天然气期货和现货价格间的相关性研究

基于天然气期货价格与现货价格序列间具有强非线性特征,本文将GARCH模型和Copula函数思想进行结合,同时考虑了天然气期货和现货价格间的时变相关结构,构建了时变Copula(GARCH-Normal、GARCH-GED和GARCH-t)模型,利用美国纽约商品交易所(NYMEX)Henry Hub交易中心天然气期货价格和现货价格数据进行实证研究。实证结果表明:GARCH-GED模型能够准确地拟合天然气期货与现货价格时间序列;时变SJC-Copula函数能够更好的描述天然气期货价格与现货价格间的相关性;天然气期货与现货价格间的相关性不是对称的,上尾的相关性小于下尾相的相关性。     

Australian electricity market and price volatility

Australian Electricity Market has experienced high price volatility since the deregulation in early 1990s. In this exploratory and preliminary analysis of 2010 data from South Australian electricity market we identify and exhibit a number of phenomena which, arguably, contribute to (A) high cost of electricity supply to consumers and (B) volatility in spot prices. These phenomena include: (i) Distinct bidding patterns of some generators occurring in trading intervals corresponding to periods of low, medium and high spot prices, (ii) Low correlation between electricity demand and spot prices on days when spot price spikes are observed, (iii) Failure of the lottery model and associated Markowitz-type optimisation approaches to adequately explain the shifting structure of generators’ bids and (iv) Unexpectedly high contribution to the consumers costs and risks from the relatively small number of trading intervals where spot price spikes were observed.     

Electricity price modeling and asset valuation: a multi-fuel structural approach

We introduce a new and highly tractable structural model for spot and derivative prices in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power and the prices of generating fuels into electricity spot prices. The stack structure allows for a range of generator efficiencies per fuel type and for the possibility of future changes in the merit order of the fuels. The derived spot price process captures important stylized facts of historical electricity prices, including both spikes and the complex dependence upon its underlying supply and demand drivers. Furthermore, under mild and commonly used assumptions on the distributions of the input factors, we obtain closed-form formulae for electricity forward contracts and for spark and dark spread options. As merit order dynamics and fuel forward prices are embedded into the model, we capture a much richer and more realistic dependence structure than can be achieved by classical reduced-form models. We illustrate these advantages by comparing with Margrabe’s formula and a simple cointegration model, and highlight important implications for the valuation of power plants.     

A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing

A mean‐reverting model is proposed for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non‐Gaussian Ornstein–Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. In a simulation example it is demonstrated that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. The pricing of European call and put options written on electricity forward contracts is also discussed.     

Spatial dependencies of wind power and interrelations with spot price dynamics

Wind power has seen strong growth over the last decade and increasingly affects electricity spot prices. In particular, prices are more volatile due to the stochastic nature of wind, such that more generation of wind energy yields lower prices. Therefore, it is important to assess the value of wind power at different locations not only for an investor but for the electricity system as a whole. In this paper, we develop a stochastic simulation model that captures the full spatial dependence structure of wind power by using copulas, incorporated into a supply and demand based model for the electricity spot price. This model is calibrated with German data. We find that the specific location of a turbine – i.e., its spatial dependence with respect to the aggregated wind power in the system – is of high relevance for its value. Many of the locations analyzed show an upper tail dependence that adversely impacts the market value. Therefore, a model that assumes a linear dependence structure would systematically overestimate the market value of wind power in many cases. This effect becomes more important for increasing levels of wind power penetration and may render the large-scale integration into markets more difficult.     

Pricing electricity derivatives within a Markov regime-switching model: a risk premium approach

In this paper we derive analytic formulas for electricity derivatives under assumption that electricity spot prices follow a 3-regime Markov regime-switching model with independent spikes and drops and periodic transition matrix. Since the classical derivatives pricing methodology cannot be used in the case of non-storable commodities, we employ the concept of the risk premium. The obtained theoretical results are then used for the European Energy Exchange data analysis. We calculate the risk premium in the case of the calibrated 3-regime MRS model. We find a time varying structure of the risk premium and an evidence for a negative risk premium (or positive forward premium), especially at short times before delivery. Finally, we use the obtained risk premium to calculate prices of European options written on spot, as well as, forward prices.     

Structural Electricity Models and Asymptotically Normal Estimators to Quantify Parameter Risk

ABSTRACT

We estimate a structural electricity (multi-commodity) model based on historical spot and futures data (fuels and power prices, respectively) and quantify the inherent parameter risk using an average value at risk approach (‘expected shortfall’). The mathematical proofs use the theory of asymptotic statistics to derive a parameter risk measure. We use far in-the-money options to derive a confidence level and use it as a prudent present value adjustment when pricing a virtual power plant. Finally, we conduct a present value benchmarking to compare the approach of temperature-driven demand (based on load data) to an ‘implied demand approach’ (demand implied from observable power futures prices). We observe that the implied demand approach can easily capture observed electricity price volatility whereas the estimation against observable load data will lead to a gap, because – amongst others – the interplay of demand and supply is not captured in the data (i.e., unexpected mismatches).     

Heavy-tails and regime-switching in electricity prices

In this paper we first analyze the stylized facts of electricity prices, in particular, the extreme volatility and price spikes which lead to heavy-tailed distributions of price changes. Then we calibrate Markov regime-switching (MRS) models with heavy-tailed components and show that they adequately address the aforementioned characteristics. Contrary to the common belief that electricity price models ‘should be built on log-prices’, we find evidence that modeling the prices themselves is more beneficial and methodologically sound, at least in case of MRS models.     

Calibration of a multifactor model for the forward markets of several commodities

We propose a model for the evolution of forward prices of several commodities, which is an extension of the factor forward model in [1, 2], to a market where multiple commodities are traded. We calibrate this model in a market where forward contracts on multiple commodities are present, using historical forward prices. First, we calibrate separately the four coefficients of each individual commodity, using an approach based on quadratic variation/covariation of forward prices. Then, with the same technique, we pass to the estimation of the mutual correlation among the Brownian motions driving the different commodities. This calibration is compared to a calibration method used by practitioners, which uses rolling time series and requires a modification of the model, but turns out to be more accurate in practice, especially with a low frequency of observed transaction. We present efficient methods to perform the calibration with both methods, as well as the calibration of the intercommodity correlation matrix. Then we calibrate our model to WTI, ICE Brent and ICE Gasoil forward prices. Finally we present a method for estimating spot volatility from forward parameters, with an application to the WTI spot volatility.     

Determination of the Probability Distribution Measures from Market Option Prices Using the Method of Maximum Entropy in the Mean

Abstract

We consider the problem of recovering the risk-neutral probability distribution of the price of an asset, when the information available consists of the market price of derivatives of European type having the asset as underlying. The information available may or may not include the spot value of the asset as data. When we only know the true empirical law of the underlying, our method will provide a measure that is absolutely continuous with respect to the empirical law, thus making our procedure model independent. If we assume that the prices of the derivatives include risk premia and/or transaction prices, using this method it is possible to estimate those values, as well as the no-arbitrage prices. This is of interest not only when the market is not complete, but also if for some reason we do not have information about the model for the price of the underlying.     

On Modelling and Pricing Rainfall Derivatives with Seasonality

Abstract

We are interested in pricing rainfall options written on precipitation at specific locations. We assume the existence of a tradeable financial instrument in the market whose price process is affected by the quantity of rainfall. We then construct a suitable ‘Markovian gamma’ model for the rainfall process which accounts for the seasonal change of precipitation and show how maximum likelihood estimators can be obtained for its parameters.

We derive optimal strategies for exponential utility from terminal wealth and determine the utility indifference price of the claim. The method is illustrated with actual measured data on rainfall from a location in Kenya and spot prices of Kenyan electricity companies.     

Two-settlement electricity markets with price caps and Cournot generation firms

We compare two alternative mechanisms for capping prices in two-settlement electricity markets. With sufficient lead time, forward market prices are implicitly capped by competitive pressure of potential entry that will occur when forward prices rise above some backstop price. Another more direct approach is to cap spot prices through a regulatory intervention. In this paper we explore the implications of these two alternative mechanisms in a two-settlement Cournot equilibrium framework. We formulate the market equilibrium as a stochastic equilibrium problem with equilibrium constraints (EPEC) capturing congestion effects, probabilistic contingencies and horizontal market power. As an illustrative test case, we use the 53-bus Belgian electricity network with representative generator costs but hypothetical demand and ownership structure. Compared to a price-uncapped two-settlement system, a forward cap increases firms’ incentives for forward contracting, whereas a spot cap reduces such incentives. Moreover, in both cases, more forward contracts are committed as the generation resource ownership structure becomes more diversified.     

Optimal electricity generation portfolios

It is common practice to base investment decisions on price projections which are gained from simulations using price processes. The choice of the underlying process is crucial for the simulation outcome. For power plants the core question is the existence of stable long-term cointegration relations. Therefore we investigate the impacts of different ways to model price movements in a portfolio selection model for the German electricity market. Three different approaches of modelling fuel prices are compared: initially, all prices are modelled as correlated random walks. Thereafter the coal price is modelled as random walk. The gas price follows the coal price through a mean-reversion process. Lastly, all prices are modelled as mean reversion processes with correlated residuals. The prices of electricity base and peak futures are simulated using historical correlations with gas and coal prices. Yearly base and peak prices are transformed into an estimated price duration curve followed by the steps power plant dispatch, operational margin and net present value calculation and finally the portfolio selection. The analysis shows that the chosen price process assumptions have significant impacts on the resulting portfolio structure and the weights of individual technologies.     

The natural hedge of a gas-fired power plant

Electricity industries worldwide have been restructured in order to introduce competition. As a result, decision makers are exposed to volatile electricity prices, which are positively correlated with those of natural gas in markets with price-setting gas-fired power plants. Consequently, gas-fired plants are said to enjoy a “natural hedge.” We explore the properties of such a built-in hedge for a gas-fired power plant via a stochastic programming approach, which enables characterisation of uncertainty in both electricity and gas prices in deriving optimal hedging and generation decisions. The producer engages in financial hedging by signing forward contracts at the beginning of the month while anticipating uncertainty in spot prices. Using UK energy price data from 2006 to 2011 and daily aggregated dispatch decisions of a typical gas-fired power plant, we find that such a producer does, in fact, enjoy a natural hedge, i.e., it is better off facing uncertain spot prices rather than locking in its generation cost. However, the natural hedge is not a perfect hedge, i.e., even modest risk aversion makes it optimal to use gas forwards partially. Furthermore, greater operational flexibility enhances this natural hedge as generation decisions provide a countervailing response to uncertainty. Conversely, higher energy-conversion efficiency reduces the natural hedge by decreasing the importance of natural gas price volatility and, thus, its correlation with the electricity price.     

Electricity price forecasting through transfer function models

Forecasting electricity prices in presentday competitive electricity markets is a must for both producers and consumers because both need price estimates to develop their respective market bidding strategies. This paper proposes a transfer function model to predict electricity prices based on both past electricity prices and demands, and discuss the rationale to build it. The importance of electricity demand information is assessed. Appropriate metrics to appraise prediction quality are identified and used. Realistic and extensive simulations based on data from the PJM Interconnection for year 2003 are conducted. The proposed model is compared with naïve and other techniques.     

中国期货市场与现货市场的价格引导与价格发现

长期以来对期货市场与现货市场价格关系的实证研究都是基于时间序列方法的研究.为了克服时间序列方法存在着的不足,将使用面板数据方法,在面板单位根检验以及面板协整检验和协整估计的基础上,构建面板误差修正模型来分析期货价格和现货价格的均衡以及相互引导关系.进一步的,在误差修正模型的基础上我们采用信息份额方法(I-S模型)和共同因子贡献法(P-T模型)分析了期货市场和现货市场的价格发现功能.通过上述研究,发现总体上讲我国大宗商品的期货价格和现货价格之间存在着长期均衡,并且表现出了相互引导互为Granger因果的关系.利用I-S模型和P-T模型测算出来的期货市场对价格形成的贡献度分别为88.17%和79.44%,这说明当前我国的期货市场总体上讲是有效率的市场.     

Optimal bidding of a virtual power plant on the Spanish day-ahead and intraday market for electricity

We develop a multi-stage stochastic programming approach to optimize the bidding strategy of a virtual power plant (VPP) operating on the Spanish spot market for electricity. The VPP markets electricity produced in the wind parks it manages on the day-ahead market and on six staggered auction-based intraday markets. Uncertainty enters the problem via stochastic electricity prices as well as uncertain wind energy production. We set up the problem of bidding for one day of operation as a Markov decision process (MDP) that is solved using a variant of the stochastic dual dynamic programming algorithm. We conduct an extensive out-of-sample comparison demonstrating that the optimal policy obtained by the stochastic program clearly outperforms deterministic planning, a pure day-ahead strategy, a benchmark that only uses the day-ahead market and the first intraday market, as well as a proprietary stochastic programming approach developed in the industry. Furthermore, we study the effect of risk aversion as modeled by the nested Conditional Value-at-Risk as well as the impact of changes in various problem parameters.     

Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps

It is known that the implied volatility skew of Forex (FX) options demonstrates a stochastic behaviour which is called stochastic skew. In this paper, we create stochastic skew by assuming the spot/instantaneous variance (InV) correlation to be stochastic. Accordingly, we consider a class of Stochastic Local Volatility (SLV) models with stochastic correlation where all drivers – the spot, InV and their correlation – are modelled by processes. We assume all diffusion components to be fully correlated, as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate the capacity of the presented approach in modelling stochastic skew.     

基于Copula-ECM-GARCH模型的农产品期货套期保值绩效研究

Copula函数具有可以准确刻画变量间的相依结构、精准描述金融时间序列"尖峰厚尾"分布特点的良好统计性质.针对传统计量模型在计算套期保值比率时存在的局限性,利用Copula函数描述变量的尾部相关性,并结合ECM-GARCH模型,对大豆、小麦、玉米三种国内农产品期货进行套期保值研究,分别计算最优的套期保值比率及其绩效,并...     

MCMC calibration of spot‐prices models in electricity markets

The calibration of some stochastic differential equation used to model spot prices in electricity markets is investigated. As an alternative to relying on standard likelihood maximization, the adoption of a fully Bayesian paradigm is explored, that relies on Markov chain Monte Carlo (MCMC) stochastic simulation and provides the posterior distributions of the model parameters. The proposed method is applied to one‐ and two‐factor stochastic models, using both simulated and real data. The results demonstrate good agreement between the maximum likelihood and MCMC point estimates. The latter approach, however, provides a more complete characterization of the model uncertainty, an information that can be exploited to obtain a more realistic assessment of the forecasting error. In order to further validate the MCMC approach, the posterior distribution of the Italian electricity price volatility is explored for different maturities and compared with the corresponding maximum likelihood estimates.