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.     

Pricing and hedging in the freight futures market

In this article, we consider the pricing and hedging of single‐route dry bulk freight futures contracts traded on the International Maritime Exchange. Thus far, this relatively young market has received almost no academic attention. In contrast to many other commodity markets, freight services are non‐storable, making a simple cost‐of‐carry valuation impossible. We empirically compare the pricing and hedging accuracy of a variety of continuous‐time futures pricing models. Our results show that the inclusion of a second stochastic factor significantly improves the pricing and hedging accuracy. Overall, the results indicate that the Schwartz and Smith ( 2000 ) two‐factor model provides the best performance. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:440–464, 2011     

Derivative pricing model and time‐series approaches to hedging: A comparison

This research compares derivative pricing model and statistical time‐series approaches to hedging. The finance literature stresses the former approach, while the applied economics literature has focused on the latter. We compare the out‐of‐sample hedging effectiveness of the two approaches when hedging commodity price risk using futures contracts. For various methods of parameter estimation and inference, we find that the derivative pricing models cannot out‐perform a vector error‐correction model with a GARCH error structure. The derivative pricing models' unpalatable assumption of deterministically evolving futures volatility seems to impede their hedging effectiveness, even when potentially foresighted optionimplied volatility term structures are employed. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:613–641, 2005     

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     

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     

Principal components analysis for correlated curves and seasonal commodities: The case of the petroleum market

This article presents a family of term structure models that can be applied to value contingent claims in multicommodity and seasonal markets. We apply the framework to the futures contracts on crude and heating oils trading on NYMEX. We show how to deal with the problem of having to value products depending on the “whole” market, such as spread options on contracts on a single commodity maturing at different times (time‐spreads) or spread options on the added value of the products derived from the raw commodity (crack spreads). Also, we show how to build term structure models for a commodity that experiences seasonality, such as heating oil. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:1019–1035, 2002     

Migration of price discovery in semiregulated derivatives markets

This study investigates the information content of futures option prices when the underlying futures price is regulated and the futures option price is not. The New York Board of Trade (NYBOT) provides the empirical setting for this regulatory mismatch. Many commodity derivatives markets regulate the prices of all derivatives on a single underlying commodity simultaneously. Some exchanges, including the NYBOT, regulate only their futures contracts, leaving the options on these futures contracts unregulated. This study takes a particular interest in the option‐implied futures price when the observed futures price is locked limit. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:209–241, 2006     

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     

Examining futures price changes and volatility on the trading day after a limit‐lock day

This paper examines the effect that price limits have on futures prices by testing what happens to price changes and volatility on the trading day following a limit‐lock day. The results show evidence that prices continue to rise on average the day after an up‐limit day. In addition, limits appear to influence price volatility for some but not all of the futures contracts. However, since the findings vary across the different commodity futures contracts, it is likely that limits do not directly impact price volatility. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:445–466, 2000     

Long‐term Futures Curves and Seasonal Structures of Wheat in the European Union and the United States

A two‐factor affine theoretical model is used to estimate the long‐term futures curves for wheat in the European Union and the United States, as represented by the Euronext and CME markets, respectively. The CME futures curve exhibits a long‐term equilibrium; in contrast, the Euronext futures curve does not show a tendency for futures to revert to a long‐term equilibrium value. The estimated seasonality is relatively similar for both markets. However, the seasonal minimum and maximum points in the futures curve occur one to two months later for Euronext compared to the CME. More importantly, the futures curve for Euronext has a much more marked seasonality than the CME futures curve. Credible intervals of the futures curves are also estimated. The width clearly increases for longer maturities, but it does so much faster for Euronext than for the CME. For long‐maturity futures, variability in the parameter estimates (as opposed to the residual errors) accounts for most of the width of the credible intervals, especially for Euronext. The proposed model can be used to price long‐term futures options, long‐term price insurance, and long‐term swaps, among other applications. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:1118–1142, 2013     

Why do expiring futures and cash prices diverge for grain markets?

In recent years, cash and futures prices have failed to converge at expiration for selected corn, soybean, and wheat commodity contracts. This lack of convergence raises questions about the effectiveness of arbitrage activities, and increases concerns about the usefulness of these contracts for hedging. We describe the delivery process for these contracts, and show that it embeds a valuable real option on the long side—the option to exchange the deliverable for another futures contract. As the relative volatility of cash and futures prices increases, this option increases in value, which disconnects the cash market from the deliverable instrument in a futures contract. Our estimates of this option's value show that it may create significant price divergence. We parameterize an option pricing model using data on these three commodities from 2000 to 2008 and show that the option model fits closely to recent episodes of non‐convergence, which lends support to the importance of real option effects. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark

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THE EFFECT OF TRADING FUTURES ON SHORT SALE CONSTRAINTS

It is commonly believed that the trading of futures on a commodity enables the market to overcome short selling constraints on the spot commodity itself. This belief is embedded in the notion that trading strategies involving futures contracts enable traders to replicate the payoffs as if they were short the spot commodity. The purpose of this paper is to investigate this common belief in a general arbitrage‐free semimartingale financial model with trading in futures and a short selling prohibition on the spot commodity. We show via various examples that, in general, this common belief is incorrect. Furthermore, we provide a set of sufficient conditions, albeit very restrictive, under which the common belief is true.     

Is it time to reduce the minimum tick sizes of the E‐mini futures?

On the Chicago Mercantile Exchange (CME), so‐called “E‐mini” index futures contracts trade on the electronic GLOBEX trading system alongside the corresponding full‐size contracts that trade on the open outcry floor. This paper finds that the current minimum tick sizes of the E‐mini S&P 500 and E‐mini Nasdaq‐100 futures contracts act as binding constraints on the bid‐ask spreads by not allowing the spreads to decline to competitive levels. We also find that, while exchange locals trade very actively on GLOBEX, they do not tend to act as liquidity suppliers. Taken together, our empirical results suggest that it is time for the CME to consider decreasing the minimum tick sizes of the S&P 500 and Nasdaq‐100 E‐mini futures contracts. A tick size reduction is likely to result in lower trading costs in the E‐mini futures markets. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:79–104, 2005     

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     

Delivery risk and the hedging role of options

Multiple delivery specifications exist on nearly all commodity futures contracts. Sellers typically are allowed to deliver any of several grades of the underlying commodity and at any of several locations. On the delivery day, the futures price as such needs not converge to the spot price of the par‐delivery grade at the par‐delivery location, thereby imposing an additional delivery risk on hedgers. This article derives the optimal hedging strategy for a risk‐averse hedger in the presence of delivery risk. In particular, it is shown that the hedger optimally uses options on futures for hedging purposes. This article provides a rationale for the hedging role of options when futures markets allow for multiple delivery specifications. © 2002 Wiley Periodicals, Inc. Jrl Fut Mark 22:339–354, 2002     

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     

The use of term structure information in the hedging of mortgage‐backed securities

This article examines the importance of term structure variables in the hedging of mortgage‐backed securities (MBS) with Treasury futures. Koutmos, G., Kroner, K., and Pericli, A. (1998) find that the optimal hedge ratio is time varying; we determine the effect of yield levels and slopes on this variation. As these variables are closely tied with mortgage refinancing, intuition suggests them to be relevant determinants of the hedge ratio. It was found that a properly specified model of the time varying hedge ratio that excludes the level and slope of the yield curve from the information set would provide similar out‐of‐sample hedging results to a model in which term structure information is included. Thus, both the level of interest rates and the slope of the yield curve are unimportant variables in determining the empirically optimal hedge ratio between MBS and Treasury futures contracts. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:661–678, 2005     

Bernoulli speculator and trading strategy risk

In a continuous‐time model of a complete information economy, we examine the case of a “pure” speculator who chooses to trade only on forward or futures contracts written on interest‐rate‐sensitive instruments. Assuming logarithmic utility, we assess whether his strategy exhibits the same structure as when he uses primitive assets only. It turns out that when interest rates follow stochastic processes, as in the model of Heath, Jarrow, and Morton (1992), where the instantaneous forward rate is driven by an arbitrary number of factors, the speculative trading strategy involving forwards exhibits an extra term vis‐a‐vis the one using futures or primitive assets. This extra term, different from a Merton–Breeden dynamic hedge, is novel and can be interpreted as a hedge against an “endogenous risk,” namely the interest‐rate risk brought about by the optimal trading strategy itself. Thus, only the strategy using futures (or the cash assets themselves) involves a single speculative term, even for the Bernoulli speculator. This result illustrates another major aspect of the marking to market feature that differentiates futures and forwards, and thus has some bearing on the issue of the optimal design of financial contracts. Real financial markets being, in fact, incomplete, the additional “endogenous” risk associated with forwards cannot be hedged perfectly. Since using futures eliminates the latter, risk‐averse agents will find them attractive in relation to forward contracts, other things being equal. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20: 507–523, 2000     

Long memory in commodity futures volatility: A wavelet perspective

The authors reexamine the volatility of agricultural commodity futures for evidence of fractional integration, providing new empirical results and extending the extant literature in important dimensions. First, they utilize two relatively new estimators based on wavelets, which are generally superior to, for example, the popular estimator by J. Geweke and S. Porter‐Hudak (GPH; 1983) and exact maximum likelihood estimators (MLEs) on the basis of mean squared error (MSE). Second, they provide simulations to contrast their point estimates with those obtained by a fractionally integrated GARCH (generalized autoregressive conditional heteroscedasticity) model. Third, they conduct a wavelet coef.cient decomposition of futures volatility. They .nd that futures volatilities display the self‐similarity property consistent with long memory and that futures volatilities exhibit persistent long memory with .nite unconditional variance. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:411–437, 2007     

Comparing alternative assumptions on the term structure of futures prices: Reply

Donald Lien and Yan Wang (this issue) suggest an alternative test for different specifications of the term structure of futures prices, as used in our recently published paper in The Journal of Futures Markets. Our paper (Y. V. Veld‐Merkoulova and F. A. de Roon, 2003) focuses on developing optimal hedging strategies in case sufficiently long‐term futures contracts are not available (or not actively traded) on the exchange. One of the preliminary steps underlying this strategy was to compare linear and log‐linear term structures of futures prices in order to choose an appropriate specification. Although this is not the main issue of our paper, it is certainly important to use the correct econometric procedure in testing alternative model specifications. The results found by Lien and Wang do not contradict our conclusion that a linear term structure of futures yields is superior to a linear term structure of futures prices. However, as we point out here, the tests suggested by Lien and Wang are not without flaws. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:1101–1104, 2004