موانع در ساختار نوسانات بازار آتی نفت خام : شواهد جدید

This paper analyses the volatility structure of commodity derivatives markets. The model encompasses hump-shaped, unspanned stochastic volatility, which entails a finite-dimensional affine model for the commodity futures curve and quasi-analytical prices for options on commodity futures. Using an extensive database of crude oil futures and futures options spanning 21 years, we find the presence of hump-shaped, partially spanned stochastic volatility in the crude oil market. The hump shaped feature is more pronounced when the market is more volatile, and delivers better pricing as well as hedging performance under various dynamic factor hedging schemes.

Commodity derivatives serve the very important role of helping to manage the volatility of commodity prices. Apart from hedgers, the volatility of commodity prices is also of keen interest to speculators, who have become more dominant in these markets in recent years, see Barone-Adesi et al. (2010) and Cifarelli and Paladino (2010). However, these derivatives have their own volatility, of which the understanding and management is of paramount importance. In this paper, we will provide a tractable model for this volatility, and carry out empirical analysis for the most liquid commodity derivative market, namely the crude oil market. The model used in this paper focuses directly on the volatility of derivatives. It is set up under the Heath et al. (1992) framework that treats the entire term structure of futures prices as the primary modelling element. Due to the standard feature that commodity futures prices are martingales under the risk-neutral measure, the model is completely identified by the volatility of futures prices and the initial forward curve. We model this volatility as a multifactor stochastic volatility, which may be partially unspanned by the futures contracts. Spot commodity prices are uniquely determined without the need to specify the dynamics of the convenience yield. Option prices can be obtained quasi-analytically1 and complex derivative prices can be determined via simulation. Commodity derivatives have been previously studied under the Heath et al. (1992) framework. However, previous works such as those of Miltersen and Schwartz (1998), Clewlow and Strickland (2000) and Miltersen (2003) as well as the convenience yield models of Gibson and Schwartz (1990) and Korn (2005) were restricted to deterministic volatility. Schwartz and Trolle (2009a) extended the literature significantly by providing empirical evidence that stochastic volatility models are superior to the deterministic volatility models as the latter are not capable of accommodating the very important feature of unspanned volatility in the commodities markets. In addition, Clewlow and Strickland (1999) have demonstrated that convenience yield models can be converted to forward models which are a class of models employed in our study. However, there are two differences between this paper and the Schwartz and Trolle (2009a) paper. First, Schwartz and Trolle (2009a) start by modelling the spot commodity and convenience yield. Convenience yield is unobservable and therefore its modelling adds complexity to model assumptions and estimation. Moreover, sensitivity analysis has to rely on applying shocks to this unobserved convenience yield, which makes it less intuitive. Second, the volatility function in the Schwartz and Trolle (2009a) paper has an exponential decaying form, predicting that long term contracts will always be less volatile than short term contracts. Our model, on the other hand, uses a hump-shaped volatility (which can be reduced to an exponential decaying one), and therefore allows for increasing volatility at the short end of the curve. The model in this paper falls under the generic framework provided by Andersen (2010) for the construction of Markovian models for commodity derivatives. As an extension to his work, we provide full results for models that allow for hump-shaped, unspanned stochastic volatility. A hump is an important factor in other markets, such as interest rate markets, see for example Litterman et al. (1991), Dai and Singleton (2000) and Bekaert et al. (2001). However, limited evidence exists in the crude oil market. As far as we are aware, this feature has only been studied in the working paper version of Schwartz and Trolle (2009a). It is reported that a hump shaped volatility function had been tried, but resulted in very similar estimates and almost indistinguishable price performance compared to the exponential volatility function. We will re-examine the volatility structure of the crude oil derivatives market. We use a larger panel dataset of crude oil futures and options traded on the NYMEX, spanning 21 years from 1 January 1990 to 31 December 2010. We find that a three-factor stochastic volatility model works well. Two of the volatility functions have a hump shape that cannot be captured by the exponential decaying specification. We also find that the hump shaped volatility matters a lot more when the market is volatile than when the market is relatively stable. The extent to which the volatility can be spanned by futures contracts varies over time, with the lowest spanning being in the recent period of 2006–2010. The fact that volatility in the market cannot be spanned by futures contracts highlights the importance of options for hedging purposes. We analyse the hedging of straddle contracts, the pricing of which is highly sensitive to volatility. Given the multifactor nature of the model, factor hedging is employed. Factor hedging has been used successfully for deterministic and local volatility models,2 such as in Clewlow and Strickland (2000) or Fan et al. (2003). We expand the method to hedge the random shocks coming from stochastic volatility. We show that the hedging performance increases dramatically when options contracts are added to the hedging instrument set. The hedging performance is measured under various different factor hedging schemes, from delta-neutral to delta-vega and delta-gamma neutral. An alternative approach to the HJM framework is modelling the spot commodity prices directly. A representative list of relevant literature would include Gibson and Schwartz (1990), Litzenberger and Rabinowitz (1995), Schwartz (1997), Hilliard and Reis (1998), Casassus and Collin-Dufresne (2005) and Fusai et al. (2008). These models have been successful in depicting essential and critical features of distinct commodity market prices, for instance, the mean-reversion of the agricultural commodity market, the seasonality of the natural gas market, the spikes and regime switching of the electricity market and the inverse leverage in the oil market. The disadvantage of the spot commodity models is the requirement to specify and estimate the unobservable convenience yield. The futures prices are then determined endogenously. In addition, some spot commodity models cannot accommodate unspanned stochastic volatility,3 a feature that can be naturally embedded in HJM models. The paper is organized as follows. Section 2 presents a generalised unspanned stochastic volatility model for pricing commodity derivatives within the HJM framework. Section 3 describes and analyses the data for crude oil derivatives and explains the estimation algorithm. Section 4 presents the results. Section 5 examines the hedging performance. Section 6 concludes. Technical details are presented in the Appendix.

A multi-factor stochastic volatility model for commodity futures curves within the Heath et al. (1992) framework is proposed. The model aims to capture themain characteristics of the volatility structure in commodity futures markets. The model accommodates exogenous stochastic volatility processes that may be partially unspanned by futures contracts. We specify a hump component for the volatility of the futures curves, which can generate a finite dimensional Markovian forward model. The resulting model is highly tractable with quasianalytical prices for European options on futures contracts. The model was fitted to an extensive database of crude oil futures prices and option prices traded in the NYSE over 21 years. We find supporting evidence for three volatility factors, two of which exhibit a hump. This provides new evidence on the volatility structure in crude oil futures markets, which has been traditionally modelled with exponentially declining volatility functions. Finally, by using hedge ratios implied by the proposed unspanned hump-shaped stochastic volatility model, the hedging performance of factor hedging schemes is examined. The results favour the proposed model compared to a model with only exponential decaying volatility. The currentwork suggests newdevelopments in commoditymarket modelling. Firstly, it will be interesting to verify the existence of humps in the volatility structure of other commodities. Our methodology is generic and can be adapted to any commodity futuresmarket. Additionally, the current model can be adjusted to accommodate stochastic convenience yield and stochastic interest rates. This direction has the potential to provide useful insights on the features of convenience yields in commodity markets.