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.     

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.     

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.     

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.     

Spot price modeling and the valuation of electricity forward contracts: The role of demand and capacity

We propose a model where wholesale electricity prices are explained by two state variables: demand and capacity. We derive analytical expressions to price forward contracts and to calculate the forward premium. We apply our model to the PJM, England and Wales, and Nord Pool markets. Our empirical findings indicate that volatility of demand is seasonal and that the market price of demand risk is also seasonal and positive, both of which exert an upward (seasonal) pressure on the price of forward contracts. We assume that both volatility of capacity and the market price of capacity risk are constant and find that, depending on the market and period under study, it could either exert an upward or downward pressure on forward prices. In all markets we find that the forward premium exhibits a seasonal pattern. During the months of high volatility of demand, forward contracts trade at a premium. During months of low volatility of demand, forwards can either trade at a relatively small premium or, even in some cases, at a discount, i.e. they exhibit a negative forward premium.     

Transmission of Stock Returns and Volatility between the U.S. and Japan: Evidence from the Stock Index Futures Markets

In this paper, we examine the nature of transmission of stock returns and volatility between the U.S. and Japanese stock markets using futures prices on the S&P 500 and Nikkei 225 stock indexes. We use stock index futures prices to mitigate the stale quote problem found in the spot index prices and to obtain more robust results. By employing a two-step GARCH approach, we find that there are unidirectional contemporaneous return and volatility spillovers from the U.S. to Japan. Furthermore, the U.S.'s influence on Japan in returns is approximately four times as large as the other way around. Finally, our results show no significant lagged spillover effects in both returns and volatility from the Osaka market to the Chicago market, while a significant lagged volatility spillover is observed from the U.S. to Japan. This revised version was published online in August 2006 with corrections to the Cover Date.     

INTRADAY MARKET BEHAVIOR AND THE EXTENT OF FEEDBACK BETWEEN S&P 500 FUTURES PRICES AND THE S&P 500 INDEX

In this study we empirically examine the intraday lead/lag relation between S&P 500 futures prices and the S&P 500 index, and whether daily market characteristics are associated with changes in the relation. We estimate daily Geweke measures of feedback and regress time series of these measures on daily price volatility and volume characteristics. Results indicate that the contemporaneous price relation is substantive and that measures of contemporaneous feedback are positively associated with the daily range of the futures price. The primary implication is that the relation between cash and futures prices becomes stronger as futures price volatility increases. As volatility increases, information is being impounded at a faster rate so that futures and equity markets operate more closely as one market. Large futures price moves, by themselves, are not responsible for breakdowns in the stock-futures price relation.     

Price dynamics in refined petroleum spot and futures markets

This article characterizes the spot and futures price dynamics of two important physical commodities, gasoline and heating oil. Using a non-linear error correction model with time-varying volatility, we demonstrate many new results. Specifically, the convergence of spot and futures prices is asymmetric, non-linear, and volatility inducing. Moreover, spreads between spot and futures prices explain virtually all spot return volatility innovations for these two commodities, and spot returns are more volatile when spot prices exceed futures prices than when the reverse is true. Furthermore, there are volatility spillovers from futures to spot markets (but not the reverse), futures volatility shocks are more persistent than spot volatility shocks, and the convergence of spot and futures prices is asymmetric and non-linear. These results have important implications. In particular, since the theory of storage implies that spreasd vary with fundamental supply and demand factors, the strong relation between spreads and volatility suggests that these fundamentals — rather than trading induced noise — are the primary determinants of spot price volatility. The volatility spillovers, differences in volatility persistence, and lead-lag relations are consistent with the view that the futures market is the primary locus of informed trading in refined petroleum product markets. Finally, our finding that error correction processes may be non-linear, asymmetric, and volatility inducing suggests that traditional approaches to the study of time series dynamics of variables that follow a common stochastic trend that ignore these complexities may be mis-specified.     

Modelling electricity prices: a time change approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.     

Model Specification and Risk Premia: Evidence from Futures Options

This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.     

Electricity Prices and Power Derivatives: Evidence from the Nordic Power Exchange

This paper examines the importance of the regular pattern in the behavior of electricity prices, and its implications for the purposes of derivative pricing. We analyze the Nordic Power Exchange's spot, futures, and forward prices. We conclude that the seasonal systematic pattern throughout the year, in particular, is of crucial importance in explaining the shape of the futures/forward curve. Moreover, in the context of the oneand two factor models analyzed in this paper, a simple sinusoidal functionis adequate in order capture the seasonal pattern of the features and forwardcurve directly implied by the seasonal behavior of spot electricity prices.     

Modelling gold futures: should the level of speculation inform our choice of variables?

Prior literature provides conflicting evidence about the impact of speculation on gold futures returns, volatility, and the relationship between market fundamentals and prices. In this paper, we exploit trade volume information to determine the most appropriate family of factors to adopt when modelling gold futures. Using the Disaggregated Commitment of Traders report, we find that extreme levels of speculation are informative in that they signify a shift in the relative modelling accuracy of macroeconomic and latent factors. A simple composite prediction framework, incorporating the changing level of speculation, empirically demonstrates the uncovered phenomenon and offers improved predictive accuracy for gold futures prices. Furthermore, our findings are shown to be robust to alternative latent and macroeconomic model specifications.     

TIME‐VARYING VOLATILITY IN CANADIAN AND U.S. STOCK INDEX AND INDEX FUTURES MARKETS: A MULTIVARIATE ANALYSIS

We use a multivariate generalized autoregressive heteroskedasticity model (M‐GARCH) to examine three stock indexes and their associated futures prices: the New York Stock Exchange Composite, S&P 500, and Toronto 35. The North American context is significant because markets in Canada and the United States share similar structures and regulatory environments. Our model allows examination of dependence in volatility as it captures time variation in volatility and cross‐market influences. Estimated time variation in volatility is significant, and the volatilities are highly positively correlated. Yet, we find that the correlation in North American index and futures markets has declined over time.     

Testing Greeks and price changes in the S&P 500 options and futures contract: A regression analysis

We use a regression model to test observed price changes with Greeks as regressors. Greeks are computed using implied volatility, price-change implied volatility and historical volatility. We find sufficient evidence to reject model Greeks as unbiased responses to underlying price as well as sufficient evidence that the American version of binomial model results in biased estimates of price changes. We use options on the S&P 500 futures contracts and their underlying. We also evaluate the frequency of “wrong signs.” Call prices and their underlying move in the opposite direction almost 10 percent of the time.     

The Samuelson hypothesis in futures markets: An analysis using intraday data

This paper considers the Samuelson hypothesis, which argues that the futures price volatility increases as the futures contract approaches its expiration. Utilizing intraday data from 20 futures markets in six futures exchanges, we find strong support for the Samuelson hypothesis in agricultural futures. However, the Samuelson hypothesis does not hold for other futures contracts. We also provide supporting evidence that the ‘negative covariance’ hypothesis is the key factor for the empirical support of the Samuelson hypothesis. In addition, our findings remain largely unaltered even after we control for seasonality and liquidity effects.     

Oil Futures Prices in a Production Economy with Investment Constraints

We document a new stylized fact, that the relationship between the volatility of oil futures prices and the slope of the forward curve is nonmonotone and has a V-shape. This pattern cannot be generated by standard models that emphasize storage. We develop an equilibrium model of oil production in which investment is irreversible and capacity constrained. Investment constraints affect firms' investment decisions and imply that the supply elasticity changes over time. Since demand shocks must be absorbed by changes in prices or changes in supply, time-varying supply elasticity results in time-varying volatility of futures prices. Estimating this model, we show it is quantitatively consistent with the V-shape relationship between the volatility of futures prices and the slope of the forward curve.     

New No-arbitrage Conditions and the Term Structure of Interest Rate Futures

Interest rate futures are basic securities and at the same time highly liquid traded objects. Despite this observation, most models of the term structure of interest rate assume forward rates as primary elements. The processes of futures prices are therefore endogenously determined in these models. In addition, in these models hedging strategies are based on forward and/or spot contracts and only to a limited extent on futures contracts. Inspired by the market model approach of forward rates by Miltersen, Sandmann, and Sondermann (J Finance 52(1); 409–430, 1997), the starting point of this paper is a model of futures prices. Using, as the input to the model, the prices of futures on interest related assets new no-arbitrage restrictions on the volatility structure are derived. Moreover, these restrictions turn out to prevent an application of a market model based on futures prices.     

Testing for constant hedge ratios in commodity markets: a multivariate GARCH approach

We develop a new multivariate generalized ARCH (GARCH) parameterization suitable for testing the hypothesis that the optimal futures hedge ratio is constant over time, given that the joint distribution of cash and futures prices is characterized by autoregressive conditional heteroskedasticity (ARCH). The advantage of the new parameterization is that it allows for a flexible form of time-varying volatility, even under the null of a constant hedge ratio. The model is estimated using weekly corn prices. Statistical tests reject the null hypothesis of a constant hedge ratio and also reject the null that time variation in optimal hedge ratios can be explained solely by deterministic seasonality and time to maturity effects.     

INDEX FUTURES OPTIONS IN AUSTRALIA -AN EMPIRICAL FOCUS ON VOLATILITY

The futures option contract on the Australian All Ordinaries Share Price Index is a relatively new hybrid security that ought to enhance the richness and potential efficiency of security markets. This paper considers the problems of valuing it using the theoretical price of a futures-style option. It was found that there was little consistency between theoretical prices using a number of historical estimates and observed market prices, either intertemporally or between in-the-money or out-of-the-money calls. Further, implied volatility was found to be a decaying function of time and, except at times of instability, did not predict the ex ante futures volatility as well as historic volatility.     

The seasonality of gold prices in China does the risk-aversion level matter?

This article aims to investigate the seasonality of gold prices at the Shanghai Gold Exchange over the 2002–2016 period. Our contributions rely in the distinction between risk-averse and risk-seeking investors regarding their investment strategies. The results show the existence of positive Monday and January effects. However, the Monday effect is more suitable to risk-seeking investors while the January effect is more suitable to risk-averse investors in bearish periods only. A robustness check shows that the Monday effect does not hold on gold futures prices. These results indicate the importance to consider the risk-aversion level of investors in seasonal investment strategies.