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.     

Evaluating Natural Resource Investments under Different Model Dynamics: Managerial Insights

We focus on factors that drive the dynamics of commodity prices. We highlight the capital budgeting implications of three highly‐cited, nested, multi‐factor models for commodity prices that have been successful in empirical investigations. Competing assumptions regarding commodity prices and their convenience yields can account for differences close to 40% on average, and in excess of 60% in cases, in the valuation of typical natural resource investments. These value differences are found to increase with the maturity and the intrinsic value of the investment, and also with the level and the volatility of the resource's convenience yield. Resources such as oil or copper, that are used for production purposes, usually exhibit high and volatile convenience yields; thus our findings should be more relevant for decision‐makers in such sectors.     

Commodity price modelling that matches current observables: a new approach

We develop a stochastic model of the spot commodity price and the spot convenience yield such that the model matches the current term structure of forward and futures prices, the current term structure of forward and futures volatilities, and the inter-temporal pattern of the volatility of the forward and futures prices. We let the underlying commodity price be a geometric Brownian motion and we let the spot convenience yield have a mean-reverting structure. The flexibility of the model, which makes it possible to simultaneously achieve all these goals, comes from allowing the volatility of the spot commodity price, the speed of mean-reversion parameter, the mean-reversion parameter, and the diffusion parameter of the spot convenience yield all to be time-varying deterministic functions.     

Optionality and Daily Dynamics of Convenience Yield Behavior: An Empirical Analysis

In this article I empirically examine the daily convenience yield behavior for six commodity markets (crude oil, heating oil, gasoline, wheat, corn, and copper). The results illustrate that convenience yield behavior can be statistically explained within an option pricing framework. However, because one of the assumptions of the standard call option formula is not fully satisfied by the observed convenience yield series, an alternative option—exchange option—may be more appropriate for modeling the daily convenience yield behavior. Furthermore, I empirically test two hypotheses on convenience yield behavior. The results confirm the assertion that the convenience yield is increasing in marginal production costs. In addition, the findings offer limited support for the hypothesis that the convenience yield is decreasing in the serial autocorrelation of spot prices. The observed switch in the sign of regression coefficients as the order of autocorrelation increases is attributed to the probable presence of mean reversion in these markets.     

多资产大宗商品期权定价研究

基于大宗商品收益率与便利收益服从均值回复过程的假设,建立带协整效应的多资产大宗商品期权定价模型,求解多资产大宗商品期权价格的解析解,将大宗商品期权定价推广到更一般情况.结果表明:标的资产收益率增加,期权价格上升,替代品期权价格下降;标的资产的便利收益增加,期权价格下降,相应替代品的期权价格上升.     

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.     

Commodity betas with mean reverting output prices

This paper provides a theoretical derivation of commodity beta (stock price sensitivity to commodity price) using a contingent-claim model. The model incorporates operating leverage, financial leverage, costly financial distress, and mean reverting commodity prices; and highlights the important role played by the speed of reversion of the commodity price. It is used to identify theoretically the main determinants of commodity beta. Commodity beta is predicted to be an increasing function of the operating and financial leverage of the firm, and a decreasing function of the company’s tax rate and the level, volatility and speed of reversion of the commodity price. Empirical tests with a sample of gold mining firms provide support for these predictions, particularly the new implications of the model (the effect of the commodity price’s speed of reversion and the company’s tax rate).     

Pricing Commodity Spread Options with Stochastic Term Structure of Convenience Yields and Interest Rates

The purpose of this research is to provide a valuation formula for commodity spread options. Commodity spread options are options written on the difference of the prices (spread) of two commodities. From the aspect of commodity contingent claims, it is considered that commodity spread options are difficult to evaluate with accuracy because of the existence of the convenience yield. Hence, the model of the convenience yield is the key factor to price commodity spread options. We use the concept of future convenience yields to develop the model that enriches the stochastic behavior of convenience yield. We also introduce Heath-Jarrow-Morton interest rate model to the valuation framework. This general model not only captures the mean reverting feature of the convenience yield, but also allows us to handle a very wide range of shape that the term structure of convenience yield can take. Therefore our model provides various types of models. The numerical analysis presented in this paper provides some unique features of commodity spread options in contrast to normal options. These characteristics have never been addressed in previous studies. Moreover, it suggests that the existing model overprice commodity spread options through neglecting the effect of interest rates.     

The Traditional Hedging Model Revisited with a Nonobservable Convenience Yield

This article examines the hedging of constrained commodity positions with futures contracts. We extend the study of Adler and Detemple (1988a, 1988b) to include a partial information framework where the convenience yield is not observable. As a consequence, futures prices depend on investor's beliefs regarding the value of the convenience yield, and every component of the hedge is impacted by these beliefs. We achieve a decomposition of the demand that clarifies the impact on the optimal hedge of the beliefs, the spot price and the risk‐free rate as well as the hedging horizon.     

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.     

Capturing the risk premium of commodity futures: The role of hedging pressure

We construct long–short factor mimicking portfolios that capture the hedging pressure risk premium of commodity futures. We consider single sorts based on the open interests of hedgers or speculators, as well as double sorts based on both positions. The long–short hedging pressure portfolios are priced cross-sectionally and present Sharpe ratios that systematically exceed those of long-only benchmarks. Further tests show that the hedging pressure risk premiums rise with the volatility of commodity futures markets and that the predictive power of hedging pressure over cross-sectional commodity futures returns is different from the previously documented forecasting power of past returns and the slope of the term structure.     

Economic linkages across commodity futures: Hedging and trading implications

We investigate cross-market trading dynamics in futures contracts written on seemingly unrelated commodities that are consumed by a common industry. On the Tokyo Commodity Exchange, we find such evidence in natural rubber (NR), palladium (PA) and gasoline (GA) futures markets. The automobile industry is responsible for more than 50% of global demand for each of these commodities. VAR estimation reveals short-run cross-market interaction between NR and GA, and from NR to PA. Cross-market influence exerted by PA is felt in longer dynamics, with PA volatility (volume) affecting NR (GA) volume (volatility). Our findings are robust to lag-specification, volatility measure, and consistent with full BEKK-GARCH estimation results. Further analysis, which benchmarks against silver futures market, TOCOM index and TOPIX transportation index, confirms that our results are driven by a common industry exposure, and not a commodity market factor. A simple trading rule that incorporates short-run GA and long-run PA dynamics to predict NR return yields positive economic profit. Our study offers new insights into how commodity and equity markets relate at an industry level, and implications for multi-commodity hedging.     

Commodity volatility breaks

Volatility is a key determinant of derivative prices and optimal hedge ratios. This paper examines whether there are structural breaks in commodity spot return volatility using an iterative cumulative sum of squares procedure and then uses GARCH (1,1) to model volatility during each regime.The main empirical finding is the very limited evidence of commodity volatility breaks during the recent financial crisis. This suggests commodity return volatility was not exceptionally high during the recent financial crisis compared to the 1985–2010 sample period as a whole. For many commodities there are multiple idiosyncratic breaks in volatility; this suggests commodity specific supply or demand factors are important determinants of volatility. The empirical results overall are consistent with the view that commodities are too diverse to be considered as an asset class. Finally, we find commodity volatility persistence remains very high for many commodity returns even after structural breaks are accounted for.     

Predictability of currency carry trades and asset pricing implications

This paper studies the time series predictability of currency carry trades, constructed by selecting currencies to be bought or sold against the US dollar, based on forward discounts. Changes in a commodity index, currency volatility and, to a lesser extent, a measure of liquidity predict in-sample the payoffs of dynamically re-balanced carry trades, as evidenced by individual and joint p-values in monthly predictive regressions at horizons up to six months. Predictability is further supported through out-of-sample metrics, and a predictability-based decision rule produces sizable improvements in the Sharpe ratios and skewness profile of carry trade payoffs. Our evidence also indicates that predictability can be traced to the long legs of the carry trades and their currency components. We test the theoretical restrictions that an asset pricing model, with average currency returns and the mimicking portfolio for the innovations in currency volatility as risk factors, imposes on the coefficients in predictive regressions.     

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.     

Decomposing swap spreads

We analyze a six-factor model for Treasury bonds, corporate bonds, and swap rates and decompose swap spreads into three components: a convenience yield from holding Treasuries, a credit risk element from the underlying LIBOR rate, and a factor specific to the swap market. The convenience yield is by far the largest component of spreads. There is a discernible contribution from credit risk as well as from a swap-specific factor with higher variability which in certain periods is related to hedging activity in the mortgage-backed security market. The model also sheds light on the relation between AA hazard rates and the spread between LIBOR rates and General Collateral repo rates and on the level of the riskless rate compared to swap and Treasury rates.     

Volatility forecasting in the Chinese commodity futures market with intraday data

Given the unique institutional regulations in the Chinese commodity futures market as well as the characteristics of the data it generates, we utilize contracts with three months to delivery, the most liquid contract series, to systematically explore volatility forecasting for aluminum, copper, fuel oil, and sugar at the daily and three intraday sampling frequencies. We adopt popular volatility models in the literature and assess the forecasts obtained via these models against alternative proxies for the true volatility. Our results suggest that the long memory property is an essential feature in the commodity futures volatility dynamics and that the ARFIMA model consistently produces the best forecasts or forecasts not inferior to the best in statistical terms.     

Can interest rate volatility be extracted from the cross section of bond yields?

Most affine models of the term structure with stochastic volatility predict that the variance of the short rate should play a ‘dual role’ in that it should also equal a linear combination of yields. However, we find that estimation of a standard affine three-factor model results in a variance state variable that, while instrumental in explaining the shape of the yield curve, is essentially unrelated to GARCH estimates of the quadratic variation of the spot rate process or to implied variances from options. We then investigate four-factor affine models. Of the models tested, only the model that exhibits ‘unspanned stochastic volatility’ (USV) generates both realistic short rate volatility estimates and a good cross-sectional fit. Our findings suggest that short rate volatility cannot be extracted from the cross-section of bond prices. In particular, short rate volatility and convexity are only weakly correlated.     

Investment shocks and the commodity basis spread

I identify a “slope” factor in the cross section of commodity futures returns: high-basis commodity futures have higher loadings on this factor than low-basis commodity futures. Combined with a level factor (an index of commodity futures), this slope factor explains most of the average excess returns of commodity futures portfolios sorted by basis. More importantly, I find that this factor is significantly correlated with investment shocks, which represent the technological progress in producing new capital. I investigate a competitive dynamic equilibrium model of commodity production to endogenize this correlation. The model reproduces the cross-sectional futures returns and many asset pricing tests.     

The Pricing Formula for Commodity-Linked Bonds with Stochastic Convenience Yields and Default Risk

At the maturity, the owner of a commodity-linked bond has the right to receive the face value of the bond and the excess amount of spot market value of the reference commodity bundle over the prespecified exercise price. This payoff structure is an important characteristic of the commodity-linked bonds. In this paper, we derive closed pricing formulae for the commodity-linked bonds. We assume that the reference commodity price and the value of the firm (bonds' issuer) follow geometric Brownian motions and that the net marginal convenience yield and interest rate follow Ornstein–Uhlenbech processes. In the appendix, we derive pricing formulae for bonds which are the same as the above commodity-linked bonds, except that the reference commodity price in the definition of the payoff at the maturity is replaced by the value of a special asset which depends on the convenience yield. This revised version was published online in August 2006 with corrections to the Cover Date.