VIX期权定价与校正

VIX期权作为波动率衍生品能为金融机构提供有效的市场风险对冲工具。文献中对VIX期权定价的实证分析误差都很大,原因在于模型的选取误差以及校正方法和样本选取不妥。通过在VIX模型中加入均值回复因素和跳因素,可以使VIX过程更加合理,也可以使VIX期权定价精度更高。通过对VIX期权市场中间报价进行校正,得到了4个文献模型的参数估计,并比较4个模型的定价精度和正向隐含波动率偏斜拟合效果。     

Double-jump diffusion model for VIX: evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyse the impact of the jump factor on VIX dynamics.     

Continuous-time VIX dynamics: On the role of stochastic volatility of volatility

This paper examines the ability of several different continuous-time one- and two-factor jump-diffusion models to capture the dynamics of the VIX volatility index for the period between 1990 and 2010. For the one-factor models we study affine and non-affine specifications, possibly augmented with jumps. Jumps in one-factor models occur frequently, but add surprisingly little to the ability of the models to explain the dynamic of the VIX. We present a stochastic volatility of volatility model that can explain all the time-series characteristics of the VIX studied in this paper. Extensions demonstrate that sudden jumps in the VIX are more likely during tranquil periods and the days when jumps occur coincide with major political or economic events. Using several statistical and operational metrics we find that non-affine one-factor models outperform their affine counterparts and modeling the log of the index is superior to modeling the VIX level directly.     

A State‐Price Volatility Index for China's Stock Market

This study derives a volatility index for China's stock market with similar properties to the Chicago Board Options Exchange Volatility Index (the ‘VIX’). A long‐term benchmark of historic volatility expectations is here presented for China from 1996 to 2011, called the ‘China‐ State‐Price Volatility (SPV)’. Construction of this index involves the use of SPV methodology, using implied volatility calculated from options on the Hang Seng China Enterprise Index (HSCEI). Historic open–high–low–close volatility on the Shanghai Composite Index (SHCI) is also used to extend the benchmark prior to the availability of HSCEI options data. The China‐SPV successfully forecasts realised volatility for the Shanghai Stock Exchange. It also serves as a ‘fear gauge’ in that it monitors daily movements of the SHCI in the same way that the VIX monitors the S&P 500 index (Whaley, 2009). The China‐SPV evidences an increasing relation with the US market in terms of the dynamic correlation of levels and changes with the VIX since 2004.     

Volatility dynamics for the S&P 500: Further evidence from non-affine,multi-factor jump diffusions

We apply Markov chain Monte Carlo methods to time series data on S&P 500 index returns, and to its option prices via a term structure of VIX indices, to estimate 18 different affine and non-affine stochastic volatility models with one or two variance factors, and where jumps are allowed in both the price and the instantaneous volatility. The in-sample fit to the VIX term structure shows that the second (stochastic long-term volatility) factor is required to fit the VIX term structure. Out-of-sample tests on the fit to individual option prices, as well as in-sample tests, show that the inclusion of jumps is less important than allowing for non-affine dynamics. The estimation and testing periods together cover more than 21 years of daily data.     

VIX option pricing and CBOE VIX Term Structure: A new methodology for volatility derivatives valuation

This study integrates CBOE VIX Term Structure and VIX futures to simplify VIX option pricing in multifactor models. Exponential and hump volatility functions with one- to three-factor models of the VIX evolution are used to examine their pricing for VIX options across strikes and maturities. The results show that using exponential volatility functions presents an effective choice as pricing models for VIX calls, whereas hump volatility functions provide efficient out-of-sample valuation for most VIX puts, in particular with deep in-the-money and deep out-of-the-money. Pricing errors for calls can be further reduced with a two-factor model.     

Local volatility of volatility for the VIX market

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models—which focus on the modeling of instantaneous variance—are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (Local variance gamma. Bloomberg Quant Research, New York, 2008) and Andreasen and Huge (Risk Mag 76–79, 2011) to a mean-reverting process.     

Pricing volatility options under stochastic skew with application to the VIX index

In recent years, there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at Chicago Board Options Exchange. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform. To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.     

Asymmetric implied market volatility and terrorist attacks

This paper studies the impact of terrorism on implied volatility in the U.S. financial market via an event study methodology. We decompose the options-based and forward looking VIX index into its negative (VIX⁻) and positive (VIX⁺) components, extracted only from put options and call options, respectively. This decomposition of the VIX index allows us to better investigate the asymmetric impact of terrorist attacks on implied volatility from the puts and calls channels separately. Our study finds evidence of a greater impact of terror detected for the puts channel of VIX, namely VIX⁻. We further show that events that occur within the U.S. appear to impact both VIX and VIX⁻ in a similar way, whereas international terrorist attacks show a greater impact on the puts component, VIX⁻. The calls component, VIX⁺, is found to be mainly detached from terrorist attacks.     

Extreme spillovers of VIX fear index to international equity markets

This study analyzes the impact of VIX spillovers on market activities during extreme market conditions in 42 international equity markets from 1998 to 2014. Specifically, tail cross-dependence suggests that a small change in VIX significantly influences global market activities during extreme market conditions. The impact of VIX is asymmetric, which is more pronounced in bearish, highly volatile, and low trading volume markets. Moreover, VIX spillovers exhibit a stronger impact on returns in developed markets and on volatility in emerging markets. In terms of geographical location, the impact of VIX spillovers is more pronounced on returns in Europe and on volatility in Latin America. These findings indicate that international investors can potentially benefit from international portfolio diversification and can serve as useful guidance to policymakers in designing appropriate policies.

     

Valuation of VIX derivatives

We conduct an extensive empirical analysis of VIX derivative valuation models before, during, and after the 2008–2009 financial crisis. Since the restrictive mean-reversion and heteroskedasticity features of existing models yield large distortions during the crisis, we propose generalisations with a time-varying central tendency, jumps, and stochastic volatility, analyse their pricing performance, and implications for term structures of VIX futures and volatility “skews.” We find that a process for the log of the observed VIX combining central tendency and stochastic volatility reliably prices VIX derivatives. We also uncover a significant risk premium that shifts the long-run volatility level.     

Option-implied volatility factors and the cross-section of market risk premia

The main goal of this paper is to study the cross-sectional pricing of market volatility. The paper proposes that the market return, diffusion volatility, and jump volatility are fundamental factors that change the investors’ investment opportunity set. Based on estimates of diffusion and jump volatility factors using an enriched dataset including S&P 500 index returns, index options, and VIX, the paper finds negative market prices for volatility factors in the cross-section of stock returns. The findings are consistent with risk-based interpretations of value and size premia and indicate that the value effect is mainly related to the persistent diffusion volatility factor, whereas the size effect is associated with both the diffusion volatility factor and the jump volatility factor. The paper also finds that the use of market index data alone may yield counter-intuitive results.     

Impact of macroeconomic announcements on implied volatility slope of SPX options and VIX

This paper examines the impact of macroeconomic announcements on the high-frequency behavior of the observed implied volatility skew of S&P 500 index options and VIX. We document that macroeconomic announcements affect VIX significantly and slope at a lesser extent. We also find evidence that good and bad announcements significantly and asymmetrically change implied volatility slope and VIX.     

Pricing and hedging contingent claims using variance and higher order moment swaps

This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and random jumps of the underlying asset price. This is done by enlarging the market with appropriate swaps whose pay-offs depend on higher order sample moments of the asset price process. Using European options and variance swaps, as well as barrier options written on the S&P 500 index, the paper provides clear cut evidence that hedging strategies employing variance and higher order moment swaps considerably improves upon the performance of traditional delta hedging strategies. Inclusion of the third-order moment swap improves upon the performance of variance swap-based strategies to hedge against random jumps. This result is more profound for short-term out-of-the money put options.     

VCRIX — A volatility index for crypto-currencies

Public interest, explosive returns, and diversification opportunities gave stimulus to the adoption of traditional financial tools to crypto-currencies. While the CRIX offered the first scientifically-backed proxy to the crypto-market (analogous to S&P 500), measuring the forward-oriented risk in the crypto-currency market posed a challenge of a different kind. Following the intuition of the “fear index” VIX for the American stock market, the VCRIX volatility index was created to capture the investor expectations about the crypto-currency ecosystem. VCRIX is built based on CRIX and offers a forecast based on the Heterogeneous Auto-Regressive (HAR) model. The HAR model was selected as the most suitable out of a horse race of volatility models, with two proxies for implied volatility, namely the 30 days mean annualized volatility and realized volatility. The model was further examined by the simulation of VIX (resulting in a correlation of 78% between the actual VIX and a “VIX” version estimated with the VCRIX technology). Trading strategies confirmed the predictive power of VCRIX and supported the selection of the 30 days means annualized volatility proxy. The best performing trading strategy with the use of VCRIX outperformed the benchmark strategy for 99.8% of the tested period and 164% additional returns. VCRIX provides forecasting functionality and serves as a proxy for the investors’ expectations in the absence of a developed crypto derivatives market. These features provide enhanced decision making capacities for market monitoring, trading strategies, and potentially option pricing.     

Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets

We estimate a flexible affine model using an unbalanced panel containing S&P 500 and VIX index returns and option prices and analyze the contribution of VIX options to the model’s in- and out-of-sample performance. We find that they contain valuable information on the risk-neutral conditional distributions of volatility at different time horizons, which is not spanned by the S&P 500 market. This information allows enhanced estimation of the variance risk premium. We gain new insights on the term structure of the variance risk premium, present a trading strategy exploiting these insights, and show how to improve S&P 500 return forecasts.     

Is VIX an investor fear gauge in BRIC equity markets?

This study examines the intertemporal relationships between CBOE market volatility index (VIX) and stock market returns in Brazil, Russia, India, and China (BRIC), and between VIX and U.S. stock market returns, to uncover if VIX serves as an investor fear gauge in BRIC and U.S. markets. We conduct the VIX-returns analysis for the 1993–2007 period.Our results suggest a strong negative contemporaneous relation between daily changes (innovations) in VIX and U.S. stock market returns. This relation is stronger when VIX is higher and more volatile. A significant negative contemporaneous relation between VIX and equity returns also exists for China and Brazil during 1993–2007 and for India during 1993–1997. Similar to the U.S. market, the immediate negative relation between the Brazilian stock returns and VIX changes is much stronger when VIX is both high and more volatile. Our results also indicate a strong asymmetric relation between innovations in VIX and daily stock market returns in U.S., Brazil, and China, suggesting that VIX is more of a gauge of investor fear than investor positive sentiment. However, the asymmetric relationship between stock market returns and VIX is much weaker when VIX is large and more volatile. These results have potential implications for portfolio diversification and for stock market and option trading timing in the equity markets of Brazil, India, and China. Overall, our results indicate that VIX is not only an investor fear gauge for the U.S. stock market but also for the equity markets of China, Brazil, and India.     

Volatility timing,sentiment, and the short-term profitability of VIX-based cross-sectional trading strategies

This paper explores the profitability of simple short-term cross-sectional trading strategies based on the implied volatility index (VIX), often referred to as an “investor fear gauge” in the stock market. These strategies involve holding sentiment-prone stocks when VIX is low and sentiment-immune stocks when VIX is high and generate significantly higher excess returns than the benchmark long–short portfolios that do not condition on VIX. We show that the profitability of our trading strategies is not subsumed by the well-known risk factors or transaction cost adjustments. Our findings are consistent with the theory of delayed arbitrage and the synchronization problem of Abreu and Brunnermeier (2002).     

Estimating and using GARCH models with VIX data for option valuation

This paper uses information on VIX to improve the empirical performance of GARCH models for pricing options on the S&P 500. In pricing multiple cross-sections of options, the models’ performance can clearly be improved by extracting daily spot volatilities from the series of VIX rather than by linking spot volatility with different dates by using the series of the underlying’s returns. Moreover, in contrast to traditional returns-based Maximum Likelihood Estimation (MLE), a joint MLE with returns and VIX improves option pricing performance, and for NGARCH, joint MLE can yield empirically almost the same out-of-sample option pricing performance as direct calibration does to in-sample options, but without costly computations. Finally, consistently with the existing research, this paper finds that non-affine models clearly outperform affine models.