The Implied Volatility Slope of VIX Options

The implied volatility slope of VIX options

Jungah Yoon1 Department of Accountancy and Finance Otago Business School, University of Otago Dunedin 9054, New Zealand [email protected]

Xinfeng Ruan Department of Accountancy and Finance Otago Business School, University of Otago Dunedin 9054, New Zealand [email protected]

Jin E. Zhang Department of Accountancy and Finance Otago Business School, University of Otago Dunedin 9054, New Zealand [email protected]

First Version: 12 October 2020 This Version: 13 November 2020

Keywords: Implied volatility, slope of IV, VIX options, VIX futures returns JEL Classiﬁcation: G12; G13

1 Corresponding author. Tel: +64 20 437 0225. The implied volatility slope of VIX options i

Abstract

In this paper, we study the slope of implied volatility (IV) curve of VIX options and its return predictability in the VIX futures market. The IV curves often provide valuable information on the market expectation of the future realized volatility. We document the shape and term structure of VIX options’ IV curves estimated with a linear function. The IV surface in the VIX options market exhibit a positive-sloping and almost linear curve, unlike the S&P 500 Index or other commodity market indices’ options, where they often show the representative smirk shape. We find a correlation between simple changes in VIX- and VVIX Index can significantly explain positive slope of IV in the VIX options market. This unique feature should be aware of by investors and volatility traders and studied further to enhance our factual knowledge. We also find the slope of VIX IV curve has statistically and negatively significant predictive power on VIX futures returns at daily frequency.

Keywords: Implied volatility, slope of IV, VIX options, VIX futures returns

JEL Classiﬁcation: G12; G13 The implied volatility slope of VIX options 1

1 Introduction

The estimated implied volatility (IV) curve by options contains information about risk-neutral moments.2 The relationship between risk-neutral moments and corresponding IV surface factors provided by the existing literature is that level, slope and curvature of the IV curves reﬂect the risk-neutral distribution of the underlying asset returns over diﬀerent horizons (Zhang and Xiang,

2008; Gehricke and Zhang, 2020). While quantifying IV smirk/curve of options has documented on few indexes’ options (Li, Gehricke, and Zhang, 2019; Aschakulporn and Zhang, 2020; Jia,

Ruan, and Zhang, 2020) and equity options (Cont and Fonseca, 2002; Bollen and Whaley, 2004;

Xing, Zhang, and Zhao, 2010; Yan, 2011; Kelly, Pástor, and Veronesi, 2016; Tian and Wu,

2020); there has been little to no studies focus on the IV curves on the VIX options, where options are written on the most critical volatility index that captures fears in the overall US market.

The shape of the implied volatility curve of volatility index options often displays demand on call or put options by investors, which can be linked to their hedging motivation (Bollen and Whaley, 2004) as well as informed trading (Rapach, Ringgenberg, and Zhou, 2016). The

VIX options have been used as one of the popular diversiﬁcation/speculation tools for equity investors since its inception. The current VIX Index is based on the S&P 500 Index (SPX) and estimates expected volatility by accounting the weighted prices of SPX options over a wide range of strike prices. The VIX Index, to this date, continues to attract many practitioners and researchers due to its practical use. In 2019, the VIX options had an average daily volume of

509,490 contracts or 127 million contracts over the entire year.

In this study, we investigate the term structure and dynamics of IV curve in the VIX options market and test the estimated IV slope as a predictor on VIX futures returns. Our study focuses on estimating the slope of the IV curve in the VIX options market by adopting a linear function.

Two factors are obtained from linear estimation, which is the level and slope. We then further develop the constant maturity IV factors to study the shape and dynamics of VIX options more comprehensively. We ﬁnd that the VIX options IV curve exhibits an upward-sloping and almost

2 The risk-neutral moments or moments of a function are quantitative estimates related to the shape of the corresponding function’s figure. The implied volatility slope of VIX options 2 a linear curve, similar to that of VXX (VIX futures exchange-traded-product; ETP) options market, however more volatile on average for shorter maturity options. As the maturity of VIX options increases, the IV curve becomes flatter (i.e., level and slope tend to decrease). According to Gehricke and Zhang (2020), quantifying specific options’ IV curve may provide a standard for creating an option pricing model founded on the empirical observations of corresponding underlying returns’ risk-neutral moments. Similar to estimation with a quadratic function, it allows us to examine an intuitive relationship between fitted IV curve parameters (level and slope) and risk-neutral moments (standard deviation and skewness).

The literature surrounding VIX derivatives has gained substantial attention since the inception of the VIX-based product in 2004 (i.e., VIX futures; ﬁrst-ever listed futures contract on

VIX Index). The VIX options convey diﬀerent information from S&P 500 index options or VIX futures, although they are directly linked to each other. The link between the parameters of

IV surface and risk-neutral moments are well-documented by existing literature. For examples,

Bakshi, Kapadia, and Madan (2003) propose a framework to retrieve risk-neutral moments from option prices and derive laws that break down individual return skewness into a systematic and idiosyncratic component. Alternatively, Zhang and Xiang (2008) provide a method in which one can easily describe the implied volatility-moneyness function and relates the coeﬃcients to the implied risk-neutral distribution. More recently, Kelly, Pástor, and Veronesi (2016) and Tian and Wu (2020) study slope of IV curves in equity options by a linear function,3 which is similar to our study.

Many existing studies have documented and confirmed that the variance risk premium is essential and useful in serving various role. Bollerslev, Tauchen, and Zhou (2009) show the variance risk premium, which is defined as the difference between implied and realized variation, can explain a notable portion of the time-series variation in post-1990 aggregate stock market returns. Xing, Zhang, and Zhao (2010) find individual stock options’ IV slope has a significant predictive ability on future cross-sectional equity returns. In the subsequent year, Chung et al. (2011) find the information content implied by the SPX options are significantly improved

3 Kelly, Pástor, and Veronesi (2016) calculate slope coefficient from the regression of options’ implied volatilities against Black and Scholes (1973)’s delta. On the other hand, Tian and Wu (2020) obtain an approximation of the risk-neutral skewness of the return distribution by the proportional implied volatility slope against the standardized moneyness measure. The implied volatility slope of VIX options 3 by the information recovered from the VIX options. Zhao, Zhang, and Chang (2013) document the returns of the swap contract and find that both skewness and kurtosis risk premiums are significantly negative. Following standard practice, Johnson (2017) studies and finds the second principal component, SLOPE, from VIX term structure summarizing nearly of all information, predicting the excess return of synthetic S&P 500 variance swaps, VIX futures, and S&P 500 straddles for all maturities and to the exclusion of the rest of the term structure. More recently,

Bardgett, Gourier, and Leippold (2019) ﬁnd that VIX options contain valuable information on the risk-neutral conditional distributions of volatility at diﬀerent time horizons which do not stem from the S&P 500 index. In addition to its return predictive ability, the slope of IV curve is also found to be an important measure in the presence of a political event, which explains as to why investors are willing to pay a higher premium during a certain period (Kelly, Pástor, and Veronesi, 2016).

Overall, this strand of literature found the information content of the VIX options and/or its term structure to be signiﬁcant in predicting future equity, asset, index returns or volatility.

However, until now there has been little eﬀort devoted to formally examing the behaviour

(characteristics) of VIX options’ IV slope, specifically. In this paper, we aim to fill this gap by taking a closer look at the information content of IV slope in the VIX options market and its implication. Our results are similar to those in Johnson (2017); however, he follows the methodology used to compute the VIX to form the VIX term structure and find SLOPE (i.e., the second principal component) summarizes nearly all information about variance risk premium.

Our methodology is more closely related to Xing, Zhang, and Zhao (2010), Kelly, Pástor, and

Veronesi (2016) and Tian and Wu (2020), who find significant predictability in individual stock options’ IV slope, the ability of IV slope in the equity options to price political uncertainty and cross-sectional variation of options IV slope reflecting investors’ expectation/sentiment on the future stock price movement, respectively.

Our study contributes to existing literature that examines the link between the VIX options market and the VIX futures market at the aggregate level. The ﬁrst contribution is to document the term structure and dynamics of IV curves in the VIX options market. We consider estimating The implied volatility slope of VIX options 4

IV curves of VIX options by fitting a linear function.4 Two IV factors are obtained from the estimation, which is level and slope. We find that the linear estimation of IV in the VIX options market has a mean R2 of 84.39% with a standard deviation of 17.34%, which implies that our model is sufficiently well-suited for the estimation. The VIX options IV curve exhibits an upward-sloping and almost linear curve, similar to that of VXX (VIX futures ETP) options market,5 however, more volatile for shorter maturity options.6 We investigate IV curve factors with constant maturity due to having multiple curves on a given day within each maturity category. We find that both the level and slope of IV have mean-reverting behaviour. The main focus, the slope of IV seems to be mean-reverting, usually displaying backward term structure.

The second contribution is to provide preliminary analysis as to where this positive slope stem from. We examine the contemporaneous relationship between monthly average slope of IV and four factors as follow: (1) correlation between simple changes in VIX and VVIX, (2) total jump risk in VIX index, which is measured by raising log changes in VIX to the power of three and separating the total jump risk into (3) positive and (4) negative jump risk, respectively.7

We find that the correlation between simple changes in VIX and VVIX index have statistically significant explanatory power on monthly slope of IV. The jump risk variables as a single factor do not have a contemporaneous relationship with slope and only become significant with the inclusion of the correlation between simple changes in VIX and VVIX. The results suggest that the correlation coefficient between two indexes are more important than the total jump, positive and negative jump risk in explaining positive slope in the VIX options market. Providing a theoretical explanation on this positive slope in the VIX options market is left for future research.

4 Only the call options are included in the estimation due to the dominant trading volume and open interest from VIX call options. For example, 69% of the trading volume of VIX options consists of a call instead of put options by 30-day rolling average trading volume, as shown in Figure 2. An upward- sloping linear curve is also observed in estimation with put options and a combination of both. In theory, in the presence of No-arbitrage principle, the IV curve obtained from the call or put are assumed be approximately the same. 5 Gehricke and Zhang (2020) employ second-order polynomial to document the IV curve in the VXX options market, which is diﬀerent from our study. Similar to the shape of IV curve in VIX options market, IV of VXX shows almost a linear curve, however, with very little convexity. 6 This is true for the entire sample period that we consider in this study, except for the year 2007-2008, where it shows occasional non-linear and negative sloping linear curve due to the global ﬁnancial crisis (i.e., United States housing bubble). 7 See Table 8 for more detail. The implied volatility slope of VIX options 5

The third contribution is to propose that an estimated IV slope by linear function captures speciﬁc information due to the following reason. When we consider linearly estimated slope of

IV, because we are separating the options into a call and put to observe their corresponding slope, it captures global slope across various moneyness of options. That is, we are no longer focusing on out-of-the-money (OTM) options (i.e., at-the-money (ATM) and in-the-money (ITM) are also considered) when estimating IV factors.8 The overall implication of global IV slope is that it has wider representation about the information contents of IV curve, in general, concerning diﬀerent moneyness of options, the information that otherwise would have been ignored. The slope of VIX IV is almost always positive over the entire sample period from February 2006 to

December 2019. The upward sloping and almost linear curve of IV is well-documented in our study.

Lastly, we ﬁnd that linearly estimated IV slope factor has statistically signiﬁcant predictability on constant 30-day daily VIX futures return both in-sample and out-of-sample analysis. More precisely, the slope of the VIX IV curve can predict negatively on next-day VIX futures returns

2 2 9 at 1% level of signiﬁcance with the R (ROS) of 0.35% (0.38%***). The results are also consistent even with the inclusion of other variables that are known to help predicting future equity returns we consider in this study.

The rest of the paper is organised as follows. Section 2 and Section 3 present the data and methodology. Section 4 provides the empirical results, and Section 5 concludes.

2 Data

2.1 VIX and VVIX

The sample period is ranging from February 2006 to December 2019 for VIX and January 2007 to December 2019 for VVIX, respectively. Time-series of VIX and VVIX index is obtain from

Chicago Board Options Exchange (CBOE) website for our sample period. The VIX index was

8 The global slope provides information about the sensitivity of the implied volatility concerning changes in moneyness. The slope of the IV curve in VIX call options and put options are referred to global slope or slope throughout the study. 9 OS denotes for out-of-sample. The result is also signiﬁcant for level factor as well at 1% level of signiﬁcance. The implied volatility slope of VIX options 6 introduced in 1993 to measure the market’s expectation of 30-day volatility implied by ATM

S&P 100 Index options prices. The CBOE updated the VIX Index to reﬂect a new way to measure expected volatility with Goldman Sachs in 2003. The VIX Index, to date, is a popular benchmark for the overall US stock market volatility.

The CBOE VVIX Index (the VVIX) represents the volatility of volatility in a sense that it estimates the expected volatility of the 30-day forward price of VIX. According to Park (2015),

VVIX is less vulnerable to computation inaccuracy than the other existing tail risk. It is because the VIX options market has pronounced liquidity than SPX options market, and less reliance on the existence of deep OTM puts that other calculation method (i.e., fear index in Bollerslev and Todorov (2011)) often relies on.

The forward price of VVIX is the price of a hypothetical VIX futures contract that has time-to-maturity equals to 30 days. The VVIX is not precisely equal to the expected volatility of the VIX, however they are very close10 as VIX futures with the same time to maturity (i.e., nearby VIX futures) track the VIX Index level. The VVIX reﬂects the price of a portfolio of

ATM and OTM VIX options. The same method used to calculate the VIX Index from SPX options is used to compute VVIX index level from VIX options.

The VVIX is a useful guide and very informative for investors, especially regarding the VIX derivatives. It provides expected volatility that helps to determine VIX options prices, expected volatility itself to nearby future. Based on most recent studies on the volatility of volatility index

ﬁnd that the VVIX implied by VIX options has predictability for tail risk hedging returns, cross- sectional, time-series of index and VIX options return (Park, 2015; Huang et al., 2019).

[Table 1 about here.]

Panel A of Table 1 reports summary statistics of daily mean, standard deviation, skewness, kurtosis, minimum and maximum of VIX, VVIX and S&P 500 index, respectively. The raw

VIX index level is around 20 on average, with a standard deviation of 9. The maximum level is around 80, which was in the 2008 ﬁnancial crisis period. The average VVIX index level is

10 We veriﬁed and conﬁmed this statement is true; daily average of VVIX index level and estimated ATM IV by our model are 89% and 90%, respectively, with 85% correlation between two. The implied volatility slope of VIX options 7 between 60 to 180, which results in an average daily level of 89 over the sample period. Daily log changes in S&P 500 Index level is close to 0.

Panel B of Table 1 reports a correlation between VIX, VVIX and log changes in S&P 500 index at daily frequency. The correlation between simple changes in VIX and log changes in

S&P 500 index over the sample period is -0.7237 and also negative for VVIX and S&P 500

(-0.5499). Most importantly, VIX and VVIX have a positive correlation of 0.2710 in the index level and 0.8067 for simple changes of two indexes. The correlations between changes in VIX,

VVIX and S&P 500 index (i.e., ∆VIX, ∆VVIX and rSP 500) are much larger in magnitude than the correlation between raw index level.

[Figure 1 about here.]

Figure 1 shows time-series of the natural logarithm of VIX, VVIX and S&P 500 index level over February 2007 to December 2019 period. The VIX and VVIX index level almost overlaps with each other between 2012 and 2018. The largest deviation between the two indexes can be observed from the 2008 to 2010 period. The index level movement between VVIX and S&P 500 seems relatively uncorrelated in comparison to VIX. Overall, it is consistent with Table 1 that

VIX and VVIX are positively correlated, whereas VIX and S&P 500 are negatively correlated on average. The time-series of VIX and SPX indexes since 1990 is around -70% and even lower in some period. For examples, in 2008 and 2018 of -83% and -88%, respectively.11

2.2 VIX futures returns

The sample period is ranging from February 2006 to December 2019. The VIX futures prices are obtained from CBOE for the sample period to compute futures returns.

The VIX futures are exchange-traded contracts on the VIX Index (which is derived from SPX options). It is the ﬁrst-ever listed futures contract on the CBOE VIX Index. It was launched in

2004 and quickly gain substantial attention from many practitioners as well as academics alike.

The VIX futures average daily volume in January 2020 was 275,380 with a month-end open interest of 431,076, which are 11% and 10% higher than in the year 2019.12

11 Please see Table 1 or refer to CBOE website for more detail. 12 See http://www.cboe.com/VIX The implied volatility slope of VIX options 8

The VIX options are introduced, in 2006, to provide market participants to hedge volatility risk of the portfolio, which is diﬀerent from the market price risk. It also enables trading activities based on one’s view of the future direction or movement of volatility. The VIX options have both monthly and weekly expirations, both of which are usually on Wednesday mornings

(Eastern Time; ET). The VIX options market is one of the most liquid options with an average daily volume of 495,229 with a month-end open interest of 6,617,689 in January 2020. It is the most popular options market by far in terms of volume and the open interest among other VIX derivatives.

[Table 2 about here.]

Table 2 shows a summary of VIX options, futures and S&P 500 index options markets. The

VIX and SPX are European style options, and both expire on the third Wednesday of each month prior to the third Friday. The common theme about these options is they all are settled at the maturity with cash, meaning that the settlement results in a cash payment, instead of settling in stocks, bonds, commodities or any other asset. Average daily trading volume and open interest are the largest in SPX, followed by VIX options and VIX futures.

[Table 3 about here.]

In Table 3 reports a summary of the VIX futures market trading activity overall and by maturity category. As the maturity of the contract increases the VIX futures prices, mean daily open interest and mean daily volume decrease. The number of observations is most extensive in the contract with maturity group 30-90 day and the least in the maturity less than 30 days.

We ﬁnd the majority of the trading volume in VIX futures coming from mid-longer maturity contracts.

We compute the returns for constant maturity VIX futures returns following Eraker and

Wu (2017) as

VF Ft+1(T1) Ft+1(T2) rT,t+1 = wt + (1 − wt) − 1, (1) Ft(T1) Ft(T2) The implied volatility slope of VIX options 9

where Ft(T ) is the day t price of a VIX futures contract with maturity date T , and Tt,1 and T are the two closest times to maturity to the target time to maturity T . w = T −Tt,2 and t,2 t Tt,1−Tt,2

(1 − wt) are corresponding weights for VIX futures price with Tt,1 and Tt,2, respectively. Note that the weight, wt is changing with time, t, since Tt,1 and Tt,2 may diﬀer by each date.

[Table 4 about here.]

Table 4 shows summary statistics of 30-day constant maturity VIX futures daily returns over the sample period. Daily VIX futures returns are negative, on average, with mean of -0.1791% with a standard deviation of 0531% and skewness of 1.0936.

2.3 VIX options

The VIX options data is obtained from OptionMetrics for our sample period. The options are European style; options that can be exercised only on the expiration date. The ﬁlters are applied to the sample following previous work by Bakshi, Cao, and Chen (1997) and Zhang and

Xiang (2008) as follow: we remove (1) option quotes with the open interest, bid price or IV is equal to zero or missing, (2) option quotes with a maturity of fewer than six days and (3) maturity groups that have less than a minimum of three observations when ﬁtting the IV curve.

The VIX options’ IV curves are positively skewed on average, unlike SPX options. The primary reason for this is that call options are more popular among investors to hedge the uncertain movement of volatility in the overall stock market in general. In other words, demand for call options in the VIX options market may results in positive skewness in its IV curve. This is consistent with Bollen and Whaley (2004), who ﬁnd that changes in IV are directly related to net buying pressure from public order ﬂow. The argument is that investors are willing to pay a higher premium for uncertainty about the future value of VIX (conversely, it is a gain for a short position).

According to Cheng (2019), the demand for hedging comes from the long side and the historical data suggest that it has been observed particularly after the 2008 ﬁnancial crisis.

Consistent with the studies mentioned above, Kelly, Pástor, and Veronesi (2016) find that the slope of IV surface is essential in the presence of downside jump risk associated with political The implied volatility slope of VIX options 10 events. They argue that, in times of political uncertainty, investors are willing to pay a higher premium for the protection against this risk. A recent study by Tian and Wu (2020) also find the cross-sectional variation of option IV skew reflects investor expectation and sentiment on the stock’s future price movement and find that it be used to predict future stock returns more accurately.

Therefore, investigating the role of VIX IV slope in pricing VIX futures returns would further add value to current literature surrounding Variance Risk Premium (VRP) and Skewness risk of returns.

[Table 5 about here.]

In Table 5, we summarize the trading activity of VIX options market by overall and by maturity category. We ﬁnd that the number of observation is lowest in more extended maturity groups than the shorter maturity groups. The mean number of strikes also has a similar pattern to the mean number of observations; in other words, most liquidity comes from shorter maturity options than more extended maturity options. Mean volume and open interest, however, decrease from maturity group less than 30 days to greater than 360 days.

[Figure 2 about here.]

Figure 2 shows the time-varying 30-day rolling average volume of VIX options over the sample period of February 2006 to December 2019. The average composition of call and put options in total trading volume is around 69% and 31%, respectively.

2.4 Other variables

Variance risk premium; VRP

Variance risk premium (VRP) is measured as the difference between the risk-neutral and objective expectations of realized variance. The expected value of variance in risk-neutral measure is calculated as the deannualized (VIX2/12) end-of-month VIX-squared, and the realized variance is the sum of squared (5-minute) log-returns of the S&P 500 index over the month. The defi- nition is following Zhou (2018), who finds predictability of VRP in bond, equity, currency and The implied volatility slope of VIX options 11 credit markets, respectively. The corresponding VRP data is available on the author’s website13 covering a comprehensive range of period; from January 1990 to December 2019.

Default risk premium; DEF

Default risk premium (DEF) is measured by the difference between Moody’s yield on Baa corporate bonds and the ten-year government bond yield. It is known to explain significantly on the cross-sectional variation of volatility risk premium. González-Urteaga and Rubio (2016) examine the determinants of the cross-sectional variation of the average volatility risk premia and find that the default premium is shown to be a key risk factor in explaining average volatility risk premium. Therefore, we check the robustness of slope by controlling DEF factor. We obtain corresponding data from the Federal Reserve Bank of St.Louis (FRED) for the sample period between February 2006 to December 2019.

Term spread; TED

Term spread (TED) is deﬁned as the diﬀerence between the three-month Treasury bill and the

Eurodollar rate. We obtain Eurodollar rate from the Federal Reserve Bank of St.Louis (FRED) for the sample period between February 2006 to December 2019. The TED spread is known as the measure of traders’ funding liquidity. The spread often widens in periods of economic crisis, as the default risk widens. The reason why this works as an indicator of credit risk is that because US T-bills are considered risk free and measure the creditworthiness of the US government.

Johnson’s SLOPE

To further examine the robustness of our slope factor, we incorporate alternative slope factor (the second principal component; SLOPE) from Johnson (2017), who ﬁnds the SLOPE summarize nearly all the information in various kinds of excess returns that he considered in his study..

The SLOPE factor is directly downloadable from the author’s website and we test predictability of our slope by controlling for Johnson’s SLOPE.14 The sample period is ranging from February

13 https://sites.google.com/site/haozhouspersonalhomepage/ 14 https://www.travislakejohnson.com/ The implied volatility slope of VIX options 12

2006 to December 2019.

3 Methodology

3.1 Measuring the slope of VIX options’ IV curve

The implied volatility-moneyness function at a ﬁxed maturity is described by a linear function as

IV (ξ) = IVAT M (1 + slope · ξ), (2) where slope measures the sensitivity of implied volatility with respect to its changes in moneyness. The level is an estimate of the at-the-money implied volatility (ATMIV). At-the-money

T T is deﬁned when K = Ft , where Ft and K denote is the implied forward price and the strike price, respectively. The regression of estimating slope factor is ﬁtted each day separately and for each maturity.

15 T The implied forward price (Ft ) is used to measure the moneyness of an option as

ln(K/F T ) ξ = √t , (3) σ¯VVIX τ where σ¯ denotes a measure of the average volatility of the VIX options.16

When estimating the IV function, OTM, ATM and ITM call options are used. The parameters in Equation (2), which has been accurately measured, would permit us to narrate the complete volatility function for a given maturity on a particular day with two parameters. To describe and explore the dynamics of the IV curve, we document these parameters across time and maturities. Our methodology is closely related to Xing, Zhang, and Zhao (2010), Kelly,

Pástor, and Veronesi (2016) and Tian and Wu (2020), who ﬁnd signiﬁcant predictability in in-

15 We download implied forward price from OptionMetrics for the sample period of February 2006 to December 2019. Subscript t denotes current time and superscript T denotes desired maturity. 16 The average volatility of VIX options can be estimated by ﬁtting market IV of VIX options against the options’ stike prices for each date and each maturity group or simply use VVIX index level. Although not reported in this study, daily mean of these two values over our sample period are 90% for ATM IV and 89% for VVIX, respectively. The implied volatility slope of VIX options 13 dividual stock options’ IV slope, the ability of IV slope in the equity options to price political uncertainty and cross-sectional variation of options IV slope reﬂecting investors’ expectation and sentiment on the stock’s future price movement, respectively.

3.2 Measuring predictability of IV slope

3.2.1 In-sample analysis

To investigate the relation between IV slope of VIX options and interpolated (30-day constant maturity) VIX futures returns, we consider the following predictive regression:

VF rT,t+1 = α + βslope + t+1, (4)

VF where rT,t+1 is the 30-day constant maturity VIX futures returns on future time, t + 1 with a target maturity T and T,t+1 is the residual. slope is our independent variable (that measures the sensitivity of VIX IV curve with respect to changes in moneyness of options) at current time t.

3.2.2 Out-of-sample analysis

In addition to the in-sample regression, we also test the out-of-sample performance of VIX IV slope. According to existing literature,17 an evaluation sample is considered as an essential parameter when analyzing the forecasting performance of a certain variable. We first divide the total sample into half so that the first half of the sample can be used as a training data set. These first half of the observations are then used for the first forecast. Proceeding in this manner through the end of the out-of-sample period, we generate a series of out-of-sample VIX

2 18 futures return forecasts. The out-of-sample R-squared (ROS) is deﬁned as

PN−1 2 (rt+1 − rˆ ) R2 = 1 − t=n t+1|t , (5) OS PN−1 2 t=n (rt+1 − r¯t+1|t)

17 See, for examples, Welch and Goyal (2008), Rapach, Strauss, and Zhou (2010), Ruan and Zhang (2018). 18 Which measures the reduction in mean squared prediction error (MSPE) for the predictive regression model relative to the historical average forecast. The implied volatility slope of VIX options 14

1 Pt where r¯t+1|t = t i=1 rt. n = ρN, where ρ is the initial estimation ratio, and N is the total number of days19 in the sample period. ρ is set to 1/2.

We test whether the predictive regression model has a signiﬁcantly lower MSPE than the historical average benchmark forecast by setting our hypothesis as

2 H0 :ROS ≤ 0

2 Ha :ROS > 0.

2 ROS > 0 indicates that rˆt+1 forecast outperforms the historical average forecast according to MSPE metric (Rapach, Strauss, and Zhou (2010; 2016)).

Additionally, we compute MSPE-adjusted statistic, ﬁrst deﬁning ft+1 following Clark and West (2007)

2 h 2 2i ft+1 = (rt+1 − r¯t+1|t) − (rt+1 − rˆt+1|t) − (¯rt+1 − rˆt+1|t) . (6)

N−1 By regressing {ft+1}t on a constant, we obtain a p-value for a one-sided (upper-tail) test with the standard normal distribution.

3.2.3 Measuring economic signiﬁcance of VIX slope

Following Rapach, Ringgenberg, and Zhou (2016), we evaluate the economic significance of out- of-sample predictability. This is done by constructing trading strategies based on VIX futures return forecasts and calculate the Certainty Equivalent Return (CER) gain for a mean-variance investor who allocates between VIX futures and risk-free assets. We first find out the optimal weight to VIX futures contract at the end of the day t as follows

1 rˆt+1 ωt = 2 , (7) γ σˆt+1

19 The total number of observation is 3,266 in this study. The implied volatility slope of VIX options 15

where γ denotes the risk-aversion of the investor, rˆt+ 1 is the return forecast of VIX futures and

2 σˆt+1 is the forecast of VIX futures return variance. Subsequently, the portfolio return at future period t + 1 is given by

P VF rt+1 = ωtrt+1 + rft+1, (8)

VF 20 where rt+1 is the VIX futures return and rft+1 is the risk-free rate. The investor could obtain an average of CER gain as

CER =r ¯P − 0.5γσ2(rP ), (9) where r¯P and σ2(rP ) are the mean and variance of portfolio returns, respectively. The CER gain for a mean-variance investor is the CER difference, which is defined as a difference between the CER of forecasts generated by the regression model and the historical average forecast benchmark. We report the results for γ = 3 and limit the weight to -0.5 to 1.5.

4 Empirical results

4.1 IV slope of VIX options

In this section, we present and analyze the dynamics of the estimated IV curve of the VIX options market.

[Table 6 about here.]

Table 6 reports summary statistics of the estimated implied volatility function of VIX call options, IV (ξ) = IVAT M (1 + slopeξ), where IV is the implied volatility and ξ is the moneyness of the option. The level (IVAT M ) is highest in shortest maturity group, <30 with 1.0912. The slope factor is also highest in the shortest maturity group with 0.2255. The slope of the IV curve across diﬀerent maturity groups is positive on average, with the overall maturity group value of

20 The Treasury yield data (risk-free rate) is from the US Department of the Treasury website. If there is no yield data on a day where there are options data, then the previous day’s value has been used as an alternative. The implied volatility slope of VIX options 16

0.1606. We find that both the level and slope of the IV curve in the VIX options market are decreasing as the maturity gets longer. The linear estimation of the IV curve factors for overall maturity group has a mean R2 (goodness of fit of the regression) of 84.39% with a standard deviation of 17.34%. The most well fitted mean R2 is coming from maturity group 30-90 and

90-180.

On average, the long-term prediction of VIX volatility by options traders are lower than the short term and that their long term volatility prediction are consistent throughout the sample.

The level coefficient is significant at the 5% level for more than 99.97% of the fitted IV curves.

This is consistent with our expectation since the exact ATM implied volatility is always positive.

Turning to the slope factor, on average, the IV curves are upward sloping as we can see from the overall mean of slope factor in Table 6. The result of estimation is diﬀerent to the ‘smirk’ or

‘smile’ shape found in many existing studies that document IV shape in stock options or stock index options (Foresi and Wu, 2005; Zhang and Xiang, 2008; Xing, Zhang, and Zhao, 2010; Yan,

2011; Kelly, Pástor, and Veronesi, 2016; Tian and Wu, 2020) but similar to the results found in Gehricke and Zhang (2020). As the maturity increases the slope becomes flatter. The term structure of the slope factor is in backwardation on average. The slope coefficient is also highly significant at the 5% level for 98.48% of fitted IV curves in our sample.

The results suggest that the term structure of the IV curve factors, both the level and slope, are in backwardation (downward sloping), on average as both values (and so are their standard deviation) decrease as the maturity gets longer.

[Figure 3 about here.]

[Figure 4 about here.]

[Figure 5 about here.]

Figures 3 to 5 plot VIX options IV curve fitted against moneyness of options on various dates for the most recent last ten years. Dates are selected to have multiple maturities to show how it changes over different maturity horizons. The circles indicate implied volatility of the market prices of call options. The solid line is the fitted estimated IV curve by a linear function. The implied volatility slope of VIX options 17

The spike chart is the trading volume for the corresponding options. Consistent with findings from Table 6, estimated IV fits reasonably well with the market IV curve. As the maturity gets longer linear model becomes more robust and conscientiously fits the VIX options market IV well. These figures illustrate that the IV curve in the VIX options market is upward slopping in almost all cases and displays straight-line rather than a curve.

4.2 VIX IV slope with constant maturity

Until now, we have been examining the term structure of VIX IV curves using maturity cate- gories. However, because there are often multiple curves on a given day within each maturity category, we investigate IV slope with constant maturity. With the constant maturity, we are able to examine the term structure and time-series of the IV curve slope covering the same horizon of traders’ expections.

To create constant maturity IV slope, we interpolate and (or) extrapolate it to speciﬁc target21 maturities as

τ τ1 τ2 slopet = slopet wt + slopet (1 − wt),

where

τ − τt,2 wt = , τt,1 − τt,2

where τ denotes the target maturity, τt,1 is the nearest maturity to the target from below, τt,2 is the nearest maturity to the target from above when interpolating and t denotes current time.

We extrapolate when the available maturities are all shorter or longer than the target maturity.

The weight, wt is changing both with time and maturities on each date.

[Table 7 about here.]

Table 7 reports summary statistics of the estimated constant 30-day maturity IV slope

21 We report the interpolated time-series of IV slope with the target maturity of 30-day. The implied volatility slope of VIX options 18 factor by a linear function.22 On average, we ﬁnd that the estimated slope to be 0.1803 with the standard deviation 0.1031. Our estimation of the VIX IV summarizes nearly all the information contained in VIX option prices and can also be used in VIX options pricing model.

[Figure 6 about here.]

Figure 6 plots the constant 30-day and 180-day interpolated slope of IV curve. The time- series of interpolated VIX slope is ranging between -0.2 to 0.4 over the sample period with the occasional falls in value between the year 2007 to 2009, aﬀected by the 2008 global ﬁnancial crisis. It seems to be that the level of both IV slope has been increasing over time until it drops sharply in the year 2019. Overall, the term structure of slope of IV curves in the VIX options market seems to exhibit a mean-reverting behaviour and in backwardation.

4.3 Explaining positive IV slope in the VIX options market

The current literature provides demand-based or informed trading-based explanation for the shape of IV curve evaluated from corresponding underlying asset’ options (Bollen and Whaley,

2004; Xing, Zhang, and Zhao, 2010; Karpoﬀ and Lou, 2010; Engelberg, Reed, and Ringgenberg,

2012; Rapach, Ringgenberg, and Zhou, 2016). However, there are no studies that provide a theoretical model or explanation on positive IV slope in the VIX options market.23 In this section, we preliminarily investigate what contributes to this positive IV slope (upward sloping) in the VIX options IV curve by examining the relationship between VIX slope and the following

3 variables. ρ(∆ VIX, ∆ VVIX); correlation between simple changes in VIX and VVIX, rVIX ; m log changes of VIX index to the power of three which measures total jump risk in VIX, Jr>0 = 1 PN 3 m 1 PN 3 N 1 rVIX |r>0; positive jump risk in VIX and Jr<0 = N 1 rVIX |r<0; negative jump risk in VIX, respectively.

22 The results on quadratic estimation can be found in Appendix A. 23 For typical market index options, observed market prices for OTM put prices (and ITM call prices) are higher than Black and Scholes (1973) prices, this stylized fact is known as the volatility smirk (Christof- fersen, Heston, and Jacobs, 2009). However, it is not true for VIX index options, due to the negative correlation; decreases in market index prices are associated with larger increases in the volatility index. In the VIX options market for the call options, higher the call price higher the premium. The high premium is to compensate for the loss in the stock market due to a negative correlation between the volatility and the returns. Theoretical explanation and the driver of positive IV slope in VIX options is an interesting topic for further research. The implied volatility slope of VIX options 19

[Table 8 about here.]

Panel A in Table 8 reports summary statistics of monthly average IV slope, correlation between simple changes in VIX and VVIX (ρ(∆ VIX, ∆ VVIX)), monthly average of log

3 changes in VIX to the power of three (rVIX ), monthly average of positive and negative jump m m 24 (Jr>0, Jr<0), respectively. The monthly average slope of IV over the sample period is at 0.1788, with a standard deviation of 0.0917. The monthly correlation between the simple changes in

VIX and VVIX is much larger in magnitude than the index level correlation coeﬃcient;25 as large as 71.90%. The jump risk factors of VIX are ranging between -0.0006 and 0.0018. Panel

B shows regression analysis on monthly average IV slope with variables mentioned above. We

ﬁnd that ρ(∆ VIX, ∆ VVIX) has signiﬁcant explanatory power on average slope of IV in the

2 VIX options market at 1% level of significance with adj-RIS of 20.30%. The jump risk variables do not have significant relationship with IV slope. However, the inclusion of negative jump risk in column (6) in addition to positive jump risk, increase the explanatory power of positive jump risk to 1.29, although rather weakly (i.e., become significant at 10% level).

Our preliminary analysis suggests that the correlation coeﬃcient between two indexes are more important than the total jump, positive and negative jump risk in explaining positive IV slope in the VIX options market.26

4.4 Predicting VIX futures return with VIX slope

Many existing studies have documented and conﬁrmed that the variance risk premium is essential and useful in serving various role (Bakshi and Kapadia, 2003; Bollerslev, Tauchen, and Zhou,

2009; Carr and Wu, 2016; Dew-Becker et al., 2017; Zhou, 2018; Da Fonseca and Xu, 2019; Cheng,

2019). Chung et al. (2011) and Bardgett, Gourier, and Leippold (2019) find the information content implied by the SPX options are significantly improved by the information recovered from the VIX options and find that VIX options contain valuable information on the risk-

24 m 1 PN 3 m 1 PN 3 3 Jumps are defined as Jr>0 = N 1 rVIX |r>0 and Jr<0 = N 1 rVIX |r<0, where rVIX is daily log changes in VIX level to the power of three. 25 Please refer to Table 1. 26 Theoretical explanation for the positive slope of VIX IV is not available in the literature, and it is in our future research interest. The implied volatility slope of VIX options 20 neutral conditional distributions of volatility at different time horizons which do not stem from the S&P 500 index. Xing, Zhang, and Zhao (2010) find individual stock options’ IV slope has a significant predictive ability on future cross-sectional equity returns. In addition to its return predictive ability, the slope of IV curve is also found to be an essential measure in the presence of a political event, which explains as to why investors are willing to pay a higher premium during a certain period (Kelly, Pástor, and Veronesi, 2016).

In this section, we investigate whether the slope of the IV curve in the VIX options market contains information related to VIX futures returns in both in-sample and out-of-sample tests.

We use daily 30-day constant maturity VIX futures returns as the independent variable and analyse the forecasting ability of interpolated daily slope of VIX options’ IV curve following the methodology described in Section 3.2.

[Table 9 about here.]

Panel A and B of Table 9 report predictive regressions for the constant 30-day daily VIX futures’ return. Panel A reports the non-overlapping univariate analysis with the level and slope factor, respectively. We assess whether the forecast performance of our predictive regression model with IV slope delivers a statistically signiﬁcant improvement in MSFE. To do this, we use statistic to test the null hypothesis set out in Clark and West (2007). The null hypothesis is that the prevailing mean MSFE is less than or equal to the predictive regression MSFE,

2 2 which we defined in Section 3.2.2 (i.e., H0 : ROS ≤ 0 against Ha : ROS > 0 ). We find that 2 2 the slope factor is significant in predicting next-day VIX futures return with adj-RIS (ROS) of 0.35% (0.38%∗∗∗). This means since the standard deviation of slope factor is equal to 0.1031, one standard deviation increase in slope factor results in a 0.24% (= 2.43 × 0.1031) decrease in next-day VIX futures return. Panel B (predictive regression with overlapping data) is consistent with Panel A, where we find negative predictability from slope factor. The significance of slope factor has decreased to (in terms of t-statistics) -5.51 from -3.15.

Our main ﬁndings add to existing literature that examines the link between the VIX options market and the VIX futures market at the aggregate level. The main implication of the results is that the slope, which is related to risk-neutral moment, skewness, in particular, should not be ignored when pricing VIX futures returns. The implied volatility slope of VIX options 21

4.5 Predicting VIX future return with other variables

In this section, we consider incorporating following variables to test robustness of our VIX

IV slope factor. Volatility of volatility index (VVIX), Variance risk premium (VRP), Default risk premium (DEF), measure of traders’ liquidity (TED) and the second principal component

(SLOPE) from Johnson (2017). The VVIX can be regarded as an alternative measure of at-the- money implied volatility. According to existing literature, the VVIX help to predict tail risk hedging returns, options and index returns (Park, 2015; Huang et al., 2019), among others. We control for this well-known VVIX index, the expected volatility of VIX that drives the price of

VIX nearby options, and examine the predictability of VIX slope. The DEF and TED are often referred to as credit risk factor. The DEF, default risk premium or default spread is defined as the difference between Moody’s yield on Baa corporate bonds and the ten-year government bond yield. It is one of the popular economic variables that have significant predictability on volatility risk premium (González-Urteaga and Rubio, 2016).

[Table 10 about here.]

Panel A and B of Table 10 report summary statistics and correlation matrix between variables that we consider in our regression specification. The mean of each variable is at daily frequency, except for variance risk premium, which is at monthly frequency. Panel A reports mean, standard deviation, median, skewness, kurtosis, minimum and maximum of each variable of our interest. Panel B reports the correlation matrix between variables. We find that the slope is very weakly correlated with other variables, except for DEF, default risk premium with the value of -0.5913. DEF, default risk premium variable is also positively correlated with TED spread and SLOPE with the value of 0.4786 and 0.1985, respectively. SLOPE and VRP are relatively highly correlated than other variables with the correlation coefficient equal to 0.6588.

The rest of the variables are either almost uncorrelated or very weakly correlated with each other. Pabel A and B of Table 10 report summary statistics and correlation matrix between variables that we consider in our regression specification. The mean of each variable is at daily frequency, except for variance risk premium, which is at monthly frequency. Panel A reports mean, standard deviation, median, skewness, kurtosis, minimum and maximum of each variable The implied volatility slope of VIX options 22 of our interest. Panel B reports the correlation matrix between variables. We find that the slope is very weakly correlated with other variables, except for DEF, default risk premium with the value of -0.5913. DEF, default risk premium variable is also positively correlated with TED spread and SLOPE with the value of 0.4786 and 0.1985, respectively. SLOPE and VRP are relatively highly correlated than other variables with the correlation coefficient equal to 0.6588.

The rest of the variables are either almost uncorrelated or very weakly correlated with each other.

[Table 11 about here.]

Table 11 reports predictive regressions on the constant 30-day daily VIX futures returns in-sample, out-of-sample and its economic signiﬁcance with several other variables. In Panel A, we report results with non-overlapping and Panel B with the overlapping data (30-day horizon).

In columns (2) - (7), we consider bivariate predictive regression with each control variable.

The results are consistent with our baseline regression, the predictive ability of IV slope factor remain signiﬁcant and negative after controlling for the alternative measure of ATM IV (VVIX), default risk premium (DEF), trader’s funding liquidity (TED), variance risk premium (VRP),

VF Johnson (2017)’s SLOPE and current VIX futures return (rt ). In column (8), we ﬁnd that the predictability of IV slope remains negative and signiﬁcant

2 2 though with a slight decrease in its t-statistic to -2.83 with adjusted-RIS (ROS) of 11.29% (9.59∗∗∗%). The predictability of IV slope is only affected mainly by DEF and SLOPE, respectively. However, the effect is rather beneficial for IV slope. For examples, the estimated coefficients for IV slope becomes more negative and significant with the inclusion of DEF (−3.07∗∗∗) and SLOPE (−3.10∗∗∗), respectively.

We assess the economic value of IV slope’s predictability from an asset allocation perspective following Rapach, Ringgenberg, and Zhou (2016). The CER gain for a mean-variance investor is the CER difference, which is defined as a difference between the CER of forecasts generated by the regression model and the historical average forecast benchmark. Panel A in Table 11 shows CER gain coming from IV slope alone, and inclusion of other variables are negative and economically large; between -1 bps and -709 bps per day. Non-negative CER gain is only observed in column (3) and (7), controlling for DEF and SLOPE, respectively. The implied volatility slope of VIX options 23

Panel B of Table 11 reports overlaping univariate, bivariate and multivariate predictive

27 2 2 regression with 30-day overlaping data horizon. Both adjusted-RIS and ROS have increased 2 enormously; with more than 10% ROS following Clark and West (2007) for all the regression 2 speciﬁcation and columns (7) having the highest value of in-sample adjusted-RIS of 50.73%. The economic value of the forecasts generated by the regression model is much larger than the historical average forecast benchmark by far; ranging between 100 to 376 bps per day. The results are consistent with Panel A that IV slope has statistically and economically signiﬁcant predictability on the next-day VIX futures returns even with the inclusion of other variables.

Overall, we ﬁnd negative and signiﬁcant predictability of IV slope on next-day VIX futures returns. The results are robust; IV slope as a sole predictor and with several other variables.

Our results are consistent with previous studies that document negative variance risk premium

(Bollerslev, Tauchen, and Zhou, 2009; Xing, Zhang, and Zhao, 2010; Zhao, Zhang, and Chang,

2013; Kelly, Pástor, and Veronesi, 2016; Johnson, 2017).

5 Conclusion

We document IV curves of VIX options and show that the slope of the VIX IV curve can robustly predict next-day VIX futures return over the 2006 to 2019 sample period. We propose a linear estimation method, which is diﬀerent from previous studies in that it is designed to capture global slope instead of local, to examine VIX options term structure. We further extend our analysis to study the constant maturity IV curve factors.

Our study contributes to existing literature that examines the link between the options market and the futures market at the aggregate level. The ﬁrst contribution is to document the term structure and dynamics of IV curves in the VIX options market. On average, the

VIX options IV curve exhibits an upward-sloping linear curve, similar to that of VXX (VIX futures ETP) options market, however more volatile for shorter maturity options. We investigate

IV curve factors with constant maturity due to having multiple curves on a given day within each maturity category. We ﬁnd that the term structure of both level and slope is usually in

27 The dependent variable, daily VIX futures return, is calculated as a rolling-average of daily VIX futures returns over 30-day horizon. The implied volatility slope of VIX options 24 backwardation and exibit mean-reverting behaviour.

The second contribution is that to provide preliminary analysis as to where this positive slope stem from. We find that the correlation between simple changes in VIX and VVIX index have statistically significant explanatory power on monthly slope of IV. The results suggest that the correlation coefficient between two indexes are more important than the total jump, positive and negative jump risk in explaining positive slope in the VIX options market. Providing a theoretical explanation on this positive slope in the VIX options market is left for future research.

The third contribution is to propose that an estimated IV slope by linear function captures speciﬁc information. The overall implication of global IV slope is that it has wider representation about the information contents of IV, in general, concerning diﬀerent moneyness of options, the information that otherwise would have been ignored. The slope of VIX IV is almost always positive over the entire sample period from February 2006 to December 2019. The upward sloping and almost linear curve of IV is well-documented in our study.

Lastly, We find the linearly estimated IV slope of VIX options can significantly predict constant 30-day daily VIX futures return both in-sample and out-of-sample analysis. More precisely, the slope of the VIX IV curve can predict negatively on next-day VIX futures returns at 1% level of significance.

Although current literature has documented evidence of VIX term structure being important in forecasting various types of returns, previous studies do not speciﬁcally focus on the IV curve slope of VIX options. We are the ﬁrst to document IV curves in VIX options market by a linear function. A linear estimation enables us to more directly study slope of the IV curves in the

VIX options market. Our results highlight that the global slope of the IV curve in the VIX options market is a nontrivial factor and is incremental to other variables in predicting VIX futures return to a certain extent. The unique feature observed in our study, the positive slope of the IV curve in the VIX options market hence should be investigated further. The implied volatility slope of VIX options 25

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Appendix

This document provides supplementary material to the paper “The implied volatility slope of

VIX options.” It provides (1) additional ﬁgures on VIX options and estimation results based on alternative way in estimating factors (Appendix A), (2) additional explanation on global/local slope (Appendix B).

A Alternative estimation of slope in VIX IV

[Table 12 about here.]

Table 12 reports the summary statistics of the estimated implied volatility function of VIX put options, IV (ξ) = IVAT M (1 + slopeξ), where IV is the implied volatility and ξ is the moneyness of the option. The regression is ﬁtted each day separately and for each maturity. The implied forward price is larger for maturity group 30-90 and 90-180 with 20.0374 and 21.2289, respectively. We ﬁnd that the implied forward price of VIX put options in monotonically decreasing as maturity gets longer. The results show that both level and slope are decreasing as the maturity gets longer. For example, slope of VIX put options IV is highest with 0.2378 in shortest maturity group, which is less than 30 day. It then drops to 0.1908, 0.1614, 0.1612 and, lastly, to 0.1101 as the maturity gets longer. Mean R2 is 88.31% for overall and highest in 30-90 maturity group.

[Figure 7 about here.]

Figure 7 plots the IV of VIX put options against the moneyness for diﬀerent maturities as at the close of 3 March 2015. The solid line indicates the estimated IV curve by a linear function.

The spike chart is the trading volume for corresponding options traded on 3 March 2015. This

ﬁgure shows that there is much less trading volume for put options in the VIX options market on this particular date. Most of the trading volume is concentrated around the ATM point; moneyness is equal to 0. The IV curve of VIX put options also displays upward sloping almost linear curve, similar to that of call options IV curve. Results of estimated level and slope factors can be found in Table 12. The implied volatility slope of VIX options 29

[Table 13 about here.]

Table 13 reports summary statistics of the estimated implied volatility function of VIX options (both call and put options inclusive). The regression is ﬁtted each day separately and for each maturity, consistent with the previous estimation. Note that using both put and call options means, we now are estimating the IV curve of VIX options with OTM options only.

We ﬁnd that overall, R2 is higher than estimation results with call and put options individually

(i.e., mean R2 is 93.87%). The largest daily average trading volume is coming from the shortest maturity group with 7,939, about twice as large as to call options’ daily trading volume. In summary, the estimated value of level and slope are similar to the estimation that uses put and call options individually. That is, both values are decreasing as the maturity gets longer. The estimation results with quadratic function is reported in Table 14, however, the values of each

IV curve factor are similar to that of Tables 12 and 13.

[Table 14 about here.]

Overall, decomposing options into call and puts provide a few critical findings. Firstly, it shows that the shape of estimated IV function, using a combination of call and put options, is identical to that of call options, that is, an almost linear curve with slight concavity. Second, the shape of the combined and call options only IV curves are indifferent, the estimated level and curvature of each function were same for all (i.e., for combined, call and put options, respectively) in terms of changes in value concerning maturity. Lastly, we find that the slope of the estimated

IV function of call (put) options is increasing as maturity gets longer as opposed to decreasing.

This is diﬀerent from what we observed in Table 14 that uses both options.

In the VIX options market for the call options, higher the call price the higher premium.

The high premium is to compensate for the loss in the stock market due to a negative correlation between the volatility and the returns. Therefore, we see OTM put options become worthless as hedging tool (in times of high volatility), thus have less trading volume or liquidity. A chance of

OTM put going into ITM is the least, and overall, slightly OTM or ATM is most liquid in the

VIX options market. The VIX ATM put or OTM call options are generally used as a hedging tool rather than speculation. ATM is the most expensive options within diﬀerent moneyness The implied volatility slope of VIX options 30 of call options rather than OTM. Call options are demanded more by market participants in comparison to put options, on average, but not necessarily OTM call options.

B Global slope and local slope

The global slope provides information about the sensitivity of the implied volatility to changes in moneyness. The linearly estimated global slope include OTM, ATM and ITM in the estimation process. The main reason for investigating global instead of local slope in the VIX options market is that the relationship between implied volatility and moneyness (or strike prices) is more close to linear than non-linear. We use call options in estimating IV factors due to larger trading volume and open interest. Our methodology is more closely align with Xing, Zhang, and

Zhao (2010), Kelly, Pástor, and Veronesi (2016) and Tian and Wu (2020), who find significant predictability in individual stock options’ IV slope, the ability of IV slope in the equity options to price political uncertainty and cross-sectional variation of options IV slope reflecting investors’ expectation and sentiment on the stock’s future price movement, respectively. The implied volatility slope of VIX options 31

Tables

Table 1: Summary statistics of VIX, VVIX and S&P 500 index. This table reports summary statistics (in Panel A) and correlation matrix (in Panel B) of VIX, VVIX and S&P 500 index, respectively. VIX and VVIX are the raw index level, ∆VIX and ∆VVIX are the simple changes in each index level and rSP 500 denotes log of changes in S&P 500 Index, respectively. The sample period is from January 2007 to December 2019.

Panel A: Summary statistics of variables

VIXVVIX ∆VIX ∆V V IX rSP 500 Mean 19.4001 88.9475 0.0031 0.0013 0.0002 Standard dev. 9.2451 13.4134 0.0809 0.0530 0.0122 Median 16.6900 87.3700 -0.0062 -0.0044 0.0006 Skewness 2.4559 1.0751 2.1560 1.6973 -0.6028 Kurtosis 11.2611 6.0470 21.1953 13.7580 12.4844 Minimum 9.1400 59.7400 -0.2957 -0.2105 -0.0947 Maximum 80.8600 180.6100 1.1560 0.5700 0.1025 Panel B: Correlation matrix between variables

VIXVVIX ∆VIX ∆V V IX rSP 500 VIX 1 VVIX 0.2710 1 ∆VIX 0.1040 0.2247 1 ∆VVIX 0.0494 0.2002 0.8067 1 rSP 500 -0.1578 -0.1890 -0.7237 -0.5499 1

Table 2: Summary of VIX options, futures and S&P 500 index options markets. This table shows a summary of VIX options, futures and S&P 500 options markets. The expiration for VIX and VX is on Wednesday that is 30 days prior to the third Friday.

Symbol VIX Index VIX futures S&P 500 Index Issuer CBOE Options Exchange CBOE Futures Exchange CBOE Options Exchange Style European Futures Eropean Expiration Wednesday Wednesday 3rd Friday Settlement Cash Cash Cash Underlying VIX futures with same maturity VIX Index SPX Multiplier $100 × Index $1,000 × Index $100 × index Average Daily Option Volume 509,490 275,380 1,116,237 Average Daily Open Interest 6,617,689 431,076 16,800,000 The implied volatility slope of VIX options 32

Table 3: Summary of VIX futures market activity. This table shows summary statistics of VIX futures daily price, open interest and volume for the sample period from March 2004 to June 2020. We use data from February 2006 to December 2019 for the VIX futures returns computation and prediction in Section 4.

By maturity (days) Overall <30 30-90 90-180 180-360 >360 >180 <180 Number of observation 32,142 3,486 28,656 21,207 10,649 374 10,562 21,493 Mean closing price 34 38 39 32 30 140 31 36 Mean settle price 35 40 40 34 35 140 38 37 Mean daily open interest 28,467 81,008 22,076 10,733 3,721 200 3,686 40,728 Median daily open interest 11,192 54,756 8,967 4,900 1,529 120 1,524 22,816 Mean daily volume 13,554 49,832 9,141 2,696 732 8 727 199,907 Median daily volume 1,514 26,688 1,167 598 172 0 171 4,597

Table 4: 30-day constant maturity VIX futures returns. This table shows summary statistics of 30-day constant maturity VIX futures returns over the sample period of February 2006 to December 2019.

Daily 30-day constant maturity VIX returns Mean -0.1791% Medain -0.5495% Standard dev. 4.0531% Minimum -25.7859% Maximum 34.8238% Skewness 1.0936 Kurtosis 9.6195 The implied volatility slope of VIX options 33

Table 5: Summary of VIX options market activity. This table shows the mean and median daily number of strikes, trading volume and open interest in the VIX options market. The statistics are shown overall and for each maturity category. The the mean and median for each market activity variable are reported in Panel A (total VIX options market activity), B (call options only) and C (put options only), respectively. Calculations are based on daily frequency for each maturity, either overall or by the maturity category. The sample period is from February 2006 to December 2019.

By maturity (days) Overall <30 30-90 90-180 180-360 >360 >180 <180 Panel A: Call options and Put options Number of observation 885,476 196,479 295,651 371,171 20,975 1,200 19,783 863,301 Mean number of strikes 45 40 49 45 25 20 24 45 Median number of strikes 48 38 52 48 26 22 25 48 Mean volume 69,154 137,356 97961 14,028 3,672 132 2,772 70,841 Median volume 18590 76460 58,124 6,010 480 35 379 19,751 Mean open interest 842,776 1,455,719 1,226,929 259,795 49,949 2,649 49,629 863,186 Median open interest 408,405 1,027,609 1,087,234 187,540 18,217 2,492 16,028 431,810 Panel B: Call options only Number of observation 500,809 103,873 160,537 223,286 12,408 705 11,462 487,696 Mean number of strikes 26 21 27 28 15 11 14 26 Median number of strikes 28 21 29 30 15 12 14 28 Mean volume 43,935 84,721 68,000 10,092 2,653 109 1,946 45,049 Median volume 10,218 29417 36,691 3,424 296 26 234 10,836 Mean open interest 551,380 924,120 867,758 181,019 33,627 2,259 34,021 565,347 Median open interest 238,042 304,690 743,693 116,477 10,444 2,071 10,158 253,037 Panel C: Put options only Number of observation 381,007 91,885 133,884 146,210 8,533 495 8,287 371,979 Mean number of strikes 20 20 23 18 11 9 11 20 Median number of strikes 20 20 2 19 11 10 11 20 Mean volume 23,762 49,650 29,088 4,014 1,175 23 931 24,311 Median volume 5,631 30,885 15,855 987 40 3 30 6,068 Mean open interest 273,365 495,540 349,366 80,084 16,135 368 15,550 279,629 Median open interest 134,766 539,168 308,946 60,143 5,263 386 4,900 142,340 The implied volatility slope of VIX options 34

Table 6: Summary of VIX call options implied volatility linear estimation. This table shows summary statistics of the estimated implied volatility function of VIX options, IV (ξ) = IVAT M (1 + slopeξ), where IV is the implied volatility, and ξ is the moneyness of the option. IVAT M and slope denote level and slope of VIX options IV curve, respectively. The regression is ﬁtted each day separately and for each maturity and the sample period is from February 2006 to December 2019.

By maturity (days) Overall < 30 30 − 90 90 − 180 180 − 360 > 360 > 180 < 180 Mean IVAT M 0.8125 1.0912 0.8461 0.6759 0.5259 0.4189 0.5083 0.8203 slope 0.1606 0.2255 0.1628 0.1321 0.1063 0.1029 0.1023 0.1621 Standard dev. IVAT M 0.2310 0.2884 0.1321 0.0866 0.0655 0.0184 0.0609 0.2287 slope 0.0719 0.0878 0.0620 0.0466 0.0464 0.0777 0.0485 0.0719 Daily R-squared Mean R2 84.39% 82.95% 85.36% 85.13% 72.23% 52.24% 68.90% 84.74% Standard dev. R2 17.34% 20.35% 16.31% 15.59% 24.15% 20.67% 24.54% 16.97%

Table 7: Summary statistics of 30-constant maturity slope of VIX IV curve. This table shows summary statistics of interpolated 30-day constant maturity IV curve slope of VIX options, slope, by linear estimation. (Interpolation method can be found in Section 3.) The sample period is from Feburary 2006 to December 2019.

Variable slope Mean 0.1803 Minimum -0.2384 Maximum 0.4434 Skewness -0.9839 Kurtosis 3.8171 Standard dev. 0.1031 The implied volatility slope of VIX options 35

Table 8: The relationship between IV slope, VIX and VVIX. Panel A reports summary statistics for the following variables. ρ(∆VIX, ∆VVIX); cor- 3 relation between simple changes in VIX and VVIX, rVIX ; log changes of VIX index to the m 1 PN 3 power of three which measures total jump risk in VIX, Jr>0 = N 1 rVIX |r>0; positive m 1 PN 3 jump risk in VIX and Jr<0 = N 1 rVIX |r<0; negative jump risk in VIX, respectively. slope denotes monthly average of 30-day constant maturity daily slope of IV. Panel B reports regression analysis on slope of IV in the VIX options market with the variables mentioned above. Note that the monthly correlation is measured with daily ∆x, where x = VIX,VVIX, which denotes simple changes ( xt−xt−1 ). In Panel B, reported βˆ co- xt−1 eﬃcients are for each variable, their Newey and West (1987) t-statistics and in-sample adjusted-R2 statistics. *,** and *** denote signiﬁcance level at 10%, 5% and 1%, respectively. The sample period is from January 2007 to December 2019.

Panel A: Summary statistics of variables 3 m m slope ρ(∆VIX, ∆VVIX) rVIX Jr>0 Jr<0 Mean 0.1788 0.7190 0.0005 0.0018 -0.0006 Standard dev. 0.0917 0.1895 0.0021 0.0059 0.0009 Median 0.2041 0.7663 0.0001 0.0005 -0.0003 Skewness -0.8554 -1.4680 9.1358 9.8738 -3.5997 Kurtosis 3.0734 5.3305 98.1241 110.6436 19.2124 Minimum -0.0921 0.0240 -0.0014 0.0000 -0.0059 Maximum 0.3088 1.0234 0.0234 0.0687 0.0000 Panel B: Univariate, bivariate and multivariate egression on slope (1) (2) (3) (4) (5) (6) (7) ρ(∆VIX, ∆VVIX) 0.22∗∗∗ 0.22∗∗∗ 0.26∗∗∗ (4.79) (4.77) (5.05) 3 ∗∗ rVIX 2.66 2.82 (1.50) (2.00) m ∗ ∗∗∗ Jr>0 0.67 1.29 1.96 (1.28) (1.77) (3.40) m ∗∗∗ Jr<0 8.76 11.99 29.37 (0.83) (1.05) (2.74) 2 adj-RIS 20.30 -0.30 20.18 -0.49 0.00 -0.07 25.42 The implied volatility slope of VIX options 36

Table 9: In-sample and out-of-sample estimation. This table shows predictive regressions for the constant 30-day daily VIX futures returns in in- and out-of-sample analysis. The table presents the estimated βˆ coefficients for slope, 2 their Newey and West (1987) t-statistics, in-sample adjusted-RIS statistics and out-of- 2 sample ROS statistics. Panel A reports regression results with non-overlapping data and B reports overlapping regression results. *,** and *** denote significance level at 10%, 2 5% and 1%, respectively. An out-of-sample ROS in Panel B is based on fitted values rˆt+1 estimated using all observations except those overlapping with rt+1. The sample period is from February 2006 to December 2019.

Panel A: Non-overlapping predictive regression ˆ 2 2 β t Adj-RIS (%) ROS (%) slope −2.43∗∗∗ (-3.15) 0.35 0.38∗∗∗ Panel B: Overlapping predictive regression ˆ 2 2 β t Adj-RIS (%) ROS (%) slope −2.53∗∗∗ (-5.51) 12.00 11.83∗∗∗ The implied volatility slope of VIX options 37

Table 10: Summary statistics and correlation matrix of the variables. This table reports daily summary statistics and correlation matrix of other variables. These include: in addition to our slope of VIX IV factor, (1) VVIX; alternative measure of ATM IV, (2) VRP ; monthly variance risk premium, defined as the different between the risk-neutral and objective expectations of realized variance, (3) DEF ; default risk premium, defined as the difference between Moody’s yield on Baa corporate bonds and the ten-year government bond yield, (4) TED; measure of traders’ funding liquidity, defined as the different between Three-month Eurodollar rate and the Three-month Treasury Bill rate and (5) SLOP E; the second principal component from Johnson (2017). The sample period is ranging from February 2006 to December 2019.

Panel A: Summary statistics of variables slope V V IX DEF T ED V RP SLOP E Mean 0.1801 88.7793 2.6922 0.1854 10.4181 0.2065 Standard dev. 0.1033 13.2247 0.8290 1.5683 23.8949 1.1055 Median 0.2022 87.3100 2.6400 1.0607 9.1481 0.0512 Skewness -0.9820 1.0243 1.6623 -1.4378 -5.4592 -2.4580 Kurtosis 3.8076 5.7350 7.3257 3.7725 57.7857 24.9093 Minimum -0.2384 59.7400 1.5300 -3.7500 -218.5638 -11.0261 Maximum 0.4434 180.6100 6.1600 1.4705 77.8161 3.7060 Panel B: Correlation matrix between variables slope V V IX DEF T ED V RP SLOP E slope 1 VVIX -0.0162 1 DEF -0.5913 0.0029 1 TED 0.0046 -0.0260 0.4786 1 VRP 0.0092 -0.2363 0.0059 0.1113 1 SLOP E -0.084 -0.1809 0.1985 0.3972 0.6588 1 The implied volatility slope of VIX options 38

Table 11: Robustness of slope as a predictor. This table shows predictive regressions on the constant 30-day daily VIX futures returns in- and out-of-sample and its economic significance with other variables. Other variables VF are VVIX, VRP , DEF , TED, SLOP E and rt , respectively. The table presents the estimated βˆ coefficients for each variable, their Newey and West (1987) t-statistics, in- 2 2 sample adjusted-RIS statistics, out-of-sample ROS statistics in Panel A and B (reports overlapping regression with 30-day horizon), respectively. *,** and *** denote significance 2 level at 10%, 5% and 1%, respectively. An out-of-sample ROS in Panel B is based on fitted values rˆt+1 estimated using all observations except those overlapping with rt+1. The CER gain for a mean-variance investor is the CER difference, which is defined as a difference between the CER of forecasts generated by the regression model and the historical average forecast benchmark. The sample period is ranging from January 2007 to December 2019.

Panel A: In-sample and out-of-sample estimations and CER gain (1) (2) (3) (4) (5) (6) (7) (8) slope −2.43∗∗∗ −2.17∗∗∗ −3.07∗∗∗ −2.47∗∗∗ −2.41∗∗∗ −3.10∗∗∗ −2.60∗∗∗ −3.15∗∗∗ (-3.15) (-3.29) (-3.89) (-3.22) (-3.59) (-4.86) (-3.22) (-2.83) VVIX 0.08∗∗∗ 0.08∗∗∗ (7.89) (6.86) DEF 0.09∗∗ -0.07 (2.05) (-0.44) TED −0.07∗ 0.17 (-1.77) (1.13) VRP −0.01∗∗∗ 0.03∗∗∗ (-3.08) (4.07) SLOP E −0.66∗∗∗ −1.07∗∗∗ (-9.19) (-7.59) VF ∗∗ ∗∗∗ rt −0.06 −0.14 (-2.34) (-6.56) 2 Adj-RIS (%) 0.35 6.34 0.43 0.38 0.86 3.47 0.64 11.29 2 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ROS (%) 0.38 6.07 0.40 0.27 0.36 1.78 0.88 9.59 CER gain (%) -0.01 -4.07 0.00 -0.02 -0.01 -0.05 0.00 -7.09 Panel B: Overlaping pregdictive regression (1) (2) (3) (4) (5) (6) (7) slope −2.53∗∗∗ −2.46∗∗∗ −2.82∗∗∗ −2.55∗∗∗ −2.51∗∗∗ −2.85∗∗∗ −1.84∗∗∗ (-5.51) (-6.73) (-7.44) (-5.60) (-6.87) (-9.54) (-5.61) VVIX 0.03∗∗∗ 0.02∗∗∗ (8.18) (8.64) DEF 0.04 −0.10∗∗ (1.52) (-2.30) TED −0.06∗∗∗ -0.05 (-2.74) (-1.15) VRP −0.01∗∗∗ 0.00 (-6.09) (0.29) SLOP E −0.32∗∗∗ −0.29∗∗∗ (-10.73) (-6.73) 2 Adj-RIS (%) 12.00 33.80 12.64 13.25 24.48 33.69 50.73 2 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ROS (%) 11.83 18.62 10.05 6.37 18.66 22.12 30.61 CER gain (%) 2.97 1.96 3.02 2.84 3.76 3.46 1.00 The implied volatility slope of VIX options 39

Table 12: Summary of VIX put options implied volatility linear estimation. This table shows summary statistics of the estimated implied volatility function of VIX put options, IV (ξ) = IVAT M (1 + slopeξ), where IV is the implied volatility and ξ is the moneyness of the option. The regression is ﬁtted each day separately and for each maturity, and the sample period is from February 2006 to December 2019.

By maturity (days) Overall < 30 30 − 90 90 − 180 180 − 360 > 360 > 180 < 180 Mean IVAT M 0.7845 1.0186 0.7982 0.6423 0.5044 0.4020 0.4921 0.7914 slope 0.1901 0.2378 0.1908 0.1614 0.1612 0.1101 0.1565 0.1908 Standard dev. IVAT M 0.2171 0.2648 0.1190 0.0755 0.0567 0.0158 0.0555 0.2149 slope 0.0730 0.0874 0.0618 0.0541 0.0730 0.0406 0.0727 0.0728 Daily R-squared Mean R2 88.31% 88.10% 89.88% 87.49% 82.13% 52.74% 79.68% 88.50% Standard dev. R2 15.12% 17.84% 11.98% 15.50% 15.32% 13.30% 16.45% 15.03%

Table 13: Summary of VIX options (calls and puts) implied volatility linear estimation. This table shows summary statistics of the estimated implied volatility function of VIX options, IV (ξ) = IVAT M (1 + slopeξ), where IV is the implied volatility and ξ is the moneyness of the option. The regression is ﬁtted each day separately and for each maturity, and the sample period is from February 2006 to December 2019.

By maturity (days) Overall < 30 30 − 90 90 − 180 180 − 360 > 360 > 180 < 180 Mean IVAT M 0.7720 1.0158 0.7907 0.6422 0.5048 0.4035 0.4950 0.7788 slope 0.2050 0.2547 0.2070 0.1794 0.1639 0.1504 0.1611 0.2061 Standard dev. IVAT M 0.2093 0.2646 0.1184 0.0750 0.0552 0.0071 0.0545 0.2072 slope 0.0698 0.0786 0.0635 0.0538 0.0638 0.0411 0.0618 0.0696 Daily R-squared Mean R2 93.87% 94.79% 94.89% 93.14% 83.33% 63.38% 81.79% 94.15% Standard dev. R2 10.65% 11.91% 8.15% 11.18% 13.47% 17.61% 13.91% 10.39% The implied volatility slope of VIX options 40

Table 14: Summary of VIX call options implied volatility function estimation. This table shows summary statistics of the estimated implied volatility function of VIX 2 call options, IV (ξ) = IVAT M (1 + slopeξ + CURV ξ ), where IV is the implied volatility and ξ is the moneyness of the option. The regression is ﬁtted separately each day and for each maturity and the sample period is from February 2006 to December 2019.

By maturity (days) Overall <30 30-90 90-180 180-360 >360 >180 <180 Mean IVAT M 0.7747 1.0394 0.7817 0.6326 0.5183 0.4335 0.5056 0.7876 slope 0.1545 0.1195 0.1533 0.1758 0.1704 0.1647 0.1668 0.1538 Standard dev. IVAT M 0.2438 0.2992 0.1306 0.0795 0.0578 0.0177 0.0556 0.2422 slope 0.0836 0.0950 0.0784 0.0742 0.0592 0.0477 0.0580 0.0846 Mean R2 89.75% 86.09% 92.15% 93.70% 88.75% 88.36% 87.97% 91.20% Standard dev. R2 14.90% 17.25% 11.96% 11.71% 18.78% 11.85% 18.95% 13.83% The implied volatility slope of VIX options 41

Figures

Time series of three log VIX, VVIX and S&P 500 index level 8 5.5 4.5 4 5 7.5 3.5 3 ln(VIX) level ln(VVIX) level 7 4.5 ln(S&P500) level 2.5 4 2 6.5 2006 2008 2010 2012 2014 2016 2018 2020 Year

ln(VIX) ln(S&P500) ln(VVIX)

Figure 1: Time-series of VIX, VVIX and S&P 500 index level. This ﬁgure plots time-varying movement of VIX, VVIX and S&P 500 index level for the period covering February 2007 and December 2019. The implied volatility slope of VIX options 42

30-day rolling average of VIX volume 1.5 1 .5 Volume million) (in 0

Jan2005 Jan2010 Jan2015 Jan2020 Date

Total Call options Put options

Figure 2: Time-varying 30-day rolling average VIX volume. This ﬁgure plots time-varying 30-day rolling average VIX options volume between March 2006 to December 2019. The implied volatility slope of VIX options 43

Maturity 19 day Maturity 47 day Maturity 75 day 1.5 1.2 50000 20000 1.5 15000 1 40000 15000 1 1 .8 10000 30000 10000 Volume Volume Volume .6 20000 Implied volatility Implied volatility Implied volatility Implied .5 .5 5000 5000 .4 10000 0 0 0 0 0 .2 -2 0 2 4 -2 0 2 4 -1 0 1 2 3 4 Moneyness Moneyness Moneyness

Maturity 110 day Maturity 138 day Maturity 173 day 1 1 .8 5000 20000 600 .7 4000 .8 .8 15000 .6 400 3000 .6 Volume Volume Volume 10000 .5 .6 2000 Implied volatility Implied volatility Implied volatility Implied 200 .4 5000 .4 1000 .4 0 0 0 .2 .3 -2 0 2 4 -2 -1 0 1 2 3 -2 -1 0 1 2 3 Moneyness Moneyness Moneyness

Figure 3: IV curve fitted against moneyness of options on 2 August 2013. This figure illustrates the implied volatility-moneyness function for different maturities as at the close of 2 August 2013. The circles indicate implied volatility of the market prices of VIX call options. The solid line indicates the estimated IV curve by a linear function. The spike chart is the trading volume for corresponding options traded on 2 August 2013.

Maturity 21 day Maturity 56 day Maturity 84 day 2 1.4 1.2 6000 50000 10000 1.2 8000 40000 1 1.5 4000 1 6000 30000 1 .8 Volume Volume Volume .8 4000 20000 Implied volatility Implied volatility Implied volatility Implied 2000 .5 .6 .6 2000 10000 0 0 0 0 .4 .4 -2 0 2 4 6 -2 0 2 4 -2 0 2 4 Moneyness Moneyness Moneyness

Maturity 112 day Maturity 147 day Maturity 175 day 1 1 1.2 2000 500 100 1 .8 .8 400 1500 300 .8 .6 .6 1000 50 Volume Volume Volume 200 Implied volatility Implied volatility Implied volatility Implied .6 .4 .4 500 100 .4 0 0 0 .2 .2 -2 0 2 4 -1 0 1 2 3 4 -1 0 1 2 3 Moneyness Moneyness Moneyness

Figure 4: IV curve fitted against moneyness of options on 27 May 2015. This figure illustratess the implied volatility-moneyness function for different maturities as at the close of 27 May 2015. The circles indicate implied volatility of the market prices of VIX call options. The solid line indicates the estimated IV curve by a linear function. The spike chart is the trading volume for corresponding options traded on 27 May 2015. The implied volatility slope of VIX options 44

Maturity 20 day Maturity 55 day Maturity 83 day 2 2.5 1.4 50000 15000 100000 1.2 2 40000 80000 1.5 10000 1 30000 60000 1.5 Volume Volume Volume .8 1 20000 Implied volatility Implied volatility Implied volatility Implied 40000 5000 1 .6 10000 20000 .5 0 0 0 .5 .4 -2 0 2 4 6 -2 0 2 4 -1 0 1 2 3 4 Moneyness Moneyness Moneyness

Maturity 111 day Maturity 146 day Maturity 174 day .9 1.2 8000 1.2 20000 5000 .8 1 4000 1 6000 15000 3000 .7 .8 .8 4000 10000 Volume Volume Volume 2000 Implied volatility Implied volatility Implied volatility Implied .6 .6 .6 5000 2000 1000 .5 0 0 0 .4 .4 -1 0 1 2 3 4 -1 0 1 2 3 4 -1 0 1 2 3 Moneyness Moneyness Moneyness

Figure 5: IV curve fitted against moneyness of options on 27 July 2017. This figure illustratess the implied volatility-moneyness function for different maturities as at the close of 27 July 2017. The circles indicate implied volatility of the market prices of VIX call options. The solid line indicates the estimated IV curve by a linear function. The spike chart is the trading volume for corresponding options traded on 27 July 2017. The implied volatility slope of VIX options 45 .4 .2 VIX SKEW 0 -.2

2005 2010 2015 2020 Date

30 Day 180 Day .4 .2 0 Difference in SKEW -.2 -.4 2005 2010 2015 2020 Date

Figure 6: Time series of interpolated slope of VIX IV. This ﬁgure plots the interpolated 30-day and-180 day slope of IV curves in the VIX options market for the sample period from February 2006 to December 2019. The implied volatility slope of VIX options 46

Maturity 15 day Maturity 43 day Maturity 78 day 3 1.2 1.5 40000 20000 15000 2.5 1 30000 15000 2 10000 1 .8 20000 10000 Volume Volume Volume 1.5 Implied volatility Implied volatility Implied volatility 5000 .6 5000 1 10000 0 0 0 .5 .5 .4 0 2 4 6 8 -1 0 1 2 3 4 -1 0 1 2 3 4 Moneyness Moneyness Moneyness

Maturity 106 day Maturity 141 day Maturity 169 day 1 1.2 1.2 150 2000 300 1 1 1500 .8 100 200 .8 .8 1000 Volume Volume Volume Implied volatility Implied volatility Implied volatility .6 50 100 .6 .6 500 0 0 0 .4 .4 .4 -1 0 1 2 3 4 -1 0 1 2 3 4 -1 0 1 2 3 4 Moneyness Moneyness Moneyness

Figure 7: IV curve fitted against moneyness of options on 3 March 2015. This figure illustrates the implied volatility curve of VIX put options against the moneyness for different maturities as at the close of 3 March 2015. The solid line indicates the estimated IV curve by a linear function. The spike chart is the trading volume for correponding options traded on 3 March 2015.