David Bates (2012)
U.S. stock market crash risk, 1926â€“2010Journal of Financial Economics, 105
Matthias Fengler (2005)
Semiparametric Modeling of Implied Volatility
David Bates (2016)
How Crashes Develop: Intradaily Volatility and Crash EvolutionRisk Management & Analysis in Financial Institutions eJournal
S. Heston (1993)
A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency OptionsReview of Financial Studies, 6
R. Kozhan, A. Neuberger, P. Schneider (2012)
The Skew Risk Premium in the Equity Index MarketTerm Structure and Volatility
Andreas Kaeck, C. Alexander (2012)
Volatility dynamics for the S&P 500: Further evidence from non-affine, multi-factor jump diffusionsJournal of Banking and Finance, 36
AÃ¯t-Sahalia (2015)
Modeling financial contagion using mutually exciting jump processesJournal of Financial Economics, 117
Oleg Bondarenko (2014)
Variance Trading and Market Price of Variance RiskAdvanced Risk & Portfolio ManagementÂ® Research Paper Series
Markus Leippold, Liuren Wu, Daniel Egloff (2007)
Variance Risk Dynamics, Variance Risk Premia, and Optimal Variance Swap InvestmentsBaruch College Zicklin School of Business Research Paper Series
Ian Martin (2011)
Simple Variance SwapsRisk Management eJournal
Guido Baltussen (2012)
Unknown Unknowns: Vol-of-Vol and the Cross Section of Stock Returns
R. Jarrow (1998)
Volatility: New Estimation Techniques for Pricing Derivatives
(2000)
Transform Analysis and Asset Pricing for Affine Jump-Diffusions
G. Jiang, Yisong Tian (2005)
The Model-Free Implied Volatility and Its Information ContentEconometric Modeling: International Financial Markets - Developed Markets eJournal
David Bates (1993)
Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark OptionsNBER Working Paper Series
Alessio Saretto, Pedro Santa-clara (2006)
Option Strategies: Good Deals and Margin CallsAmerican Finance Association Meetings (AFA)
J. MencÃa, Enrique Sentana (2010)
Valuation of Vix DerivativesMicroeconomics: General Equilibrium & Disequilibrium Models of Financial Markets eJournal
M. Broadie, M. Johannes, Mikhail Chernov (2007)
Understanding Index Option ReturnsDerivatives
(2009)
The CBOE Volatility Index (R) â€“ VIX (R)
Andras Fulop, Junye Li, Jun Yu (2014)
Self-Exciting Jumps, Learning, and Asset Pricing ImplicationsERN: Other Econometrics: Applied Econometric Modeling in Financial Economics (Topic)
P. Carr, Liuren Wu (2011)
Leverage Effect, Volatility Feedback, and Self-Exciting Market DisruptionsJournal of Financial and Quantitative Analysis, 52
(1996)
SVCJ (stochastic volatility and contemporaneous jump model of Eraker
BjÃ¸rn Eraker (2004)
Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option PricesJournal of Finance, 59
Chris Bardgett, Elise Gourier, Markus Leippold (2013)
National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No . 870 Inferring Volatility Dynamics and Risk Premia from the S & P 500 and VIX Markets
D. Madan, G. Bakshi (2000)
Spanning and Derivative-Security ValuationRobert H. Smith School of Business Research Paper Series
Andreas Kaeck, C. Alexander (2013)
Continuous-time VIX dynamics: On the role of stochastic volatility of volatilityInternational Review of Financial Analysis, 28
Joost Driessen, Pascal Maenhout, G. Vilkov (2006)
The Price of Correlation Risk: Evidence from Equity OptionsDerivatives eJournal
M. Johannes, BjÃ¸rn Eraker, Nicholas Polson (2000)
The Impact of Jumps in Volatility and ReturnsChicago Booth RPS: Econometrics & Statistics (Topic)
Nikunj Kapadia, G. Bakshi, D. Madan (2000)
Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity OptionsJournal of Financial Abstracts eJournal
G. Bakshi, C. Cao, Zhiwu Chen (1997)
Empirical Performance of Alternative Option Pricing ModelsJournal of Finance, 52
Yacine Ait-Sahalia, Julio Cacho-Diaz, R. Laeven (2010)
Modeling Financial Contagion Using Mutually Exciting Jump ProcessesMicroeconomics: General Equilibrium & Disequilibrium Models of Financial Markets eJournal
Jun Pan (2002)
MIT Sloan School of Management
(2010)
Dissecting the Pricing of Equity In34I use 100,000 simulation runs for these graphs
David Bates (1998)
Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options
Dmitriy Muravyev, Neil Pearson (2019)
Option Trading Costs Are Lower than You ThinkERN: Options (Topic)
A. Neuberger (2012)
Realized SkewnessDerivatives eJournal
C. Alexander, Dimitris Korovilas (2013)
Volatility Exchange-Traded Notes: Curse or Cure?The Journal of Alternative Investments, 16
Daniel Egloff, Markus Leippold, Liuren Wu (2010)
The Term Structure of Variance Swap Rates and Optimal Variance Swap InvestmentsJournal of Financial and Quantitative Analysis, 45
C. Jones (2003)
The dynamics of stochastic volatility: evidence from underlying and options marketsJournal of Econometrics, 116
P. Carr, D. Madan, M. Stanley (2001)
Option Pricing, Interest Rates and Risk Management: Towards a Theory of Volatility Trading
Tyler Shumway, Joshua Coval (2000)
Expected Option ReturnsThe Stephen M. Ross School of Business at the University of Michigan Research Paper Series
(2015)
Log-normal stochastic volatility model: Pricing of vanilla options and econometric estimation
P. Carr, Liuren Wu (2009)
Variance Risk PremiumsReview of Financial Studies, 22
Peter Christoffersen, Kris Jacobs, Karim Mimouni (2007)
Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns and Option PricesDerivatives eJournal
Abstract This article explores the premium for bearing the variance risk of the VIX index, called the variance-of-variance risk premium. I find that during the sample period from 2006 until 2014 trading strategies exploiting the difference between the implied and realized variance of the VIX index yield average excess returns of − 24.16% per month, with an alpha of − 16.98% after adjusting for Fama–French and Carhart risk factors as well as accounting for variance risk (both highly significant). The article provides further evidence of risk premium characteristics using corridor variance swaps and compares empirical results with the predictions of reduced-form and structural benchmark models. The heteroskedasticity of equity index returns is one of the most prominent stylized facts in the finance literature. Empirical and theoretical studies over the last decades have investigated the effects of stochastic volatility on a wide range of financial applications such as derivative pricing or investment decisions (see, for instance, Bakshi, Cao, and Chen, 1997; Liu and Pan, 2003). The random nature of variance also raises the question whether investors demand a premium for holding variance-sensitive assets. By comparing the prices of synthetic variance swaps with realized variances, Carr and Wu (2009) conclude that the market demands a significant premium for bearing the variance risk of S&P 500 index returns. Following this important finding, theoretical as well as empirical studies have contributed substantially to the understanding of higher-order risk, related premia, and their wider economic implications (see, for instance, Neuberger, 2012; Kozhan, Neuberger, and Schneider, 2013; Martin, 2013; Bondarenko, 2014). As a consequence of the importance of stochastic volatility for market participants, the VIX index (published by CBOE1) has become a major benchmark in the finance industry as well as in academic research. The index can be interpreted as a measure of option-implied volatility of S&P 500 index returns and also serves as an approximation of 30-day variance swap rates (CBOE, 2009). The VIX is not a traded instrument, but to provide investors with direct access to volatility risk, futures and options on the VIX index have been successfully launched in 2004 and 2006, respectively. VIX options also provide investors with exposure to the volatility of the VIX process.2 Empirical findings in Mencía and Sentana (2013) suggest that this volatility-of-volatility (henceforth, vol-of-vol) is time-varying and an important risk factor in explaining the market prices of VIX options. Kaeck and Alexander (2013) model the variance of the VIX process directly and show that such feature is also important for explaining many time-series properties of the index. Baltussen, van Bekkum, and van der Grient (2014) find that vol-of-vol calculated from implied volatility measures is a predictor of future stock returns. In this article, I study the so-called variance-of-variance risk premium (VVP) which is defined as the difference between the (ex ante) risk-neutral variance and the (ex post) realized variance of the VIX index over a specified time horizon.3 Such analysis is important for at least two reasons: first, it provides empirical evidence whether investors demand risk premia related to the variability of the VIX index, and such results may serve as an important reference for market participants exposed to vol-of-vol risk. Second, option pricing applications require an understanding of whether such a premium exists in the market. Recent research in this strand of the literature is based on the assumption that no such risk premia exist (see Mencía and Sentana, 2013). In calculating model-free risk premia, this article follows recent theoretical developments in the definition of realized variance and builds on a framework that is free of jump and discretization biases (for a detailed discussion, see Neuberger (2012) and Bondarenko (2014)). Using option data from April 2006 until August 2014, I find that the difference between the implied and realized variance of the VIX is significant and investment strategies designed to exploit this yield an average monthly return of −24.16%. More than three-quarters of this return cannot be explained by standard risk factors, culminating in a highly significant alpha of −17.78% to −16.98% per month, depending on the exact model specification.4 Following this main result, various aspects of the VVP are explored in more detail: What drives the VVP? Is there a term structure of variance-of-variance risk? How does the VVP compare to other VIX option strategies? What is the relationship between the variance risk premium (VP) and the VVP? To study the first question, I investigate the explanatory power of standard asset pricing risk factors and find that the market return exhibits the highest explanatory power (with a highly significant and negative effect). The momentum and size factors have no significant relationship with VVP returns, whereas the book-to-market factor can explain some of the return variation. After controlling for variance risk, market index returns become insignificant, whereas the regression alpha remains unaltered. I then follow ideas in Andersen and Bondarenko (2010) and dissect the realized variance into up-variance and down-variance measures. Empirical results indicate that monthly risk premia are statistically significant, independent of whether the variance was accumulated in an up or down corridor. Up-variance trades provide a lower monthly return of −30.73% whereas the down-variance contributes only −15.00%, both statistically significant. Alphas for these trades are also significant with values between −21.30% and −16.65%. Interestingly, the explanatory power of standard risk factors differs markedly, with an R2 of only 6% during periods of stagnant volatility compared to almost 60% during periods of upward moving volatility. To address the second question, I study the return of VVP investment strategies over different holding periods and find that the return of a monthly variance-of-variance contract is indistinguishable from the holding period return of trades with a 2-, 3-, or 4-month horizon, all yielding holding period returns of less than −20%. Interestingly, standard equity risk factors show a stronger relation to longer-term investments, providing evidence that short-term variance-of-variance risk premia provide more market-independent sources of risk. To understand the contribution of the variance risk along the term structure, I study the return of option trading strategies that liquidate longer-term investments early, and hence are designed to depend on the realized variance of longer-term VIX futures prior to their maturity. While I do not find any significant alpha for such investment strategies, returns on these investments are measured with considerable noise which may have an adverse effect on the power of these tests. How compatible are these empirical findings with the prediction of standard option pricing models? To address this questions, I show that the size and sign of the monthly premium can be generated in extensions of VIX option pricing models introduced in Mencía and Sentana (2013) and Bardgett, Gourier, and Leippold (2013). Reconciling variance and variance-of-variance risk premia in a single model requires the separation of volatility and vol-of-vol risk. I provide new evidence on model specifications that allow to model both empirical features simultaneously. Finally, I compare VVP trades with two other sets of option strategies. First, I compare VVP to other simple VIX option trades such as selling out-of-the-money (OTM) options or at-the-money (ATM) straddles. While some of these have high absolute returns over the sample period, I find no evidence that any of these returns are significantly different from zero or exhibit significant alphas. This provides not only insights into the nature of variance risk but also shows that VVP contracts may be interesting trading strategies for VIX option investors. In addition, this article compares the VVP with the variance risk premium implied in S&P 500 index options. VVP investments provide interesting return characteristics beyond those of the variance risk premium of S&P 500 index returns. The remainder of the article is structured as follows: Section 1 presents the methodological framework and Section 2 introduces the dataset. The main empirical results are provided in Section 3, and model-based evidence of VVP is presented in Section 4. Section 5 concludes. 1. Methodology The variance risk premium is defined as the difference between the realized variance of a financial instrument and its (ex ante) risk-neutral expectation. A significant difference between these two variance measures indicates that investors require a risk premium to hold variance-sensitive assets. A large body of literature examines such risk premia by employing high-frequency returns and an approximation of the risk-neutral characteristic that ignores jump risk (see Carr and Wu, 2009). Following Neuberger (2012) and Bondarenko (2014), the realized variance over a partition Π={t=t0<…<tn=T} of the interval from t to T is defined as RVt,TΠ=2∑i=1n(riF− log [1+riF]), (1) where riF=Fti,T/Fti−1,T−1 is the simple return of a futures contract Ft,T (with fixed maturity T) between two points in the partition. Neuberger (2012) shows that this nonstandard definition can be regarded as a generalized variance measure and that its risk-neutral expectation can be calculated in a model-free way if Ft,T follows a martingale. Therefore, using this definition ensures that results are not affected by jump or discretization biases.5 I follow standard practice and work with daily returns, that is, the points in the partition are comprised of trading days between t and T. To simplify notation, τ≡T−t. The risk-neutral expectation of realized variance, as defined above, can be calculated from observed vanilla option prices. Following Bakshi and Madan (2000) and Bondarenko (2014), this expectation is given by IVt,T≡Etℚ[RVt,TΠ]=2×Etℚ[FT,TFt,T−1− log [FT,TFt,T]]=2B(t,T) ∫0∞k−2M(k,t,T) dk, (2) where Etℚ[·] is the time-t risk-neutral expectation, M(K,t,T) is the time-t market price of an OTM option with strike K and maturity date T and B(t, T) denotes the price of a zero bond maturing at time T with a notional of one.6 I interpolate between observed strikes using a simple cubic spline and extrapolate the observed implied volatility curves by setting values beyond the quoted strike range equal to the last observed implied volatility, a procedure that has become standard in the related literature. Conceptually similar to the literature, the integral is approximated using small step sizes in the strike dimension. However, rather than keeping the step size constant over the whole strike range (like a fraction of an index point), I apply a simple adaptive numerical integration routine which samples more frequently in a range where the integrand changes rapidly. This simple procedure alleviates the discretization bias that can otherwise arise in markets with pronounced skews (see Jiang and Tian, 2005). To shed light on whether risk premia differ during alternative VIX index regimes, I also consider corridor variance swaps. This analysis follows ideas in Andersen and Bondarenko (2010) and relies on the computation of synthetic corridor variance swaps similar to Carr and Madan (1998). I define the implied corridor variance with down and up barriers, denoted Bd and Bu, as IVt,TBd,Bu=2B(t,T)∫BdBuk−2M(k,t,T) dk. (3) The realized leg is given by RVt,TΠ,Bd,Bu=∑i=1ng(Fti,T)−g(Fti−1,T)−g′(Fti−1,T)×(Fti,T−Fti−1,T), (4) with g(x)={2×(− log Bu−xBu+1)ifx>Bu−2 log xifx∈[Bd,Bu]2×(− log Bd−xBd+1)ifx<Bd and g′(x) denotes the partial derivative of g(x). This definition converges to the realized variance in Equation (1) for Bd = 0 and Bu→∞. The realized corridor variance is now determined by the level of the futures prices at the partition points, as well as the corridor barriers Bu and Bd. First, if both futures prices are within the corridor, the contribution to the realized corridor variance coincides with that of a standard variance swap. Second, if the futures price jumps into the corridor from one partition point to the next, the contribution to realized variance is adjusted and only the distance from the barrier level to the new futures price enters the return calculation. Third, if the futures prices are below or above the corridor at two consecutive partition points, then the price movement does not enter the realized corridor variance. And fourth, if the futures price jumps across the corridor between two partition points, then the return between the two barriers is adjusted by the distance of the final value from the near barrier.7 The contract defined in Equation (4) is a special case of the generalized variance contract. Bondarenko (2014) shows that the definition of this contract retains the virtue of discretization and model invariance. Calculating investment returns for variance-of-variance swaps with different corridors allow us to dissect the pricing of variance-of-variance risk and to disentangle the types of risk that are priced in VIX options. To specify the barrier levels over the sample period, I follow Andersen and Bondarenko (2010) and construct the ratio Rt,T(K)≡P(K,t,T)/(P(K,t,T)+C(K,t,T)), which can be calculated from market prices of VIX call and put options, denoted as C(K,t,T) and P(K,t,T), respectively.8 This function is monotonically increasing in the strike with Rt,T(0)=0 and Rt,T(∞)=1 for all t and T and its inverse Kq=Rt,T−1(q) can be used to define the barriers of corridor variance swaps in a time-consistent way. For the main empirical results, I focus on breaking the variance into down- and up-variance measures which are labeled VVP0,50 (with Bd = 0 and Bu=Ft0,T) and VVP50,100 (with Bd=Ft0,T and Bu=∞). To check the robustness of these results I also break the range into three groups: up-variance (VVP67,100 for which Bd=K0.67 and Bu=∞), center-variance (VVP33,67, for which Bd=K0.33 and Bu=K0.67), and down-variance (VVP0,33, for which Bd = 0 and Bu=K0.33). 2. Data VIX futures data are obtained from the CBOE website. VIX futures commenced trading on March 26, 2004 and the data cover daily price information until August 2014. Typically on each trading day during the sample, between three and six different maturities are traded. Settlement prices are used for the calculation of realized statistics (defined above). Furthermore, VIX option data are obtained from Market Data Express from April 2006 until August 2014. VIX options also trade for several maturities on every trading day, typically with monthly expiries for upcoming months and a quarterly cycle thereafter. The maturity date of VIX futures and options is the Wednesday, 30 days prior to the third Friday in the following month, which is the S&P 500 index option (ticker: SPX) expiry date. This guarantees that, at expiry, the options used for calculation of the VIX index have exactly 30 days to maturity. I apply standard filters to the raw data to ensure that empirical results are not affected by illiquid quotes or obvious recording errors. In particular, I discard options that violate standard no-arbitrage relationships. After applying these filters, the database used in this study contains more than half a million VIX option quotes.9 For the comparison of variance risk premia and variance-of-variance risk premia, I also collect data on S&P 500 index options from OptionMetrics for the period from January 1996 until August 2014 (the longest sample period available at the time of writing). S&P 500 index options are European-style contracts and are among the most liquidly traded equity derivative instruments. On every trading day in the sample, I back out implied futures prices from put–call parity to circumvent the estimation of a dividend yield (using the ATM option pair for which the absolute price difference between put and call price is the smallest). I then apply, as before, a range of standard filters to remove quotes that fall outside standard arbitrage bounds (see Bakshi, Cao, and Chen, 1997). As a substitute for the unobservable risk-free rate of interest, I use interpolated rates from the OptionMetrics zero-curve file. 3. Empirical Results 3.1 Preliminary Data Analysis I use the term VVIX for normalized implied variance measures of the VIX which are defined as τ−1 × IVt,T (using VIX options for the calculation of the implied characteristic IVt,T).10 These measures can be constructed for different time horizons and this article first focuses on 45, 90, and 135 days to maturity. To this end, I calculate VVIX for all available VIX option maturity dates and for each trading day in the sample. I then interpolate using a shape-preserving cubic interpolation method to construct time series with constant time to maturity.11 I deviate from the standard 30 days to maturity as the shortest maturity to guarantee that no extrapolation is required and no options with less than 1 week to maturity are used in the index construction. Since long-term options can be fairly illiquid, I do not extend the analysis beyond 135 days.12 Figure 1 displays the evolution of the indices over the sample period. There are several noteworthy features. First (and unsurprisingly), the variance of the VIX is time-varying. Second, spikes in the short-term VVIX are observed relatively frequently. The events around the Lehman default and the subsequent financial market crisis led to an increase in the volatility of the VIX index, but contrary to many other financial variables the spikes during this period are not exceptional. The most extreme vol-of-vol levels are in fact observed in 2007 and 2010. Third, the speed of mean reversion is stronger than for the VIX index itself and the first-order autocorrelation of the indices increases with their maturity. The short-term index exhibits a first-order autocorrelation of 0.9559, whereas the coefficient increases to 0.9805 for the 135-day index. And fourth, a downward sloping term structure of VVIX indices is observed throughout the whole sample period where the long-term VVIX index is consistently lower than the short-term index. This feature can be explained by the fact that the volatility of VIX futures increases for shorter maturity futures contracts (see Alexander and Korovilas, 2013). Figure 1 View largeDownload slide VVIX indices. This figure shows the evolution of VVIX indices for three different constant maturities (45, 90, and 135 days). The indices measure the variance of VIX futures implied in the VIX option prices and are constructed according to (T−t)−1 IVt,T where IVt,T is defined in Equation (2). Figure 1 View largeDownload slide VVIX indices. This figure shows the evolution of VVIX indices for three different constant maturities (45, 90, and 135 days). The indices measure the variance of VIX futures implied in the VIX option prices and are constructed according to (T−t)−1 IVt,T where IVt,T is defined in Equation (2). Table I provides summary statistics for the three VVIX indices. Panel A confirms that the term structure of VVIX indices is downward sloping with mean, standard deviation, and skewness monotonically decreasing for increasing index maturities. Panel B focuses on properties of (daily) first-order differences. It is evident that the index with the shortest time to maturity is the most volatile with a standard deviation several times the standard deviation of the 135-day index (0.05 compared to 0.01). All indices are positively skewed and show high levels of kurtosis. As a result, first-order differences in the indices are highly non-normal (which unreported tests confirm at the highest significance levels). Columns (6)–(9) of Table I provide results of a standard principal component analysis (PCA). These results indicate that the first principal component covers 91% of the variation in the multivariate system, whereas further 7% are attributed to the second principal component. These findings compare to other term structures, such as interest rates, implied volatilities, or VIX indices. In Panel C, I normalize the first-order changes by their sample standard deviations, hence the PCA for these time series is based on the correlation rather than the covariance matrix. The interpretation of the components as shift, tilt, and curvature is evident from these results. In addition, normalizing the time series leads to a stronger influence of the second and third component which now cover 19% of the variation. For completeness, Panel D reports the results for changes in the logarithm of the VVIX. Previous findings (including positive skewness and excess kurtosis) are confirmed with only minor quantitative differences. Table I Summary statistics and PCA This table reports the summary statistics and results from a PCA of VVIX indices with three maturities (45, 90, and 135 days) over the sample period from April 2006 until August 2014. Columns (2)–(5) report the mean, standard deviation (Std), skewness (Skew), and kurtosis and columns (6)–(8) report the eigenvectors of the three principal components. Column (9) (%) summarizes the amount of variation explained by the first (first line), second (second line), and third (third line) principal component. Results in Panel A are based on levels, Panel B on daily first differences, in Panel C these differences are normalized by their sample standard deviation. Panel D is based on first differences of the log of the index values. Maturity (days) Mean Std Skew Kurtosis PC(1) PC(2) PC(3) (%) Panel A: levels 45 0.66 0.18 0.87 3.98 0.82 0.56 −0.11 0.95 90 0.47 0.11 0.38 2.69 0.48 −0.57 0.67 0.05 135 0.37 0.07 0.36 2.91 0.31 −0.61 −0.73 0.00 Panel B: changes 45 0.00 0.05 1.09 12.74 −0.93 −0.37 −0.00 0.91 90 0.00 0.02 0.62 11.22 −0.33 0.82 −0.48 0.07 135 0.00 0.01 0.99 15.26 −0.18 0.44 0.88 0.02 Panel C: normalized changes 45 0.00 1.00 1.09 12.74 0.57 0.68 0.46 0.81 90 0.00 1.00 0.62 11.22 0.59 0.05 −0.80 0.12 135 0.01 1.00 0.99 15.26 0.57 −0.73 0.38 0.07 Panel D: log changes 45 0.00 0.07 0.97 8.70 0.83 0.55 −0.09 0.83 90 0.00 0.05 0.49 6.97 0.46 −0.58 0.67 0.12 135 0.00 0.04 0.65 10.77 0.32 −0.60 −0.73 0.05 Maturity (days) Mean Std Skew Kurtosis PC(1) PC(2) PC(3) (%) Panel A: levels 45 0.66 0.18 0.87 3.98 0.82 0.56 −0.11 0.95 90 0.47 0.11 0.38 2.69 0.48 −0.57 0.67 0.05 135 0.37 0.07 0.36 2.91 0.31 −0.61 −0.73 0.00 Panel B: changes 45 0.00 0.05 1.09 12.74 −0.93 −0.37 −0.00 0.91 90 0.00 0.02 0.62 11.22 −0.33 0.82 −0.48 0.07 135 0.00 0.01 0.99 15.26 −0.18 0.44 0.88 0.02 Panel C: normalized changes 45 0.00 1.00 1.09 12.74 0.57 0.68 0.46 0.81 90 0.00 1.00 0.62 11.22 0.59 0.05 −0.80 0.12 135 0.01 1.00 0.99 15.26 0.57 −0.73 0.38 0.07 Panel D: log changes 45 0.00 0.07 0.97 8.70 0.83 0.55 −0.09 0.83 90 0.00 0.05 0.49 6.97 0.46 −0.58 0.67 0.12 135 0.00 0.04 0.65 10.77 0.32 −0.60 −0.73 0.05 Table I Summary statistics and PCA This table reports the summary statistics and results from a PCA of VVIX indices with three maturities (45, 90, and 135 days) over the sample period from April 2006 until August 2014. Columns (2)–(5) report the mean, standard deviation (Std), skewness (Skew), and kurtosis and columns (6)–(8) report the eigenvectors of the three principal components. Column (9) (%) summarizes the amount of variation explained by the first (first line), second (second line), and third (third line) principal component. Results in Panel A are based on levels, Panel B on daily first differences, in Panel C these differences are normalized by their sample standard deviation. Panel D is based on first differences of the log of the index values. Maturity (days) Mean Std Skew Kurtosis PC(1) PC(2) PC(3) (%) Panel A: levels 45 0.66 0.18 0.87 3.98 0.82 0.56 −0.11 0.95 90 0.47 0.11 0.38 2.69 0.48 −0.57 0.67 0.05 135 0.37 0.07 0.36 2.91 0.31 −0.61 −0.73 0.00 Panel B: changes 45 0.00 0.05 1.09 12.74 −0.93 −0.37 −0.00 0.91 90 0.00 0.02 0.62 11.22 −0.33 0.82 −0.48 0.07 135 0.00 0.01 0.99 15.26 −0.18 0.44 0.88 0.02 Panel C: normalized changes 45 0.00 1.00 1.09 12.74 0.57 0.68 0.46 0.81 90 0.00 1.00 0.62 11.22 0.59 0.05 −0.80 0.12 135 0.01 1.00 0.99 15.26 0.57 −0.73 0.38 0.07 Panel D: log changes 45 0.00 0.07 0.97 8.70 0.83 0.55 −0.09 0.83 90 0.00 0.05 0.49 6.97 0.46 −0.58 0.67 0.12 135 0.00 0.04 0.65 10.77 0.32 −0.60 −0.73 0.05 Maturity (days) Mean Std Skew Kurtosis PC(1) PC(2) PC(3) (%) Panel A: levels 45 0.66 0.18 0.87 3.98 0.82 0.56 −0.11 0.95 90 0.47 0.11 0.38 2.69 0.48 −0.57 0.67 0.05 135 0.37 0.07 0.36 2.91 0.31 −0.61 −0.73 0.00 Panel B: changes 45 0.00 0.05 1.09 12.74 −0.93 −0.37 −0.00 0.91 90 0.00 0.02 0.62 11.22 −0.33 0.82 −0.48 0.07 135 0.00 0.01 0.99 15.26 −0.18 0.44 0.88 0.02 Panel C: normalized changes 45 0.00 1.00 1.09 12.74 0.57 0.68 0.46 0.81 90 0.00 1.00 0.62 11.22 0.59 0.05 −0.80 0.12 135 0.01 1.00 0.99 15.26 0.57 −0.73 0.38 0.07 Panel D: log changes 45 0.00 0.07 0.97 8.70 0.83 0.55 −0.09 0.83 90 0.00 0.05 0.49 6.97 0.46 −0.58 0.67 0.12 135 0.00 0.04 0.65 10.77 0.32 −0.60 −0.73 0.05 3.2 Risk Premia and the Performance of VIX Option Strategies I first explore the size and significance of the VVP over the sample period. The left graph in Figure 2 compares monthly realized variances of nearby VIX futures to their implied counterparts; the right graph provides the corresponding (excess) returns which are defined as RVt,TΠ/IVt,T−1. Each month the return is calculated such that T is the expiry date of VIX options in the next calendar month and t is the trading day following the expiry date of the current month. Since VIX futures maturities coincide with these expiry dates, this approach requires no interpolation of futures prices or VVIX indices at different maturities. Similar to standard variance swap investments, for most months the realized variance is below its implied characteristic, hence returns are often highly negative (see Bondarenko, 2014). Only occasionally, especially during excessively volatile market regimes, realized characteristics exceed their implied counterparts and large positive returns can be realized. Figure 2 View largeDownload slide Variance-of-variance risk premia. The left graph shows the monthly realized variances RVt,TΠ of VIX futures in the last month before their expiry (defined in Equation (1)) and the implied characteristic IVt,T constructed from VIX options as in Equation (2). The right graph provides the corresponding (excess) returns which are defined as RVt,TΠ/IVt,T−1. Each month the return is calculated such that T is the expiry date of the VIX options. Figure 2 View largeDownload slide Variance-of-variance risk premia. The left graph shows the monthly realized variances RVt,TΠ of VIX futures in the last month before their expiry (defined in Equation (1)) and the implied characteristic IVt,T constructed from VIX options as in Equation (2). The right graph provides the corresponding (excess) returns which are defined as RVt,TΠ/IVt,T−1. Each month the return is calculated such that T is the expiry date of the VIX options. Table II provides summary statistics for the VVP. The average monthly return for the VVP trade described above is −24.16%, and despite the relatively short sample period this return is highly statistically significant with a t-statistic of −3.39. I compare these returns to other popular option strategies commonly applied to equity index options (see Coval and Shumway, 2001; Broadie, Chernov, and Johannes, 2009) and select buying OTM VIX options with fixed moneyness levels, buying the closest to the money option and an ATM VIX straddle. Average monthly returns for such strategies (for moneyness levels between 0.8 and 1.2) are also reported in Table II.13 Selling OTM call and put options can generate large absolute returns, however none of these are significantly different from zero with relatively low t-statistics throughout. A simple ATM straddle yields a return of roughly −4.45% per month, but only with a t-statistic of −0.44. In addition to a pure return analysis, I also report standard risk-adjusted performance measures for all strategies (Sharpe ratio, Sortino ratio, Stutzer index14). These risk-adjusted measures confirm that the VVP trade is the most successful strategy after adjusting for (potentially non-normal) risk. Table II Variance-of-variance risk premium and VIX option returns This table reports the statistics for the VVP, other VIX option strategies, and the market return over the sample period from April 2006 until August 2014. VVP is measured as the monthly return defined by RVt,TΠ/IVt,T−1, where IVt,T is the implied variance and RVt,TΠ the realized daily variance over a monthly period. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. For comparison, the table also reports the return statistics of simple VIX option strategies such as buying OTM options with different moneyness levels and an ATM straddle. Delta-hedged option strategies are labeled DH. Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP −24.16 (−3.39) −0.34 −0.48 −0.31 Market 0.88 (1.68) 0.17 −0.24 0.16 OTM – 0.8 1.45 (0.06) 0.01 −0.01 0.01 OTM – 0.9 24.86 (1.52) 0.15 −0.17 0.16 ATM – 1 14.04 (1.31) 0.13 −0.16 0.13 OTM – 1.1 −26.94 (−0.97) −0.10 −0.29 −0.10 OTM – 1.2 −25.28 (−0.75) −0.08 −0.26 −0.07 ATM straddle −4.45 (−0.44) −0.04 −0.10 −0.04 ATM – 1 (DH) −15.10 (−1.02) −0.10 −0.19 −0.10 ATM straddle (DH) −9.87 (−1.85) −0.19 −0.25 −0.18 Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP −24.16 (−3.39) −0.34 −0.48 −0.31 Market 0.88 (1.68) 0.17 −0.24 0.16 OTM – 0.8 1.45 (0.06) 0.01 −0.01 0.01 OTM – 0.9 24.86 (1.52) 0.15 −0.17 0.16 ATM – 1 14.04 (1.31) 0.13 −0.16 0.13 OTM – 1.1 −26.94 (−0.97) −0.10 −0.29 −0.10 OTM – 1.2 −25.28 (−0.75) −0.08 −0.26 −0.07 ATM straddle −4.45 (−0.44) −0.04 −0.10 −0.04 ATM – 1 (DH) −15.10 (−1.02) −0.10 −0.19 −0.10 ATM straddle (DH) −9.87 (−1.85) −0.19 −0.25 −0.18 Table II Variance-of-variance risk premium and VIX option returns This table reports the statistics for the VVP, other VIX option strategies, and the market return over the sample period from April 2006 until August 2014. VVP is measured as the monthly return defined by RVt,TΠ/IVt,T−1, where IVt,T is the implied variance and RVt,TΠ the realized daily variance over a monthly period. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. For comparison, the table also reports the return statistics of simple VIX option strategies such as buying OTM options with different moneyness levels and an ATM straddle. Delta-hedged option strategies are labeled DH. Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP −24.16 (−3.39) −0.34 −0.48 −0.31 Market 0.88 (1.68) 0.17 −0.24 0.16 OTM – 0.8 1.45 (0.06) 0.01 −0.01 0.01 OTM – 0.9 24.86 (1.52) 0.15 −0.17 0.16 ATM – 1 14.04 (1.31) 0.13 −0.16 0.13 OTM – 1.1 −26.94 (−0.97) −0.10 −0.29 −0.10 OTM – 1.2 −25.28 (−0.75) −0.08 −0.26 −0.07 ATM straddle −4.45 (−0.44) −0.04 −0.10 −0.04 ATM – 1 (DH) −15.10 (−1.02) −0.10 −0.19 −0.10 ATM straddle (DH) −9.87 (−1.85) −0.19 −0.25 −0.18 Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP −24.16 (−3.39) −0.34 −0.48 −0.31 Market 0.88 (1.68) 0.17 −0.24 0.16 OTM – 0.8 1.45 (0.06) 0.01 −0.01 0.01 OTM – 0.9 24.86 (1.52) 0.15 −0.17 0.16 ATM – 1 14.04 (1.31) 0.13 −0.16 0.13 OTM – 1.1 −26.94 (−0.97) −0.10 −0.29 −0.10 OTM – 1.2 −25.28 (−0.75) −0.08 −0.26 −0.07 ATM straddle −4.45 (−0.44) −0.04 −0.10 −0.04 ATM – 1 (DH) −15.10 (−1.02) −0.10 −0.19 −0.10 ATM straddle (DH) −9.87 (−1.85) −0.19 −0.25 −0.18 One caveat of the alternative option strategies presented above is that they may be exposed to both price and variance risk (see Broadie, Chernov, and Johannes, 2009). ATM straddles held over a monthly period, for instance, are delta-neutral only at the inception of the trade. Since the underlying VIX index is very volatile (Table I), straddles may move in- or out-of the money during a trading month and their P&Ls may be affected by both price and volatility risk. To study whether this affects the conclusions, I calculate the returns of previously reported option strategies and delta hedge their exposure to the underlying VIX index. To do so, I calculate the Black delta of the option position (i.e., the sensitivity with respect to the VIX futures contract) on each trading day during the month and offset the delta risk by trading in VIX futures contracts. The returns for two delta-hedged option strategies are presented in Table II. Delta-hedged straddle returns are now closer to the VVP return with a monthly average of −9.87% and a t-statistic of −1.85. The remaining difference between the performance of the two strategies may have several potential reasons. First, delta-hedged returns require a model assumption (the Black model in this case), and hence their returns may not be completely insensitive to underlying price movements. Second, although (approximately) delta-hedged, the variance risk exposure of such strategies is time-varying. And third, a daily hedging strategy may not completely remove all underlying price risk if the underlying is very volatile (i.e., has high gamma). Buying ATM options and delta-hedging their VIX futures exposure leads to a return of −15.10% (t-statistic: −1.02), whereas the return of the unhedged position is positive.15 These empirical results suggest that the synthetic variance-of-variance swap reliably captures the VVP and also compares well to competing option strategies. To understand whether the risk premium is regime specific, I extend the results to corridor variance swaps. Figure 3 provides up- and down-variances and corresponding investment returns (which are denoted VVP0,50 and VVP50,100) for the entire sample period. In general, both up- and down-variance trades are highly skewed and leptokurtic, with the up-trade providing more extreme returns than the down-trade. In particular, the realized up-variance measure is equal to zero for quite a few months in the sample, especially following a peak in the VVIX index. On the other hand, during few months the realized up-variance exceeds the implied characteristic several times, culminating in extreme positive returns for the strategy. The down-trade was highly lucrative to writers until 2011 and only after this, occasional extreme positive outliers are observed. Figure 3 View largeDownload slide Realized versus implied characteristics for corridor variance-of-variance swaps. The top left graph shows the monthly realized up-variances with barriers Bd=Ft0,T and Bu=∞ and the corresponding implied characteristics. The implied characteristic constructed from VIX options are calculated using Equation (3). The top right graph provides the corresponding (excess) returns which are defined as RVt,TΠ,Bd,Bu/IVt,TBd,Bu−1. Each month the return is calculated such that T is the expiry date of VIX options. The bottom left and right graphs correspond to the down-variance with Bd = 0 and Bu=Ft0,T. Figure 3 View largeDownload slide Realized versus implied characteristics for corridor variance-of-variance swaps. The top left graph shows the monthly realized up-variances with barriers Bd=Ft0,T and Bu=∞ and the corresponding implied characteristics. The implied characteristic constructed from VIX options are calculated using Equation (3). The top right graph provides the corresponding (excess) returns which are defined as RVt,TΠ,Bd,Bu/IVt,TBd,Bu−1. Each month the return is calculated such that T is the expiry date of VIX options. The bottom left and right graphs correspond to the down-variance with Bd = 0 and Bu=Ft0,T. Table III reports average returns and several performance measures for different corridor variance trades over the sample period from 2006 until 2014. Overall, the up-trade VVP50,100 provides an average monthly return of −30.73% with a t-statistic of −2.58. The average return of the down-trade VVP0,50 is −15.00% per month, also significant with a t-statistic of −2.11. These findings suggest that during the sample the exposure to both up- and down-variance is compensated and that VIX futures return variance in both corridors contributes to the overall premium. The return for an up-trade is considerably larger; nevertheless, I find no statistical significance for return differences of VVP0,50 and VVP50,100. The results for the second partition of the distribution confirms the two main findings. First, variance exposure is priced in all corridors. And second, the highest average return is observed for trading up-variance with VVP67,100 exhibiting an average monthly return of −37.93% and a t-statistic of −2.41, whereas the return of the down-variance contract is merely −21.64% (t-statistic of −2.64). As before, I find no statistical differences between average returns for different corridors. Table III Corridor variance-of-variance risk premia This table reports the return statistics for the variance-of-variance risk premium VVPBd,Bu for different corridors Bd and Bu over the sample period from April 2006 until August 2014. The risk premium is measured as the monthly return defined by RVt,TΠ,Bd,Bu/IVt,TBd,Bu−1, where Bd and Bu are specified as described in Section 1. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP0,50 −15.00 (−2.11) −0.21 −0.31 −0.20 VVP50,100 −30.73 (−2.58) −0.26 −0.41 −0.24 VVP0,33 −21.64 (−2.64) −0.26 −0.35 −0.25 VVP33,67 −14.95 (−2.06) −0.21 −0.31 −0.20 VVP67,100 −37.93 (−2.41) −0.24 −0.44 −0.22 Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP0,50 −15.00 (−2.11) −0.21 −0.31 −0.20 VVP50,100 −30.73 (−2.58) −0.26 −0.41 −0.24 VVP0,33 −21.64 (−2.64) −0.26 −0.35 −0.25 VVP33,67 −14.95 (−2.06) −0.21 −0.31 −0.20 VVP67,100 −37.93 (−2.41) −0.24 −0.44 −0.22 Table III Corridor variance-of-variance risk premia This table reports the return statistics for the variance-of-variance risk premium VVPBd,Bu for different corridors Bd and Bu over the sample period from April 2006 until August 2014. The risk premium is measured as the monthly return defined by RVt,TΠ,Bd,Bu/IVt,TBd,Bu−1, where Bd and Bu are specified as described in Section 1. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP0,50 −15.00 (−2.11) −0.21 −0.31 −0.20 VVP50,100 −30.73 (−2.58) −0.26 −0.41 −0.24 VVP0,33 −21.64 (−2.64) −0.26 −0.35 −0.25 VVP33,67 −14.95 (−2.06) −0.21 −0.31 −0.20 VVP67,100 −37.93 (−2.41) −0.24 −0.44 −0.22 Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP0,50 −15.00 (−2.11) −0.21 −0.31 −0.20 VVP50,100 −30.73 (−2.58) −0.26 −0.41 −0.24 VVP0,33 −21.64 (−2.64) −0.26 −0.35 −0.25 VVP33,67 −14.95 (−2.06) −0.21 −0.31 −0.20 VVP67,100 −37.93 (−2.41) −0.24 −0.44 −0.22 3.3. Alphas and Risk Factor Sensitivities To test how much of the excess return is due to the correlation with well-known risk factors, I study the following, standard regression model: rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit (5) where rit is the excess return (in month t) of the strategy under consideration (denoted i), rm is the market return, rsmb denotes the return of the size portfolio (SMB), rhml denotes the return of the book-to-market portfolio (HML), rumd is the return of the momentum portfolio (UMD), and rvixf denotes the simple return of monthly VIX futures.16 The choice of risk factors is partly motivated by standard asset pricing models, and partly by the importance to account for variance risk that may be correlated with VVP. The risk-free rate rf as well as all other historical factor returns are downloaded from Kenneth French’s website. I routinely adjust for heteroskedasticity and autocorrelation in the error terms εit. Table IV reports the regression results for two sets of risk factors. First, I estimate model (5) with only the market return (thus setting βsmb,i=βhml,i=βumd,i=βvixf,i=0), and second with no restrictions on the beta coefficients. I find a significant negative relationship between VVP and the market index with a highly negative beta of −7.27 (t-statistic: −5.08). This finding is similar in magnitude to the variance risk premium regressions in Carr and Wu (2009), Kozhan, Neuberger, and Schneider (2013), and Bondarenko (2014), who find market betas between −4.51 and −8.53 for their regressions of the variance risk premium return on market excess returns. The CAPM alpha of VVP is −17.78% and highly statistically significant with a t-statistic of −2.98. The R2 of the regressions is 28.51%. The second regression which includes Fama–French and Carhart risk factors as well as VIX futures returns provides interesting differences. Most importantly, the stock market return becomes an insignificant determinant of VVP after controlling for variance risk (I confirm this by running additional regressions using all risk factors but rvixf). This suggests that variance and variance-of-variance trading strategies have a similar dependence on stock market returns which cancels out if both variables are included in the regression.17 SMB and UMD are unrelated to VVP, the only other risk factor with a significant coefficient at the 5% level is HML with a t-statistic of −2.02. The alpha remains high after accounting for the additional risk factors with −16.98%, with a t-statistic of −3.28.18 Table IV Alphas and risk factor sensitivities This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit, where rit is the monthly excess return of VIX option strategies listed in column (1). Columns (2)–(7) report the intercept (alpha, αi) and the coefficients for the following risk factors: the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), the momentum factor (UMD, βumd,i), and the return of VIX Futures (VIXF, βvixf,i). Column (8) reports the R2 of the regression. The variance-of-variance risk premium trade is labeled VVP. The sample period is from April 2006 until August 2014. Strategy alpha MktRP SMB HML UMD VIXF R2 VVP −17.78 −7.27 0.29 (−2.98) (−5.08) −16.98 −0.22 −1.17 −2.46 −1.26 1.75 0.45 (−3.28) (−0.16) (−0.50) (−2.02) (−1.35) (5.25) VVP0,50 −16.65 2.58 0.03 (−2.17) (2.65) −17.63 −0.66 2.49 −1.48 0.79 −0.81 0.09 (−2.39) (−0.27) (0.59) (−0.76) (0.75) (−2.30) VVP50,100 −21.30 −14.71 0.36 (−2.54) (−6.40) −17.15 −0.96 −2.33 −3.31 −2.54 3.33 0.59 (−2.49) (−0.49) (−0.68) (−1.28) (−2.31) (7.08) VVP0,33 −25.02 5.27 0.10 (−2.96) (3.76) −26.27 1.16 3.24 −3.57 0.53 −1.05 0.18 (−3.18) (0.42) (0.82) (−1.49) (0.52) (−2.15) VVP33,67 −12.75 −3.43 0.05 (−1.78) (−2.83) −12.74 −4.59 0.91 0.78 −0.37 −0.16 0.06 (−1.81) (−1.99) (0.30) (0.37) (−0.26) (−0.36) VVP67,100 −25.57 −19.27 0.36 (−2.49) (−5.73) −19.86 −0.08 −3.68 −2.72 −2.87 4.64 0.60 (−2.59) (−0.03) (−0.80) (−0.78) (−1.54) (8.22) Strategy alpha MktRP SMB HML UMD VIXF R2 VVP −17.78 −7.27 0.29 (−2.98) (−5.08) −16.98 −0.22 −1.17 −2.46 −1.26 1.75 0.45 (−3.28) (−0.16) (−0.50) (−2.02) (−1.35) (5.25) VVP0,50 −16.65 2.58 0.03 (−2.17) (2.65) −17.63 −0.66 2.49 −1.48 0.79 −0.81 0.09 (−2.39) (−0.27) (0.59) (−0.76) (0.75) (−2.30) VVP50,100 −21.30 −14.71 0.36 (−2.54) (−6.40) −17.15 −0.96 −2.33 −3.31 −2.54 3.33 0.59 (−2.49) (−0.49) (−0.68) (−1.28) (−2.31) (7.08) VVP0,33 −25.02 5.27 0.10 (−2.96) (3.76) −26.27 1.16 3.24 −3.57 0.53 −1.05 0.18 (−3.18) (0.42) (0.82) (−1.49) (0.52) (−2.15) VVP33,67 −12.75 −3.43 0.05 (−1.78) (−2.83) −12.74 −4.59 0.91 0.78 −0.37 −0.16 0.06 (−1.81) (−1.99) (0.30) (0.37) (−0.26) (−0.36) VVP67,100 −25.57 −19.27 0.36 (−2.49) (−5.73) −19.86 −0.08 −3.68 −2.72 −2.87 4.64 0.60 (−2.59) (−0.03) (−0.80) (−0.78) (−1.54) (8.22) Table IV Alphas and risk factor sensitivities This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit, where rit is the monthly excess return of VIX option strategies listed in column (1). Columns (2)–(7) report the intercept (alpha, αi) and the coefficients for the following risk factors: the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), the momentum factor (UMD, βumd,i), and the return of VIX Futures (VIXF, βvixf,i). Column (8) reports the R2 of the regression. The variance-of-variance risk premium trade is labeled VVP. The sample period is from April 2006 until August 2014. Strategy alpha MktRP SMB HML UMD VIXF R2 VVP −17.78 −7.27 0.29 (−2.98) (−5.08) −16.98 −0.22 −1.17 −2.46 −1.26 1.75 0.45 (−3.28) (−0.16) (−0.50) (−2.02) (−1.35) (5.25) VVP0,50 −16.65 2.58 0.03 (−2.17) (2.65) −17.63 −0.66 2.49 −1.48 0.79 −0.81 0.09 (−2.39) (−0.27) (0.59) (−0.76) (0.75) (−2.30) VVP50,100 −21.30 −14.71 0.36 (−2.54) (−6.40) −17.15 −0.96 −2.33 −3.31 −2.54 3.33 0.59 (−2.49) (−0.49) (−0.68) (−1.28) (−2.31) (7.08) VVP0,33 −25.02 5.27 0.10 (−2.96) (3.76) −26.27 1.16 3.24 −3.57 0.53 −1.05 0.18 (−3.18) (0.42) (0.82) (−1.49) (0.52) (−2.15) VVP33,67 −12.75 −3.43 0.05 (−1.78) (−2.83) −12.74 −4.59 0.91 0.78 −0.37 −0.16 0.06 (−1.81) (−1.99) (0.30) (0.37) (−0.26) (−0.36) VVP67,100 −25.57 −19.27 0.36 (−2.49) (−5.73) −19.86 −0.08 −3.68 −2.72 −2.87 4.64 0.60 (−2.59) (−0.03) (−0.80) (−0.78) (−1.54) (8.22) Strategy alpha MktRP SMB HML UMD VIXF R2 VVP −17.78 −7.27 0.29 (−2.98) (−5.08) −16.98 −0.22 −1.17 −2.46 −1.26 1.75 0.45 (−3.28) (−0.16) (−0.50) (−2.02) (−1.35) (5.25) VVP0,50 −16.65 2.58 0.03 (−2.17) (2.65) −17.63 −0.66 2.49 −1.48 0.79 −0.81 0.09 (−2.39) (−0.27) (0.59) (−0.76) (0.75) (−2.30) VVP50,100 −21.30 −14.71 0.36 (−2.54) (−6.40) −17.15 −0.96 −2.33 −3.31 −2.54 3.33 0.59 (−2.49) (−0.49) (−0.68) (−1.28) (−2.31) (7.08) VVP0,33 −25.02 5.27 0.10 (−2.96) (3.76) −26.27 1.16 3.24 −3.57 0.53 −1.05 0.18 (−3.18) (0.42) (0.82) (−1.49) (0.52) (−2.15) VVP33,67 −12.75 −3.43 0.05 (−1.78) (−2.83) −12.74 −4.59 0.91 0.78 −0.37 −0.16 0.06 (−1.81) (−1.99) (0.30) (0.37) (−0.26) (−0.36) VVP67,100 −25.57 −19.27 0.36 (−2.49) (−5.73) −19.86 −0.08 −3.68 −2.72 −2.87 4.64 0.60 (−2.59) (−0.03) (−0.80) (−0.78) (−1.54) (8.22) Table IV also reports alphas for the corridor variances trades, and there are some interesting findings. First, up-variance swap returns can be much better explained by standard risk factors than the down-variance returns. This is reflected by the fact that the R2s are substantially higher for the up-trade with levels between 59% and 60% (for the most general risk factor regressions), whereas standard risk factors explain less than 18% of the variance for the down-trade. A potential explanation for this finding is that risk factors become more correlated during periods of market turmoil. This strong asymmetry can also be interpreted as evidence that the explanatory power of standard risk factors, and the market return in particular, are regime-dependent.19 Second, the results for the center-variance indicate that standard risk factors are unsuccessful in explaining the return variation for center-corridor swaps. Also note that up- and down-corridor trades have alphas of similar size, suggesting that the additional return of the up-variance trade can be mainly attributed to the correlation with the market return. The only category for which alphas are only significant at a 10% level is the VVP33,67 which exhibits values around −13%. 3.4 Term Structure of Variance of Variance Risk Premia Empirical results presented so far, as well as the empirical findings in the related literature, concentrate on 1-month investment returns. Longer holding periods may provide additional insights into the pricing of vol-of-vol risk and this term structure of risk premia is the focal point of this section.20 To provide preliminary results, I first construct return time series as follows: every month I select the trading day after the VIX option expiry and calculate variance-of-variance contract returns with 2, 3, and 4 months to expiry. As for the monthly holding period, the investment dates are chosen in a way that no interpolation of VIX futures or VVIX indices is required, as contracts are held until expiration. Table V reports average returns and corresponding t-statistics of VVP trades with different holding periods.21 The average risk premium over different holding periods is very stable and ranges from −20.44% for the 2-month investment to −23.01% for the 3-month investment. The premium remains highly significant for all investment horizons. For completeness, I also report the term structure of corridor variances and find that for longer-term investment horizons, down-variance provides insignificant risk premia, while center-variance trades remain significant. Table V Variance-of-variance risk premium (monthly term structure) This table reports the average variance-of-variance risk premia defined as RVt,TΠ/IVt,T−1, where IVt,T is the implied variance and RVt,TΠ the realized daily variance. Risk premia are reported for a holding period of 2, 3, and 4 months. Every month the day after the VIX option expiry is selected and variance-of-variance contract returns are calculated from VIX options and futures with 2, 3, and 4 months to expiry. The table also report results for corridor variance contracts VVPBd,Bu where Bd and Bu denote the lower and upper bound of the corridor. 2 months 3 months 4 months Mean return (t-stat) Mean return (t-stat) Mean return (t-stat) VVP −20.44 (−3.25) −23.01 (−4.48) −21.45 (−4.12) VVP0,50 −6.04 (−0.84) −8.08 (−1.19) −11.13 (−1.70) VVP50,100 −31.60 (−2.73) −33.98 (−3.23) −28.94 (−2.56) VVP0,33 1.45 (0.13) 1.91 (0.16) 3.30 (0.31) VVP33,67 −19.89 (−3.30) −22.99 (−3.76) −32.68 (−5.95) VVP67,100 −32.78 (−1.91) −35.46 (−2.31) −23.06 (−1.38) 2 months 3 months 4 months Mean return (t-stat) Mean return (t-stat) Mean return (t-stat) VVP −20.44 (−3.25) −23.01 (−4.48) −21.45 (−4.12) VVP0,50 −6.04 (−0.84) −8.08 (−1.19) −11.13 (−1.70) VVP50,100 −31.60 (−2.73) −33.98 (−3.23) −28.94 (−2.56) VVP0,33 1.45 (0.13) 1.91 (0.16) 3.30 (0.31) VVP33,67 −19.89 (−3.30) −22.99 (−3.76) −32.68 (−5.95) VVP67,100 −32.78 (−1.91) −35.46 (−2.31) −23.06 (−1.38) Table V Variance-of-variance risk premium (monthly term structure) This table reports the average variance-of-variance risk premia defined as RVt,TΠ/IVt,T−1, where IVt,T is the implied variance and RVt,TΠ the realized daily variance. Risk premia are reported for a holding period of 2, 3, and 4 months. Every month the day after the VIX option expiry is selected and variance-of-variance contract returns are calculated from VIX options and futures with 2, 3, and 4 months to expiry. The table also report results for corridor variance contracts VVPBd,Bu where Bd and Bu denote the lower and upper bound of the corridor. 2 months 3 months 4 months Mean return (t-stat) Mean return (t-stat) Mean return (t-stat) VVP −20.44 (−3.25) −23.01 (−4.48) −21.45 (−4.12) VVP0,50 −6.04 (−0.84) −8.08 (−1.19) −11.13 (−1.70) VVP50,100 −31.60 (−2.73) −33.98 (−3.23) −28.94 (−2.56) VVP0,33 1.45 (0.13) 1.91 (0.16) 3.30 (0.31) VVP33,67 −19.89 (−3.30) −22.99 (−3.76) −32.68 (−5.95) VVP67,100 −32.78 (−1.91) −35.46 (−2.31) −23.06 (−1.38) 2 months 3 months 4 months Mean return (t-stat) Mean return (t-stat) Mean return (t-stat) VVP −20.44 (−3.25) −23.01 (−4.48) −21.45 (−4.12) VVP0,50 −6.04 (−0.84) −8.08 (−1.19) −11.13 (−1.70) VVP50,100 −31.60 (−2.73) −33.98 (−3.23) −28.94 (−2.56) VVP0,33 1.45 (0.13) 1.91 (0.16) 3.30 (0.31) VVP33,67 −19.89 (−3.30) −22.99 (−3.76) −32.68 (−5.95) VVP67,100 −32.78 (−1.91) −35.46 (−2.31) −23.06 (−1.38) Table VI presents estimations of regression model (5) applied to the returns calculated over longer-investment horizons.22 These results confirm the findings in Table IV, in particular I find that VVP trades have significant alphas up to a 4-month investment horizon. Alphas are only marginally lower than the alphas for 1-month investments and range from −11.85% for the 2-month strategy (t-statistic: −2.27) to −15.97% for the 3-month returns (t-statistic: −4.03). Interestingly, the R2 increases with the holding period and reaches 61.42% for 4-month returns. This confirms that short-term, 1-month trades have different investment characteristics and that longer-term investments are better explained by standard risk factors such as the market return of equities and/or the return of VIX futures. Other risk factors play a minor role in explaining long-term VVP investments, with SMB, HML, and UMD all insignificant for longer-term horizons. Table VI Alphas and risk factor sensitivities (term structure) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit, where rit is the excess return for the variance-of-variance contracts in held over different holding periods from 2 to 4 months. Columns (2)–(7) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), the momentum factor (UMD, βumd,i), and the return of VIX futures (VIXF, βvixf,i). Column (8) reports the R2 of the regression. Panel A reports results for a holding period of 2 months, Panel B for 4-month holding period returns. All t-statistics are adjusted for heteroskedasticity and autocorrelation. Strategy alpha MktRP SMB HML UMD VIXF R2 Panel A: 2-month investments VVP −13.56 −5.06 0.33 (−2.32) (−6.05) −11.86 −0.15 −0.27 −0.03 −0.67 1.32 0.51 (−2.27) (−0.13) (−0.14) (−0.03) (−1.12) (5.37) Panel B: 3-month investments VVP −14.86 −3.50 0.39 (−2.72) (−5.57) −15.97 0.39 0.49 −0.28 0.32 1.07 0.58 (−4.03) (0.51) (0.55) (−0.45) (1.14) (6.23) Panel C: 4-month investments VVP −11.95 −3.19 0.45 (−2.08) (−6.72) −14.04 0.01 −0.06 −0.02 0.32 0.92 0.61 (−3.17) (0.01) (−0.08) (−0.04) (1.56) (5.81) Strategy alpha MktRP SMB HML UMD VIXF R2 Panel A: 2-month investments VVP −13.56 −5.06 0.33 (−2.32) (−6.05) −11.86 −0.15 −0.27 −0.03 −0.67 1.32 0.51 (−2.27) (−0.13) (−0.14) (−0.03) (−1.12) (5.37) Panel B: 3-month investments VVP −14.86 −3.50 0.39 (−2.72) (−5.57) −15.97 0.39 0.49 −0.28 0.32 1.07 0.58 (−4.03) (0.51) (0.55) (−0.45) (1.14) (6.23) Panel C: 4-month investments VVP −11.95 −3.19 0.45 (−2.08) (−6.72) −14.04 0.01 −0.06 −0.02 0.32 0.92 0.61 (−3.17) (0.01) (−0.08) (−0.04) (1.56) (5.81) Table VI Alphas and risk factor sensitivities (term structure) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit, where rit is the excess return for the variance-of-variance contracts in held over different holding periods from 2 to 4 months. Columns (2)–(7) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), the momentum factor (UMD, βumd,i), and the return of VIX futures (VIXF, βvixf,i). Column (8) reports the R2 of the regression. Panel A reports results for a holding period of 2 months, Panel B for 4-month holding period returns. All t-statistics are adjusted for heteroskedasticity and autocorrelation. Strategy alpha MktRP SMB HML UMD VIXF R2 Panel A: 2-month investments VVP −13.56 −5.06 0.33 (−2.32) (−6.05) −11.86 −0.15 −0.27 −0.03 −0.67 1.32 0.51 (−2.27) (−0.13) (−0.14) (−0.03) (−1.12) (5.37) Panel B: 3-month investments VVP −14.86 −3.50 0.39 (−2.72) (−5.57) −15.97 0.39 0.49 −0.28 0.32 1.07 0.58 (−4.03) (0.51) (0.55) (−0.45) (1.14) (6.23) Panel C: 4-month investments VVP −11.95 −3.19 0.45 (−2.08) (−6.72) −14.04 0.01 −0.06 −0.02 0.32 0.92 0.61 (−3.17) (0.01) (−0.08) (−0.04) (1.56) (5.81) Strategy alpha MktRP SMB HML UMD VIXF R2 Panel A: 2-month investments VVP −13.56 −5.06 0.33 (−2.32) (−6.05) −11.86 −0.15 −0.27 −0.03 −0.67 1.32 0.51 (−2.27) (−0.13) (−0.14) (−0.03) (−1.12) (5.37) Panel B: 3-month investments VVP −14.86 −3.50 0.39 (−2.72) (−5.57) −15.97 0.39 0.49 −0.28 0.32 1.07 0.58 (−4.03) (0.51) (0.55) (−0.45) (1.14) (6.23) Panel C: 4-month investments VVP −11.95 −3.19 0.45 (−2.08) (−6.72) −14.04 0.01 −0.06 −0.02 0.32 0.92 0.61 (−3.17) (0.01) (−0.08) (−0.04) (1.56) (5.81) To gain a better understanding of the term premia, I construct a second set of returns. I retain a monthly investment horizon, but enter every month a variance-of-variance contract with 2 (3 or 4) months to maturity, which is then sold 1 month later (and hence it is not held until its maturity). The return of such strategy only depends on the realized variance over the first month, as risks thereafter are fully hedged. Denoting the time of investment as t0, the end of the holding period as th and the maturity as T, the (excess) return of such strategy is defined as 1IVt0,T×(e−rf×(T−th)×RVt0,thΠ+IVth,T)−1. (6) I first provide empirical evidence of the average term premium in Table VII and use the notation VVPx→y for the premium of a contract initiated x months before maturity and held for y months. As shown in Table VII, these returns are substantially lower with values increasing from −4.55% for VVP2→1 to −0.70% for VVP4→1. Corresponding t-statistics suggest that the investment returns are not significantly different from zero with t-statistics between −0.29 ( VVP4→1) and −1.14 ( VVP2→1). Table VIII presents corresponding alphas and unsurprisingly, given earlier results, I find that none of the investment returns provide significant alpha values. Two further (untabulated) robustness checks highlight the high premium in the first month: (i) investment returns for longer holding periods, such as VVP4→2 or VVP3→2, also have insignificant alphas, and (ii) alphas for long–short strategies consisting of a long contract in VVP1→1 and a short position in longer-dated contracts (such as VVP4→1, VVP3→1, or VVP2→1) have alphas (and Sharpe ratios) almost identical to the short-term investment highlighting the difference in the VIX variance risk at different horizons. Table VII Variance-of-variance risk premium (holding periods) This table reports the statistics for the monthly variance-of-variance risk premium calculated from longer-term investments. VVPx→y denotes the premium of a contract initiated x months before maturity and held for y months over the sample period from April 2006 until August 2014. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP2→1 −4.55 (−1.14) −0.11 −0.20 −0.11 VVP3→1 −3.16 (−1.06) −0.11 −0.19 −0.10 VVP4→1 −0.70 (−0.29) −0.03 −0.05 −0.03 Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP2→1 −4.55 (−1.14) −0.11 −0.20 −0.11 VVP3→1 −3.16 (−1.06) −0.11 −0.19 −0.10 VVP4→1 −0.70 (−0.29) −0.03 −0.05 −0.03 Table VII Variance-of-variance risk premium (holding periods) This table reports the statistics for the monthly variance-of-variance risk premium calculated from longer-term investments. VVPx→y denotes the premium of a contract initiated x months before maturity and held for y months over the sample period from April 2006 until August 2014. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP2→1 −4.55 (−1.14) −0.11 −0.20 −0.11 VVP3→1 −3.16 (−1.06) −0.11 −0.19 −0.10 VVP4→1 −0.70 (−0.29) −0.03 −0.05 −0.03 Strategy Mean return (t-stat) Sharpe Sortino Stutzer VVP2→1 −4.55 (−1.14) −0.11 −0.20 −0.11 VVP3→1 −3.16 (−1.06) −0.11 −0.19 −0.10 VVP4→1 −0.70 (−0.29) −0.03 −0.05 −0.03 Table VIII Alphas and risk factor sensitivities (term structure) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit, where rit is the excess return of different variance-of-variance contracts in column (1). VVPx→y denotes the premium of a contract initiated x months before maturity and held for y months over the sample period from April 2006 until August 2014. Columns (2)–(7) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), the momentum factor (UMD, βumd,i), and the return of VIX Futures (VIXF, βvixf,i). Column (8) reports the R2 of the regression. Strategy alpha MktRP SMB HML UMD VIXF R2 VVP2→1 −0.49 −4.63 0.36 (−0.17) (−5.81) 1.13 −2.45 1.61 −1.42 −1.89 1.13 0.53 (0.38) (−2.41) (1.16) (−1.44) (−2.91) (2.83) VVP3→1 −0.03 −3.54 0.40 (−0.01) (−7.25) 0.39 −2.06 0.57 −1.65 −1.27 0.94 0.54 (0.21) (−2.79) (0.53) (−1.81) (−2.58) (3.26) VVP4→1 1.38 −2.54 0.36 (0.79) (−6.69) 1.47 −1.38 0.98 −1.59 −0.82 0.85 0.50 (0.86) (−2.57) (1.15) (−2.35) (−2.18) (3.22) Strategy alpha MktRP SMB HML UMD VIXF R2 VVP2→1 −0.49 −4.63 0.36 (−0.17) (−5.81) 1.13 −2.45 1.61 −1.42 −1.89 1.13 0.53 (0.38) (−2.41) (1.16) (−1.44) (−2.91) (2.83) VVP3→1 −0.03 −3.54 0.40 (−0.01) (−7.25) 0.39 −2.06 0.57 −1.65 −1.27 0.94 0.54 (0.21) (−2.79) (0.53) (−1.81) (−2.58) (3.26) VVP4→1 1.38 −2.54 0.36 (0.79) (−6.69) 1.47 −1.38 0.98 −1.59 −0.82 0.85 0.50 (0.86) (−2.57) (1.15) (−2.35) (−2.18) (3.22) Table VIII Alphas and risk factor sensitivities (term structure) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+βvixf,i rtvixf+εit, where rit is the excess return of different variance-of-variance contracts in column (1). VVPx→y denotes the premium of a contract initiated x months before maturity and held for y months over the sample period from April 2006 until August 2014. Columns (2)–(7) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), the momentum factor (UMD, βumd,i), and the return of VIX Futures (VIXF, βvixf,i). Column (8) reports the R2 of the regression. Strategy alpha MktRP SMB HML UMD VIXF R2 VVP2→1 −0.49 −4.63 0.36 (−0.17) (−5.81) 1.13 −2.45 1.61 −1.42 −1.89 1.13 0.53 (0.38) (−2.41) (1.16) (−1.44) (−2.91) (2.83) VVP3→1 −0.03 −3.54 0.40 (−0.01) (−7.25) 0.39 −2.06 0.57 −1.65 −1.27 0.94 0.54 (0.21) (−2.79) (0.53) (−1.81) (−2.58) (3.26) VVP4→1 1.38 −2.54 0.36 (0.79) (−6.69) 1.47 −1.38 0.98 −1.59 −0.82 0.85 0.50 (0.86) (−2.57) (1.15) (−2.35) (−2.18) (3.22) Strategy alpha MktRP SMB HML UMD VIXF R2 VVP2→1 −0.49 −4.63 0.36 (−0.17) (−5.81) 1.13 −2.45 1.61 −1.42 −1.89 1.13 0.53 (0.38) (−2.41) (1.16) (−1.44) (−2.91) (2.83) VVP3→1 −0.03 −3.54 0.40 (−0.01) (−7.25) 0.39 −2.06 0.57 −1.65 −1.27 0.94 0.54 (0.21) (−2.79) (0.53) (−1.81) (−2.58) (3.26) VVP4→1 1.38 −2.54 0.36 (0.79) (−6.69) 1.47 −1.38 0.98 −1.59 −0.82 0.85 0.50 (0.86) (−2.57) (1.15) (−2.35) (−2.18) (3.22) Term-structure investment returns in Equation (6) depend on both the realized variance as well as the prevailing future spot rate IVth,T. While term-structure strategies provide insignificant alphas, these results may be driven by the additional noise that results from the dependence on a future implied variance. To single out the contribution of RVt0,thΠ one could use the forward implied variance to hedge the risk after th. This way the price of the hedge would be known at time t, and the return would not be calculated on the spot implied variance but only on a fraction that relates to the chosen investment horizon. Such strategy would reduce the noise in the investment returns as the only random return component is RVt,thΠ. Unfortunately, it is not straightforward to calculate forward rates in this set-up as only one option expiry for each underlying VIX futures contract is available, and no simple model-free relationship between VIX futures with different maturity exists. Therefore, the empirical findings, especially the insignificant term premia, have to be interpreted with the caveat in mind that this measure of term premia may be relatively noisy. 3.5 Comparing Variance of Variance Risk Premia with Variance Risk Premia Earlier results demonstrate that part of the VVP can be explained by the returns of VIX futures. The variance contract calculated from S&P 500 index options (VP) provides a second important benchmark as the construction of VVP and VP returns is conceptually similar. In this section, I address two related issues. First, I provide a simple comparison of the two risk premia over the sample period. I then use VP in the risk factor regressions to provide a robustness check for whether previous results are sensitive to how variance returns are measured. To this end, I first construct VP returns for the S&P 500 index following the procedure outlined in Section 1. Average risk premia and performance statistics are reported in Table IX. Panel A summarizes the premia for the complete OptionMetrics sample from January 1996 until August 2014. I find an average risk premium of −20.77%, confirming results in Kozhan, Neuberger, and Schneider (2013) whose sample lasts until January 2012. More interestingly, Panel B details the VP return for the same period that was used for VVP premium estimates. It is evident that over the shorter sample, the VP is reduced to −16.68% with a t-statistic of −2.03. Table IX Variance risk premium This table reports the return statistics for the VP calculated from S&P 500 index options from April 2006 until August 2014. VP is measured as the monthly return defined by RVt,TΠ/IVt,T−1, where IVt,T is the implied variance and RVt,TΠ the realized daily variance of S&P 500 index futures over a monthly period. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. Strategy Mean return (t-stat) Sharpe Sortino Stutzer Panel A: January 1996 to August 2014 VP −20.77 (−4.70) −0.31 −0.48 −0.27 Panel B: April 2006 to August 2014 VP −16.68 (−2.03) −0.20 −0.38 −0.19 Strategy Mean return (t-stat) Sharpe Sortino Stutzer Panel A: January 1996 to August 2014 VP −20.77 (−4.70) −0.31 −0.48 −0.27 Panel B: April 2006 to August 2014 VP −16.68 (−2.03) −0.20 −0.38 −0.19 Table IX Variance risk premium This table reports the return statistics for the VP calculated from S&P 500 index options from April 2006 until August 2014. VP is measured as the monthly return defined by RVt,TΠ/IVt,T−1, where IVt,T is the implied variance and RVt,TΠ the realized daily variance of S&P 500 index futures over a monthly period. Columns (4)–(6) report the Sharpe ratio, the Sortino ratio, and the Stutzer index. Strategy Mean return (t-stat) Sharpe Sortino Stutzer Panel A: January 1996 to August 2014 VP −20.77 (−4.70) −0.31 −0.48 −0.27 Panel B: April 2006 to August 2014 VP −16.68 (−2.03) −0.20 −0.38 −0.19 Strategy Mean return (t-stat) Sharpe Sortino Stutzer Panel A: January 1996 to August 2014 VP −20.77 (−4.70) −0.31 −0.48 −0.27 Panel B: April 2006 to August 2014 VP −16.68 (−2.03) −0.20 −0.38 −0.19 For completeness, Table X shows the alphas and sensitivities with respect to the Fama–French risk factors. There are two noteworthy results. First, VP alphas over the 2006–14 sample period are insignificant with t-statistics between −1.46 and −1.27, whereas VVP alphas over the same sample period are highly significant. Second, the variation of the VP explained by the market return and other Fama–French risk factors is substantially higher than the variation explained in VVP. This indicates that VVP provides returns that exhibit more market-independent behavior. Table X Alphas and risk factor sensitivities (variance risk premium) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+εit, where rit is the return of the variance contract in column (1). VP is the standard variance contract. Columns (2)–(6) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), and the momentum factor (UMD, βumd,i). Column (7) reports the R2 of the regression. Panel A reports results for a holding period of 2 months, Panel B and C focus on 3 and 4 months. All t-statistics are corrected for heteroskedasticity and autocorrelation. Strategy alpha MktRP SMB HML UMD R2 Panel A: January 1996 to August 2014 VP −16.20 −7.23 0.34 (−3.69) (−5.26) −13.43 −7.68 −2.96 −3.05 −2.13 0.39 (−2.90) (−5.34) (−2.41) (−2.05) (−3.05) Panel B: April 2006 to August 2014 VP −9.78 −10.37 0.50 (−1.46) (−5.68) −8.61 −12.95 2.55 4.51 −1.80 0.56 (−1.27) (−6.06) (1.01) (1.47) (−1.91) Strategy alpha MktRP SMB HML UMD R2 Panel A: January 1996 to August 2014 VP −16.20 −7.23 0.34 (−3.69) (−5.26) −13.43 −7.68 −2.96 −3.05 −2.13 0.39 (−2.90) (−5.34) (−2.41) (−2.05) (−3.05) Panel B: April 2006 to August 2014 VP −9.78 −10.37 0.50 (−1.46) (−5.68) −8.61 −12.95 2.55 4.51 −1.80 0.56 (−1.27) (−6.06) (1.01) (1.47) (−1.91) Table X Alphas and risk factor sensitivities (variance risk premium) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+εit, where rit is the return of the variance contract in column (1). VP is the standard variance contract. Columns (2)–(6) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), and the momentum factor (UMD, βumd,i). Column (7) reports the R2 of the regression. Panel A reports results for a holding period of 2 months, Panel B and C focus on 3 and 4 months. All t-statistics are corrected for heteroskedasticity and autocorrelation. Strategy alpha MktRP SMB HML UMD R2 Panel A: January 1996 to August 2014 VP −16.20 −7.23 0.34 (−3.69) (−5.26) −13.43 −7.68 −2.96 −3.05 −2.13 0.39 (−2.90) (−5.34) (−2.41) (−2.05) (−3.05) Panel B: April 2006 to August 2014 VP −9.78 −10.37 0.50 (−1.46) (−5.68) −8.61 −12.95 2.55 4.51 −1.80 0.56 (−1.27) (−6.06) (1.01) (1.47) (−1.91) Strategy alpha MktRP SMB HML UMD R2 Panel A: January 1996 to August 2014 VP −16.20 −7.23 0.34 (−3.69) (−5.26) −13.43 −7.68 −2.96 −3.05 −2.13 0.39 (−2.90) (−5.34) (−2.41) (−2.05) (−3.05) Panel B: April 2006 to August 2014 VP −9.78 −10.37 0.50 (−1.46) (−5.68) −8.61 −12.95 2.55 4.51 −1.80 0.56 (−1.27) (−6.06) (1.01) (1.47) (−1.91) As a robustness check, I alter model specification (5) and include VP as an explanatory variable instead of rvixf (see Table XI).23 VP returns are more variable than VIX futures returns, and hence the beta coefficient for VP is lower than for rvixf, albeit with very similar significance levels. VP returns are able to explain slightly more of the variation in VVP with an R2 of 61% for the most general regression model. Overall, the earlier findings are confirmed by these additional results, in particular Table XI shows that the alpha of VVP remains significant when including VP as an explanatory variable. Alphas are now −15.55% (CAPM regression, t statistic: −3.07) and −15.53% (Fama–French–Carhart regression, t-statistic: −3.18), and hence similar to earlier reported values. I also follow Kozhan, Neuberger, and Schneider (2013) and estimate the regression equation as part of a seemingly unrelated regression (SUR), where the second regression equation is identical to the VP regression of this section, augmented by the VVP as an explanatory variable. For both the CAPM and the Fama–French–Carhart factors, the estimates of VVP alphas remain highly significant and this does not change my conclusions.24 Table XI Variance-of-Variance contract alpha (after adjusting for variance risk) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βvp,i rtvp+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+εit, where rit is the return of the variance-of-variance contract. Columns (2) to (7) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the return of the variance contract (VP, βvp,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), and the momentum factor (UMD, βumd,i). Column (8) reports the R2 of the regression. Strategy alpha MktRP VP SMB HML UMD R2 VVP −15.55 −0.63 0.50 0.60 (−3.07) (−0.58) (4.91) −15.53 0.36 0.53 −0.39 −4.23 −0.55 0.61 (−3.18) (0.24) (5.42) (−0.16) (−2.52) (−0.61) Strategy alpha MktRP VP SMB HML UMD R2 VVP −15.55 −0.63 0.50 0.60 (−3.07) (−0.58) (4.91) −15.53 0.36 0.53 −0.39 −4.23 −0.55 0.61 (−3.18) (0.24) (5.42) (−0.16) (−2.52) (−0.61) Table XI Variance-of-Variance contract alpha (after adjusting for variance risk) This table reports the estimation results for the regression model rit=αi+βm,i (rtm−rft)+βvp,i rtvp+βsmb,i rtsmb+βhml,i rthml+βumd,i rtumd+εit, where rit is the return of the variance-of-variance contract. Columns (2) to (7) report the alpha and beta coefficients for the following risk factors: alpha (αi), the market risk premium (MktRP, βm,i), the return of the variance contract (VP, βvp,i), the size factor (SMB, βsmb,i), the book-to-market factor (HML, βhml,i), and the momentum factor (UMD, βumd,i). Column (8) reports the R2 of the regression. Strategy alpha MktRP VP SMB HML UMD R2 VVP −15.55 −0.63 0.50 0.60 (−3.07) (−0.58) (4.91) −15.53 0.36 0.53 −0.39 −4.23 −0.55 0.61 (−3.18) (0.24) (5.42) (−0.16) (−2.52) (−0.61) Strategy alpha MktRP VP SMB HML UMD R2 VVP −15.55 −0.63 0.50 0.60 (−3.07) (−0.58) (4.91) −15.53 0.36 0.53 −0.39 −4.23 −0.55 0.61 (−3.18) (0.24) (5.42) (−0.16) (−2.52) (−0.61) 3.6 Transaction Costs Empirical features that rely on mid prices of option quotes may not be exploitable after accounting for transaction costs. Driessen, Maenhout, and Vilkov (2009), for instance, show that the correlation risk premium calculated from option mid-quotes is not robust to high equity option bid–ask spreads. Santa-Clara and Saretto (2009) investigate the effect of trading frictions on various S&P 500 index option strategies. This section discusses whether transaction costs prevent investors from exploiting the high VVP premium. Option market transaction costs are substantially higher than in equity markets. Broadie, Chernov, and Johannes (2009) report bid–ask spreads of the order between 3% and 10% depending on the moneyness of the contracts, for VIX options trading costs are even higher with average bid–ask spreads of the order of 20% of the mid-price. I follow closely Carr and Wu (2009). To exploit the profitability of the strategy, I assume that investors short the realized variance of the VIX index on a monthly basis by entering a VVP trade the day after the monthly VIX option expiry and holding the position until maturity (as in Section 3.2). To do so, I replicate the payoff of a VVP trade with time T cashflow equal to P & Lt,T=N($)×[IVt,TΠ,Bd,Bu−RVt,TΠ,Bd,Bu]. (7) where N($) is the notional of the contract and IV and RV are defined above. Bondarenko (2014) shows that the realized leg of a (generalized) variance contract can be perfectly replicated as follows: RVt,TΠ,Bd,Bu=∫BdBug″(k)M(k,t,T) dk−∑i=1n[g′(Fti,T)−g′(Ft0,T)]×(Fti,T−Fti−1,T) (8) where g is defined in Section 1. Entering a short position in RV, therefore, requires a short position in VIX options at the forward cost IVt,TΠ,Bd,Bu. I take into account the cost of the replication by calculating the option integral using bid prices and hence IVt,TΠ,Bd,Bu (and the profitability) is lowered compared to the assumption of a mid-quote trade.25 I provide average P&Ls for a range of different strategies: (i) with constant notional of N($)=100 or N($)=100×[IVt,TΠ,Bd,Bu]−1, (ii) for standard VVP trades with Bd = 0 and Bu=∞ and corridor variance swaps that use only the available range of options at the initiation of the contract (that is Bd corresponds to the lowest strike available whereas Bu corresponds to the highest strike available), and (iii) I use partitions of 1 day or 1 week. Table XII reports the average monthly P&L for various strategies over the sample period, with and without adjusting for transaction cost. Overall, I find that all strategies provide a positive performance despite the substantial transaction cost in the VIX option market. The performance of the absolute strategy with Bd = 0 and Bu=∞ and a realized variance based on daily returns provides a positive profit with a t-statistic of 2.32. Most strategies (especially for the weekly realized variance measure) are statistically significant at 1% level, and all strategies remain significant at the 10% level. Results are also consistent with the intuition that the profitability of a strategy with N($)=100×[IVt,TΠ,Bd,Bu]−1 should be lowered by approximately half of the average 20% bid–ask spread as this is the difference in the cost compared to trading at mid-prices (24.16% versus 14.01%). Table XII Variance-of-Variance Risk Premium (transaction cost) This table reports average monthly variance-of-variance contract returns after accounting for transaction cost. Panel A reports the average P&L and corresponding t-statistics for a daily rebalancing interval, for strategies with a fixed notional N($)=100 and for N($)=100×[IVt,TΠ,Bd,Bu]−1. Short (Mid) is the short VVP trade based on mid-option quotes, whereas Short (TC) accounts for bid–ask spreads. Corridor variance swaps are calculated using the smallest and largest available strike as Bd and Bu. Panel B reports empirical results for a weekly rebalancing scheme. Notional N($)=100 N($)=100×[IVt,TΠ,Bd,Bu]−1 Average P&L t-stat Average P&L t-stat Panel A: daily rebalancing Short (Mid) 1.70 (3.93) 24.16 (3.39) Short (TC) 1.00 (2.33) 14.01 (1.74) Short (Mid) corridor swap 1.61 (3.73) 23.04 (3.18) Short (TC) corridor swap 0.99 (2.31) 13.88 (1.71) Panel B: weekly rebalancing Short (Mid) 2.39 (4.80) 35.10 (4.44) Short (TC) 1.69 (3.44) 26.88 (3.08) Short (Mid) corridor swap 2.30 (4.64) 34.17 (4.26) Short (TC) corridor swap 1.68 (3.43) 26.76 (3.05) Notional N($)=100 N($)=100×[IVt,TΠ,Bd,Bu]−1 Average P&L t-stat Average P&L t-stat Panel A: daily rebalancing Short (Mid) 1.70 (3.93) 24.16 (3.39) Short (TC) 1.00 (2.33) 14.01 (1.74) Short (Mid) corridor swap 1.61 (3.73) 23.04 (3.18) Short (TC) corridor swap 0.99 (2.31) 13.88 (1.71) Panel B: weekly rebalancing Short (Mid) 2.39 (4.80) 35.10 (4.44) Short (TC) 1.69 (3.44) 26.88 (3.08) Short (Mid) corridor swap 2.30 (4.64) 34.17 (4.26) Short (TC) corridor swap 1.68 (3.43) 26.76 (3.05) Table XII Variance-of-Variance Risk Premium (transaction cost) This table reports average monthly variance-of-variance contract returns after accounting for transaction cost. Panel A reports the average P&L and corresponding t-statistics for a daily rebalancing interval, for strategies with a fixed notional N($)=100 and for N($)=100×[IVt,TΠ,Bd,Bu]−1. Short (Mid) is the short VVP trade based on mid-option quotes, whereas Short (TC) accounts for bid–ask spreads. Corridor variance swaps are calculated using the smallest and largest available strike as Bd and Bu. Panel B reports empirical results for a weekly rebalancing scheme. Notional N($)=100 N($)=100×[IVt,TΠ,Bd,Bu]−1 Average P&L t-stat Average P&L t-stat Panel A: daily rebalancing Short (Mid) 1.70 (3.93) 24.16 (3.39) Short (TC) 1.00 (2.33) 14.01 (1.74) Short (Mid) corridor swap 1.61 (3.73) 23.04 (3.18) Short (TC) corridor swap 0.99 (2.31) 13.88 (1.71) Panel B: weekly rebalancing Short (Mid) 2.39 (4.80) 35.10 (4.44) Short (TC) 1.69 (3.44) 26.88 (3.08) Short (Mid) corridor swap 2.30 (4.64) 34.17 (4.26) Short (TC) corridor swap 1.68 (3.43) 26.76 (3.05) Notional N($)=100 N($)=100×[IVt,TΠ,Bd,Bu]−1 Average P&L t-stat Average P&L t-stat Panel A: daily rebalancing Short (Mid) 1.70 (3.93) 24.16 (3.39) Short (TC) 1.00 (2.33) 14.01 (1.74) Short (Mid) corridor swap 1.61 (3.73) 23.04 (3.18) Short (TC) corridor swap 0.99 (2.31) 13.88 (1.71) Panel B: weekly rebalancing Short (Mid) 2.39 (4.80) 35.10 (4.44) Short (TC) 1.69 (3.44) 26.88 (3.08) Short (Mid) corridor swap 2.30 (4.64) 34.17 (4.26) Short (TC) corridor swap 1.68 (3.43) 26.76 (3.05) While it is common in the related literature to rely on bid/ask prices provided by options exchanges, there is evidence in the literature that such approach is likely to be very conservative. Carr and Wu (2009) point out that bid–ask spreads in the broker dealer market for variance swaps on the S&P 500 index are much lower than the spread calculated from bid–ask spreads of quoted S&P 500 index options. If option bid and ask prices lead to substantially different bid–ask swap rates, this indicates either a significant underestimation of the hedging costs by broker dealers or that the hedging cost may be somewhat lower than what is implied in option quotes. In a recent paper, Muravyev and Pearson (2015) find support for the latter. They argue that by timing option trades the effective bid–ask spread may be substantially reduced, and conclude that “the quoted spread overstates the cost of taking liquidity by a factor of almost two” (p. 29). It is therefore possible that VIX options investors may reduce transaction costs compared to the assumptions used in this article. 4. Variance-of-Variance Premium in Benchmark Models This section provides simulation-based evidence. My aim is to investigate whether option pricing models employed in the literature are able to explain the size of the risk premium, as well as other facets of the data presented in previous sections. 4.1 Structural Benchmark Models As the main benchmark, I employ an extension of the two-factor stochastic variance model proposed in Duffie, Pan, and Singleton (2000); Egloff, Leippold, and Wu (2010); and Bates (2012). The simulation results in this section rely on estimated parameters from Bardgett, Gourier, and Leippold (2013), and therefore I follow closely their specification of asset price dynamics. The model is labeled SVSMRJ (stochastic volatility, stochastic mean reversion, jumps) and it is assumed that under the risk-neutral pricing measure ℚ the S&P 500 index St evolves according to the following stochastic differential equations: dStSt=(r−q−λtsvψℚ) dt+vt dWts,ℚ+(eξts,ℚ−1) dNtsv,ℚ, (9) dvt=κvℚ(mt−vt) dt+σvvt(ρ dWts,ℚ+1−ρ2 dWtv,ℚ)+ξtv,ℚ dNtsv,ℚ, (10) dmt=&