U.S. stock market crash risk, 1926–2010

This paper examines how well alternate time-changed Lévy processes capture stochastic volatility and the substantial outliers observed in U.S. stock market returns over the past 85 years. The autocorrelation of daily stock market returns varies substantially over time, necessitating an additional state variable when analyzing historical data. I estimate various one- and two-factor stochastic volatility/Lévy models with time-varying autocorrelation via extensions of the Bates (2006) methodology that provide filtered daily estimates of volatility and autocorrelation. The paper explores option pricing implications, including for the Volatility Index (VIX) during the recent financial crisis.     

Forecasting volatility of the U.S. oil market

We examine the information content of the CBOE Crude Oil Volatility Index (OVX) when forecasting realized volatility in the WTI futures market. Additionally, we study whether other market variables, such as volume, open interest, daily returns, bid-ask spread and the slope of the futures curve, contain predictive power beyond what is embedded in the implied volatility. In out-of-sample forecasting we find that econometric models based on realized volatility can be improved by including implied volatility and other variables. Our results show that including implied volatility significantly improves daily and weekly volatility forecasts; however, including other market variables significantly improves daily, weekly and monthly volatility forecasts.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

The impacts of futures trading on volatility and volatility asymmetry of Bitcoin returns

The impact of Bitcoin futures introduction on the underlying Bitcoin volatility has been a controversial topic. Conflicting results had been obtained from different sample periods and methodologies. To address this debate, this study examines the impact of futures trading on volatility and volatility asymmetry of Bitcoin returns in the short and long run. Using exponential GARCH models, we introduce a dummy in the variance equation to capture the changes in the volatility after the introduction of Bitcoin futures. We find that after the introduction, spot return volatility decreases in the short run, but increases in the long run. Besides, in the short run, there exists an inverse leverage effect before and after the introduction; in the long run, the inverse leverage effect before the introduction changes to a usual level effect after the introduction. Finally, we examine whether greater futures trading activity, proxied by trading volume and open interest, is associated with greater Bitcoin volatility. To do so, we decompose each proxy into expected and unexpected components and document that, in the long run, Bitcoin volatility covaries positively with unexpected futures trading volume, but negatively with unexpected futures open interest.     

News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns

This paper models components of the return distribution, which are assumed to be directed by a latent news process. The conditional variance of returns is a combination of jumps and smoothly changing components. A heterogeneous Poisson process with a time‐varying conditional intensity parameter governs the likelihood of jumps. Unlike typical jump models with stochastic volatility, previous realizations of both jump and normal innovations can feed back asymmetrically into expected volatility. This model improves forecasts of volatility, particularly after large changes in stock returns. We provide empirical evidence of the impact and feedback effects of jump versus normal return innovations, leverage effects, and the time‐series dynamics of jump clustering.     

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.     

An empirical comparison of continuous-time models of implied volatility indices

We explore the ability of alternative popular continuous-time diffusion and jump-diffusion processes to capture the dynamics of implied volatility indices over time. The performance of the various models is assessed under both econometric and financial metrics. To this end, data are employed from major European and American implied volatility indices and the rapidly growing CBOE volatility futures market. We find that the addition of jumps is necessary to capture the evolution of implied volatility indices under both metrics. Mean reversion is of second-order importance though. The results are consistent across the various metrics, markets, and construction methodologies.     

基于高频数据的股指期货与现货市场波动溢出和信息传导研究

本文利用沪深300指数和当月股指期货连续合约的高频数据,采用非参数方法估计日度股票指数和股指期货的整体波动、连续性波动和跳跃,发现两个市场波动成分存在双向的格兰杰因果关系,但是期货市场的跳跃并不会影响后续股票市场的跳跃。此外,已实现相关系数在股指期货上市初期表现出了较大的变动,整体表现出了较强的联动趋势。最后,日内高频价格之间存在稳定的协整关系,两个市场存在双向的信息传导,股指期货的价格发现功能得到发挥。     

The Impact of Jumps in Volatility and Returns

This paper examines continuous‐time stochastic volatility models incorporating jumps in returns and volatility. We develop a likelihood‐based estimation strategy and provide estimates of parameters, spot volatility, jump times, and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes, and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997, and 1998. Using formal and informal diagnostics, we find strong evidence for jumps in volatility and jumps in returns. Finally, we study how these factors and estimation risk impact option pricing.     

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.     

Dynamic hedging with futures: a copula-based GARCH model with high-frequency data

Modeling the joint distribution of spot and futures returns is crucial for establishing optimal hedging strategies. This paper proposes a new class of dynamic copula-GARCH models that exploits information from high-frequency data for hedge ratio estimation. The copula theory facilitates constructing a flexible distribution; the inclusion of realized volatility measures constructed from high-frequency data enables copula forecasts to swiftly adapt to changing markets. By using data concerning equity index returns, the estimation results show that the inclusion of realized measures of volatility and correlation greatly enhances the explanatory power in the modeling. Moreover, the out-of-sample forecasting results show that the hedged portfolios constructed from the proposed model are superior to those constructed from the prevailing models in reducing the (estimated) conditional hedged portfolio variance. Finally, the economic gains from exploiting high-frequency data for estimating the hedge ratios are examined. It is found that hedgers obtain additional benefits by including high-frequency data in their hedging decisions; more risk-averse hedgers generate greater benefits.     

On the realized volatility of the ECX CO2 emissions 2008 futures contract: distribution, dynamics and forecasting

This article documents the conditional and unconditional distributions of the realized volatility for the 2008 futures contract in the European climate exchange (ECX), which is valid under the EU emissions trading scheme (EU ETS). Realized volatility measures from naive, kernel-based and subsampling estimators are used to obtain inferences about the distributional and dynamic properties of the ECX emissions futures volatility. The distribution of the daily realized volatility in logarithmic form is shown to be close to normal. The mixture-of-normals hypothesis is strongly rejected, as the returns standardized using daily measures of volatility clearly departs from normality. A simplified HAR-RV model (Corsi in J Financ Econ 7:174–196, 2009) with only a weekly component, which reproduces long memory properties of the series, is then used to model the volatility dynamics. Finally, the predictive accuracy of the HAR-RV model is tested against GARCH specifications using one-step-ahead forecasts, which confirms the HAR-RV superior ability.     

Jump robust two time scale covariance estimation and realized volatility budgets

We estimate the daily integrated variance and covariance of stock returns using high-frequency data in the presence of jumps, market microstructure noise and non-synchronous trading. For this we propose jump robust two time scale (co)variance estimators and verify their reduced bias and mean square error in simulation studies. We use these estimators to construct the ex-post portfolio realized volatility (RV) budget, determining each portfolio component’s contribution to the RV of the portfolio return. These RV budgets provide insight into the risk concentration of a portfolio. Furthermore, the RV budgets can be directly used in a portfolio strategy, called the equal-risk-contribution allocation strategy. This yields both a higher average return and lower standard deviation out-of-sample than the equal-weight portfolio for the stocks in the Dow Jones Industrial Average over the period October 2007–May 2009.     

Futures volatility forecasting based on big data analytics with incorporating an order imbalance effect

Future markets play vital roles in supporting economic activities in modern society. For example, crude oil and electricity futures markets have heavy effects on a nation’s energy operation management. Thus, volatility forecasting of the futures market is an emerging but increasingly influential field of financial research. In this paper, we adopt big data analytics, called Extreme Gradient Boosting (XGBoost) from computer science, in an attempt to improve the forecasting accuracy of futures volatility and to demonstrate the application of big data analytics in the financial spectrum in terms of volatility forecasting. We further unveil that order imbalance estimation might incorporate abundant information to reflect price jumps and other trading information in the futures market. Including order imbalance information helps our model capture underpinned market rules such as supply and demand, which lightens the information loss during the model formation. Our empirical results suggest that the volatility forecasting accuracy of the XGBoost method considerably beats the GARCH-jump and HAR-jump models in both crude oil futures market and electricity futures market. Our results could also produce plentiful research implications for both policy makers and energy futures market participants.     

Investor Attention and Time‐varying Comovements

This paper analyses the effect of an increase in market‐wide uncertainty on information flow and asset price comovements. We use the daily realised volatility of the 30‐year treasury bond futures to assess macroeconomic shocks that affect market‐wide uncertainty. We use the ratio of a stock's idiosyncratic realised volatility with respect to the S&P500 futures relative to its total realised volatility to capture the asset price comovement with the market. We find that market volatility and the comovement of individual stocks with the market increase contemporaneously with the arrival of market‐wide macroeconomic shocks, but decrease significantly in the following five trading days. This pattern supports the hypothesis that investors shift their (limited) attention to processing market‐level information following an increase in market‐wide uncertainty and then subsequently divert their attention back to asset‐specific information.     

The hedging effect of green bonds on carbon market risk

This paper explores effective hedging instruments for carbon market risk. Examining the relationship between the carbon futures returns and the returns of four major market indices, i.e., the VIX index, the commodity index, the energy index and the green bond index, we find that the connectedness between the carbon futures returns and the green bond index returns is the highest and this connectedness is extremely pronounced during the market's volatile period. Further, we develop and evaluate hedging strategies based on three dynamic hedge ratio models (DCC-APGARCH, DCC-T-GARCH, and DCC-GJR-GARCH models) and the constant hedge ratio model (OLS model). Empirical results show that among the four market indices the green bond index is the best hedge for carbon futures and performs well even in the crisis period. The paper also provides evidence that the dynamic hedge ratio models are superior to the OLS model in the volatile period as more sophisticated models can capture the dynamic correlation and volatility spillover between the carbon futures and market index returns.     

A High‐Frequency Investigation of the Interaction between Volatility and DAX Returns

One of the most noticeable stylised facts in finance is that stock index returns are negatively correlated with changes in volatility. The economic rationale for the effect is still controversial. The competing explanations have different implications for the origin of the relationship: Are volatility changes induced by index movements, or inversely, does volatility drive index returns? To differentiate between the alternative hypotheses, we analyse the lead‐lag relationship of option implied volatility and index return in Germany based on Granger causality tests and impulse‐response functions. Our dataset consists of all transactions in DAX options and futures over the time period from 1995 to 2005. Analyzing returns over 5‐minute intervals, we find that the relationship is return‐driven in the sense that index returns Granger cause volatility changes. This causal relationship is statistically and economically significant and can be clearly separated from the contemporaneous correlation. The largest part of the implied volatility response occurs immediately, but we also observe a smaller retarded reaction for up to one hour. A volatility feedback effect is not discernible. If it exists, the stock market appears to correctly anticipate its importance for index returns.     

Valuation of commodity derivatives in a new multi-factor model

This paper extends existing commodity valuation models to allow for stochastic volatility and simultaneous jumps in the spot price and spot volatility. Closed-form valuation formulas for forwards, futures, futures options, geometric Asian options and commodity-linked bonds are obtained using the Heston (1993) and Bakshi and Madan (2000) methodology. Stochastic volatility and jumps do not affect the futures price at a given point in time. However, numerical examples indicate that they play important roles in pricing options on futures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Intraday jumps and US macroeconomic news announcements

This paper applies recent non-parametric intraday jump detection procedures to investigate the presence and importance of intraday jumps in US futures markets. More importantly, the paper investigates the extent to which statistically significant intraday jumps are associated with US macroeconomic news announcements. Jumps are prevalent, large and contribute heavily to total daily price variation. Approximately one third of jumps correspond to US macroeconomic news announcements, with pure announcement effects causing large increases in the absolute sizes of jumps and the informational surprise of the announcement explaining large proportions of the jumps. The statistical and economic significance of news-related jumps is confirmed by results that show higher volatility persistence, predictability of lower frequency returns, larger effects on microstructure variables, jump clustering and co-jumps from these jumps versus non-news-related jumps, although there are some interesting variations across asset classes.     

Foreign Exchange Volatility Is Priced in Equities

This paper finds that standard asset pricing models fail to explain the significantly negative delta hedging errors that occur as a result of the purchase of options on foreign exchange futures. Foreign exchange volatility does influence stock returns, however. The volatility of the JPY/USD exchange rate predicts the time series of stock returns and is priced in the cross‐section of stock returns.