THE STOCHASTIC VOLATILITY MODEL OF BARNDORFF‐NIELSEN AND SHEPHARD IN COMMODITY MARKETS

We consider the non‐Gaussian stochastic volatility model of Barndorff‐Nielsen and Shephard for the exponential mean‐reversion model of Schwartz proposed for commodity spot prices. We analyze the properties of the stochastic dynamics, and show in particular that the log‐spot prices possess a stationary distribution defined as a normal variance‐mixture model. Furthermore, the stochastic volatility model allows for explicit forward prices, which may produce a hump structure inherited from the mean‐reversion of the stochastic volatility. Although the spot price dynamics has continuous paths, the forward prices will have a jump dynamics, where jumps occur according to changes in the volatility process. We compare with the popular Heston stochastic volatility dynamics, and show that the Barndorff‐Nielsen and Shephard model provides a more flexible framework in describing commodity spot prices. An empirical example on UK spot data is included.     

Regime switching and cointegration tests of the efficiency of futures markets

Many researchers have found that spot and futures prices are not cointegrated in some commodity markets, or they are cointegrated but not with a cointegrating vector (1, −1). One interpretation is that disturbances to excess returns have a unit root persistence, which implies that spot and futures prices do not move together one-for-one in the long run. To provide an alternative explanation for this finding, this article proposes a regime switching model of spot prices that can be viewed in the same framework as Fama and French (1988). Based on this model, Monte Carlo experiments are performed to show that tests for cointegration and estimates of the cointegrating vector are likely to be biased when a sample contains infrequent changes in regime. Taking these shifts into account, the null hypothesis that spot and futures prices are cointegrated and move together one-for-one in the long run can no longer be rejected. © 1998 John Wiley & Sons, Inc. Jrl Fut Mark 18:871–901, 1998     

Do futures prices help forecast the spot price?

This study proposes a futures‐based unobserved components model for commodity spot prices. Prices quoted at the same time incorporate the same information, but are affected differently, resulting in the different shapes of futures curves. This model utilizes information from part of the futures curve to improve forecasting accuracy of the spot price. Applying this model to oil market data, I find that the model forecasts outperform the literature benchmark (the no‐change forecast) and futures prices forecasts in multiple dimensions, with smaller average error variation over the sample period and higher chance of smaller absolute error in each period.     

Mean Reversion and the Comovement of Equilibrium Spot and Futures Prices: Implications from Alternative Data‐Generating Processes

The comovements of spot and futures prices are characterized by six binary variables, including the term structure curvature of futures prices. These variables are used to uniquely identify 48 possible comovement patterns. Among them, 24 cases are associated with mean reversion, which is defined as a state when spreads between futures and spot prices are shrinking. These pattern frequencies are then calculated on a daily basis with the futures prices of 10 commodities, including precious metal, agricultural, and financial commodities. The results are further compared to simulation output from three data‐generating processes: a bivariate pure random walk, a mixed random walk with first‐order autoregression (AR(1)), and an error‐correction representation. The mean‐reverting frequencies for all 10 commodities are about 50%. Around half of the time, spot and futures prices are moving toward each other, and the rest of the time they move in the same direction. The symmetry of these results implies that the existence of substantial shocks originated from futures markets; thus, this is consistent with the risk premium view of futures trading. Also, although all simulation models produce similar mean‐reversion frequencies, the patterns of comovements of spot and futures prices are different, and the price dynamics depend heavily on whether the market is dominant contango or backwardation. Furthermore, the error‐correction model outperforms the random‐walk model for agricultural commodities, and the mixed random walk with AR(1) is hardly distinguishable from the pure random walk. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:769–796, 2001     

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     

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     

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 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.     

A STRUCTURAL RISK‐NEUTRAL MODEL FOR PRICING AND HEDGING POWER DERIVATIVES

We develop a structural risk‐neutral model for energy market modifying along several directions the approach introduced in Aïd et al. In particular, a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contracts on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closed‐form formulae for futures prices and semiexplicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.     

A first look at the empirical relation between spot and futures electricity prices in the United States

In this article we investigate the statistical properties of wholesale electricity spot and futures prices traded on the New York Mercantile Exchange for delivery at the California–Oregon Border. Using daily data for the years 1998 and 1999, we find that many of the characteristics of the electricity market can be viewed to be broadly consistent with efficient markets. The futures risk premium for 6‐month futures contracts is estimated to be 0.1328% per day or about 4% per month. Using a GARCH specification, we estimate minimum variance hedge ratios for electricity futures. Finally, we study the dynamic relation between spot and futures prices using an Exponential GARCH model and between the spot and futures returns series using a vector autoregression. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:931–955, 2003     

The forward pricing function of the shipping freight futures market

This article investigates the unbiasedness hypothesis of futures prices in the freight futures market. Being the only market whose underlying asset is a service, it sets it apart from other markets investigated so far in the literature. Cointegration techniques, employed to examine this hypothesis, indicate that futures prices one and two months before maturity are unbiased forecasts of the realized spot prices, whereas a bias exists in the three-months futures prices. This mixed evidence is in agreement with studies in other markets and suggests that the acceptance or rejection of unbiasedness depends on the idiosyncrasies of the market under investigation and on the time to maturity of the contract. Despite the existence of a bias in the three-months prices, futures prices for all maturities are found to provide forecasts of the realized spot prices that are superior to forecasts generated from error correction, ARIMA, exponential smoothing, and random walk models. Hence it appears that users of the BIFFEX market receive accurate signals from the futures prices (regarding the future course of cash prices) and can use the information generated by these prices to guide their physical market decisions. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 353–376, 1999     

Hedging multiple price and quantity exposures

We examined the general hedging problem faced by a global portfolio manager or a pure exporting multinational firm. Most hedging models assume that these economic agents hold only a single asset in the spot market and are exposed only to a single source of price–quantity uncertainty. Such models are less relevant to many financial and exporting firms that face multiple sources of risk. In this study, we developed a general hedging model that explicitly recognizes that these hedgers are faced with multiple price and quantity uncertainties. Our model takes advantage of the full correlation structure of changes in spot prices, quantities, and forward prices. We performed simulations of the hedging model for a firm with two pairs of price and quantity exposures to demonstrate potential gains in hedging efficiency and effectiveness. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:145–172, 2001     

A NOTE ON NONAFFINE SOLUTIONS OF TERM STRUCTURE EQUATIONS WITH APPLICATIONS TO POWER EXCHANGES

A method based on transformations of well-known solutions of term structure equations is presented in order to incorporate Martin Barlow's spot price model for electricity into a model for future prices on electricity. The setting for the evolution of term structures is chosen in the spirit of Da Prato and Zabczyk.     

Asymmetric volatility of basis and the theory of storage

The theory of storage states that the marginal convenience yield on inventory falls at a decreasing rate as inventory increases. Previous literature has tested this hypothesis using the so‐called “direct test” approach, which employs a direct measurement of inventory levels, or the “indirect test” approach, which examines the relative variation of spot and futures prices and the relative variation of negative basis to positive basis as alternative proxies for inventory levels. The rationale behind the indirect test is based on the hypothesis that futures prices are less variable than spot prices when inventory is low, and have similar variability when inventory is high. The authors propose a “unified test” of the theory of storage that incorporates aspects of both direct and indirect tests in an ARMAX‐asymmetric GARCH model framework. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:399–418, 2005     

The relationship between spot and futures prices: Evidence from the crude oil market

This article examines the relationship between the spot and futures prices of WTI crude oil using a sample of daily data. Linear causality testing reveals that futures prices lead spot prices, but nonlinear causality testing reveals a bidirectional effect. This result suggests that both spot and futures markets react simultaneously to new information. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 175–193, 1999     

我国黄金期货市场套期保值功能的实证研究

本文选取2008年1月9日至2010年12月31日之间期货价格和现货价格数据,运用传统回归模型(OLS)、双变量向量自回归模型(B-VAR)、误差修正套期保值模型(ECM)、误差修正GARCH模型(EC-GARCH)对样本数据进行平稳性和协整关系检验,在估计最小风险套期保值比率的基础上发现:(1)我国黄金期货市场运行三年多来,通过黄金期货市场进行套期保值是有效的,可以较为明显地降低参与者面临的价格波动风险;(2)在具体进行套期保值操作时,应该根据套期保值时限长短的不同和预期效果的差异,采用不同的模型来合理确定自身的套期保值比例。在此基础上,本文提出了相关政策建议。     

Asymmetric information in commodity futures markets: Theory and empirical evidence

This article examines the theoretical and empirical implications of asymmetric information in commodity futures markets. In particular, it formulates and tests a theoretical model that recognizes two distinct categories of traders: hedgers, who participate in both spot and futures markets, and speculators, who participate only in the futures market. Speculators are assumed to possess differential information about the realized values of selected random variables. Multiperiod futures market equilibria are derived under competitive conditions, and the ability of futures markets to forecast changes in equilibrium spot market prices are examined. The key variable is shown to be the randomness and informational asymmetry in the aggregate supply by participating hedgers in the spot market, whose absence turns out to be the major determinant of the revelation of informational asymmetry. Moreover, under the assumption of independence of error forecasts for prices and spot market supplies, it is shown that futures market equilibrium ends up with linear expressions for prices and futures contract volumes. These linear expressions are then used to develop empirically testable models. The main empirical implications in these models revolve around the role of the basis as a predictor of future spot price changes. The paper provides an empirical investigation of these implications, using three commodities traded on the Winnipeg Commodity Exchange (WCE). © 1998 John Wiley & Sons, Inc. Jrl Fut Mark 18:803–825, 1998     

Extreme volatility,speculative efficiency,and the hedging effectiveness of the oil futures markets

This study investigates the efficiency of the New York Mercantile Exchange (NYMEX) Division light sweet crude oil futures contract market during recent periods of extreme conditional volatility. Crude oil futures contract prices are found to be cointegrated with spot prices and unbiased predictors of future spot prices, including the period prior to the onset of the Iraqi war and until the formation of the new Iraqi government in April 2005. Both futures and spot prices exhibit asymmetric volatility characteristics. Hedging performance is improved when asymmetries are accounted for. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:61–84, 2007     

A note on the superiority of the OLS hedge ratio

Suppose that spot and futures prices are generated from an error‐correction model. This note demonstrates that, although the OLS model is misspecified, it provides a hedge ratio that usually outperforms the hedge ratio derived from the correct error‐correction model. The opposite result is possible only when the postsample incurs a major structural change from the estimation sample. ©2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:1121–1126, 2005     

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.