Description of Colombian Electricity Pricing Derivatives

Electricity markets are becoming a popular field of research amongst academics because of the lack of appropriate models for describing electricity price behavior and pricing derivatives instruments. Models for price dynamics must consider seasonality and spiky behavior of jumps which seem hard to model by standard jump process. Without good models for electricity price dynamics, it is difficult to think about good models for futures, forward, swaps and option pricing. In this paper we attempt to introduce an algorithm for pricing derivatives to intuition from Colombian electricity market. The main ambition of this study is fourfold: 1) First we begin our approach through to simple stochastic models. for electricity pricing. 2) Next, we derive analytical formulas for prices of electricity derivatives with different derivatives tools. 3) Then we extent short of the model for price risk in the electricity spot market 4) finally we construct the model estimation under the physical measures for Colombian electricity market. And this paper end with conclusion.


Introduction
Deregulation of electricity markets has led to a considerable increase in risk accepted by market participants. The often unexpected, extreme spot price changes range even two orders of greatness and can cause severe financial problems to the efficacies that buy electricity in the wholesale market and deliver it to consumers at fixed prices. The utilities and other power market companies essential to hedge against this price risk. A forthright way to do it is to use derivatives, like forwards and options.
Here, we use the latter approach and describe the spot price dynamic models to review the electricity pricing with details of how to implement the pricing of electricity market a After specifying a model we must choose for derivatives pricing methodology. However, such approach fails in case of electricity due to extremely limited storage possibilities. purposes, the electricity market is a highly incomplete market and pure arbitrage option pricing methods fail for most structured products.
Previous work has been focused mainly on either of the two following approaches:  Market models for futures prices: As an alternative of modeling the spot price and deriving futures prices, the futures prices themselves are demonstrated. This method goes back to Black's model (1976), where a single futures contract is measured. Ideas from the Heath -Jarrow -Morton theory for interest rates  are used in Clewlow and Strickland (1999), Box and Jenkins (1976), Koekebakker and Ollmar (2001) and Manoliu and Tompaidis (2002) to model the dynamics of the complete futures price curve. Such models have the benefit that the market can be considered as being complete and standard riskneutral pricing may be used. Risk-neutral parameters can often be roundabout from traded options on futures prices. The weakness of such approaches is that futures prices do not expose the information about price behavior on an hourly or even daily time scale.  Spot price models: This model aims at apprehending the hourly price behavior by fitting their model to historical spot price data. Since there is no arbitrage kith and kin between spot prices and futures prices, in additional conventions must be made to use this model for pricing derivatives. Generally, this is done either by assuming the balanced expectation hypothesis As done e.g. in Tseng and Barz (2002) to price generation assets, or by standardizing a market price of risk for each factor and then the altering to an equivalent martingale measure * P under which the relation * , =E ,   t T T t F S F holds.
Maximum, the models for the spot market engagement at least two risk factors: first one, seizing the short-term hourly price dynamics characterized by mean reversion and tremendously high volatility, and the other factor demonstrating long-term price behavior observed in the futures market. Since there are no liquidly traded derivatives on a daily or hourly time gauge that have a strong dependence on the short-term risk factor, it is exceptionally difficult to guesstimate the short-term market price of risk.
Through this paper we will denote t S the spot market price at time t. Since are working in a predetermined interest rate framework, we will not differentiate between forward and futures prices. Therefore, single hour futures prices at time t for delivery at time T are Or, in a discrete time setting by The simplest model considering mean-reverting behavior is given by an Ornstein-Uhlenbeck process. Here the price process t S is a diffusion process satisfying the stochastic differential equation Where   t W is a standard Brownian motion,  the volatility of the development, and  the velocity with which the process reverts to its long term mean a .
In electricity markets prices show powerfully mean reverting behavior so that estimations for  are quite large. Representative characteristic times for mean reversion are within a few days. Therefore, this model has the main disadvantage that futures prices are nearly constant over time, since under the assumption of Koekebakker and Ollmar (2001) the futures price is given by For this motive, several authors proposed a two-factor model, see e.g. , Pilipovic (1988) and Schwartz and Smith (2000). In Pilipovic (1988) where   t X is an Ornstein -Uhlenbeck process accountable for the short-term variation and   t Y is a Brownian motion recitation the long-term dynamics. From this model, we will present in and can be considered as an extension of the ideas of Schwartz and Smith (2000).
All models are measured so far did not contemplate seasonality. Some authors purely abandonment this serious difficulty. Others propose to use deterministic seasonality designated by sinusoidal functions, see Barlow (2002), , Roncoroni (2003), andSmith (2000). In Knittel and Roberts (2001) it is suggested to use equation (5) with a long run mean t a describing the seasonal patterns. A general deterministic seasonality is proposed in Knittel and Roberts (2001) and . Here, the spot price is modeled as with an arbitrary deterministic function   ft and a mean-reverting stochastic process t X In our approach, the deterministic component   ft is specified by the load forecast `t and additional stochastic behavior is introduced using SARIMA models for the time series of load and prices.
Many authors are attempted different way to account for price spikes. One possibility to manage with spikes is the overview of jump terms, see Box and Jenkins (1976), Knittel and Roberts (2001) and . The main disparagement for these models is that under the typical assumption of a jump-diffusion model a huge upward jump is not necessarily tailed by a great downward jump. Therefore, some authors suggest hidden Markov models, known as Markovian regime-switching models, where it is defined that upward jumps are tracked by downward jumps. Such models have been reflected e. g. in De Jong, and Huisman (2002), Deng (2000), , Huisman, and Mahieu (2001) and Kholodnyi (2000). Regime-switching models are very instinctive candidates for electricity price models since there are some clear physical reasons for switches of regimes such as forced outages of significant power plants. On the other hand, it gives the impression to be problematic to combine regime-switching with seasonality. In Additional, another approach is interested by the economic background for price spikes. Prices are originated mainly by supply and demand (load). Therefore, the non-linear relation between load and price should be considered in the model. This non-linear renovation is called the 'power stack function' in Eydeland, and Geman (1999) and Skantze, Gubina, and Ilic, (2000). They recommend an exponential function for that determination. A similar model for spot prices has recently been measured in Black (1976), where the relation is suggested with t X being an Ornstein-Uhlenbeck process, and f a power function. We also choose an method based on the power stack function, since there is a natural understanding of this non-linear transform in terms of the merit order curve.

Electricity options
The power industry had been exploiting the idea of options through surrounded terms and conditions in many supplies and obtaining contracts for decades, deprived of obviously recognizing and valuing the options until the commencement of the electricity industry rearrangement in the U.K., the U.S., and the Nordic countries in the 1990s. The development of the electricity wholesale markets, and the dissemination of option pricing and risk management techniques have shaped electricity options not only based on the underlying price attribute (with plain vanilla electricity call and put options), but also other attributes like volume, delivery location and timing, quality, and fuel type.
Fundamentally, a counterparty of each financial option can be shaped in the domain of electricity options and by substituting the underlying in a financial option with electricity market Hull (2000) for introduction to various kinds of financial options (American Style, European Style, Exchange Traded Options, Over the Counter Options, Option Type by Underlying Security and Option Type by Expiration). At this point, we entitle a sample of electricity options that are frequently utilized in risk management applications in generation and distribution sectors. These options are generally had short-to medium maturity times such as months or a couple of years. Options with maturity times longer than three years are usually entrenched in long-term supply or purchase contracts, which are termed as organized transactions. Now, we turn to pricing of a European call option written on the electricity spot price. Recall, that the European option is a contract that gives the buyer the right to buy/sell the underlying commodity at some future date t (called maturity) at a certain price K (called the strike price). First, we find the pricing measure  Q . Like Merton (1976) in the context of jump-diffusion processes we assume that the dynamics of spikes and drops are the same in the actual and pricing measures. We start with finding the spot price dynamics under  horizon long enough to contain all maturities of derivatives quoted in the market, and introduce a new process  t W : with the filtration W t F , being the natural filtration of the process t W .
Now, the base regime process b t X , can be rewritten as: and the expected future spot price is given by: The function ) (t  can be calibrated to the market forward prices so that e.g. by using some fitting procedure (like the least squares minimization). Alternatively, one can find the risk premium and then use the relation between the market price of risk ) (t  and the risk premium: which is a simple consequence of the fact that ) (15) and Ito's lemma. Now, the price of a European call option written on the electricity spot price can be derived.
Option price formula. If the electricity spot price t P is given by the MRS model then the price of a European call option written on t P with strike price K and maturity T is equal to: is the cumulative distribution function of the log-normal distribution with parameters  and 2  .
Here, we assume that the option is settled in an infinitesimal period     T T, . However, in practice, the electricity spot price usually corresponds to a delivery during some period (e.g. an hour, a day) and, hence, the maturity of the option should be specified on the same timescale. On the other hand, the examined spot price quotations usually represent some delivery period. For example, if the measured data is quoted daily, then the maturity of the option would be also given in daily timescale and would correspond to daily delivery.  -ISSN: 2746-136X. Vol 2, No.3, September 2021 Published by:

International Journal of Finance Research
Page 199 An Electricity forward contract is a customized contract between two parties to buy or sell an asset at a specified price on a future date. A forward contract can be used for hedging or speculation, although its non-standardized nature makes it particularly apt for hedging. Electricity forward contracts characterize the obligation to buy or sell a fixed amount of electricity at a pre-specified agreed price, known as the forward price, at specific time in the future (called maturity or expiration time). In other words, electricity forwards are convention tailored supply contracts between a buyer and a seller, where the buyer is obligated to take power and the seller is obligated to supply. The payoff of a forward contract auspicious to deliver one unit of electricity at price F at a future time T is: Where T S is the electricity spot price at time T. Although the payoff function (1) appears to be the same as for any financial forwards, electricity forwards differ from other financial and commodity forward contracts in that the underlying electricity is a different commodity at different times. The settlement price T S is usually calculated based on the average price of electricity over the delivery period at the maturity time T.
Undoubtedly, the most popular electricity derivatives are the forward contracts. Recall that a forward contract is an agreement to buy (sell) a with specific amount of the underlying (here MWh of electricity) at a specified forward date. Settlement of the contract can be specified with physical delivery of electricity or with only financial clearing. Both types of settlement are in the following called delivery date. Denote the price at time t of a forward contract with a delivery at time T by T t f . There is no investment at the of entering a forward contract, the expected future payoff under the pricing measure should fulfill: what implies that Observe, that now we define the price of a forward contract at any future date t . This is motivation by the circumstance that the valuation at time 0 of an option written on a forward contract requires the knowledge about the forward price dynamics at the option's maturity t e-ISSN: 2746-136X.

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Published by: Page 200 Forward price formula. If the electricity spot price t P is given by the MRS model, then the price at time t of a forward contract written on t P with a delivery at time T is given by the following formula Note that in the above formula On the other hand, if at time t a spike or a drop occurred then and again this expectation is dependent on When deriving the forward price dynamics, we must remember that the belongings of the obtained model should observe with the observed market prices. One of the most pronounced features of the market forward prices is the detected term structure of volatility, called the Samuelson effect. Exactly, the volatility of the forward prices is moderately law for distant delivery periods, however, it increases quickly with impending maturity of the contracts.
Here, the forward price volatility is described by the part (22). Hence, it is specified by the volatility of the spot price base regime scaled with ) ( t T e   and the corresponding probability of switching to the base regime. Observe that the scaling factor ) ( t T e   exhibits the Samuelson effect as it increases to 1 with t approaching maturity time T . Moreover, the forward price volatility, again due to the scaling factor, is lower than the spot price volatility. This is following the behavior of the market spot and forward prices. Electricity forward contracts listed on energy exchanges are usually settled during a certain period (a week, a month, a year, etc.). Denote the price at time t of a forward contract

OPTIONS WRITTEN ON ELECTRICITY FORWARD CONTRACTS
Lastly, we derive an unambiguous formula for a European call option written on a forward contract delivering electricity during a specified period. Observe, that the forward price depends on the spot price at time t and, consequently, also on the state process value at time t . Now, we will consider an option written on an electricity forward contract with settlement during a specified period, as it is the most popular specification of electricity options on energy exchanges. For example, in the European Energy Exchange (EEX) market there are options written on forward contracts with monthly, quarterly, and yearly settlement periods.
The maturity of such options is set to the quarter business day before the beginning of the underlying contract's settlement period.  -ISSN: 2746-136X. Vol 2, No.3, September 2021 Published by: Page 203 Colombian competitive electricity market. It makes it compulsory to develop an appropriate risk management to maximize agents' benefits and minimize the corresponding uncertainty upon them. Among the risks that firms must handle with, are macroeconomic risk, price risk, market risk, credit risk, regulatory risk, country risk, and quantity risk. The last one is a non-tradable risk, and an implicit feature of electricity, which has to do with the amount of energy that will be demanded in the future, which follows a stochastic process as well as the spot price. This situation directly impacts the firm's revenues and makes it necessary to include it when designing the hedging portfolio.
Here, we developed a financial theory for research studies to find how to address this problem. In Liu, and Wu (2007) electricity risk management is handled by a multi-market trading approach, while other references like Huisman, Mahieu, and Schlichter (nd) get emphasis on best risk management trough forward on-peak and off-peak contracts. Nevertheless, the electricity derivatives are progressively studied and used around the world to manage the financial risks and resource appropriateness of power markets, like it is advocated in Chao, and Wilson (nd). Particularly Villareal, and Soto (nd) proposed a financial call option to hedge against critic hydrologic scenarios in the Colombian electricity system by ensuring generation adequacy.
Further, derivative instruments have been developed to also handle the quantity risk, like swing options, weather derivatives, interruptible contracts, among other instruments named in Deng, and Oren(2006). The features of these derivative products make them be usually traded over the counter (OTC); therefore, they are low liquidity instruments, which is why they are not regarded in this paper. However, references Oum, Oren, and Deng (2006) and Prokopczuk, Rachev, Schindlmayr, and Truck,(nd) study quantity risk.
Oum, Y., Oren, S. (nd) deal with the static hedging problem of a LSE who has to serve an uncertain electricity demand q at a regulated fixed price r in a single period from 0 to 1.
Besides, the LSE procures the electricity to serve his customers, from the wholesale market at a spot price p . Hence the profit of the LSE would be: To protect himself against price risk, the LSE can take a long position in 9  q forward contracts at a fixed forward price F . However, the LSE will face another risk that arises  -ISSN: 2746-136X. Vol 2, No.3, September 2021 Published by:

International Journal of Finance Research
Page 204 from the fact that demand q may vary from the expected value at time 0 to the actually realized value at time 1. Then, if the actual demand realized by the retailer agent is q q   , then, the share of the profit in Chao and Wilson (nd), that is at risk is:   q p r    , where q  could be different to zero (gains/losses).
To deal with this hedging problem, the authors derive the optimal hedging portfolio as a function of spot price p . That is: where   p x is the optimal hedging portfolio as function of the spot price p .
Regarding the LSE's preferences and risk aversion profile, it is necessary to identify its utility function U over total profit Y . In turn, because of the positive correlation between price and demand, which will be evident later in the paper for the Colombian electricity market, there exists a joint probability function   q p f , defined on the probability measure P , which characterizes the behavior of p and q at time 1. On the other side, Q is a riskneutral probability measure by which the hedging instruments are priced and   p g is the probability density function of p under Q . Keeping this in mind, in the optimization problem is formulated as follows: denote expectations under probability measures P and Q , respectively. In (30) the constraint implies that the hedge portfolio   p x is self-financing, that is, the LSE can borrow funds in the money market, in order to purchase the derivative instruments needed to obtain the maximum expected utility over the total profit   q p Y , .
This constraint also means that there are not arbitrage opportunities through this hedging portfolio, under a constant risk-free rate. The optimization process yields as a result the optimal payoff function   p x  .
Through an extension of the fundamental calculus theorem, it is demonstrated that any twice continuously differentiable function can be written as follows, for fixed value F : The errors of this replicating strategy are calculated depending on the range in which spot price is realized at time 1.

MODEL ESTIMATION UNDER THE PHYSICAL MEASURE
The non-storability feature of electricity along with the steeply rising supply and the inelastic demand curve, both schematically represented in Figure 1, makes the electricity price p and the electricity demand q to be positive correlated. It happens in this way in the Colombian electricity market, since the dispatch is carried out in order of merit, so, when demand rises during on-peak hours, it forces the system to put in operation a more expensive generation resource, increasing the spot price as well.
Additional to the fact explained above, throughout summer seasons, the hydropower plants which are technologies with lower variable costs, reduce their power production due to natural water inflows shortages and of course to a diminishment in water reserves into reservoirs. This turns out in a higher marginal system cost, so an increment in demand translates into an increase in the spot price. This increment in the spot price reaches larger values during summer seasons than during winter seasons. Therefore, the correlation e- ISSN: 2746-136X. Vol 2, No.3, September 2021 Published by: Page 206 between p and q is bigger during dry periods in the Colombian wholesale electricity market.
The earlier reasons justify the usage of the price-quantity hedging strategy presented, to design suitable derivative instruments to be used by the agents of the electricity market.
The advantage offered by the financial derivative products, arises from the fact that is likely to achieve the maximum expected value of total profit   q p Y , , together with the minimum deviation of this profit (which means less uncertainty over LSE's total profit), after building a hedging portfolio with those instruments. To do so, a utility function to represent the risk aversion preferences of the agents like a mean-variance utility function is a good candidate to accomplish both goals at the same time. Here it is utilized the same mean-variance utility function which has been used in financial hedging literature to deal with non-tradable risk: where a represents the agent's risk aversion coefficient. Certainly, maximizing the expected value of the utility function in (33) , is equivalent to maximize the expected value of   q p Y , and to minimize the variance of (29), that is the advantage provided by a price-quantity hedging portfolio.
In equation (33) it is presented the optimal hedging function constrained to the meanvariance utility function. The arithmetic procedure to find this expression is clearly developed. Published by: Page 208 soon in Colombia. Since there are no financial options to replicate the optimal hedging function on the current electricity market. Now, for the empirical analysis and the estimation procedure we use data from the Colombian electricity markets daily market representative rate value (in Colombian Pesos) the period from January 1, 1995, to December 2 2013 with 6853 observations. The fig 2 time series of the resulting price for the above Colombian electricity daily price.

Conclusion
From the above discussion, in this paper we attempted to introduce an algorithm for pricing derivatives to intuition from Colombian electricity market. Initially we started our approach through to simple stochastic models for electricity pricing. And we derived analytical formulas for prices of electricity derivatives with different derivatives tools. 3) Then we extended short of the model for price risk in the electricity spot market. Finally, we constructed the model estimation under the physical measures for Colombian electricity market. And this paper end with conclusion.