Stochastic Convenience Yield and the Pricing of Oil Contingent Claims

This paper develops and empirically tests a two-factor model for pricing financial and real assets contingent on the price of oil. The factors are the spot price of oil and the instantaneous convenience yield. The parameters of the model are estimated using weekly oil futures contract prices from January 1984 to November 1988, and the model's performance is assessed out of sample by valuing futures contracts over the period November 1988 to May 1989. Finally, the model is applied to determine the present values of one barrel of oil deliverable in one to ten years time.     

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.     

Commodity derivatives valuation with autoregressive and moving average components in the price dynamics

In this paper, we develop a continuous time factor model of commodity prices that allows for higher-order autoregressive and moving average components. We document the need for these components by analyzing the convenience yield’s time series dynamics. The model we propose is analytically tractable and allows us to derive closed-form pricing formulas for futures and options. Empirically, we estimate a parsimonious version of the general model for the crude oil futures market and demonstrate the model’s superior performance in pricing nearby futures contracts in- and out-of-sample. Most notably, the model substantially improves the pricing of long-horizon contracts with information from the short end of the futures curve.     

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.     

Markov models for commodity futures: theory and practice

The objective of this paper is to develop a generic, yet practical, framework for the construction of Markov models for commodity derivatives. We aim for sufficient richness to permit applications to a broad variety of commodity markets, including those that are characterized by seasonality and by spikes in the spot process. In the first, largely theoretical, part of the paper we derive a series of useful results concerning the low-dimensional Markov representation of the dynamics of an entire term structure of futures prices. Extending previous results in the literature, we cover jump-diffusive models with stochastic volatility as well as several classes of regime-switching models. To demonstrate the process of building models for a specific commodity market, the second part of the paper applies a selection of our theoretical results to the exercise of constructing and calibrating derivatives trading models for USD natural gas. Special attention is paid to the incorporation of empirical seasonality effects in futures prices, in implied volatilities and their ‘smile’, and in correlations between futures contracts of different maturities. European option pricing in our proposed gas model is closed form and of the same complexity as the Black–Scholes formula.     

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。     

Futures and realized cash or settle prices for gold,silver, and copper

This article examines the relation between 15-, 30-, 45-, and 60-day gold, silver, and copper futures, and their realized cash or delivery settle prices, for deliveries on the first, middle, and last business day (FD, MD, and LD, respectively) of the delivery month. Samples of futures prices for gold, silver, and copper, and the realized cash or delivery settle prices, based on fixed maturities for a cross-section of contracts are used, because time-series over fixed maturities are unfeasible owing to the restrictive maturity periods in these contracts. This study contrasts with similar studies for forwards, which are not hampered by this problem (e.g., Baillie and Bollerslev in 1989), studies that use contemporaneous time-series of futures and cash prices to obtain optimal hedge ratios (e.g., Ghosh and Clayton in 1996), and studies that examine for parity between different futures prices (e.g., Franses and Kofman in 1991). Near-term gold, closest to delivery silver and all copper futures (the latter only with settle prices at delivery) are good predictors of the future cash price, except for silver and copper deliveries on the last day of delivery months. These results are consistent with previous studies on short-term cash (Ma in 1985) and futures (Ma and Soenen in 1988) parity between gold and silver, and with Gross's (1988) results for copper. The longer-term gold (45- and 60-day) and silver (30-, 45-, and 60-day) futures reject an unbiased expectations hypothesis. This result is consistent with Leistikow (1990) because cash prices respond less than futures prices in these instances.     

Long term spread option valuation and hedging

This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are in dynamic equilibrium (i.e., cointegrated). The spread exhibits properties different from its two underlying commodity prices and should therefore be modelled directly. This approach offers significant advantages relative to the traditional two price methods since the correlation between two asset returns is notoriously hard to model. In this paper, we propose a two factor model for the spot spread and develop pricing and hedging formulae for options on spot and futures spreads. Two examples of spreads in energy markets – the crack spread between heating oil and WTI crude oil and the location spread between Brent blend and WTI crude oil – are analyzed to illustrate the results.     

Cross-sectional tests of deterministic volatility functions

We study the cross-sectional performance of option pricing models in which the volatility of the underlying stock is a deterministic function of the stock price and time. For each date in our sample of FTSE 100 index option prices, we fit an implied binomial tree to the panel of all European style options with different strike prices and maturities and then examine how well this model prices a corresponding panel of American style options. We find that the implied binomial tree model performs no better than an ad-hoc procedure of smoothing Black–Scholes implied volatilities across strike prices and maturities. Our cross-sectional results complement the time-series findings of Dumas et al. [J. Finance 53 (1998) 2059].     

人民币汇率对国际原油定价的影响和机制

本文基于动态条件相关混频数据抽样模型,研究人民币汇率收益率与国际原油收益率的相关性.结果表明:人民币收益率与国际原油收益率存在一定程度负相关关系,但负相关程度不高;中国原油进口量和制造业采购经理指数上升能提升人民币和原油收益率的负相关程度,面额效应的作用机制存在;国际期货市场投机活跃程度增强能提升人民币和原油收益率的负...     

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.     

Cojumping: Evidence from the US Treasury bond and futures markets

The basis between spot and future prices will be affected by jump behavior in each asset price, challenging intraday hedging strategies. Using formal cojumping tests this paper considers the cojumping behavior of spot and futures prices in high frequency US Treasury data. Cojumping occurs most frequently at shorter maturities and higher sampling frequencies. We find that the probability of cojumping is altered by the presence of an anticipated macroeconomic news announcement. The probability of cojumping is particularly affected by news surprises in non-farm payrolls, CPI, GDP and retail sales. However, the two cojumping tests are also more likely to provide contradictory results in the presence of surprises in non-farm payrolls. On these occasions the market does not clearly signal its short term pricing behavior.     

PRICING CURRENCY OPTIONS UNDER STOCHASTIC INTEREST RATES AND JUMP‐DIFFUSION PROCESSES

We investigate the effects of stochastic interest rates and jumps in the spot exchange rate on the pricing of currency futures, forwards, and futures options. The proposed model extends Bates's model by allowing both the domestic and foreign interest rates to move around randomly, in a generalized Vasicek term‐structure framework. Numerical examples show that the model prices of European currency futures options are similar to those given by Bates's and Black's models in the absence of jumps and when the volatilities of the domestic and foreign interest rates and futures price are negligible. Changes in these volatilities affect the futures options prices. Bates's and Black's models underprice the European currency futures options in both the presence and the absence of jumps. The mispricing increases with the volatilities of interest rates and futures prices. JEL classification: G13     

Hedging Long-Term Forwards with Short-Term Futures: A Two-Regime Approach

In this paper we investigate Metallgesellschafts problem of hedging long-term forwards with short-term futures. Very different hedging strategies have been proposed in the literature. We attribute these differences to the underlying valuation approaches for oil futures and empirically compare five model-based hedging strategies. In particular, we consider a strategy which results from a two-regime pricing model. This continuous-time equilibrium model reflects the observation that prices of oil futures exhibit a very different behavior for low and high oil prices. Our empirical study shows that time diversification is the dominant effect for an effective hedging of long-term oil forwards with short-term futures. JEL classification G13, G30     

Arbitrage-Free Valuation of Exhaustible Resource Firms

We provide an arbitrage-free valuation of exhaustible resource firms through extending the Gibson and Schwartz (1990) model and also the Jamshidian and Fein (1990) solution to valuing an entire petroleum firm based on quoted oil futures. Our solutions are compared to accounting, traditional finance and to stockmarket valuations on a daily basis. An alternative expression of the valuations relative to stockmarket prices is in terms of the time varying implied 'market price' of convenience yield risk. Initial illustrations show that the implied convenience yield risk is not necessarily consistent between stockmarket and derivative market participants. Finally, we calculate the sensitivities of petroleum firm values to changes in oil prices, the convenience yield observable on NYMEX, and oil price volatilities. These partial derivatives show some of the complexities in the dynamic hedging process of using the contingent claims approach to valuing (and hedging) real assets.     

Business Cycles and the Behavior of Metals Prices

The theory of storage says that the marginal convenience yield on inventory falls at a decreasing rate as inventory increases. The authors test this hypothesis by examining the relative variation of spot and futures prices for metals. As the hypothesis implies, futures prices are less variable than spot prices when inventory is low, but spot and futures prices have similar variability when inventory is high. The theory of storage also explains inversions of “normal” futures-spot price relations around business-cycle peaks. Positive demand shocks around peaks reduce metal inventories and, as the theory predicts, generate large convenience yields and price inversions.     

Local parametric analysis of derivatives pricing and hedging

A novel methodology for the analysis of derivatives pricing in incomplete markets is tested empirically. The methodology generates hedge ratios and derivatives prices. They are estimated from the correlation structure between the local co-movements of securities prices. First, the hedge ratios from a parsimonious complete-market model are estimated by fitting locally the changes in the derivatives and the underlying securities prices. Second, derivatives prices are obtained from the locally estimated hedge ratios. The methodology, referred to as local parametric estimation, is tested on a dataset of DAX index options and futures transactions from the computerized German Futures Exchange.     

Oil convenience yields estimated under demand/supply shock

This paper extends the call option model of Milonas and Thomadakis (1997) to estimate oil convenience yields with futures prices. We define the business cycle of a seasonal commodity with demand/supply shocks and find that the convenience yield for crude oil exhibits seasonal behavior. The convenience yield for West Texas Intermediate (WTI) crude oil is the highest in the summer, while that for Brent crude oil is the highest in the winter. This implies that WTI crude oil is more sensitive to high summer demand and that Brent crude oil is more sensitive to shortages in winter supply. Convenience yields are negatively related to the inventory level of the underlying crude oil and positively related to interest rates due to the business cycle. We also show that convenience yields may explain price spread between WTI crude oil and Brent crude oil. Our computed convenience yields are consistent with Fama and French (1988) in that oil prices are more volatile than futures prices at low inventory level, verifying the Samuelson (1965) hypothesis that future prices are less variables than spot prices at lower inventory levels.

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.     

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.