ask spread in the closest to expiration contract that has at least 10-business days until expiration. The greeks are calculated with the Black (1976) model adjusted to a business day basis. The closest to the money (ATM) strike and the two lower (ATM-1 and ATM-2) and two higher (ATM+1 and ATM+2) strikes are examined. The P&Ls from the greeks are cumulated over the holding period with the greeks recalculated each day. The results are shown for the whole sample period from January 2007 through March 2014 and when the trades are not entered if ex-ante volatility premiums are above their highest decile cutoffs.

The whole sample period

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 1,032.39 881.67 735.87 610.33 460.33 (.117) (.148) (.159) (.179) (.240) Return (p-value) .386 .323 .261 .221 .171 (.088) (.121) (.130) (.146) (.202) Top decile 6350 6900 4800 4200 3400 Bot. Decile -2500 -2600 -2200 -1775 -1400 Pct. Gains 48% 49% 48% 43% 41% Avg. Gain 4205.95 3729.55 3234.09 2982.50 2571.05 Avg. Loss -1865.22 -1842.39 -1554.17 -1214.42 -1025.00 Delta P&L 1173.04 1005.41 891.57 692.15 556.79 Gamma P&L 349.98 430.78 441.01 445.43 421.41 Vega P &L -42.38 -41.76 -41.59 7.09 32.08 Theta P&L -346.85 -464.57 -515.98 -562.35 -551.83 NOBS 44 45 46 46 46

With trades not entered if ex-ante volatility premiums are above their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 1,172.30 760.26 607.93 506.41 375 (.100) (.222) (.231) (.282) (.341) Return (p-value) .382 .264 .197 .176 .129 (.077) (.156) (.161) (.211) (.261) Top decile 8,325.00 6,900.00 4,200.00 5,550.00 4,400.00 Bot. Decile -3,050.00 -3,125.00 -2,200.00 -1,775.00 -1,425.00 Pct. Gains 51% 51% 49% 41% 41% Avg. Gain 4,178.95 3,411.25 2,945.00 2,998.44 2,429.69 Avg. Loss -2,001.39 -2,030.26 -1,617.86 -1,227.17 -1,054.35 Delta P&L 1,253.55 849.95 762.59 635.51 479.19 Gamma P&L 336.73 435.62 411.57 428.85 -579.16 Vega P &L -16.89 -43.66 -48.41 15.39 55.71 Theta P&L -351.22 -499.43 -542.60 -584.09 -579.16 NOBS 37 39 41 39 39

28

Exhibit 7. Five-business day P&Ls from buying VIX put options when the spread between VIX futures and 10- day lagged S&P 500 volatility is greater than 10 percentage points. Option transactions are assumed to take place at the mid-point of the closing bid-ask spread in the closest to expiration contract that has at least 10-business days until expiration. The greeks are calculated with the Black (1976) model adjusted to a business day basis. The closest to the money (ATM) strike and the two lower (ATM-1 and ATM-2) and two higher (ATM+1 and ATM+2) strikes are examined. The P&Ls from the greeks are cumulated over the holding period with the greeks recalculated each day. The results are shown for the whole sample period from January 2007 through March 2014 and when the trades are not entered if ex-ante volatility premiums are above their highest decile cutoffs.

The whole sample period

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 144.09 229.52 305.85 377.93 437.77 (.053) (.021) (.018) (.017) (.016) Return (p-value) .052 .085 .112 .138 .159 (.036) (.011) (.009) (.009) (.008) Top decile 875 1150 1650 1875 2100 Bot. Decile -525 -775 -1000 -1300 -1550 Pct. Gains 54% 56% 60% 61% 62% Avg. Gain 583.00 814.15 1010.27 1221.05 1375.00 Avg. Loss -366.28 -526.22 -732.24 -920.95 -1072.22 Delta P&L 157.10 232.67 319.52 401.00 480.14 Gamma P&L 136.23 160.57 163.24 153.25 133.58 Vega P &L 65.60 119.05 163.24 143.31 153.22 Theta P&L -217.89 -277.55 -315.94 -325.32 -317.85 NOBS 93 94 94 94 94

With trades not entered if ex-ante option volatility premiums are above their highest quartile cutoffs

ATM-2 ATM-1 ATM ATM+1 ATM+2

$ P&L (p-value) 89.50 149.51 456.85 583.85 555.60 (.168) (.146) (.008) (.004) (.010) Return (p-value) .041 .070 .154 .207 .188 (.077) (.061) (.005) (.001) (.006) Top decile 650.00 975.00 1850.00 2,450 2,400.00 Bot. Decile -437.5 -725.00 -775.00 -1,025 -1,000.00 Pct. Gains 58% 59% 60% 66% 64% Avg. Gain 365.52 606.67 1,194.59 1,340.70 1,396.51 Avg. Loss -291.67 -503.57 -635.00 -895.45 -951.04 Delta P&L 93.30 189.34 463.44 582.87 586.31 Gamma P&L 135.81 138.98 168.64 161.08 134.86 Vega P &L 68.01 106.42 158.33 172.43 169.44 Theta P&L -220.65 -263.90 -330.98 -334.64 -322.30 NOBS 50 51 62 65 67

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