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A robust time-varying regime-switching model for price dynamics of hourly spot price of electricity on the electricity market is developed. We propose a two-state Markov Regime Switching (MRS) model that gives weight to the existence of different variance for each regime. Our model is tractable as it integrates the main features exhibited in the hourly spot price dynamics on the electricity market. The parameters of our hourly spot price of electricity market model are estimated using the Expectation Maximization algorithm. Based on this model, an efficient and tractable pricing technique can be developed to price the dynamics of the hourly spot price of electricity.

Electricity, among other commodities, is one of the most important blessings science has given to the world. It is an essential commodity for social and economic development of developing countries. Most small household and manufacturing industries depend on electricity for their activities. According to the World Banks Global Tracking Framework (GTF), released in April 2017, 1.06 billion people live without electricity―a negligible improvement since 2012 (http://www.worldbank.org/en/topic/energy/overview, accessed on 02/09/2017) and this impedes the growth of countries economy due to the over-reliance of most activities on electricity. Crousillat, Hamilton, and Antmann (2010) stated that “eventhough electricity alone is not sufficient to spur economic growth, it is certainly necessary for human development” [

The electricity market facilitates the purchase of electricity through bids to buy; sales, through offers to sell; and short-term trades. In the early 1990s, the deregulation of the energy market (electricity market in our case) started in some countries (among others were the United Kingdom, Australia, and Norway) and gradually spread out to the European Union and the United States. This has created competitive markets that boost wholesale trading in most countries. This deregulation instigated substantial elements of risk such as uncertain demand, price risk, and volumetric risk; the principal of them being electricity price volatility. In result of this, there is the need to understand and model the spot price dynamics of the electricity market accurately to aid in an efficient pricing of electricity spots.

The spot price dynamics of electricity show signs of strong seasonality, high volatility, and generally unexpected extreme changes known as “spike” or “jumps” [

Ethier and Mount (1998) presented MRS models to electricty prices [

that the mean-reverting rate of the long-run price level and the normal periods are not the same. Weron, Bierbrauer, and Trück (2004) modelled spot electricity prices by reviewing different electricity spot models [

In this paper, we develop a robust two state regime-switching model with time-varying volatility for the price dynamics of the electricity spot price on the electricity market. The model is mathematically tractable to represent well the characteristics of the spot price dynamics of the electricity market.

Suppose in a two independent state regime switching, each state undergoes discrete shifts between states S t of the process. Then S t follows a first order Markov chain with the transition matrix:

P = [ ℙ ( S t = 1 | S t − 1 = 1 ) ℙ ( S t = 1 | S t − 1 = 2 ) ℙ ( S t = 2 | S t − 1 = 1 ) ℙ ( S t = 2 | S t − 1 = 2 ) ]

P = [ P 11 P 12 P 21 P 22 ] = [ 1 − P 12 P 12 P 21 1 − P 21 ]

The transition matrix P contains the probabilities P i j ( i , j = 0 , 1 ) = P { S t = j | S t − 1 = 1 } , 0 ≤ P i j ≤ 1 , ∑ P i j = 1 and satisfying P i 0 + P i 1 = 1 , ∀ i = 0 , 1

Keeping the stylized features of the spot price dynamics of electricity in mind, we propose a two-state Markov regime-Switching model with base regime driven by a mean-reverting process and a shifted regime driven by a Brownian- “Jump” process. In both regimes, we assume that the volatility of the current spot price is dependent on the current spot price level X t . The “jump” behaviour is as a result of an “extreme” Brownian motion with a greater extreme drift and volatility than the standard mean-reverting regime. The “jump” regime is modelled with a simple Itô process. Given a time interval [ 0, T ] at a finite time horizon [ T < ∞ ] , assume there is trading activities in the electricty market. Suppose, given a probability space ( Ω , F , ℙ )

X ( t ) = { X t , 1 : d X t , 1 = ( α 1 − λ X t , 1 ) d t + σ 1 X t γ d W t M , if X ( t ) is in regime 1 X t , 2 : d X t , 2 = μ 2 d t + σ 2 X t γ d W t J , if X ( t ) is in regime 2 (1)

where λ is the mean-reversion rate of the base regime, α 1 λ is the long-term

mean for the spot price reverting to, σ 1 X t γ and σ 2 are the daily price volatility of the base and shifted regimes respectively, W t M and W t J are the standard Brownian motion and a Brownian-”Jump” process respectively, T t is the temperature at time t. Let p 1 and 1 − p 1 be the probabilities that the process is in regime 1 and 2, respectively. The base regime model is based on the mean-reverting constant elasticity variance developed by [

X ( t ) = { X t , 1 : d X t , 1 = ( α 1 − λ X t , 1 ) d t + σ 1 X t ,1 d W t M , if X ( t ) is in regime 1 X t , 2 : d X t , 2 = α 2 d t + σ 2 X t , 2 d W t J , if X ( t ) is in regime 2 (2)

By the application of Itô’s lemma to model (2), the integral form to the base regime and shifted regime is explicitly given in integral form as

X t , 1 = α 1 λ + e − λ t [ X 0 , 1 − α 1 λ ] + σ 1 ∫ t 0 X s , 1 e − λ ( t − s ) d W s M (3)

X t , 2 = X 0 , 2 + μ 2 t + ∫ 0 t σ 2 d W s J (4)

The EM algorithm was first introduced by Dempster, Laird, and Rubin (1977) [

The discretized version of model 1 in the base and shifted regime is respectively given as

X t + 1 , 1 = α 1 + X t , 1 ( 1 − λ ) + σ 1 X t , 1 ϵ t + 1 , 1 (5)

X t + 1 , 2 = α 2 + X t , 2 + σ 2 X t , 2 ϵ t + 1 , 2 (6)

where ϵ t + 1 ~ N ( 0,1 ) . Let F t k X be the vector of past k + 1 last values of (5) and (6), i.e. F t k X = ( X t 0 , X t 1 , X t 2 , ⋯ , X t k ) . Also, let H + 1 be the size of the past data and Ψ be the equivalent increasing pattern of time at which the data is recorded, i.e. Ψ = { t j ; 0 = t 0 ≤ t 1 ≤ t 2 ≤ ⋯ ≤ t H − 1 ≤ t H = T } .

As stated earlier, the regime switching model is latent, hence the inference of the regimes are given by the equations below:

∀ i = { 1 , 2 } , k = 1 , 2 , 3 , ⋯ , H , and n = number of iterations.

A i , t k n = ℙ ( S t k = i | F t k X ; Θ ^ ( n ) ) = ℙ ( S t k , X t k | F t k − 1 X ; Θ ^ ( n ) ) f ( X t k | F t k − 1 X ; Θ ^ ( n ) ) (7)

= ℙ ( S t k = i | F t k − 1 X ; Θ ^ ( n ) ) f ( X t k | S t k = i ; F t k − 1 X ; Θ ^ ( n ) ) ∑ i = 1 2 ℙ ( S t k = i | F t k − 1 X ) f ( X t k | S t k = i ; F t k − 1 X ; Θ ^ ( n ) ) (8)

with

the process in regime i. From (2) and (5), the base regime has a conditional Gaussian distribution with mean

The probability density functions (pdf) of the base and shifted regimes are respectively given as

We compute the maximum likelihood estimates

The transition probabilities

From (11) and (12), the log-likelihood functions of the base and shifted regimes are given respectively as

From (15), each of the parameter in the base regime can be estimated by differentiating the log-likelihood with respect to that parameter.

where

From (16), each of the parameter in the shifted regime can be estimated by differentiating the log-likelihood with respect to that parameter.

Historical electricity hourly spot price on the NordPool market is used, specifically we took an hourly data of Oslo. The data set consist of 764 hours spanning from 01/03/2017-31/03/2017. From

The estimated results of the model is found in

significant probability of 0.1529, the hourly price will remain in the shifted regime.

In this paper, a two-state Markov Regime switching model for the dynamics of the hourly spot price of electricity is developed. It is clear from the illustrated

Skewness | Kurtosis |
---|---|

3.9056 | 24.6572 |

Parameter | |||||||
---|---|---|---|---|---|---|---|

Estimate | 10.9876 | 1.7671 | 6.8814 | 0.8471 | 0.1529 | 7.0913 | 4.0936 |

NIG and Gamma distributions can capture the extreme and skewed features of the hourly spot price of the NordPool Electricity data.

We thank Pan African University, Instutute of Basic Sciences Technology and Innovation for their support for this research.

Gyamerah, S.A. and Ngare, P. (2018) Regime-Switching Model on Hourly Electricity Spot Price Dynamics. Journal of Mathematical Finance, 8, 102-110. https://doi.org/10.4236/jmf.2018.81008