Testing mean reversion in financial market volatility: Evidence from S&P 500 index futures

This article presents a comprehensive study of continuous time GARCH (generalized autoregressive conditional heteroskedastic) modeling with the thintailed normal and the fat‐tailed Student's‐t and generalized error distributions (GED). The study measures the degree of mean reversion in financial market volatility based on the relationship between discrete‐time GARCH and continuoustime diffusion models. The convergence results based on the aforementioned distribution functions are shown to have similar implications for testing mean reversion in stochastic volatility. Alternative models are compared in terms of their ability to capture mean‐reverting behavior of futures market volatility. The empirical evidence obtained from the S&P 500 index futures indicates that the conditional variance, log‐variance, and standard deviation of futures returns are pulled back to some long‐run average level over time. The study also compares the performance of alternative GARCH models with normal, Student's‐ t, and GED density in terms of their power to predict one‐day‐ahead realized volatility of index futures returns and provides some implications for pricing futures options. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1–33, 2008     

Impact of international energy prices on China's industries

This study examines how returns and volatility of future contracts for Brent crude oil (Brent), West Texas Intermediate crude oil (WTI), Henry Hub natural gas, and Newcastle thermal coal impacts industries in China. Using the firm-level data of 3,750 stock listings across both Shanghai and Shenzhen stock exchanges, segregated into 138 subindustries under the Global Industry Classification Standard, this study finds evidence that crude oil futures have the most significant influence. Further analysis suggests that stock returns of oil-related companies are more closely align to Brent and WTI's futures returns following China's key oil pricing reform on March 27, 2013. Overall, Chinese industries are also more exposed to global crude oil futures volatility after this event.     

Optimal futures hedging for energy commodities: An application of the GAS model

This paper applies generalized autoregressive score-driven (GAS) models to futures hedging of crude oil and natural gas. For both commodities, the GAS framework captures the marginal distributions of spot and futures returns and corresponding dynamic copula correlations. We compare within-sample and out-of-sample hedging effectiveness of GAS models against constant ordinary least square (OLS) strategy and time-varying copula-based GARCH models in terms of volatility reduction and Value at Risk reduction. We show that the constant OLS hedge ratio is not inherently inferior to the time-varying alternatives. Nonetheless, GAS models tend to exhibit better hedging effectiveness than other strategies, particularly for natural gas.     

The realized volatility of FTSE‐100 futures prices

Five‐minute returns from FTSE‐100 index futures contracts are used to obtain accurate estimates of daily index volatility from January 1986 to December 1998. These realized volatility measures are used to obtain inferences about the distributional and autocorrelation properties of FTSE‐100 volatility. The distribution of volatility measured daily is similar to lognormal while the volatility time series has persistent positive autocorrelation that displays long‐memory effects. The distribution of daily returns standardized using the measures of realized volatility is shown to be close to normal, unlike the unconditional distribution. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:627–648, 2002     

Return distributions and volatility forecasting in metal futures markets: Evidence from gold,silver, and copper

The characterization of return distributions and forecast of asset‐price variability play a critical role in the study of financial markets. This study estimates four measures of integrated volatility—daily absolute returns, realized volatility, realized bipower volatility, and integrated volatility via Fourier transformation (IVFT)—for gold, silver, and copper by using high‐frequency data for the period 1999 through 2008. The distributional properties are investigated by applying recently developed jump detection procedures and by constructing financial‐time return series. The predictive ability of a GARCH (1,1) forecasting model that uses various volatility measures is also examined. Three important findings are reported. First, the magnitude of the IVFT volatility estimate is the greatest among the four volatility measures. Second, the return distributions of the three markets are not normal. However, when returns are standardized by IVFT and realized volatility, the corresponding return distributions bear closer resemblance to a normal distribution. Notably, the application of financial‐time sampling technique is helpful in obtaining a normal distribution. Finally, the IVFT and realized volatility proxies produce the smallest forecasting errors, and increasing the time frequency of estimating integrated volatility does not necessarily improve forecast accuracy. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:55–80, 2011     

Returns and volatility in the Kuala Lumpur crude

This article investigates the determinants of daily returns and volatility in the Kuala Lumpur crude palm oil futures market over the period 1980 to 1994. We find significant evidence of month and open interest effects in returns and also find strong evidence of daily, monthly, yearly, volume and open interest effects in volatility when ARCH/ GARCH models are used to estimate volatility. © 1998 John Wiley & Sons, Inc. Jrl Fut Mark 18:985–999, 1998     

Long memory models for daily and high frequency commodity futures returns

The volatility of daily futures returns for six important commodities are found to be well described as FIGARCH, fractionally integrated processes, whereas the mean returns exhibit very small departures from the martingale difference property. Several years of high frequency intraday commodity futures returns are also found to have very similar long memory in volatility features as the daily returns. Semiparametric local Whittle estimation of the long memory parameter in absolute returns also finds very significant long memory features. Estimating the long memory parameter across many different data sampling frequencies provides consistent estimates of the long memory parameter, suggesting that the series are self‐similar. The results have important implications for empirical work using commodity futures price data. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:643–668, 2007     

Minimum capital requirement calculations for UK futures

Key to the imposition of appropriate minimum capital requirements on a daily basis is accurate volatility estimation. Here, measures are presented based on discrete estimation of aggregated high‐frequency UK futures realizations underpinned by a continuous time framework. Squared and absolute returns are incorporated into the measurement process so as to rely on the quadratic variation of a diffusion process and be robust in the presence of fat tails. The realized volatility estimates incorporate the long memory property. The dynamics of the volatility variable are adequately captured. Resulting rescaled returns are applied to minimum capital requirement calculations. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:193–220, 2004     

Volatility dynamics of NYMEX natural gas futures prices

We examine the volatility dynamics of NYMEX natural gas futures prices via the partially overlapping time‐series model of Smith (2005. Journal of Applied Econometrics, 20, 405–422). We show that volatility exhibits two important features: (1) volatility is greater in the winter than in the summer, and (2) the persistence of price shocks and, hence, the correlations among concurrently traded contracts, displays substantial seasonal and cross‐sectional variation in a way consistent with the theory of storage. We demonstrate that, by ignoring the seasonality in the volatility dynamics of natural gas futures prices, previous studies have suggested sub‐optimal hedging strategies. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:438–463, 2008     

Return and volatility transmission between China's and international crude oil futures markets: A first look

We examine return and volatility transmission between the newly established crude oil futures in China and international major crude oil futures markets using intraday data. For the first time, we document evidence for cointegration relationships among these oil futures markets. Both China's and Oman's oil futures markets react to deviations from their long-run equilibrium with West Texas Intermediate and Brent oil futures. There is also new evidence for asymmetric volatilities and correlations across these oil futures markets. Furthermore, the Chinese oil futures have stronger linkages with the international major futures markets than Oman futures.     

An N‐factor Gaussian model of oil futures prices

This article studies the ability of an N‐factor Gaussian model to explain the stochastic behavior of oil futures prices when estimated with the use of all available price information, as opposed to traditional approaches of aggregating data for a set of maturities. A Kalman filter estimation procedure that allows for a time‐dependent number of daily observations is used to calibrate the model. When applied to all daily oil futures price transactions from 1992 to 2001, the model performs very well, requiring at least three factors to explain the term structure of futures prices, but four factors to fit the volatility term structure. The model also performs very well for daily copper futures transactions from 1992 to 2001 and for out‐of‐sample daily oil futures transactions from 2002 to 2004. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:243–268, 2006     

Memory in returns and volatilities of futures' contracts

Various authors claim to have found evidence of stochastic long‐memory behavior in futures’ contract returns using the Hurst statistic. This paper reexamines futures’ returns for evidence of persistent behavior using a biased‐corrected version of the Hurst statistic, a nonparametric spectral test, and a spectral‐regression estimate of the long‐memory parameter. Results based on these new methods provide no evidence for persistent behavior in futures’ returns. However, they provide overwhelming evidence of long‐memory behavior for the volatility of futures’ returns. This finding adds to the emerging literature on persistent volatility in financial markets and suggests the use of new methods of forecasting volatility, assessing risk, and optimizing portfolios in futures’ markets. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:525–543, 2000     

Forecasting oil price movements: Exploiting the information in the futures market

Relying on the cost of carry model, the long‐run relationship between spot and futures prices is investigated and the information implied in these cointegrating relationships is used to forecast out of sample oil spot and futures price movements. To forecast oil price movements, a vector error correction model (VECM) is employed, where the deviations from the long‐run relationships between spot and futures prices constitute the equilibrium error. To evaluate forecasting performance, the random walk model (RWM) is used as a benchmark. It was found that (a) in‐sample, the information in the futures market can explain a sizable portion of oil price movements; and (b) out‐of‐sample, the VECM outperforms the RWM in forecasting price movements of 1‐month futures contracts. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:34–56, 2008     

A maximal affine stochastic volatility model of oil prices

This study develops and estimates a stochastic volatility model of commodity prices that nests many of the previous models in the literature. The model is an affine three‐factor model with one state variable driving the volatility and is maximal among all such models that are also identifiable. The model leads to quasi‐analytical formulas for futures and options prices. It allows for time‐varying correlation structures between the spot price and convenience yield, the spot price and its volatility, and the volatility and convenience yield. It allows for expected mean‐reversion in the short term and for an increasing expected long‐term price, and for time‐varying risk premia. Furthermore, the model allows for the situation in which options' prices depend on risk not fully spanned by futures prices. These properties are desirable and empirically important for modeling many commodities, especially crude oil. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:101–133, 2010     

Volatility spillover effects and cross hedging in corn and crude oil futures

Using a volatility spillover model, we find evidence of significant spillovers from crude oil prices to corn cash and futures prices, and that these spillover effects are time‐varying. Results reveal that corn markets have become much more connected to crude oil markets after the introduction of the Energy Policy Act of 2005. Furthermore, when the ethanol–gasoline consumption ratio exceeds a critical level, crude oil prices transmit positive volatility spillovers into corn prices and movements in corn prices are more energy‐driven. Based on this strong volatility link between crude oil and corn prices, a new cross‐hedging strategy for managing corn price risk using oil futures is examined and its performance is studied. Results show that this cross‐hedging strategy provides only slightly better hedging performance compared with traditional hedging in corn futures markets alone. The implication is that hedging corn price risk in corn futures markets alone can still provide relatively satisfactory performance in the biofuel era. © 2010 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.     

Regime‐switching in stock index and Treasury futures returns and measures of stock market stress

We investigate bivariate regime‐switching in daily futures‐contract returns for the US stock index and ten‐year Treasury notes over the crisis‐rich 1997–2005 period. We allow the return means, volatilities, and correlation to all vary across regimes. We document a striking contrast between regimes, with a high‐stress regime that exhibits a much higher stock volatility, a much lower stock–bond correlation, and a higher mean bond return. The high‐stress regime is associated with higher average values of stock‐implied volatility, stock illiquidity, and stock and bond futures trading volume. The lagged implied volatility from equity‐index options is useful in modeling the time‐varying transition probabilities of the regime‐switching process. Our findings support the notions that: (1) stock market stress can have a material influence on Treasury bond pricing, and (2) the diversification benefits of combined stock–bond holdings tend to be greater during times with relatively high stock market stress. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:753–779, 2010     

Revisiting the finite mixture of Gaussian distributions with application to futures markets

We present a new estimation method for Gaussian mixture modeling, namely, the kurtosis‐controlled expectation‐maximization (EM) algorithm, which overcomes the limitations of the usual estimation techniques via kurtosis control and kernel splitting. Our simulation study shows that the dynamic allocation of kernels according to the value of the total kurtosis measure makes the proposed kurtosis‐controlled EM algorithm an efficient method for Gaussian mixture density estimation. This algorithm yielded considerable improvements over the classical EM algorithm. We then used the discrete Gaussian mixture framework to account for the observed thick‐tailed distributions of futures returns and applied the kurtosis‐controlled EM algorithm to estimate the distributions of real (agricultural, metal, and energy) and financial (stock index and currency) futures returns. We proved that this framework is perfectly adapted to capturing the departures from normality of the observed return distributions. Unlike in previous studies, we found that a two‐component Gaussian mixture is too poor a model to accurately capture the distributional properties of returns. Similar results have been obtained for stocks, indexes, currencies, interest rates, and commodities. This has important implications in many financial studies using Gaussian mixtures to incorporate the thickness of the tails of the distributions in the computation of the value at risk or to infer implied risk‐neutral densities from option prices, to name but a few. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21: 347–376, 2001     

Global Volatility and Forex Returns in East Asia*

During 2001–2010, increases in mature market volatility were associated with declines in forex returns for East Asian economies, consistent with an overall ‘flight to safety’ effect. Estimates from GARCH models suggest that a 10 percentage point increase in mature market equity volatility generated an exchange rate depreciation of up to 3/4 percent. This sensitivity rose during a more tranquil subsample for some countries, reflecting their greater integration with global financial markets. Long‐run forex volatility increased in Asian economies after 2008, reflecting the global reach of the financial crisis in mature markets. Unconditional standard deviations estimated from these models provide operational measures of ‘long‐term’ and ‘excess’ volatility in forex markets.     

A cointegrated commodity pricing model

We propose a commodity pricing model that extends the Gibson–Schwartz two‐factor model to incorporate the effect of linear relations among commodity spot prices, and provide a condition under which such linear relations represent cointegration. We derive futures and call option prices for the proposed model, and indicate that, unlike in Duan and Pliska (2004), the linear relations among commodity prices should affect commodity derivative prices, even when the volatilities of commodity returns are constant. Using crude oil and heating oil market data, we estimate the model and apply the results to the hedging of long‐term futures using short‐term ones.