Time-variations in commodity price jumps

In this paper, we study jumps in commodity prices. Unlike assumed in existing models of commodity price dynamics, a simple analysis of the data reveals that the probability of tail events is not constant but depends on the time of the year, i.e. exhibits seasonality. We propose a stochastic volatility jump–diffusion model to capture this seasonal variation. Applying the Markov Chain Monte Carlo (MCMC) methodology, we estimate our model using 20 years of futures data from four different commodity markets. We find strong statistical evidence to suggest that our model with seasonal jump intensity outperforms models featuring a constant jump intensity. To demonstrate the practical relevance of our findings, we show that our model typically improves Value-at-Risk (VaR) forecasts.     

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.     

The Stochastic Seasonal Behaviour of Natural Gas Prices

Previous studies have explored the seasonal behaviour of commodity prices as a deterministic factor. This paper goes further by proposing a general (n+2m)‐factor model for the stochastic behaviour of commodity prices, which nests the deterministic seasonal model by Sorensen (2002) . We consider seasonality as a stochastic factor, with n non‐seasonal and m seasonal factors. The non‐seasonal factors are as defined in Schwartz (1997) , Schwartz and Smith (2000) and Cortazar and Schwartz (2003) . The seasonal factors are trigonometric components generated by stochastic processes. The model has been applied to the Henry Hub natural gas futures contracts listed by NYMEX. We find that models allowing for stochastic seasonality outperform standard models with deterministic seasonality. We obtain similar results with other energy commodities. Moreover, we find that stochastic seasonality implies that the volatility of futures returns follows a seasonal pattern. This result has important implications in terms of option pricing.     

中国大宗商品期货市场定价机制研究

大宗商品期货市场是我国资本市场的重要组成部分,其定价有效性关系到投资者套期保值和价格发现等功能的实现。本文对国际前沿研究中常用的定价因子进行全面系统梳理,并对这些因子对我国商品期货合约收益率的解释和预测能力进行检验。在此基础上,本文构建了适用于我国大宗商品期货市场的包含市场、基差以及基差动量的三因子定价模型。进一步研究表明,基于大宗商品存储理论和现货存货数据构建的投资组合收益率可以被本文三因子模型有效解释,验证了经典的存储理论在我国的适用性。此外,本文对基差与基差动量两个重要因子的经济学意义进行了阐释。本文研究为进一步厘清大宗商品期货市场定价机制提供了一定参考。     

A continuous time equilibrium model of forward prices and futures prices in a multigood economy

This paper is a theoretical investigation of equilibrium forward and futures prices. We construct a rational expectations model in continuous time of a multigood, identical consumer economy with constant stochastic returns to scale production. Using this model we find three main results. First, we find formulas for equilibrium forward, futures, discount bond, commodity bond and commodity option prices. Second, we show that a futures price is actually a forward price for the delivery of a random number of units of a good; the random number is the return earned from continuous reinvestment in instantaneously riskless bonds until maturity of the futures contract. Third, we find and interpret conditions under which normal backwardation or contango is found in forward or futures prices; these conditions reflect the usefulness of forward and futures contracts as consumption hedges.     

Long Memory In Futures Prices

This paper tests for fractional roots in the futures prices for selected commodities, foreign currencies, and stock indexes. The fractional testing method is the spectral regression method suggested by Geweke and Porter-Hudak (1983). The empirical results suggest the presence of a fractional exponent in the differencing process for several commodity and foreign currency futures prices. The returns series for these commodities and currencies exhibit long range positive dependence. However, differencing of exact order one is sufficient for the stock index futures prices. Implications are drawn concerning theoretical and econometric modeling and price forecasting.     

The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging

In this article we compare three models of the stochastic behavior of commodity prices that take into account mean reversion, in terms of their ability to price existing futures contracts, and their implication with respect to the valuation of other financial and real assets. The first model is a simple one-factor model in which the logarithm of the spot price of the commodity is assumed to follow a mean reverting process. The second model takes into account a second stochastic factor, the convenience yield of the commodity, which is assumed to follow a mean reverting process. Finally, the third model also includes stochastic interest rates. The Kalman filter methodology is used to estimate the parameters of the three models for two commercial commodities, copper and oil, and one precious metal, gold. The analysis reveals strong mean reversion in the commercial commodity prices. Using the estimated parameters, we analyze the implications of the models for the term structure of futures prices and volatilities beyond the observed contracts, and for hedging contracts for future delivery. Finally, we analyze the implications of the models for capital budgeting decisions.     

A two-factor cointegrated commodity price model with an application to spread option pricing

In this paper, we propose an easy-to-use yet comprehensive model for a system of cointegrated commodity prices. While retaining the exponential affine structure of previous approaches, our model allows for an arbitrary number of cointegration relationships. We show that the cointegration component allows capturing well-known features of commodity prices, i.e., upward sloping (contango) and downward sloping (backwardation) term-structures, smaller volatilities for longer maturities and an upward sloping correlation term structure. The model is calibrated to futures price data of ten commodities. The results provide compelling evidence of cointegration in the data. Implications for the prices of futures and options written on common commodity spreads (e.g., spark spread and crack spread) are thoroughly investigated.     

Limits to arbitrage and hedging: Evidence from commodity markets

We build an equilibrium model of commodity markets in which speculators are capital constrained, and commodity producers have hedging demands for commodity futures. Increases in producers' hedging demand or speculators' capital constraints increase hedging costs via price-pressure on futures. These in turn affect producers' equilibrium hedging and supply decision inducing a link between a financial friction in the futures market and the commodity spot prices. Consistent with the model, measures of producers' propensity to hedge forecasts futures returns and spot prices in oil and gas market data from 1979 to 2010. The component of the commodity futures risk premium associated with producer hedging demand rises when speculative activity reduces. We conclude that limits to financial arbitrage generate limits to hedging by producers, and affect equilibrium commodity supply and prices.     

No-arbitrage conditions for storable commodities and the modeling of futures term structures

One distinguishable feature of storable commodities is that they relate to two markets: cash market and storage market. This paper proves that, if no arbitrage exists in the storage-cash dual markets, the commodity convenience yield has to be non-negative. However, classical reduced-form models for futures term structures could allow serious arbitrages due to the high volatility of the convenience yield. To avoid negative convenience yield, this paper proposes a semi-affine arbitrage-free model, which prices futures analytically and fits futures term structures reasonably well. Importantly, our model prices commodity-related contingent claims (such as calendar spread options) quite differently with classical models.     

On the seasonality in the implied volatility of electricity options

Seasonality is an important topic in electricity markets, as both supply and demand are dependent on the time of the year. Clearly, the level of prices shows a seasonal behaviour, but not only this. Also, the price fluctuations are typically seasonal. In this paper, we study empirically the implied volatility of options on electricity futures, investigate whether seasonality is present and we aim at quantifying its structure. Although typically futures prices can be well described through multi-factor models including exponentially decreasing components, we do not find evidence of exponential behaviour in our data set. Generally, a simple linear shape reflects the squared volatilities very well as a curve depending on the time to maturity. Moreover, we find that the level of volatility exhibits clear seasonal patterns that depend on the delivery month of the futures. Furthermore, in an out-of-sample analysis we compare the performance of several implementations of seasonality in the one-factor framework.     

Commodity markets dynamics: What do cross-commodities over different nearest-to-maturities tell us?

In this paper we investigate cross-commodity futures markets connectedness over different nearest-to-maturities. We thus implement time and time-frequency estimations for two constructed baskets of commodities, classified based on common delivery months. Using daily data spanning the period 1995–2020, we provide a set of stylized facts on the extent to which commodity markets are integrated or segmented. More specifically, our results show that the total connectedness is broadly insensitive to maturity. However, after 2008 financial crisis, the connectedness among commodity futures prices increases when the maturity increases. Furthermore, the overall connectedness amplifies during crises periods compared to tranquil periods. Moreover, certain pairwise markets are comparatively highly linked such as crude oil and heating oil, wheat and corn, corn and soybean, and soybean and soybean oil. The results also demonstrate that crude oil and heating oil are net transmitters all the time and across maturities, while natural gas, gold, and wheat are net receivers all the time and across maturities. More interestingly, the frequency decomposition reveals that most of periods of high total connectedness are driven mostly by high frequency components, which may indicate that commodity markets process information rapidly, except for the COVID-19 crisis period where total connectedness has been driven by lower frequency components.     

A Model of Financialization of Commodities

We analyze how institutional investors entering commodity futures markets, referred to as the financialization of commodities, affect commodity prices. Institutional investors care about their performance relative to a commodity index. We find that all commodity futures prices, volatilities, and correlations go up with financialization, but more so for index futures than for nonindex futures. The equity‐commodity correlations also increase. We demonstrate how financial markets transmit shocks not only to futures prices but also to commodity spot prices and inventories. Spot prices go up with financialization, and shocks to any index commodity spill over to all storable commodity prices.     

Seasonality and the valuation of commodity options

Price movements in many commodity markets exhibit significant seasonal patterns. However, given an observed futures price, a deterministic seasonal component at the price level is not relevant for the pricing of commodity options. In contrast, this is not true for the seasonal pattern observed in the volatility of the commodity price. Analyzing an extensive sample of soybean, corn, heating oil and natural gas options, we find that seasonality in volatility is an important aspect to consider when valuing these contracts. The inclusion of an appropriate seasonality adjustment significantly reduces pricing errors in these markets and yields more improvement in valuation accuracy than increasing the number of stochastic factors.     

The role of speculation in international futures markets on commodity prices

The role of futures contracts on spot prices has been one of the key focus areas of research since the recent surge in commodity prices and increase in the volatility of commodity returns. However, no consensus arises from this literature, and hence it is difficult to link the use of futures contracts in agricultural commodities by non-hedgers and the growing food insecurity within developing countries. The purpose of this paper is to highlight causal relationships from futures contracts to spot prices of underlying assets, namely agricultural commodities. As research that focus on exchange-traded funds do not provide any clear conclusions, we focus on the imbalance between short- and long-open positions, this imbalance being caused by the exchange traded funds’ participation in futures markets. In this paper, we estimate relationships between financial variables including indicators for speculation in futures markets and the returns of cocoa, corn, soybean, wheat, coffee, rice, and sugar on a weekly basis from 1998 to 2013. Significant results lead to Granger-causality tests that in turn validate the hypothesis of a positive impact of speculation in futures markets to returns on the underlying commodities.     

国外著名商品指数编制比较及其对我国的启示

商品期货指数,是商品期货的金融化,是期货市场发展到一定阶段的必然产物。为了更好编制和开发我国商品指数及其衍生品,本文对国外著名商品指数的编制方法从编制目的和原则、品种选择、权重设计、合约选择、指数计算等五个方面进行全方位比较研究,进而提出编制我国相关商品指数的总体思路。     

A four-factor stochastic volatility model of commodity prices

The number of factors driving the uncertain dynamics of commodity prices has been a central consideration in financial literature. While the majority of empirical studies relies on the assumption that up to three factors are sufficient to explain all relevant uncertainty inherent in commodity spot, futures, and option prices, evidence from Trolle and Schwartz (Rev Financ Stud 22(11):4423–4461, 2009b) and Hughen (J Futures Mark 30(2):101–133, 2010) indicates a need for additional risk factors. In this article, we propose a four-factor maximal affine stochastic volatility model that allows for three independent sources of risk in the futures term structure and an additional, potentially unspanned stochastic volatility process. The model principally integrates the insights from Hughen (2010) and Tang (Quant Finance 12(5):781–790, 2012) and nests many well-known models in the literature. It can account for several stylized facts associated with commodity dynamics such as mean reversion to a stochastic level, stochastic volatility in the convenience yield, a time-varying correlation structure, and time-varying risk-premia. In-sample and out-of-sample tests indicate a superior model fit to futures and options data as well as lower hedging errors compared to three-factor benchmark models. The results also indicate that three factors are not sufficient to model the joint dynamics of futures and option prices accurately.     

Pricing rainfall futures at the CME

Many business people such as farmers and financial investors are affected by indirect losses caused by scarce or abundant rainfall. Because of the high potential of insuring rainfall risk, the Chicago Mercantile Exchange (CME) began trading rainfall derivatives in 2011. Compared to temperature derivatives, however, pricing rainfall derivatives is more difficult. In this article, we propose to model rainfall indices via a flexible type of distribution, namely the normal-inverse Gaussian distribution, which captures asymmetries and heavy-tail behaviour. The prices of rainfall futures are computed by employing the Esscher transform, a well-known tool in actuarial science. This approach is flexible enough to price any rainfall contract and to adjust theoretical prices to market prices by using the calibrated market price of risk. The empirical analysis is conducted with US precipitation data and CME futures data providing first results on the market price of risk for rainfall derivatives.     

Calibrating a market model with stochastic volatility to commodity and interest rate risk

Based on the multi-currency LIBOR Market Model, this paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the ‘orthonormal Procrustes’ problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.     

A new methodology for studying intraday dynamics of Nikkei index futures using Markov chains

The empirical study of intraday patterns of stock trading volatilities and bid–ask spreads in the literature depends on assumptions of specific price generating process and may therefore not be robust to distributional assumptions. By creating discrete states that conform more naturally to the way prices are actually quoted in the derivatives and assets markets, we employ a new methodology of Markov chains for studying the intraday dynamics of derivative prices. We apply the method to study the intraday behavior of the Nikkei index futures prices, trading volumes, and spreads. We find some interesting results such as higher probabilities of transitions between larger volatilities at the opening and closing times. The volatility at lunch break is strikingly low. Contrary to most of the literature, the Nikkei intraday bid–ask spread does not show a U-shaped pattern. We offer some explanations.