Approximation for convenience yield in commodity futures pricing

The pricing of commodity futures contracts is important both for professionals and academics. It is often argued that futures prices include a convenience yield, and this article uses a simple trading strategy to approximate the impact of convenience yields. The approximation requires only three variables—underlying asset price volatility, futures contract price volatility, and the futures contract time to maturity. The approximation is tested using spot and futures prices from the London Metals Exchange contracts for copper, lead, and zinc with quarterly observations drawn from a 25‐year period from 1975 to 2000. Matching Euro‐Market interest rates are used to estimate the risk‐free rate. The convenience yield approximation is both statistically and economically important in explaining variation between the futures price and the spot price after adjustment for interest rates. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:1005–1017, 2002     

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     

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     

A dimension-invariant cascade model for VIX futures

We propose a new stochastic volatility model by allowing for a cascading structure of volatility components. The model, under a minor assumption, allows us to add as many components as desired with no additional parameters, effectively defeating the curse of dimensionality often encountered in traditional models. We derive a semi-closed-form solution to the VIX futures price, and find that our six-factor model with only six parameters can closely fit spot VIX and VIX futures prices from 2004 to 2015 and produce out-of-sample pricing errors of magnitudes similar to those of in-sample errors.     

Time‐varying market price of risk in the crude oil futures market

In this study, a three‐factor model of crude oil prices is estimated, which incorporates a time‐varying market price of risk. The model is able to accurately capture the term structure of futures prices with evidence suggesting that risk premiums in the crude oil market are time‐varying. Using the cross‐section of futures prices, we estimate a time‐series of the market price of risk in the crude oil market implied by the model. We find that the risk premiums in the crude oil market are driven by the same risk factors as equity and bond markets. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:779–807, 2011     

Do volatility determinants vary across futures contracts? Insights from a smoothed Bayesian estimator

We apply a new Bayesian approach to multiple‐contract futures data. It allows the volatility of futures prices to depend upon physical inventories and the contract's time to delivery—and it allows those parametric effects to vary over time. We investigate price movements for lumber contracts over a 13‐year period and find a time‐varying negative relationship between lumber inventories and lumber futures price volatility. The Bayesian approach leads to different conclusions regarding the size of the inventory effect than does the standard method of parametric restrictions across contracts. The inventory effect is smaller for the most recent contracts when the inventory levels are larger. In contrast, the Bayesian approach does not lead to substantively different conclusions about the time‐to‐delivery effect than do traditional classical methods. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:257–277, 2010     

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     

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 Valuation of Equity Futures on the Tokyo Stock Exchange: 1920–1923

The futures price of an asset should depend on the spot price of that asset, the interest rate, cash flows during the contract term, the convenience yield, and storage costs. Despite many tests of the spot–future relation for commodities in historical periods, there have been no tests of this historical relation for equities. We price single‐stock equity futures on the Tokyo Stock Exchange between 1920 and 1923 and find that mispricing is considerably worse than in contemporary U.S. markets, after adjusting for (unavoidable) asynchronous data issues. The main cause of the mispricing is short‐sales constraints, rather than investor naivety. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark     

Causality in the VIX futures market

This study examines the price‐discovery function and information efficiency of a fast growing volatility futures market: the Chicago Board of Option Exchange VIX futures market. A linear Engle–Granger cointegration test with an error correction mechanism (ECM) shows that during the full sample period, VIX futures prices lead spot VIX index, which implies that the VIX futures market has some price‐discovery function. But a modified Baek and Brock nonlinear Granger test detects bi‐directional causality between VIX and VIX futures prices, suggesting that both spot and futures prices react simultaneously to new information. Quarter‐by‐quarter investigations show that, on average, the estimated parameters are not significantly different from zero, thus providing further evidence supporting information efficiency in the VIX futures market. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

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.     

A No‐Arbitrage Fractional Cointegration Model for Futures and Spot Daily Ranges

The no‐arbitrage relation between futures and spot prices implies an analogous relation between futures and spot daily ranges. The long‐memory features of the range‐based volatility estimators are analyzed, and fractional cointegration is tested in a semi‐parametric framework. In particular, the no‐arbitrage condition is used to derive a long‐run relationship between volatility measures and to justify the use of a fractional vector error correction model (FVECM) to study their dynamic relationship. The out‐of‐sample forecasting superiority of FVECM, with respect to alternative models, is documented. The results highlight the importance of incorporating the long‐run equilibrium in volatilities to obtain better forecasts, given the information content in the volatility of futures prices. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:77–102, 2013     

Hedging long‐term commodity risk

This study focuses on the problem of hedging longer‐term commodity positions, which often arises when the maturity of actively traded futures contracts on this commodity is limited to a few months. In this case, using a rollover strategy results in a high residual risk, which is related to the uncertain futures basis. We use a one‐factor term structure model of futures convenience yields in order to construct a hedging strategy that minimizes both spot‐price risk and rollover risk by using futures of two different maturities. The model is tested using three commodity futures: crude oil, orange juice, and lumber. In the out‐of‐sample test, the residual variance of the 24‐month combined spot‐futures positions is reduced by, respectively, 77%, 47%, and 84% compared to the variance of a naïve hedging portfolio. Even after accounting for the higher trading volume necessary to maintain a two‐contract hedge portfolio, this risk reduction outweighs the extra trading costs for the investor with an average risk aversion. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:109–133, 2003     

Drift matters: An analysis of commodity derivatives

This article presents a reduced‐form, two‐factor model to price commodity derivatives, which generalizes the model by Schwartz and Smith (2000). The model allows for two mean‐reverting stochastic factors and therefore implies that spot and futures prices can be stationary. An empirical study for the crude oil market tests the new model. Out‐of‐sample pricing and hedging results for futures and forwards show that the new model dominates the nonstationary model by Schwartz and Smith in the following sense: It works equally well for short‐term contracts but leads to major improvements for long‐term contracts. This finding is particularly relevant for typical applications like the valuation of commodity‐linked real assets with long maturities. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:211–241, 2005     

Valuing real options using implied binomial trees and commodity futures options

A real option on a commodity is valued using an implied binomial tree (IBT) calibrated using commodity futures options prices. Estimating an IBT in the absence of spot options (the norm for commodities) allows real option models to be calibrated for the first time to market‐implied probability distributions for commodity prices. In addition, the existence of long‐dated futures options means that good volatility estimates may now be incorporated into capital budgeting evaluations of real options projects with long planning horizons. An example is given using gold futures options and a real option to extract gold from a mine. A detailed out‐of‐sample test is included that shows how IBT option pricing errors evolve on subtrees emanating from future levels of the underlying asset. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:203–226, 2007     

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     

Natural gas prices and the gas storage report: Public news and volatility in energy futures markets

This study examines the short‐term volatility of natural gas prices through an examination of the intraday prices of the nearby natural gas futures contract traded on the New York Mercantile Exchange. The influence on volatility of what many regard as a key element of the information set influencing the natural gas market is investigated. Specifically, we examine the impact on natural gas futures price volatility of the Weekly American Gas Storage Survey report compiled and issued by the American Gas Association during the period January 1, 1999 through May 3, 2002 and the subsequent weekly report compiled and issued by the U.S. Energy Information Administration after May 6, 2002. We find that the weekly gas storage report announcement was responsible for considerable volatility at the time of its release and that volatility up to 30 minutes following the announcement was also higher than normal. Aside from these results, we document pronounced price volatility in this market both at the beginning of the day and at the end of the day and offer explanations for such behavior. Our results are robust to the manner in which the mean percentage change in the futures price is estimated and to correlation of these changes both within the day and across days. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:283–313, 2004     

The pricing of electricity futures: Evidence from the European energy exchange

This study investigates the pricing of electricity futures at the European Energy Exchange (EEX) over the period 2002 through 2004. To calculate theoretical contract values, the reduced‐form models of J. J. Lucia and E. S. Schwartz (2002) are used, and a thorough empirical analysis by means of an out‐of‐sample test is conducted for both one‐ and two‐factor models, incorporating a constant non‐zero price of risk. Although the models are proven to capture all basic spot market characteristics and provide an accurate in‐the‐sample fit to observed futures prices, the forecasting performance is subject to biases. For instance, it was found that the relative mispricing depends on both the spot price level and the remaining time‐to‐maturity of the futures contracts. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:387–410, 2007     

Cointegration,unbiased expectations,and forecasting in the BIFFEX freight futures market

The relationship between freight cash and futures prices is investigated using cointegration econometrics. Results illustrate that the BIFFEX futures market is unbiased, and hence efficient for the current, one, two, and quarterly contract horizons. Since the futures contract is based on an index of various shipping routes, which has undergone several changes since its inception, stability in the relationship between the spot and futures rates is investigated using rolling cointegration techniques. Results indicate that the futures contract appears to have become more efficient over time in predicting the spot rate, and that the decrease in trading volume found in the BIFFEX market is not driven by a lack of efficiency in this market. Rather, the decrease in futures trading might be attributed to the growth rate of the freight forward market. This article incorporates the long‐run cointegrating relationships between cash and futures prices in a forecasting model and compares the forecasting performance of this model with several alternatives. It is found that while the futures price is the best predictor of future spot rates for the current‐month contract, time‐series models can outperform the futures contract at longer contract horizons. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:545–571, 2000.     

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.