The forecast quality of CBOE implied volatility indexes

We examine the forecast quality of Chicago Board Options Exchange (CBOE) implied volatility indexes based on the Nasdaq 100 and Standard and Poor's 100 and 500 stock indexes. We find that the forecast quality of CBOE implied volatilities for the S&P 100 (VXO) and S&P 500 (VIX) has improved since 1995. Implied volatilities for the Nasdaq 100 (VXN) appear to provide even higher quality forecasts of future volatility. We further find that attenuation biases induced by the econometric problem of errors in variables appear to have largely disappeared from CBOE volatility index data since 1995. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:339–373, 2005     

The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

Improving volatility prediction and option valuation using VIX information: A volatility spillover GARCH model

We develop a new generalized autoregressive conditional heteroskedasticity (GARCH) model that accounts for the information spillover between two markets. This model is used to detect the usefulness of the CBOE volatility index (VIX) for improving the performance of volatility forecasting and option pricing. We find the significant ability of VIX to predict stock volatility both in-sample and out-of-sample. VIX information also helps to greatly reduce the option pricing error. The proposed volatility spillover GARCH model performs better than the related approaches proposed by Kanniainen et al. (2014, J Bank Finance, 43, pp. 200-211) and P. Christoffersen et al. (2014, J Financ Quant Anal, 49, pp. 663–697).     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

Pricing and hedging S&P 500 index options with Hermite polynomial approximation: empirical tests of Madan and Milne's model

The universal use of the Black and Scholes option pricing model to value a wide range of option contracts partly accounts for the almost systematic use of Gaussian distributions in finance. Empirical studies, however, suggest that there is an information content beyond the second moment of the distribution that must be taken into consideration.This article applies a Hermite polynomial-based model developed by Madan and Milne (1994) to an investigation of S&P 500 index option prices from the CBOE when the distribution of the underlying index is unknown. The model enables us to incorporate the non-normal skewness and kurtosis effects empirically observed in option-implied distributions of index returns. Out-of-sample tests confirm that the model outperforms Black and Scholes in terms of pricing and hedging. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 735–758, 1999     

The Term Structure of VIX

In this study, we extend the Chicago Board Options Exchange volatility index, VIX, from 30‐day to any arbitrary time‐to‐maturity, and study the term structure of VIX. We propose new concepts of instantaneous and long‐term squared VIXs as the limits at the short and long ends of the term structure respectively. Modeling the volatility process with instantaneous and long‐term squared VIXs, we establish a parsimonious approach to capture information contained in the term structure of VIX. Our study provides an efficient setup to further study the pricing of VIX derivatives and their relation with S&P 500 options.     

The information content of the S&P 500 index and VIX options on the dynamics of the S&P 500 index

Given that both S&P 500 index and VIX options essentially contain information about the future dynamics of the S&P 500 index, in this study, we set out to empirically investigate the informational roles played by these two option markets with regard to the prediction of returns, volatility, and density in the S&P 500 index. Our results reveal that the information content implied from these two option markets is not identical. In addition to the information extracted from the S&P 500 index options, all of the predictions for the S&P 500 index are significantly improved by the information recovered from the VIX options. Our findings are robust to various measures of realized volatility and methods of density evaluation. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

VIX futures pricing with conditional skewness

We develop a closed‐form VIX futures valuation formula based on the inverse Gaussian GARCH process by Christoffersen et al. that combines conditional skewness, conditional heteroskedasticity, and a leverage effect. The new model outperforms the benchmark in fitting the S&P 500 returns and the VIX futures prices. The fat‐tailed innovation underlying the model substantially reduced pricing errors during the 2008 financial crisis. The in‐ and out‐of‐sample pricing performance indicates that the new model should be a default modeling choice for pricing the medium‐ and long‐term VIX futures.     

Analysis of Bitcoin prices using market and sentiment variables

This paper proposes an empirical model for analysing the dynamics of Bitcoin prices. To do this, we consider a vector error correction model over two overlapping periods: 2010–17 and 2010–19. Price discovery is achieved through the Gonzalo–Granger permanent‐transitory decomposition. The pricing factors are endogenous linear combinations of the S&P 500 index, gold price, a Google search variable associated to Bitcoin and a fear index proxied by the FED Financial Stress Index. Our empirical analysis shows that during the first period, a linear combination of four pricing factors describes the efficient Bitcoin price. The S&P 500 index and Google searches have a positive effect whereas gold prices and the fear index have a negative effect. In contrast, during the second period, the efficient price behaves idiosyncratically and can be only rationalised by individuals' search for information on the cryptocurrency. These findings provide empirical evidence on the presence of a correction in Bitcoin prices during the period 2018–19 uncorrelated to market fundamentals. We also show that standard empirical asset pricing models perform poorly for explaining Bitcoin prices.     

Conditional Volatility and the GARCH Option Pricing Model with Non‐Normal Innovations

On the basis of the theory of a wedge between the physical and risk‐neutral conditional volatilities in Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2010), we develop a modification of the GARCH option pricing model with the filtered historical simulation proposed in Barone‐Adesi, G., Engle, R. F., & Mancini, L. (2008). The one‐day‐ahead conditional volatilities under physical and risk‐neutral measures are the same in the previous model, but should have been allowed to be different. Using extensive data on S&P 500 index options, our approach, which employs one‐day‐ahead risk‐neutral conditional volatility estimated from the cross‐section of the option prices (in contrast to the existing GARCH option pricing models), maintains theoretical consistency under conditional non‐normality, and improves the empirical performances. Remarkably, the risk‐neutral volatility dynamics are stable over time in this model. In addition, the comparison between the VIX index and the risk‐neutral integrated volatility economically validates our approach. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:1–28, 2013     

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.     

Option prices for risk-neutral density estimation using nonparametric methods through big data and large-scale problems

Option pricing theory determines the structure of call and put option pricing functions. In nonparametric risk-neutral density estimation based on kernel functions, local constraints cannot induce a second derivative function that must integrate one. Convexity and monotonicity of pricing functions also cannot be enforced. A large-scale (optimization) approach is proposed for the risk-neutral density estimation, imposing an enlarged set of no-arbitrage constraints. We considered simulations using Heston's model and hypergeometric functions. The method is applied to samples of intraday data from VIX and S&P500 indexes.     

Equity volatility,bond yields,and yield spreads

This study examines the impact of implied and contemporaneous equity market volatility on Treasury yields, corporate bond yields, and yield spreads over Treasuries. The CBOE VIX is the measure of implied volatility, and the measure of contemporaneous volatility is constructed using intraday squared S&P 500 returns. We find that bond yields and spreads respond to changes in equity market volatility in a manner consistent with a flight‐to‐quality effect. Both short‐ and long‐term Treasury yields fall in response to increases in implied volatility, and the yield curve flattens modestly. Yields on short‐term investment grade bonds fall in response to contemporaneous volatility shocks, while long‐term spreads on low‐quality issues widen. This indicates that investors “look ahead” in anticipation of changes in equity market volatility but respond more strongly to changes in contemporaneous market activity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

The new market for volatility trading

This study 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 average VIX futures prices is upward sloping, whereas 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 with a stochastic long‐term mean level. Using daily calibrated long‐term mean and VIX, the model gives good predictions of VIX futures prices under normal market situation. These parameter estimates could be used to price VIX options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 30:809–833, 2010     

Volatility as an asset class: Holding VIX in a portfolio

Hedging market downturns without sacrificing upside has long been sought by investors. If VIX was directly investable, adding it as a hedge to the S&P 500 would result in significantly improved performance over the equity only portfolio. However, tradable VIX products do not provide the hedge or returns investors seek over long-term horizons. Alternatively, deconstructing VIX to find the key S&P 500 options which drive VIX movements leads to a synthetic VIX portfolio that provides a more effective hedge. Using these options captures correlations and returns similar to VIX, and combined with the S&P 500, outperforms the buy-and-hold index portfolio.     

A new simple square root option pricing model

This study derives a simple square root option pricing model using a general equilibrium approach in an economy where the representative agent has a generalized logarithmic utility function. Our option pricing formulae, like the Black–Scholes model, do not depend on the preference parameters of the utility function of the representative agent. Although the Black–Scholes model introduces limited liability in asset prices by assuming that the logarithm of the stock price has a normal distribution, our basic square root option pricing model introduces limited liability by assuming that the square root of the stock price has a normal distribution. The empirical tests on the S&P 500 index options market show that our model has smaller fitting errors than the Black–Scholes model, and that it generates volatility skews with similar shapes to those observed in the marketplace. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

The CBOE S&P 500 three‐month variance futures

In this article, we study the market of the Chicago Board Options Exchange S&P 500 three‐month variance futures that were listed on May 18, 2004. By using a simple mean‐reverting stochastic volatility model for the S&P 500 index, we present a linear relation between the price of fixed time‐to‐maturity variance futures and the VIX². The model prediction is supported by empirical tests. We find that a model with a fixed mean‐reverting speed of 1.2929 and a daily‐calibrated floating long‐term mean level has a good fit to the market data between May 18, 2004, and August 17, 2007. The market price of volatility risk estimated from the 30‐day realized variance and VIX² has a mean value of −19.1184. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:48–70, 2010     

Pricing efficiency of the S&P 500 index market: Evidence from the Standard & Poor's Depositary Receipts

Standard & Poor's Depositary Receipts (SPDRs) are exchange traded securities representing a portfolio of S&P 500 stocks. They allow investors to track the spot portfolio and better engage in index arbitrage. We tested the impact of the introduction of SPDRs on the efficiency of the S&P 500 index market. Ex‐post pricing efficiency and ex‐ante arbitrage profit between SPDRs and futures were also examined. We found an improved efficiency in the S&P 500 index market after the start of SPDRs trading. Specifically, the frequency and length of lower boundary violations have declined since SPDRs began trading. This result is consistent with the hypothesis that SPDRs facilitate short arbitrage by simplifying the process of shorting the cash index against futures. Tests of pricing efficiency comparing SPDRs and futures suggested that index arbitrage using SPDRs as a substitute for program trading in general results in losses. Although short arbitrages earn a small profit on average, gains are statistically insignificant. A trade‐by‐trade investigation showed that prices are instantaneously corrected after the presence of mispricing signals, introducing substantial risk in arbitraging. Evidence in general supported pricing efficiency between SPDRs and the S&P 500 index futures—both ex‐post and ex‐ante. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:877–900, 2002     

Too many options? Theory and evidence on option exchange design

A model of option exchange design is proposed and tested. The model allows investors to choose among several exchange‐traded options based on a trade‐off between standardization costs and liquidity/transaction costs. It employs a spatial economics approach to provide results for the existence of markets for particular option contracts on the exchange, a comparison of exchange design by a social planner and a profit‐maximizing monopolist (corresponding to the idea that most derivatives exchanges centralize the design and creation of option contracts), and comparative statics that can potentially aid decision makers in the design of option exchanges. In the empirical work, open interest is analyzed for Chicago Board Options Exchange (CBOE) options on the stocks in the S&P 100 index. In accordance with the model's predictions, open interest forms a previously undocumented seesaw pattern across strike prices, clustering around certain strike prices, and dropping off for the adjacent strike prices. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:533–570, 2006     

STOCHASTIC VOLATILITY MODELS AND THE PRICING OF VIX OPTIONS

In this paper, we examine and compare the performance of a variety of continuous‐time volatility models in their ability to capture the behavior of the VIX. The “3/2‐ model” with a diffusion structure which allows the volatility of volatility changes to be highly sensitive to the actual level of volatility is found to outperform all other popular models tested. Analytic solutions for option prices on the VIX under the 3/2‐model are developed and then used to calibrate at‐the‐money market option prices.