Stochastic models of natural gas prices

Jurnal Karya Asli Lorekan Ahli Matematik Vol. 6 No.1 (2013) Page 050-065. Jurnal Karya Asli Lorekan Ahli Matematik STOCHASTIC MODELS OF NATURAL GAS PRICES Leyla Ranjbari, Arifah Bahar and Zainal Abdul Aziz Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310, UTM Johor Bahru, Johor, Malaysia Abstract : The paper is a survey on some recent literature in natural gas spot modelling without plunging into calibration, spot-futures and spot-forward dynamics. This work, based on the fact that the crucial property of spot price in energy markets, as a commodity market, is mean-reverting. We observe that the natural gas spot modelling can be divided into essential categories, i.e. mean-reversion models and regime-switching models. We then examine their historical extensions in the form of new techniques. In the former models, one-factor, twofactor and three-factor spot mean-reverting models as well as their extensions are resulted from splitting the longrun mean into two stochastic and deterministic components. We also consider the Levy diffusion based on alphastable process and normal inverse Gaussian process as well as affine structure for seasonality term. As the latter category, one-factor regime-switching model is considered, which consists of two regimes either mean-reverting process or geometric Brownian motion with positive/negative drift. Key Words: Natural gas, multi-factor mean reverting, one-factor regime-switching spot model, alpha-stable Levy process, normal inverse Gaussian processes, forward curves, future curves. 1. Introduction Over the past two decades, natural gas plays a very essential role in the energy market due to the cleanest burning fossil fuel as well as growing concern over air pollution control, which gives the market further growth potential. The gas market has experienced radical changes in North America and Europe over a deregulation period and elimination of stage monopolies and the corresponding financial markets have also gradually furnished in the last decades. The futures market successfully adapted itself to this dynamic and highly competitive market resulted from counterparty performance risk, as a solution for essential risk management necessity. Consequently, the successful futures market then became the foundation for many other forms of derivatives trade, such as options and swaps. Now, the energy market has become fairly liberal market. Moreover, a number of fundamental price drivers, such as issues of extraction, storage, transportation, weather, policies, technological advances, etc., cause extremely complex gas prices behavior.The natural gas spot prices have several important properties. First, the natural gas spot prices have been historically considered to be a mean-reverting process. This means that the prices move up and down frequently, but oscillate around an equilibrium level (long-run mean) from the point of long term view. This is just the effect of mean reversion, i.e. the prices mean revert to a long-term mean. The mean-reversion behavior of natural gas prices is related to their reactions to events such as floods, summer heat waves, and other news-making events, which can create new and unexpected supply-and-demand imbalances in the market. Such as the temperatures reverting to their average seasonal levels, the natural gas prices tend to cause the natural gas prices to come back to their typical levels. Another very important characteristic of gas spot prices is strong cyclic in nature over a year due to seasonal variation in supply and demand. Seasonality results from mainly demand fluctuations. This is largely due to natural gas being a main source of heating homes and businesses. Heat usage increases in the winter and goes down in the summer, so due to the market forces of supply and demand, the price of natural gas has a general upward price movement in the winter and downward movement in the summer. This seasonality is also seen in the price of natural gas forwards/futures. On the other hand, the difficulty of storage and the limitation of transmission capacity make the supply side not elastic enough to match the suddenly increased demand side very quickly. Hence, seasonal © 2013 Jurnal Karya Asli Lorekan Ahli Matematik Published by Pustaka Aman Press Sdn. Bhd. Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz fluctuation of gas prices is the inevitable result of the seasonal imbalances between demand and supply. The seasonality effects can be seen not only through historical spot prices, but also through futures and forward prices. In section 2, we shall review the literature on natural gas spot prices. Ultimately, section 3 involves the summary and future work. 2. Stochastic Models of Natural Gas Spot Prices In this work, we shall present different spot price models used in natural gas spot pricing in two separate frameworks: mean-reversion models and regime-switching models, and then explain other new price-modelling techniques based on these two categories. The typical feature of many commodities such as natural gas is that of mean reversion and this is captured by an OrnsteinUhlenbeck (OU) process. The commonly OU process is used in a single-factor model with a Wiener process as the risk term. In modelling the forward curves, Schwartz [16] showed that this is insufficient due to a cost of carry and its effects on the drift term. To overcome this drift adjustment, convenience yield was taken into models. The interest rate also was considered a third stochastic factor by Schwartz [16], but this did not yield any qualitative advantage over the two factor model. Thus in commodity modelling literature, stochastic interest rates are rarely ever considered. In Pilipovic [12, 13] a long-run stochastic mean also is proposed in the second factor model. Now in the case of natural gas, there is seasonality exhibited in the price dynamics. Xu [18] modified Philipovic's model [12, 13] to include seasonality via a positioning term, which is a sum of two sinusoids with different periods, whose parameters are obtained from the forward curve. Chen and Forsyth [5] used a one-factor regime-switching model to simulate the natural gas price that supposedly imitates the two-factor convenience yield model from Gibson-Schwartz [7]. The PDE method of pricing by Chen and Forsyth [5] is much more efficient in computing the value of storage. Hikspoors and Jaimungal [8] proposed a two-factor model with stochastic long-run meanreversion and a seasonal component gt in spot price process. To capture the appropriate mathematical models for gas prices, there is an important test that spot models must withstand, i.e., how well does the model fit the futures curve. Since the futures price is equal to the discounted expected value of the spot price at its expiry, the proposed spot model must be able to match the futures curve when taking its expectation. Thus in the literature, it’s important that the spot price process chosen has an explicit form for its expectation, so as to determine the futures price. We shall introduce a series of stochastic processes because of the random behaviour of gas prices) appropriate for natural gas spot prices based on available literature. 2.1 A one-factor model - Ornstein-Uhlenbeck process (OU) The commonly used process to model natural gas behaviour is the mean-reverting OrnsteinUhlenbeck (OU) process. This is the most popular one-factor model in natural gas spot simulation. The OU process is defined by where the speed of mean reversion is, is the value that the spot price reverts to, is the diffusion term and is a Wiener process or Brownian motion. The expectation, variance and covariance of are 51 Jurnal KALAM Vol. 6, No. 1, Page 050-065 This process is used as the standard spot price model for pricing the natural gas storage in Boogert and Jong [1], Chen and Forsyth [5], Thompson, Davison et al. [17] and Bringedal [2]. The disadvantage of this approach is that the spot price evolution can't be accurately accounted for. But the advantage is the ease of calibration, and the simple form for the futures price, which follows from equation (2) and (3). The futures price is given by where is the futures price at time t, for a contract expiring at time T given that the spot price is St. The appropriate model for industry practitioners, who have to take positions every day with respect to injection and withdrawal of gas from the storage, is that of Li [9] which is relatively simple but can directly simulate the expected spot price process with respect to the futures price. He takes the spot price to take the following process. Here ST is the spot price at time T in the future, So is the spot price on the valuation date (current) and is the spot price volatility. is the price of the futures contract as of today based on a expiry date i. Since the futures price is the expected value of the spot price, for every subsequent month, to begin with the futures price expiring in month, the spot price process is set for the month. This approach facilitates the expected spot dynamics and includes the forward curve with less computationally expensive. 2.2 Multi-factor models in commodity pricing In this section, a variety of two-factor models with their third-factor extensions are described. The most popular two-factor model and the first in the class of convenience yield models is that of Gibson-Schwartz [7]. To represent the existing two driving noise terms in market movements, twofactor models can be used to indicate these two random sources of volatility. In Carmona [4], a stochastic market price of risk term is introduced to fit the implied convenience yield for different maturities. We shall now discuss the necessity of convenience yield in energy market before discussion about the models based on convenience yield. 2.2.1 Why is convenience yield necessity in energy markets? If supply constraints show shock, then demand exerts its own fundamental price drivers. In energies, demand drivers introduce the issues of convenience yield and seasonality that have no parallel in money markets. Sometimes, due to irregular market movements such as an inverted market, the holding of an underlying good or security may become more profitable than owning the contract or derivative instrument, due to its relative scarcity versus high demand. The amount of benefit or premium associated with holding an underlying product or physical good, rather than the contract or derivative product is called convenience yield. The convenience yield is derived by the difference between the first purchase price of the underlying asset and its price after the shock. In other words, when an asset is easy to come by, an investor doesn’t have need to own the actual asset at that time, and can buy or sell as he please. When there is shortage of the particular asset, it is better to own the asset rather than to own its contract or have to purchase it during the shortage period because it is likely to be at a higher price due to the demand. In energy related assets, storage costs of 52 Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz energy along with the other industrial management costs influence the true value of the assets. Although the holder of the energy contract has the option of consumption flexibility and has no risk in the event of commodity shortage, cost flow implied by the storage expenses affects on the holder’s decision to postpone consumption. This driven net flow is the convenience yield and is represented by δ where δ= Convenience yield = Benefit of direct access - cost of carry The standard pricing for forward contracts with maturity T in markets is that discounted the spot price, i.e. where T is the time of exercise, r is the riskless-interest rate St is the spot price and F(t, T) is the price of the forward at time t with exercise at time T. However, the forward contract price that includes the convenience yield is obtained by where δ is the convenience yield. Q is the risk-neutral measure, so this implies that St can be inferred as a drift correction term in the spot price process. 2.2.2 Gibson-Schwartz model The first spot convenience yield model was introduced by Gibson and Schwartz in 1990. The spot price has the convenience yield added to the drift and is assumed to be a mean-reverting process that drives the geometric Brownian motion commodity spot price . Let be a probability space under a filtration . According to the GibsonSchwartz model, under the risk-neutral measure Q, where and are correlated Wiener processes with . In many energy commodities, mean-reverting process is typically accounted for spot prices of energy assets, but in 1990 Gibson [7] argued that the convenience yield affects the spot price process and induces mean-reversion to it. Unlike interest rate models, it makes sense that convenience yields can take positive or negative values so the model proposed seems logical. Schwartz [16] in 1997 compared the one, two and three-factor spot models in calibrating forward curves. The one-factor model just has the mean-reverting Ornstein-Uhlenbeck spot price process, the two-factor model is the above model and the three-factor model has stochastic interest rates. It is shown that there is no qualitative improvement in assuming a stochastic interest rate, so stochastic interest rates are not included in the paper. In Schwartz [16], it is shown that the futures price for the above spot prices is where 53 Jurnal KALAM Vol. 6, No. 1, Page 050-065 We can see the affine form in with respect to convenience yield in the futures price model in equation (12). Runggaldier [15] in 2003 developed another two-factor model with introducing another OU process for the spot price and exerting an affine structure with respect to market price of risk , the differential between the actual return that an asset pays vs. the risk-free rate, normalized by the asset’s volatility. The market price of risk , which follows the following OU process (16) and can take positive or negative values, is the risk-neutral measure of the spot price process, i.e. (15) In 2004, Carmona [3] used Runggaldier’s idea to improve the Gibson-Schwartz two-factor model, for better calibration of the futures curve. The Wiener process of the spot price process is substituted by equation (15), and the extra stochastic factor is added as third factor to the model, equation (10) and (11). (18) This three-factor model of the spot price process provides another approach to model the forward curve, in a form of the stochastic differential equation (20). And make an explicit form for the futures price due to the affine nature of . Carmona three-factor model, (17) through (19), is specific to natural gas or energy and can be used for general commodities. For energy commodities, there are the strong seasonal forces of supply and demand not only in the spot price process but also in futures and forward curves which create complicated characteristics for them. We shall now discuss models where a seasonality term was introduced and characterized in the movement of natural gas. The estimation of parameters of the seasonality term is typically done through the forward curves. 54 Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz Figure 1.Average spot price of natural gas Figure 2. The seasonality of the spot price on the same time scale as the seasonality Figure 3. Natural gas futures prices 55 Jurnal KALAM Vol. 6, No. 1, Page 050-065 Figures 1 through 3 are examples of the behaviour of natural gas spot and futures prices, and show a general trend of high gas prices in the winter months and lower gas prices in the summer months on both spot and futures prices. 2.2.3 The Pilipovic two-factor model for energy Dragana Pilipovic in 1997 wrote the first book in Energy Risk [12], where she presented a two-factor model for energy taking into consideration the complex spot-futures dynamics, the futures curve with respect to the spot price, which long-run mean is implied with the futures curve. She presented the following two-factor model (22) and are uncorrelated standard Brownian motions, spot price process as a meanwhere reverted risk-adjusted process and the long-run mean as a geometric Brownian motion (GBM). If is a Wiener process, which drives the spot process then becomes a riskadjusted Wiener process, which will drives . Notice that by and application of Girsanov’s theorem the dependency of Wiener process , on the market price of risk, defined on the same probability space , can be absorbed into an equivalent martingale measure. The Wiener process under the equivalent martingale Q are given by so that .A risk-neutral (adjusted) measure Q is any probability measure, equivalent to the market measure P, which makes all discounted bond prices martingales. In energy market with seasonality, the underlying price refers to the spot price with the seasonality factors taken away: is the underlying spot price at time t. Sometimes the processes of the spot prices stripped of seasonality because we need to strip the effect of seasonality out of price data in order to analyze the underlying price behaviour. The removing the seasonality from price data allows to model the seasonality separately from modelling the underlying price processes, however, seasonality should be modelled as a stochastic process. Note that there are three seasonal factors: summer seasonality, winter seasonality, third seasonality in order to capture any additional repetitive annual event behaviour, such as an additional peaking or hump behaviour in the summer or winter, for example. Thus, the seasonality terms will be defined the same way for all models. However, spot price removed from the seasonality effects, will be defined uniquely by each model being tested. The calibration of the model parameters and the seasonality parameters will be performed simultaneously. For each model calibrating will be ended up with specifying the model parameters, and all the seasonality parameters. Sometimes, there is seasonality of seasonality, that is, not only are there annual summer and winter seasonality patterns, but also 10-year and even 100-year cycles. where 56 Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz Ultimately, if there is seasonality in the futures curve, as in natural gas, the futures curve should be modelled by first stripping off the seasonality such as seasonal hump in the fall, i.e., is underlying (UND) futures prices. The spot process of which is an exponential affine form also has an explicit form for its futures price shown by Pilipovic [12, 13]: where 2.2.4 Xu's generalization of Pilipovic's model Xu [18] added a seasonality term to Pilipovic's model, equations (21) and (22). He exclusively studied the natural gas following model was proposed by Xu: For and constant , Xu’s model would be equivalent to Pilipovic's model. He considered the above model when is constant, i.e., one-factor model, and studied the model with and without seasonality. He concluded that the models concluding seasonality terms performed the best. 2.2.5 Hikspoors and Jaimungal's model In 2007, Hikspoors et. al. [8] proposed a class of models with long-run mean and a seasonal component gt. They present the following two-factor model where is a stochastic process satisfied on the observed equation Both of and are OU processes with the advantage of being able to estimate the conditional probabilities. They also developed a three-factor model with additional stochastic volatility, with the spot price process given by: 57 Jurnal KALAM Vol. 6, No. 1, Page 050-065 , , They extend their two and three-factor models to include jumps, such that The jump component satisfies where is a compound Poisson process. Instead of inserting the jump term in , jump term is included directly to the spot price dynamics of . That means, a jump is randomly added to commodity prices rather than a jump exerts the whole prices alter with its changing, and then the price returns back to its original state. This accounts as an advantages of the model because the typical behaviour of commodity prices have that of spikes in prices and typically returning to its regular level. Quan [8] in 2006 studied exclusively on one and two-factor model with affine jump-diffusion with and without seasonality. Since Nedunthally’s models are accounted for Quan’s models, they are not mentioned in this paper. 2.2.6 Eydeland and Wolyniec's model In Eydeland [6], a model is introduced to capture the entire forward curve. The forward equation is determined by the Schwartz model in a form of an HJM (Heath-Jarrow-Morton) model, which is induced as an underlying forward curve without seasonality to follow an interest rate type model. The forward process generally has the following multifactor form: are correlated Wiener processes. where For commodities, the forward curve model that could be simplified to The above equations propose a very different approach without worried about the actual spot price process. They try to capture the dynamics of the futures curve which actually has a very erratic behaviour in gas due to its dependence on long and short term supply and demand. 2.2.7 Nedunthally’s models. Nedunthally [10] introduce Levy processes into spot modeling of gas spot prices, and claims that the one-factor that Ornstain-Uhlenbech (OU) processes with the normal inverse Gaussian (NIG) 58 Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz process is the most effective in the class of one-factor models. Since there are few jumps in natural gas spot prices, so it is suitable to use a stable process to model the spot price. The advantage of the alpha-stable levy process is ability to model the skewness and kurtosis. He also presents a new two factor model that assigns an affine structure for its seasonality term, the model being an extension to Pilipovic’s two-factor model. He also shows that the normal inverse Gaussian (NIG) process as a computationally feasible case of the generalized hyperbolic (GH) process which manages to retain properties of asset returns such as semi-heavy tails. He discusses about the important criteria for modeling natural gas spot prices and how expected value of the spot price process at different times in the future must be consistent with that of the futures curve. He also introduces two different one-factor models based on an alpha-stable process and NIG process, and take advantage of the fact that Ornstain-Uhlenbech (OU) processes based on stable or NIG processes have an explicit solutions. The parameters of the seasonality term are obtained from a combination of spot and futures prices, which is used in Levy based OU and Cox-Ingersoll-Ross (CIR) processes to match the futures price. To calibrate the twofactor models, he uses linear regression to strip the underlying futures curve and then uses maximum likelihood to estimate the parameters. His work deals with one-factor Levy-based stochastic models of the following form where is a Levy process, is the seasonality parameter and is the natural gas price process. If , it be an OU process and for it be CIR type process. Determine the spot price processes that satisfy the following boundary condition: where is the price of the contract at time t with the date maturity being T when the spot price at time t is given s. The value of a futures contract at maturity T tells about the markets expectation of the spot price at time T under the risk-neutral measure Q. Where is a seasonality term whose parameters are calibrated by Observations show that this seasonality is not constant, but an affine form for the coefficient allows capturing the amplitude of the seasonality. That is where , i.e. has an affine structure, or where 59 Jurnal KALAM Vol. 6, No. 1, Page 050-065 2.2.8 Parsons’s Models Parsons [11] develops a two-factor tree model which consistently captures large amounts of optionality on both fast and slow-cycle leases, based on assuming the strong mean-reverting in U.S. natural gas spot prices. Also he drives the discrete-time spot price process in order to be applied in tree method in order to model natural gas storage value, which is outside of this work concern. He shows that according to principal components analysis, presuming two factors of risk explain approximately 95% of movements in U.S. natural gas forward prices. Since forward prices are merely expected spot prices in risk-neutral measure, he claims a two-factor prices model as a natural starting point for the spot price process. His model is very similar to the one in Pilipovic [12] with a meanreverting spot price and a geometric Brownian motion long-run mean. In Parsons pricing model [11], a two-component long-run mean is assumed with one component as a mean-reverting process, and another component as a deterministic process, as follows: Where Gas-daily (spot) price at time t Stochastic component of the long-run mean at time t Deterministic component at time t Long-run mean of the process Mean-reversion speed of the process Mean-reversion speed of the process Deterministic volatility of the process at time t Deterministic volatility of the process at time t Independent Brownian motion of the process Independent Brownian motion of the process The model introduces the stochastic component to the long-run mean to overcome the shortfalls of the one-factor price model and to be more realistic from the analysis of mean-reversion. Moreover, forward curve can bend and shift parallel in the model in the reason of resulting less correlation between spot and forward prices. Furthermore, with introducing the deterministic component to the long-run mean, the model allows seasonality to be incorporated into the spot prices as well as to facilitate calibration. Deterministic volatilities in both and processes exert to capture seasonality as well. By applying Ito’s lemma to (55) to find the solution for then inserting into (54) to solve , the spot price process for is obtained the below, where The gas-daily (spot) price at time 60 Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz 2.2.9 One-Factor Regime Switching Model Although one-factor mean-reverting models typically are used in literature of natural gas storage evaluation, they do not give us enough efficiency in practice. On the other hand, multi-factor models are computationally expensive. In 2007, Chan and Forthys [5] introduced a one-factor regime switching model to evaluate the gas storage facilities. This model seems to work almost as well as two-factor models with respect to fitting forward curves. The model to be proposed has two regimes, i.e. mean-reverting process (MR) and geometric Brownian motion process (GBM), and switches between a combination of MR and GBM processes. Therefore, this produce several model resulted from variations of regimes, i.e. MRMR, MRGBM, GBMMR, and GBMGBM. Which they were interested in three variations: MRMR, MRGBM and GBMGBM. In MRMR model, the processes in both regimes are mean-reverting. An MRGBM shows the mean-reverting process in one regime and GBM with positive drift in another regime. In GBMGBM variation, the process in one regime follows GBM with a positive drift while that in another regime is GBM with negative drift. Moreover, they used futures curves and options on futures to obtain the models parameter, in particular used the options on futures to find the volatility parameter. When regimes switch between MR and MR's equilibrium price, reproduces dynamics of Xu [18], which includes seasonality and mean reverting long-run mean, in section 2.2.4 the equations (27) through (31).However, when regimes switch between GBM and GBM (with different signs of drifts), reproduces Gibson-Schwartz [16], which extends the typical mean-reverting OU model with adding additional stochastic factor of convenience yield, in section 2.2.2 the equations (10) and (11). Their examined the three MRMR, MRGBM and GBMGBM as well as other models for calibration spot-forward dynamics. Their results showed that the MRMR and MRGBM variations of the regime-switching model are capable of fitting the market gas forward curves more accurately than the MR model. And, the GBMGBM does not appear to be consistent with market data. The switch between two regimes can be modelled by a two-state continuous-time Markov chain , taking two values 0 or 1. The value of indicates the regime in which the riskdenote probability of shifting from regime 0 to adjusted gas spot price resides at time t. let regime 1 over a small time interval dt, and let be the probability of switching from regime 1 to regime 0 over dt. Then can be represented by and are the independent Poisson where t− is the time infinitesimally before t, and and , respectively.In the regime-switching model, the risk-adjusted processes with intensity natural gas spot price is modeled by an SDE given by As indicated in equations (62-63), within a regime the gas spot price follows the process (64-65) with parameters (but the signs of and are not constrained). where is the mean-reverting rate, is the long-term equilibrium price, is the volatility, 61 Jurnal KALAM Vol. 6, No. 1, Page 050-065 is an increment of the standard Gauss-Wiener process, is a time-dependent term so that is the price change at time t contributed by the seasonality effect. Note that multiplying with P guarantees the price of natural gas always stays positive, is the annual seasonality parameter, is a reference time satisfying is the annual seasonality centering parameter for t0. We define (66) where is a constant time adjustment parameter obtained through calibration; is the distance between the reference time and the first date in January in the year of . Thus, by calibrating the value of , we are able to determine the evolution of the annual seasonality effect over time. is the semi annual seasonality parameter, is the semiannual seasonality centering parameter for . Similar to the definition of , we define (67) where the constant time adjustment parameter is obtained from a calibration process. Meanwhile, the stochastic factors for the two regimes are perfectly correlated. Note that we assume that the centring parameters and , as given in equations (66-67), respectively, are identical for two regimes in order to reduce the number of calibrated parameters. This simple model is considered by several authors ([12] and [18]), although the seasonality feature is handled in a slightly different manner. Remark 2.1 (Effect of the seasonality term on gas price dynamics). We can rewrite equation Since according to equation (65), if then there exists certain periods of time within which . In this case, if P is large and in equation (68), then the process (64) becomes a GBM process with positive drift rate due to the strong seasonality effect. At other times, the process is mean-reverting. Note that the deseasoned process (i.e., setting in SDE (64)) is a mean-reverting process. Remark 2.2 (Mean-reverting or GBM-like process). From the model (62-63), the deseasoned spot price in regime can follow either a mean-reverting process or a GBM-like process by setting and , then the deseasoned gas price (obtained parameter values. If we choose in SDE (62)) follows a mean-reverting process from setting the seasonality term with equilibrium level If we set 62 and mean-reversion rate . in equation (62), then the deseasoned gas price SDE becomes Leyla Ranjbari, Arifah Bahar & Zainal Abdul Aziz This is a GBM-like process. Specifically, if the drift coefficient standard GBM process, i.e., gas price P will drift up at a rate . the gas price will drift down at a rate , then SDE (71) is a at time t; if ,then 2.2.10 Variations of the regime-switching model As indicated in Remark 2.2, the deseasoned spot price in each regime can follow either a mean-reverting process or a GBM-like process. Consequently, there exist many possible variations of the regime-switching model by choosing different combinations of the stochastic processes in two regimes. We are interested in the following three variations, i.e. MRMR, MRGBM, GBMGBM variations which are described the following: MRMR variation The processes in both regimes are mean-reverting with different equilibrium levels, i.e., in SDE (62). In this variation, the equilibrium level of the gas spot price , which thus creates a sort of mean-reverting effect on the switches between two constants, equilibrium level. This simulates the behaviour of the equilibrium price in the two-factor model proposed by Xu[18], where the gas spot price P follows a one-factor mean-reverting process and its equilibrium price evolves over time according to the other one-factor mean-reverting process. MRGBM variation The process in one regime is mean-reverting while the other regime is a GBM process with a positive drift, i.e., in SDE (62). The mean-reverting regime represents the normal price dynamics, and the GBM regime can be regarded as the sudden drifting up of the gas price driven by exogenous events. GBMGBM variation The processes in both regimes are GBM processes with a positive drift in one regime and a negative drift in the other, i.e., in SDE (62). This simulates the behaviour of the two-factor model in Schwartz [7], where the risk adjusted commodity spot price process is modelled by a GBM-like process given by Here is the constant riskless interest rate; is the instantaneous convenience yield, following an Ornstein-Uhlenbeck mean-reverting process. The drift coefficient can switch between positive and negative values during a time interval since the value of is stochastic and may change signs during the interval. Thus the gas price p will either drift up or drift down at any time depending on the sign of . According to (70), the gbmgbm variation can produce behaviour similar to the SDE (71). 3. Summary and Conclusion The paper accounts for an overview on natural gas spot modelling without diving into calibration, spot-futures and spot-forward dynamics. In this work, based on mean-reversion property of spot price in energy markets, the spot modelling is divided into two categories: mean-reversion models and regime-switching models. The historical extensions or new techniques based on these two categories are explained which some of them emphasizing on fitting the forward curve, as examples [11] and [6], and some others attempt to capture spot-futures dynamics such as [1,2,3,5,7,9,10 and 16] 63 Jurnal KALAM Vol. 6, No. 1, Page 050-065 and others capture both the forward and futures fitting, as [3,4,12 and 13]. This paper presented different spot prices models for natural gas in two types: mean-reverting models and regimeswitching models. According to the literature on gas spot price modeling, there are some historical ways and ideas to extend these two type models. For mean-reverting models, there are some strategies to extend as the following: 1. Modify the first factor or spot price equation [8,9,10,11,14 and 15] 2. Manipulate the second factors a. Change the second factor: [12, 13 and 15] b. Split into some components:[11] 3. Add the seasonality terms to spot price model: [18] and [8]. a. Sinusoidal term b. Co-sinusoidal term c. Linear combination of sinusoidal and co-sinusoidal functions d. Exponential term 4. Use sinusoidal or co-sinusoidal with different periods 5. Add some factor:[3,4,16 and 18] 6. Add jump process: [14] 7. Use the Levy process instead of Wiener diffusion: [10] For the second type model, Chen and Forsyth [5], due to the novelty, there is not any modification or extension yet; therefore it would be known later in some future work. Since natural gas is the most volatile markets, it would be expected that newer models would have to be developed in order to respond for the complexity in market behaviours. 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