Using no arbitrage principle, we derive a relationship between the drift term of risk-neutral dynamics for instantaneous variance and the term structure of forward variance curve. We show that the forward variance curve can be derived from options market. Based on the variance term structure, we derive a no arbitrage pricing model for VIX futures pricing. The model is the first no arbitrage model combining options market and VIX futures market. The model can be easily generalized to price other volatility derivatives.
In this paper we analyze CBOE VIX futures price time series data from Mar. 2004 to Feb. 2008. We derive a new pricing framework for VIX futures that is convenient to study variance term structure dynamics. Our main contribution to existing literature is the identification of the number of factors implicit in VIX futures term structure. We find that three-factor model is ideal to characterize the variance term structure. We further construct and estimate structured two- and three-factor models to identify the components and find similar results.
We analyze pricing models for VIX derivatives which account for the theoretical link to stock options, taking Log-VIX models as a benchmark. We focus on up to three risk factors to model variance risk. To assess the performance of the models, we do not only look at the pricing errors, but also at the level and dynamics of the VIX' risk-neutral moments which vary considerably over time. We find that both model classes, consistent- and Log-VIX models, can reproduce the empirical patterns if three risk factors are included. In both approaches, a stochastic central tendency is of first order importance to capture the term structure of VIX futures prices, i.e. the first moment of the risk-neutral distribution. A stochastic vol-of-vol then helps to match the prices of VIX options, i.e. the higher order moments. Finally, variance jumps add the finishing touches to the model performance. All in all, consistency comes at notable costs in-sample, while out-of-sample performances are close. We find that the main difference between both model classes is the ability to capture the second moment of the VIX risk-neutral distribution.
Pricing Models of Volatility Products and Exotic Variance Derivatives summarizes most of the recent research results in pricing models of derivatives on discrete realized variance and VIX. The book begins with the presentation of volatility trading and uses of variance derivatives. It then moves on to discuss the robust replication strategy of variance swaps using portfolio of options, which is one of the major milestones in pricing theory of variance derivatives. The replication procedure provides the theoretical foundation of the construction of VIX. This book provides sound arguments for formulating the pricing models of variance derivatives and establishes formal proofs of various technical results. Illustrative numerical examples are included to show accuracy and effectiveness of analytic and approximation methods. Features Useful for practitioners and quants in the financial industry who need to make choices between various pricing models of variance derivatives Fabulous resource for researchers interested in pricing and hedging issues of variance derivatives and VIX products Can be used as a university textbook in a topic course on pricing variance derivatives
This paper analyses the new market for trading volatility; VIX futures. We first use market data to establish the relationship between VIX futures prices and the index itself. We observe that VIX futures and VIX are highly correlated; the term structure of VIX futures price is upward sloping while the term structure of VIX futures volatility is downward sloping. To establish a theoretical relationship between VIX futures and VIX, we model the instantaneous variance using a simple square root mean-reverting process. Using daily calibrated variance parameters and VIX, the model gives good predictions of VIX futures prices. These parameter estimates could be used to price VIX options.
The shape of the VIX term structure conveys information about the price of variance risk rather than expected changes in the VIX, a rejection of the expectations hypothesis. A single principal component, Slope, summarizes nearly all this information, predicting the excess returns of S&P 500 variance swaps, VIX futures, and S&P 500 straddles for all maturities and to the exclusion of the rest of the term structure. Slope's predictability is incremental to other proxies for the conditional variance risk premia, is economically significant, and can only partially be explained by variations in observable risk measures.
Leverage Python for expert-level volatility and variance derivative trading Listed Volatility and Variance Derivatives is a comprehensive treatment of all aspects of these increasingly popular derivatives products, and has the distinction of being both the first to cover European volatility and variance products provided by Eurex and the first to offer Python code for implementing comprehensive quantitative analyses of these financial products. For those who want to get started right away, the book is accompanied by a dedicated Web page and a Github repository that includes all the code from the book for easy replication and use, as well as a hosted version of all the code for immediate execution. Python is fast making inroads into financial modelling and derivatives analytics, and recent developments allow Python to be as fast as pure C++ or C while consisting generally of only 10% of the code lines associated with the compiled languages. This complete guide offers rare insight into the use of Python to undertake complex quantitative analyses of listed volatility and variance derivatives. Learn how to use Python for data and financial analysis, and reproduce stylised facts on volatility and variance markets Gain an understanding of the fundamental techniques of modelling volatility and variance and the model-free replication of variance Familiarise yourself with micro structure elements of the markets for listed volatility and variance derivatives Reproduce all results and graphics with IPython/Jupyter Notebooks and Python codes that accompany the book Listed Volatility and Variance Derivatives is the complete guide to Python-based quantitative analysis of these Eurex derivatives products.
In this chapter, we consider volatility swap, variance swap and VIX future pricing under different stochastic volatility models and jump diffusion models which are commonly used in financial market. We use convexity correction approximation technique and Laplace transform method to evaluate volatility strikes and estimate VIX future prices. In empirical study, we use Markov chain Monte Carlo algorithm for model calibration based on S&P 500 historical data, evaluate the effect of adding jumps into asset price processes on volatility derivatives pricing, and compare the performance of different pricing approaches.
A guide to using the VIX to forecast and trade markets Known as the fear index, the VIX provides a snapshot of expectations about future stock market volatility and generally moves inversely to the overall stock market. Trading VIX Derivatives will show you how to use the Chicago Board Options Exchange's S&P 500 volatility index to gauge fear and greed in the market, use market volatility to your advantage, and hedge stock portfolios. Engaging and informative, this book skillfully explains the mechanics and strategies associated with trading VIX options, futures, exchange traded notes, and options on exchange traded notes. Many market participants look at the VIX to help understand market sentiment and predict turning points. With a slew of VIX index trading products now available, traders can use a variety of strategies to speculate outright on the direction of market volatility, but they can also utilize these products in conjunction with other instruments to create spread trades or hedge their overall risk. Reviews how to use the VIX to forecast market turning points, as well as reveals what it takes to implement trading strategies using VIX options, futures, and ETNs Accessible to active individual traders, but sufficiently sophisticated for professional traders Offers insights on how volatility-based strategies can be used to provide diversification and enhance returns Written by Russell Rhoads, a top instructor at the CBOE's Options Institute, this book reflects on the wide range of uses associated with the VIX and will interest anyone looking for profitable new forecasting and trading techniques.
An important component of theoretical CBOE Volatility Index (VIX) futures prices is a term correcting for the negative convexity of the square root function by subtracting from the forward-starting variance swap rate an estimate of the future volatility of VIX futures prices. In the same fashion that an index option's traditional implied volatility can be viewed as an aggregate market consensus of future realized volatility, this convexity value can be viewed as an aggregate market consensus of future volatility of volatility. This article examines the predictive properties and features of this convexity adjustment needed to value VIX futures prices by extracting it from the relationship between observed VIX futures prices and the corresponding spot option market prices used to compute the forward-starting variance swap rate. The authors find that implied convexity levels can indeed be used to forecast the future volatility of VIX futures prices, even though implied convexity consistently underestimates future realized VIX futures variance. They also show that implied convexity can at times violate strict theoretical conditions by being negative, although we are able to rule out arbitrage opportunities. Finally, they examine the properties of this implied convexity adjustment, both as a time series and with respect to various market volatility factors with which they find positive and statistically significant relations.
