عرضه و تقاضا بر اساس مدل نوسانات قیمت انرژی

This paper proposes a new volatility model for energy prices using the supply–demand relationship, which we call a supply and demand based volatility model. We show that the supply curve shape in the model determines the characteristics of the volatility in energy prices. It is found that the inverse Box–Cox transformation supply curve reflecting energy markets causes the inverse leverage effect, i.e., positive correlation between energy prices and volatility. The model is also used to show that an existing (G)ARCH-M model has the foundations on the supply–demand relationship. Additionally, we conduct the empirical studies analyzing the volatility in the U.S. natural gas prices.

High volatility in price returns often appears in deregulated energy markets. The market participants such as energy producers and distributors always face such high volatility risks from energy markets. As a simple example to illustrate the influence of the volatility, let us think of a thermal power plant procuring natural gas as the fuel from the spot market. Since the natural gas prices are volatile due to the supply and demand in the market, the risk manager in charge of the procurement of natural gas needs to capture the volatility as accurately as possible by using an energy volatility model. The volatility models have been introduced into energy markets because of the needs from energy market participants as described in the above example. Although a lot of volatility models both in continuous and discrete time were developed in financial markets, they are directly applied to the volatility models in energy markets without any adjustment for the energy characteristics. The continuous time models in stock markets, such as Heston model introduced by Heston (1993) and C.E.V. (Constant Elasticity of Variance) models developed by Cox (1975) and extended by Emanuel and MacBeth (1982), are directly used for the models in energy markets (e.g., Eydeland and Wolyniec (2003)). Similarly, the discrete time models such as ARCH, GARCH, and ARCH-M models in Engle (1982), Bollerslev (1986), and Engle et al. (1987) are employed in energy market models in Duffie et al. (1999), Pindyck (2004), and Deaves and Krinsky (1992), respectively. Energy markets may utilize the same volatility models as financial markets. However, the volatility in energy markets is not necessarily the same as the volatility in stock markets. For instance, the “inverse leverage effect”, i.e., volatility increases in prices, often appears in energy markets, while the analyses in stock markets illustrate the opposite relationship between the volatility and the prices. Energy price models have been developed as asset pricing models in commodity markets. Gibson and Schwartz (1990) propose a two-factor model for crude oil in which the log of spot price follows a normal process and the convenience yield follows a mean reverting process. Brennan (1991) also models commodity spot prices and convenience yields as separate stochastic processes with a constant correlation. Schwartz (1997) compares one-, two-, and three-factor models in which the log of spot price follows a simple stochastic process to describe commodity prices. One relevant paper in this class is Kolos and Ronn (2008) where energy forward price returns precisely demonstrate volatility-in-mean effect. While these models work well to describe energy prices, supply and demand relationship of energy is not explicitly structured in the models. This paper proposes the SDV model for energy prices that can accurately demonstrate the characteristics of volatility in energy prices using the supply–demand relationship. We show time series of demand and price for natural gas in the U.S. in Fig. 1. The figure seems to suggest that prices increase in demand, especially in recent years because peaks of prices correspond to peaks of demand. In order to express the curve using a simple model, we introduce an equilibrium price model determined by the inelastic demand curve fluctuating stochastically and the upward-sloping supply curve fixed in a short period of time, what we call a supply and demand based volatility (SDV) model. We show that the SDV model provides twofold characteristics of the volatility in energy price returns. First, the volatility is time varying owing to both the upward-sloping supply curve and demand shocks. Second, the mean of price returns changes with the volatility, which we call “volatility-in-mean effect”, due to the drift term of price returns represented by the function of the volatility. Then, we specify the model by employing the inverse Box–Cox transformation supply curve that can exhibit the drastic slope changes with an appropriate function parameter. It is found that the model can produce both the inverse leverage effect and volatility-in-mean effect. In addition, getting an idea from the volatility-in-mean effect, we investigate the relationship between the discrete time SDV model and (G)ARCH-M model. We show that an existing (G)ARCH-M model has the foundations on the supply–demand relationship. We conduct the empirical analyses on the volatility in the U.S. natural gas prices by using the SDV model and the implications from the model. First, in an effort to examine the existence of the inverse leverage effect in the U.S. natural gas market, we identify the model parameters of the inverse Box–Cox supply curve employing nonlinear least squares. Both monthly equilibrium prices and demands are used for the identification, supposing that the sum of consumption and storage demands approximates to the equilibrium demand. The results illustrate that the extremely large change of the gradient causes the inverse leverage effect in the natural gas market. Second, we examine the existence of the volatility-in-mean effect in the natural gas market by using GARCH(1,1)-M model linked to the discrete time SDV model. It is found that the estimation results support the existence of volatility-in-mean effect. Finally, we empirically investigate the validity of two assumptions in this paper. The first assumption is that the sum of consumption and storage demands approximates to the equilibrium demand. The second assumption is that demand is inelastic to prices in the SDV model. We show that both assumptions hold as the first order approximation. The remainder of this paper is organized as follows. Section 2 proposes a new model for time-varying volatility in energy prices, what we call SDV model and then investigates the relationship between the SDV model and existing (G)ARCH-M models. Section 3 conducts empirical studies on the volatility in the U.S. natural gas prices in attempts to examine the existence of the inverse leverage and volatility-in-mean effects as the characteristics of the price volatility. Section 4 addresses the validity of the assumption on the equilibrium demand. Section 5 identifies the simultaneous equations for the supply and demand curves of the natural gas market by implementing the nonlinear two-stage least square estimation in order to assess the validity of the demand inelasticity assumption. Section 6 concludes and offers the directions for our future research.

This paper has proposed a volatility model, which we call a supply and demand based volatility (SDV) model for energy prices characterized by the supply–demand relationship. We have illustrated that it can produce both the “time-varying volatility” and the “volatility-in-mean effect” that characterize energy prices. Additionally, it has been shown that the SDV model can produce the “inverse leverage effect” often seen in energy markets, supposing that the inverse Box–Cox transformation function represents the supply curve. Moreover, this paper has shown that the existing (G)ARCH-M model has the foundations on the supply–demand relationship, being derived from the discrete time SDV model with some approximation. The empirical studies have analyzed the volatility in the U.S. natural gas prices using the SDV model. First, to examine the existence of the inverse leverage effect: volatility increases in prices, we estimated the inverse Box–Cox transformation parameter for the supply curve using the historical prices and demands. The negative estimate implied that the U.S. natural gas possesses the effect. It is consistent with energy market observations in Eydeland and Wolyniec (2003). Second, we showed that volatility-in-mean effect is statistically significant in GARCH(1,1)-M model. It supports the volatility-in-mean models for energy prices. Finally in order to enhance the SDV model, we have empirically examined the validity of two model assumptions using the natural gas data. First, we have shown that as the first order approximation the sum of consumption and storage demands approximates to the equilibrium one by comparing the inverse Box–Cox transformation parameter estimated from the extended Kalman filter with that from the nonlinear least squares. Second, we have shown that the demand is inelastic to the prices by conducting the identification of simultaneous equations for the supply and demand curves with the nonlinear two-stage least square estimation. For future research, we will conduct the empirical studies using more frequent natural gas prices like weekly and daily ones. The applications to other energy prices like crude oil and heating oil are possible extensions for our further studies.