TY - JOUR
AB - 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<…Bu−2 log xifx∈[Bd,Bu]2×(− log Bd−xBd+1)ifxK ( Ft,T