Robust estimation of the optimal hedge ratio

When using derivative instruments such as futures to hedge a portfolio of risky assets, the primary objective is to estimate the optimal hedge ratio (OHR). When agents have mean‐variance utility and the futures price follows a martingale, the OHR is equivalent to the minimum variance hedge ratio,which can be estimated by regressing the spot market return on the futures market return using ordinary least squares. To accommodate time‐varying volatility in asset returns, estimators based on rolling windows, GARCH, or EWMA models are commonly employed. However, all of these approaches are based on the sample variance and covariance estimators of returns, which, while consistent irrespective of the underlying distribution of the data, are not in general efficient. In particular, when the distribution of the data is leptokurtic, as is commonly found for short horizon asset returns, these estimators will attach too much weight to extreme observations. This article proposes an alternative to the standard approach to the estimation of the OHR that is robust to the leptokurtosis of returns. We use the robust OHR to construct a dynamic hedging strategy for daily returns on the FTSE100 index using index futures. We estimate the robust OHR using both the rolling window approach and the EWMA approach, and compare our results to those based on the standard rolling window and EWMA estimators. It is shown that the robust OHR yields a hedged portfolio variance that is marginally lower than that based on the standard estimator. Moreover, the variance of the robust OHR is as much as 70% lower than the variance of the standard OHR, substantially reducing the transaction costs that are associated with dynamic hedging strategies. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:799–816, 2003     

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     

Realized volatility and correlation in energy futures markets

Using high‐frequency returns, realized volatility and correlation of the NYMEX light, sweet crude oil, and Henry‐Hub natural gas futures contracts are examined. The unconditional distributions of daily returns and daily realized variances are non‐Gaussian, whereas the distributions of the standardized returns (normalized by the realized standard deviation) and the (logarithms of) realized standard deviations appear approximately Gaussian. The (logarithms of) standard deviations exhibit long‐memory, but the realized correlation between the two futures does not, implying rather weak inter‐market linkage in the long run. There is evidence of asymmetric volatility for natural gas but not for crude oil futures. Finally, realized crude oil futures volatility responds with an increase in the weeks immediately before the OPEC events recommending price increases. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:993–1011, 2008     

Information content of extended trading for index futures

The recent extension of trading hours for Hang Seng Index Futures provides an opportunity to examine whether extended futures trading contains useful information about spot returns. Using the weighted price contribution measure, we find that pre‐open futures trades are associated with significant price discovery. We extend the model from T. Hiraki, E. D. Maberly, and N. Takezawa (1995) and adjust for the existence of a pre‐open trading session and the overnight trading of cross‐listed shares in London. Our results indicate that extended trading for index futures contains useful information in explaining subsequent spot returns during the trading day. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:861–886, 2004     

The economic value of volatility transmission between the stock and bond markets

This study has two main objectives. Firstly, volatility transmission between stocks and bonds in European markets is studied using the two most important financial assets in these fields: the DJ Euro Stoxx 50 index futures contract and the Euro Bund futures contract. Secondly, a trading rule for the major European futures contracts is designed. This rule can be applied to different markets and assets to analyze the economic significance of volatility spillovers observed between them. The results indicate that volatility spillovers take place in both directions and that the stock‐bond trading rule offers very profitable returns after transaction costs. These results have important implications for portfolio management and asset allocation. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1066–1094, 2008     

The lead–lag relationship between equities and stock index futures markets around information releases

This paper documents a strengthening in the lead of stock index futures returns over stock index returns around macroeconomic information releases. Some evidence of a strengthening in feedback from the equities market to the futures market and weakening in the lead of the futures market around major stock‐specific information releases is also provided. This is consistent with the hypothesis that investors with better marketwide information prefer to trade in stock index futures while investors with stock‐specific information prefer to trade in underlying stocks. A small weakening in the contemporaneous relationship between stock index futures returns and stock index returns around both types of releases is also documented. This is consistent with disintegration in the relationship between the two markets associated with noise induced volatility. One by‐product of this study is new comparative evidence on the performance of adjustments for infrequent trading of index stocks based on a commonly used ARMA technique versus recalculation of the stock index using quote midpoints. The results suggest that the quote midpoint index performs at least as well as the ARMA adjusted index across the entire sample period, as well as around the different types of information releases. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:467–487, 2000     

Bias and backwardation in natural gas futures prices

This paper tests the fair‐game efficient‐markets hypothesis for the natural gas futures prices over the period 1990 through 2003. We find evidence consistent with the Keynesian notion of normal backwardation. Regressing the future spot prices on the lagged futures prices and using the Stock‐Watson (1993) procedure to correct for the correlation between the error terms and the futures prices, we find that natural gas futures are biased predictors of the corresponding future spot prices for contracts ranging from 3 to 12 months. These results cast a serious doubt on the commonly held view that natural gas futures sell at a premium over the expected future spot prices, and that this bias is due to the systematic risk of the futures price movements represented by a negative “beta.” We also find evidence for the Samuelson effect. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:281–308, 2005     

Stock index futures trading and volatility in international equity markets

This article examines stock market volatility before and after the introduction of equity‐index futures trading in twenty‐five countries, using various models that account for asynchronous data, conditional heteroskedasticity, asymmetric volatility responses, and the joint dynamics of each country's index with the world‐market portfolio. We found that futures trading is related to an increase in conditional volatility in the United States and Japan, but in nearly every other country, we found either no significant effect or a volatility‐dampening effect. This result appears to be robust to model specification and is corroborated by further analysis of the relationship between volatility, trading volume, and open interest in stock futures. An increase in conditional covariance between country‐specific and world returns at the time of futures listing is also documented. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:661–685, 2000     

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     

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     

Return and Volatility Dynamics in the Spot and Futures Markets in Australia: An Intervention Analysis in a Bivariate EGARCH‐X Framework

This article provides evidence of linkages between the equity market and the index futures market in Australia, where the futures market has experienced a major structural event due to the futures contract respecification. A bivariate Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model is developed that includes a cointegrating residual as an explanatory variable for both the conditional mean and the conditional variance. The conditional mean returns from both markets are influenced by the long‐run equilibrium relationship, and these markets are informationally linked through the second moments. The crossmarket spillovers exhibit asymmetric behavior in that the volatility responses to past standardized innovations are different for market advances and market retreats. An intervention analysis shows that some of the parameters describing the return‐generating process have shifted after the contract respecification by the futures exchange. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:833–850, 2001     

Modeling discontinuous periodic conditional volatility: Evidence from the commodity futures market

This paper examines a wide variety of models that allow for complex and discontinuous periodic variation in conditional volatility. The value of these models (including augmented versions of existing models) is demonstrated with an application to high frequency commodity futures return data. Their use is necessary, in this context, because commodity futures returns exhibit discontinuous intraday and interday periodicities in conditional volatility. The former of these effects is well documented for various asset returns; however, the latter is unique amongst commodity futures returns, where contract delivery and climate are driving forces. Using six years of high‐frequency cocoa futures data, the results show that these characteristics of conditional return volatility are most adequately captured by a spline‐version of the periodic generalized autoregressive conditional heteroscedastic (PGARCH) model. This model also provides superior forecasts of future return volatility that are robust to variation in the loss function assumed by the user, and are shown to be beneficial to users of Value‐at‐Risk (VaR) models. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:805–834, 2004     

The limits to stock index arbitrage: Examining S&P 500 futures and SPDRS

This study examines factors affecting stock index spot versus futures pricing and arbitrage opportunities by using the S&P 500 cash index and the S&P 500 Standard and Poor's Depository Receipt (SPDR) Exchange‐Traded Fund (ETF) as “underlying cash assets.” Potential limits to arbitrage when using the cash index are the staleness of the underlying cash index, trading costs, liquidity (volume) issues of the underlying assets, the existence of sufficient time to execute profitable arbitrage transactions, short sale restrictions, and the extent to which volatility affects mispricing. Alternatively, using the SPDR ETF as the underlying asset mitigates staleness and trading cost problems as well as the effects of volatility associated with the staleness of the cash index. Minute‐by‐minute prices are compared over different volatility levels to determine how these factors affect the limits of S&P 500 futures arbitrage. Employing the SPDR as the cash asset examines whether a liquid tradable single asset with low trading costs can be used for pricing and arbitrage purposes. The analysis examines how long mispricing lasts, the impact of volatility on mispricing, and whether sufficient volume exists to implement arbitrage. The minute‐by‐minute liquidity of the futures market is examined using a new transaction volume futures database. The results show that mispricings exist regardless of the choice of the underlying cash asset, with more negative mispricings for the SPDR relative to the S&P 500 cash index. Furthermore, mispricings are more frequent in high‐ and mid‐volatility months than in low‐volatility months and are associated with higher volume during high‐volatility months. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1182–1205, 2008     

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     

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.     

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     

The drift factor in biased futures index pricing models: A new look

The presence of bias in index futures prices has been investigated in various research studies. Redfield ( 11 ) asserted that the U.S. Dollar Index (USDX) futures contract traded on the U.S. Cotton Exchange (now the FINEX division of the New York Board of Trade) could be systematically arbitraged for nontrivial returns because it is expressed in so‐called “European terms” (foreign currency units/U.S. dollar). Eytan, Harpaz, and Krull ( 4 ) (EHK) developed a theoretical factor using Brownian motion to correct for the European terms and the bias due to the USDX index being expressed as a geometric average. Harpaz, Krull, and Yagil ( 5 ) empirically tested the EHK index. They used the historical volatility to proxy the EHK volatility specification. Since 1990, it has become more commonplace to use option‐implied volatility for forecasting future volatility. Therefore, we have substituted option implied volatilities into EHK's correction factor and hypothesized that the correction factor is “better” ex ante and therefore should lead to better futures model pricing. We tested this conjecture using twelve contracts from 1995 through 1997 and found that the use of implied volatility did not improve the bias correction over the use of historical volatility. Furthermore, no matter which volatility specification we used, the model futures price appeared to be mis‐specified. To investigate further, we added a simple naïve δ based on a modification of the adaptive expectations model. Repeating the tests using this naïve “drift” factor, it performed substantially better than the other two specifications. Our conclusion is that there may be a need to take a new look at the drift‐factor specification currently in use. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:579–598, 2002     

The stock closing calland futures price behavior: Evidence from the Taiwan futures market

This study examines the behavior of futures prices around stock market close before and after changes to the batching period of the stock closing call. On July 1, 2002, the Taiwan Stock Exchange expanded the length of the batching period roughly 10‐fold, from an average of 30 seconds to 5 minutes. This change presents an opportunity to analyze how a stock closing method affects the behavior of index futures prices. Empirical results indicate that an increase in the length of the batching period affects the return volatility and trading volume of index futures contracts around stock market close. Furthermore, preclose stock returns have a great impact on extended futures returns when the batching period of the stock closing call is long. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:1003–1019, 2007     

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