Term Structure Of VIX Futures Research Articles

Consistent time‐homogeneous modeling of SPX and VIX derivatives

AbstractThis paper shows how to recover a stochastic volatility model (SVM) from a market model of the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore are better suited for pricing VIX futures and VIX derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Therefore, consistent modeling for both SPX and VIX should involve an SVM that can be obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function by solving an inverse problem where the input is the VIX function given by a market model. Analysis will show conditions necessary for there to be a unique solution to this inverse problem. The models are consistent if the recovered volatility function is non‐negative. Examples are presented to illustrate the theory, to highlight the issue of negativity in solutions, and to show the potential for inconsistency in non‐Markov settings.

Trading Signals in VIX Futures

ABSTRACT We propose a new approach for trading VIX futures. We assume that the term structure of VIX futures follows a Markov model. Our trading strategy selects a position in VIX futures by maximizing the expected utility for a day-ahead horizon given the current shape and level of the term structure. Computationally, we model the functional dependence between the VIX futures curve, the VIX futures positions, and the expected utility as a deep neural network with five hidden layers. Out-of-sample backtests of the VIX futures trading strategy suggest that this approach gives rise to reasonable portfolio performance, and to positions in which the investor will be either long or short VIX futures contracts depending on the market environment.

Dynamics in the VIX Complex

This paper provides a characterization of the dynamic interactions in the VIX complex, composed of the VIX itself, the term structure of VIX futures, and VIX ETPs. I investigate a model that summarizes the VIX futures term structure using latent factors (level, slope, and curvature) and expand it with the VIX and VIX futures demand stemming from VIX ETPs. I find evidence of VIX ETPs impacting the VIX futures term structure, but no evidence of any impacts on the VIX.

VIX VERSUS VXX: A JOINT ANALYTICAL FRAMEWORK

We propose a framework for modeling in a consistent manner the VIX index and the VXX, an exchange-traded note written on the VIX. Our study enables to link the properties of VXX to those of the VIX in a tractable way. In particular, we quantify the systematic loss observed empirically for VXX when the VIX futures term-structure is in contango and we derive option prices, implied volatilities and skews of VXX from those of VIX in infinitesimal developments. We also perform a calibration on real data which highlights the flexibility of our model in fitting the futures and the vanilla options market of VIX and VXX. Our framework can be used to model other exchange-traded notes on the VIX as well as any market where exchange-traded notes have been introduced on a reference index, hence providing tools to better anticipate and quantify systematic behavior of an exchange-traded note with respect to the underlying index.

A Note of Trading the Term Structure of VIX Futures

The term structure of VIX futures contains a very strong signal of dealer risk appetite. Unlike balance sheet quantities, this feature is available at very high frequencies. Here we exhibit two systematic strategies to mine the attendant risk premium from the term structure of expected volatility. We optimize our two hyper-parameters by OOS cross-validation. We compare our strategies to holding the S&P 500, selling short-term vol un-hedged, and a portfolio that sells short-term vol and hedges by going long on medium-term vol. We find that our strategies allow us to harvest a considerable portion of the risk premium associated with the balance sheet management of market-based intermediaries. Both in-sample and OOS, the risk-adjusted returns on our strategies are at least twice as high as the three benchmarks.

VIX futures term structure and the expectations hypothesis

Tests of the expectations hypothesis reveal that the slope of the VIX futures term structure predicts the direction but not the magnitude of the evolution of the short-end of the curve, but predicts neither the direction nor the magnitude of short-term changes in the long-end of the curve. Relative value seeking spread trades, constructed to exploit such violations, deliver excess returns with annualized Sharpe ratios equal or greater than those of volatility-writing strategies deployed by VIX ETN's for a majority of the 32 spread trade combinations tested. I demonstrate that profits from beta-neutral variations of the spread trades, which are not compensation for taking on equity downside risk by design, are propagated by inflows of capital into VIX futures markets, after controlling for factors that measure changes in the availability of hedge fund capital, risk appetite, and momentum. At the heart of profits, and by extension the term structure anomalies, is a disproportionally elevated basis propagated by long VIX demand that enters the futures market through ETN channels.

VIX Futures as a Market Timing Indicator

Our work relates to the literature supporting that the VIX also mirrors investor sentiment and, thus, contains useful information regarding future S&P500 returns. The objective of this empirical analysis is to verify if the shape of the volatility futures term structure has signaling effects regarding future equity price movements, as several investors believe. Our findings generally support the hypothesis that the VIX term structure can be employed as a contrarian market timing indicator. The empirical analysis of this study has important practical implications for financial market practitioners, as it shows that they can use the VIX futures term structure not only as a proxy of market expectations on forward volatility, but also as a stock market timing tool.

Derivatives pricing when supply and demand matter: Evidence from the term structure of VIX futures

AbstractWe decompose the VIX futures term structure into systematic components driving the VIX and idiosyncratic components reflecting demand by various types of futures end‐users. We model two distinct channels by which trading activity manifests itself into futures prices: a contemporaneous “level effect” across the term structure due to the aggregate size of nondealer net demand and a mean‐reverting “roll effect” due to large trades in specific contracts. The observed futures term structure was, on average, higher and steeper than it would have been in the absence of the observed nondealer demand, but the impact varies in sign and magnitude over time.

Selling VIX Futures and Options for Portfolio Return Enhancement

The term structure of VIX futures is generally upward sloping. The persistent VIX contango may result in abnormally strong performance for VIX futures selling or VIX call writing strategies. However, the high volatility of volatility and significant jump risk may expose short uncovered VIX positions to extreme tail risk. In this paper we consider the performance of VIX futures and call option selling in a portfolio context in 2008 and 2016 as well as over a 10+ year period beginning in 2006. In addition, we consider alternative strategies including VIX futures spread strategies, and the dynamically de-levering VIX futures selling strategies represented by the VPD and VPN indexes. The VPD Index had annual returns that were much higher than those of the S&P 500 Index in 2009 and 2016 (years without large spikes in VIX), but the VPD returns were lower than those of the S&P 500 in 2008 and 2018, years with significant upward spikes in the VIX Index. While the goal of the study is not to make a particular recommendation for a VIX-based investment strategy, meaningful portfolio return enhancements are possible over particular time periods with small allocations to VIX selling. A large allocation to uncovered selling of VIX instruments could have substantial volatility jump risk exposure which can result in economically meaningful and potentially catastrophic losses. However, the VPN Index - which sells VIX futures, holds money market instruments and buys VIX calls – had lower volatility and less severe maximum drawdowns than the stock and commodity benchmark indexes studied in the period of analysis.

The Portfolio Diversification Potential of Long VIX Futures and Options Strategies

It is well established that the VIX Index tends to be negatively correlated with equity markets. This suggests that VIX futures and options may have the potential to provide significant diversification benefits for traditional portfolios. However, since the term structure of VIX futures is generally upward sloping, long VIX futures positions can place a significant drag on portfolio performance. In this paper we consider the performance of strategies that buy VIX futures or VIX call options in a portfolio context in 2008 and 2016, as well as over a 10+ year period from 2006 to 2017. In addition, we consider alternative strategies including long S&P 500® protective put strategies and the dynamic S&P 500 plus VIX call buying strategy of the VXTH index. Meaningful portfolio diversification benefits for risk-averse investors are possible over particular time periods with small allocations to long VIX futures or call options, but there can be substantial portfolio drag if large allocations are made over long time periods during which there are flat to rising stock markets.

Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models

This paper shows how to recover stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. This paper's main result is a method for the recovery of a stochastic volatility function as the output of an inverse problem, with the inputs given by a VIX futures market model. Analysis will show that some conditions need to be met in order for there to not be any inter-model arbitrage or mis-priced derivatives. Given these conditions the inverse problem can be solved. Several models are analyzed and explored numerically to gain a better understanding of the theory and its limitations.

U.S. Stock Returns and VIX Futures Curve

The authors investigate the question of whether the VIX futures term structure curve can detect inflection points in the U.S. stock market. They find that transformations of the slope of the VIX futures term structure, such as exponential moving average of past slopes, curvature, and polynomial fitting, outperform the simple slope and can be used as factors to improve probabilistic models that predict downturns in the U.S. stock market.

VIX Futures As a Market Timing Indicator

This study contributes to the age-old question of whether stock market returns are predictable, by studying the relationship of VIX futures term structure and future S&P500 returns. The objective of this empirical analysis is to verify if the shape of the volatility futures term structure has signaling effects regarding future equity prices movements, as several investors believe. Our findings generally support the hypothesis that the term structure of VIX futures can be employed as a contrarian market timing indicator for the equity market. The empirical analysis of this study has important practical implications for financial market practitioners, as it shows that they can use the VIX futures term structure not only as a proxy of market expectations on forward volatility, but also as a stock market predictive tool.

VIX vs VXX: A Joint Analytical Framework

We propose a framework for modeling in a consistent manner the VIX index and the VXX, an exchange-traded note written on the VIX. Our study enables to link the properties of VXX to those of the VIX in a tractable way. In particular, we quantify the systematic loss observed empirically for VXX when the VIX futures term-structure is in contango and we derive option prices, implied volatilities and skews of VXX from those of VIX in infinitesimal developments. We also perform a calibration on real data which highlights the flexibility of our model in fitting the futures and the vanilla options market of VIX and VXX. Our framework can be used to model other exchange-traded notes on the VIX as well as any market where exchange-traded notes have been introduced on a reference index, hence providing tools to better anticipate and quantify systematic behavior of an exchange-traded note with respect to the underlying index.

VIX Christmas Effect

An anomaly may exist in the U.S. stock market in which the VIX trades lower during the first 17 to 22 days of December. This is potentially due to a Christmas effect, in which trading activity slows before Christmas. The authors test this hypothesis using two approaches, one from the spot VIX and the other from the VIX futures term structure. In addition, the authors test the holiday explanation, in which the VIX declines prior to holidays because trading activity declines.

Pure jump models for pricing and hedging VIX derivatives

Recent non-parametric statistical analysis of high-frequency VIX data (Todorov and Tauchen, 2011) reveals that VIX dynamics is a pure jump semimartingale with infinite jump activity and infinite variation. To our best knowledge, existing models in the literature for pricing and hedging VIX derivatives do not have these features. This paper fills this gap by developing a novel class of parsimonious pure jump models with such features for VIX based on the additive time change technique proposed in Li et al. (2016a, 2016b). We time change the 3/2 diffusion by a class of additive subordinators with infinite activity, yielding pure jump Markov semimartingales with infinite activity and infinite variation. These processes have time and state dependent jumps that are mean reverting and are able to capture stylized features of VIX. Our models take the initial term structure of VIX futures as input and are analytically tractable for pricing VIX futures and European options via eigenfunction expansions. Through calibration exercises, we show that our model is able to achieve excellent fit for the VIX implied volatility surface which typically exhibits very steep skews. Comparison to two other models in terms of calibration reveals that our model performs better both in-sample and out-of-sample. We explain the ability of our model to fit the volatility surface by evaluating the matching of moments implied from market VIX option prices. To hedge VIX options, we develop a dynamic strategy which minimizes instantaneous jump risk at each rebalancing time while controlling transaction cost. Its effectiveness is demonstrated through a simulation study on hedging Bermudan style VIX options.

The Relationship between VIX Futures Term Structure and S&P500 Returns

The current paper tests and documents the relationship between the term structure of VIX futures and the underlying equity returns. Furthermore, it investigates the signaling effects of VIX futures term structure in respect to future stock index movements. The objective of this empirical analysis is to verify if a steep upward-sloping term structure indicates a late phase of a bullish trend and conversely if an extreme negative term structure suggests an over-sold market, as certain market participants believe.The empirical findings of this study suggest that there is a strong statistical significant positive contemporaneous relationship between the changes in the VIX futures term structure and the returns of the underlying equity index. Finally, the econometric analysis lends some support to the hypothesis that the term structure of VIX futures can be used as a contrarian indicator for investing in the equity market.

Modeling and Trading the VIX and VSTOXX with Futures, Options and the VXX

Understanding the behavior of the VIX and developing trading strategies for VIX related products is the leitmotif of the Sibyl-Working papers. In the previous papers a signal based approach was used. These strategies are the cash cows of the Sibyl-Fund. This working paper develops first a mean-reverting logarithmic model for the VIX. In the next step the relation between the VIX and VIX Futures is modeled. VIX Options are evaluated by Monte-Carlo Simulation of Option-Trading Strategies. The model is also applied to the VIX based ETN VXX and VXX Options and extended to the VSTOXX and VSTOXX Futures.The performance of several model-based trading strategies is backtested. The paper presents not only good but also bad or dubious trading ideas. The signal based strategies exploit the term-structure of VIX Futures. The model exploits additionally the mean-reversion of volatility. This improves – at least in the backtest – the performance. The paper discusses in appendix B the critique on the backtest methodology formulated in the paper Pseudo-Mathematics and Financial Charlatanism. The (well known) critique has to be taken serious, but the suggested solution is of no practical use.Version 1 adds a completely different VXX trading strategy which was suggested by a reviewer.

Does Historical VIX Term Structure Contain Valuable Information for Predicting VIX Futures?

We suggest that the term structure of VIX futures shows a clear pattern of dependence on the current level of VIX index. At the low levels of VIX (below 20), the term structure is highly upward sloping, while at the high VIX levels (over 30) it is strongly downward sloping. We use these features to predict future VIX futures prices more precisely. We begin by introducing some quantitative measures of volatility term structure ( VTS ) and volatility risk premium ( VRP ). We use them further to estimate the distance between the actual value and the fair (model) value of the VTS . We find that this distance has significant predictive power for volatility futures and index futures and we use this feature to design simple strategies to invest in VIX futures.

VIX Futures Basis Trading: The Calvados-Strategy 2.0

I developed in a previous working paper the Sidre and Most-Strategy. The strategy relies on the typical termstructure of VIX futures. The Calvados is a refined and condensed version of these strategies. The starting point was a paper of Simon and Campasano. The authors demonstrate that the VIX futures basis does not have significant forecast power for the change in the VIX spot index, but does have forecast power for subsequent VIX futures returns. It is especially profitable to short VIX futures contracts when the basis is in contango. The original Calvados working paper presented improved metrics and parameter settings of the Simon&Campasano approach. The current working paper improves the original work in several points and extends the historic backtest.The overall patterns of the original results are reassured and improved upon. The Calvados is traded in the Sybil-Fund. It is so far the pick of the bunch. One gets a lot of fun for a medium dose of risk.