The VIX and Future Information

This article illustrates the volatile nature of the premiums and discounts embedded in the prices of VIX (Chicago Board Options Exchange Market Volatility Index) futures contracts. The fact that the underlying VIX index cannot be traded leads VIX futures to be priced more on expectations of market participants than on a typical cost-of-carry relationship. As they near expiration, VIX futures, in the aggregate, tend to trade at an increased premium, when trading in contango, and at an increased discount, when trading in backwardation. In addition, the premium in these contracts tends to peak as the VIX index nears a low, and the discount in the contracts tends to bottom as the index nears a high.

In this study we present a closed‐form, exact solution for the pricing of VIX futures in a stochastic volatility model with simultaneous jumps in both the asset price and volatility processes. The newly derived formula is then used to show that the well‐known convexity correction approximations can sometimes lead to large errors. Utilizing the newly derived formula, we also conduct an empirical study, the results of which demonstrate that the Heston stochastic volatility model is a good candidate for the pricing of VIX futures. While incorporating jumps into the underlying price can further improve the pricing of VIX futures, adding jumps to the volatility process appears to contribute little improvement for pricing VIX futures. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark

The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew

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.

PurposePrevious studies have shown the VIX futures tend to roll-down the term structure and converge towards the spot as they grow closer to maturity. The purpose of this paper is to propose an approach to improve the volatility index fear factor-level (VIX-level) prediction.Design/methodology/approachFirst, the authors use a forward-looking technique, the Heath–Jarrow–Morton (HJM) no-arbitrage framework to capture the convergence of the futures contract towards the spot. Second, the authors use principal component analysis (PCA) to reduce dimensionality and save substantial computational time. Third, the authors validate the model with selected VIX futures maturities and test on value-at-risk (VAR) computations.FindingsThe authors show that the use of multiple factors has a significant impact on the simulated VIX futures distribution, as well as the computations of their VAR (gain in accuracy and computing time). This impact becomes much more compelling when analysing a portfolio of VIX futures of multiple maturities.Research limitations/implicationsThe authors’ approach assumes the variance to be stationary and ignores the volatility smile. Nevertheless, they offer suggestions for future research.Practical implicationsThe VIX-level prediction (the fear factor) is of paramount importance for market makers and participants, as there is no way to replicate the underlying asset of VIX futures. The authors propose a procedure that provides efficiency to both pricing and risk management.Originality/valueThis paper is the first to apply a forward-looking method by way of a HJM framework combined with PCA to VIX-level prediction in a portfolio context.

By means of a data set previously unavailable for academic research, actual trading histories of individual futures traders are examined. With this more detailed data, the author is able to (1) test the risk/return hypothesis directly; (2) include a much larger segment of the market than before; and (3) use actual instead of hypothetical t rading strategies. It is shown that the commercial (hedging) traders are most profitable, while noncommercial (speculative) traders earn n egative or zero profits. Because speculators are not receiving reward s for the risks they willingly absorb, the theory of normal backwarda tion and its extension can be rejected. Copyright 1987 by University of Chicago Press.

Asymmetric Effects of Volatility Risk on Stock Returns: Evidence from VIX and VIX Futures

The paper analyses how volatility derivatives on the volatility index VIX can be used as trading and risk management tools for investors and trad-ers. Volatility and the different types of volatility are discussed. It elabo-rates upon assumptions of option pricing models and specifies which complications accompany the determination of volatility. The weaknesses of the Black-Scholes-Merton model are illuminated and the difference between the model assumptions regarding volatility and market reality is identified. Using the skew- and term-curve-effect, the paper demonstrates how volatility behaves in reality towards other model parameters. In terms of pure volatility trading, the volatility derivatives are presented and analysed in terms of their merits and fields of application. Additionally, the stylized facts about volatility are considered. The paper shows how VIX futures and options can hedge equity portfolios and when they are superior to traditional hedging alternatives and compares the outcome of a VIX hedging strategy with a Buy & Hold strategy of the S & P 500 index over a time period of 20 years.

Using the Clegg–Krauss framework, this paper first examines a partial cointegration relationship between stock index futures and VIX futures prices and then constructs a hedging strategy based upon this relationship. This paper argues that the stock index futures and the VIX futures are both affected by unobservable investor sentiment and thus the price series should be modelled by the partial cointegration relationship. Our empirical results validate a partial cointegration relationship between stock index and VIX futures prices. Based upon the partial cointegration relationship between stock index futures and VIX futures prices, we demonstrate that the proposed strategy outperforms conventional strategies, e.g., OLS, VAR, and VECM, in terms of tail risk reduction and expected utility, especially when the length of the hedge horizon increases. In addition, the partial cointegration-based strategy becomes more dominant when the stock index futures price is near its historical high. Overall, our empirical evidence of the hedging effectiveness for different hedge horizons and market timing with VIX futures provides valuable information for practitioners in risk management.

VIX as a Variance Swap

Realized semivariance, computed from intraday positive/negative squared returns, provides an accurate measure of the upside/downside variations of stock returns. This article investigates the role of realized semivariance in pricing the CBOE VIX and VIX futures, using a realized semivariance-based model. We obtain the closed-form pricing formula for the VIX index and VIX futures prices, and show that the new model provides superior pricing performance, both in-sample and out-of-sample. We further analytically derive the pricing formulas for the upside/downside components of the VIX (risk-neutral semivariance). Such a decomposition shows that the information gains from the conventional unsigned realized variance are concentrated on pricing the downside part of the VIX; the new realized semivariance-based model provides a larger and more balanced improvement for both the upside and downside components of the VIX. Our results provide strong evidence that the spread between upside/downside variance is the main driver of the asymmetry in return distributions.

Understanding the dynamics of realized volatility is critical to effective investment management and portfolio risk management. Volatility measures can reach levels well above the median and are trending and mean reverting. Tradable volatility exposure is now available with VIX futures and exchange-traded products benchmarked to VIX futures indices. These tools are useful for modifying portfolio risk based on expected S&P 500 Index volatility rather than shifting fixed-income allocations.

In this chapter, we give an overview on applications of change of time methods considered in this book in Chapters 4–8 These applications include yet another (among many) derivation of the Black-Scholes formula; the derivation of option pricing formula for a mean-reverting asset in energy finance; pricing of variance, volatility, covariance, and correlation swaps for the classical Heston model; pricing of variance and volatility swaps in energy markets; pricing of financial and energy derivatives with multifactor Levy models; and pricing of variance and volatility swaps and hedging of volatility swaps for the delayed Heston model. This chapter not only describes the applications of the change of time method but also constitutes the ultimate difference between Barndorff-Nielsen-Shiryaev’s book (2010) and present book.

Modelling and trading the U.S. implied volatility indices. Evidence from the VIX, VXN and VXD indices

Volatility, Variance and Covariance Swaps for BN-S Type Levy Process Driven Financial Markets