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Asian options pricing in the day-ahead electricity market
Accepted Manuscript Title: Asian options pricing in the day-ahead electricity market Author: Viviana Fanelli Lucia Maddalena Silvana Musti PII: DOI: Reference:
S2210-6707(16)30137-8 http://dx.doi.org/doi:10.1016/j.scs.2016.06.025 SCS 459
To appear in: Received date: Revised date: Accepted date:
3-9-2015 20-4-2016 29-6-2016
Please cite this article as: Fanelli, Viviana., Maddalena, Lucia., & Musti, Silvana., Asian options pricing in the day-ahead electricity market.Sustainable Cities and Society http://dx.doi.org/10.1016/j.scs.2016.06.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Asian options pricing in the day-ahead electricity market Viviana Fanelli1 Lucia Maddalena2 Silvana Musti3
1
Dipartimento di Economia, Management e Diritto d'Impresa, Università degli Studi di Bari, [email protected] Dipartimento di Economia, Università degli Studi di Foggia, [email protected]
2
3
Dipartimento di Economia, Università degli Studi di Foggia, [email protected]
1
Highlights
We implement an electricity market equilibrium model. We model electricity prices by using the interest rate theory. We price Asian options in the day-ahead electricity market. We verify that agent provisional ability enhances the market efficiency by decreasing prices.
Abstract The purpose of this paper is to show that by using the toolkit of interest rate theory, it is possible to evaluate option prices through an electricity market equilibrium model. Options represent an adequate instrument to manage price risk faced by electricity producers that are related to both pool price and unit availability uncertainty. We have priced path dependent put and call Asian options by using arithmetic average, verifying that the possibility of market agents to forecast the future demand contributes to the market efficiency making option price level lower. Keywords: Electricity price modeling; Asian options; electricity producer; risk; equilibrium model.
1. Introduction During the last decades, dramatic changes to the structure of the power sector have taken place: a growing number of countries have deregulated their monopolistic electricity markets. They become, therefore, competitive markets, where energy prices are derived by the interaction of supply and demand. A fundamental role is played by the electricity demand side: the electric load management (ELM) . Many researchers have focused on the demand side management in order to develop techniques and methods to improve the efficiency of the power infrastructure by controlling and coordinating the operation of power loads (see [28], [29], [30], [31]). A complete and exhaustive literature review on this topic is given by Benetti et al. ([3]). This new context, along with the physical characteristics of the electricity, has generated new price dynamics, never seen before, neither on the financial markets nor on the commodities markets. The most notable electricity characteristic is its non-storability, or at least the relevant costs associated with the storage. Therefore, with the exception of hydropower, electricity should be generated exactly at the time of its request for consumption. The offer turns out to be completely inelastic to price changes, and also prices show very high volatilities and sudden changes of price levels, which are called spikes. Market operators, both producers and consumers, therefore suffer the exposure to energy price uncertainty. In Valenzuela, Thimmapuram, and Kim [37], a multi-agent architecture is used to 2
simulate the energy market, and to study the behaviour of customers under dynamic-pricing conditions. Several approaches have been proposed in the literature, focusing on demand side management and demand response, in order to control and change the power consumption habits: Gelazanskas and Gamage [16], Strbac [35], Aalami et al. [1], Ashok and Banerjee [2], Finn et al. [13] Risk management techniques become fundamental tools to quantify and cope with price risk generated from such uncertainty. Electricity derivatives aiming at handling the exposure of market agents to different kinds of risk, have emerged in the restructured electricity production market. The financial derivatives contribute to control risks through adequate hedging strategies. In particular, forward contracts are agreements in order to buy or sell a fixed amount of electricity at a settled price throughout a future time period. On one hand, this contract allows power producers to hedge price risk; on the other hand, the forward contract is mandatory and the power producer is obliged to supply the agreed quantity throughout the established time interval. Therefore, the forward contract reduces the price risk, but increases the probability of financial losses due to unexpected unit shutdowns, exposing the producer to the so called availability risk [33]. An alternative to these derivatives is offered to the producers by selling or buying their production through electricity options. An option is a contract that gives the holder of the option the right to buy (call option) or sell (put option) e specified energy amount during a certain future time period and at a fixed price (strike). The holder can decide whether or not to exercise the option, depending on the price behavior and the availability of the generating units. Nonetheless, while a forward contract is costless, the subscriber of the option has to pay an initial adequate cost, option price. In [33] the authors describe electricity options as instruments to manage the two main risks faced by power producers: price and production availability risk. They describe a multi-stage stochastic programming model that enables a risk averse power producer to decide its optimal portfolio of forward contracts and options taking into account the pool price volatility. The option price obviously plays a crucial role in these hedging strategies and in this paper we evaluate numerically European and Asian call and put options. Asian options are derivative contracts which have, as underlying variable, the average price of the asset over a period of time. Due to this definition, Asian options display a lower volatility compared to the standard European ones and therefore they show lower prices ([24],[34]). These path dependent options appear to be particularly appropriate to the electricity market, where the 3
contracts are written to give electrical power continuously over a period of time. It appears therefore reasonable for this market to refer to the average price over a period of time instead of price at the maturity of the option. The pricing of Asian Options cannot be obtained with analytical methods but
solely by developing numerical approximations [23]. The procedure is
computationally intensive following different methods: the authors in [36], [32] and [27] approximate the average distribution by fitting integer moments. Rogers and Shi [34] and Vecer [38] give a numerical solution of a rescaled version of the pricing partial differential equation, while Geman and Yor [17] apply numerical inversion of the Laplace transform, and Fusai [14] and Cai and Kou [5] apply numerical inversion of the double transform. Linetsky [27] makes use of the eigenfunction method and Zhang [40] applies the perturbation approach. The values are monitored on the time horizon between the settlement day and the contract maturity. The first option, depending on a price average, has been traded in Tokyo in 1987, giving the name of this option type. For this reason, both researchers and practitioners have concentrated on the electricity price evolution study, by setting up models able to catch the main characteristics of price evolution, in order to price derivative products and manage price risk. The present work follows the research line which refers to the modeling framework of Heath, Jarrow and Morton (hereafter HJM) [19] that, using only few stochastic factors and the initial price curve as given, models futures prices under some equivalent martingale measure in a no-arbitrage environment. Clewlow and Strickland [8] have been the first researchers to introduce this approach to the energy market. Bjerksund et al. [4] and Koekebakker et al. [25] model a continuum of instantaneous-delivery forward contracts under risk neutral probability measure. Fanelli et al. [10] model electricity prices using seasonal path dependent volatility. Hinz [18] gives an interesting interpretation of the electricity market and demonstrates that it is possible to create a market framework where the existence of a risk neutral measure is guaranteed: the energy cannot be stored, but it can be produced. Therefore, the producer can put himself in the condition of having the ability to produce electricity, creating a sort of “electricity storability”. According to this perspective, the electricity market becomes more complex and has to be considered as composed of both power electricity and agreements on power production capacities. The market reaches the equilibrium and determines the price process for all tradable assets both physical (production capacity agreements) and financial (future electricity prices). The equilibrium existence gives an economic interpretation of the martingale measure Q: it is 4
equivalent to the market measure P, such that equilibrium asset prices under P are given by their future revenues, expected with respect to Q. Price dynamics can be, therefore, described directly under the equivalent martingale measure Q and it becomes possible to price all contracts by using classical no arbitrage models of financial mathematics. In addition, The HJM [19] interest rate framework we have developed, does not need the market price of risk as input, but only the volatility function ([7], [8], [9], [10], [11]). This important feature allows us to bypass the market price of risk as input to price electricity derivatives (Weron [39]): we have obtained from market data the volatility parameters representing the input of the equilibrium model. According to this approach, Fanelli et al. ([12]) have obtained the electricity price evolution in the electricity market by applying the HJM term structure model to the Currency Change Electricity Market. The option pricing has never been obtained in the existing literature by combining the HJM framework with the equilibrium model approach. The advantage of this research is to combine the benefits of the free volatility formulation of the HJM model, with the stochastic electricity demand dynamics of the equilibrium model. Market structure is analyzed in both aspects, allowing us to efficiently price options whose evaluation in most cases results numerically difficult to handle. Our contribution to the existing literature consists in furnishing a range of average option pricing in order to complete former studies about hedging strategies in the electricity market. In the present work, we further develop our research on electricity price evolution in the no arbitrage HJM approach, by connecting it with the Hinz ([20]) equilibrium model. The explicit dynamics of the spot price is obtained with a precise parameters evaluation by collecting market data as input in the HJM model. The calibration of the model on the market information is therefore twofold: on one side the explicit dynamics of the spot price is calibrated on the market with the market data by use of the volatility parameters. This represents the contribution of our work to the existing literature on the equilibrium model ([15],[20]). On the other side, the equilibrium model allows us to price options taking into account the electricity level request at each single day of the time interval of the implementation. To our knowledge, there are no published papers pricing Asian options taking account both of the market price volatility and the daily demand fluctuation: the Asian Option prices confirm that the demand function reflects in the results level and in most cases the forecast capability of the agents contributes to market efficiency. Indeed, having as input the spot prices obtained with the methodology introduced by Fanelli et al. ([12]), we numerically implement the Hinz ([20]) equilibrium model to numerically estimate Asian 5
and European Options on the day-ahead electricity market. The rest of the paper is organized as follows: in paragraph 2 we describe the equilibrium model that we are implementing in its singleperiod and multi-period version. In paragraph 3 we expose the methodology to obtain the day ahead price series with the equilibrium model. In paragraph 4 we expose the input data and the numerical estimates obtained. Results and conclusions are exposed in paragraph 5.
2. Equilibrium model We consider at this stage the one period version of the model. The market is composed by N agents acting simultaneously as retailer as well as producers of electricity. In the single period model there are two dates: 𝑇0 represents today and 𝑇1 represents tomorrow, the delivery time. At time 𝑇0 each agent does not know the energy request he will be asked for the following day 𝑇1 . This amount represents an exogenous random variable. The agent energy request can be satisfied both by purchasing today on the day ahead market with price 𝑝 and by tomorrow production, by using his own power plants. The choice about how to satisfy the request depends on the economic valuation based on prices and quantities. The profit and loss function for agent i is the following (Hinz [20]):
𝑓 𝐺𝑖 (𝑝, 𝑞𝑖 , 𝑄̃𝑖 , 𝑝̃ 𝑠 ) ≔ 𝑝𝑖𝑟 𝑄̃𝑖 + 𝑝𝑏 (𝑞𝑖 − 𝑄̃𝑖 )+ − 𝑃𝑖 − 𝑝𝑞𝑖 − 𝑚𝑖𝑛{𝑝𝑖𝑣 , 𝑝̃ 𝑠 }𝑚𝑖𝑛 {(𝑄̃𝑖 − 𝑞𝑖 )+ , 𝑐𝑖 } +
−𝑝̃ 𝑠 (𝑄̃𝑖 − 𝑞𝑖 − 𝑐𝑖 )+ whose variables are defined in Table 1.
Each agent at time 𝑇0 establishes the quantity of energy to buy on the day-ahead market and the interaction among decisions of the N agents determines the price 𝑝. The random variables are ̃𝑖 and 𝑝̃ 𝑠 . The profit and loss function is concave in respect to the represented by the quantities 𝑄 control variable 𝑞𝑖 and it guarantees the optimal solution for every agent maximizing his expected utility 𝔼[𝒰𝑖 (𝐺𝑖 (𝑝, 𝑞𝑖 ,∙,∙))] subject to the global production capacity constraints. Indeed, every agent can sell at least his production, buy at least the all system production and the net balance of all quantities has to be null: 6
𝑞𝑖 ∈ [−𝑐𝑖 , ∑ 𝑐𝑗 ] 𝑗≠𝑖
∑𝑁 𝑖=1 𝑞𝑖 = 0. The author in [20] demonstrates that the equilibrium price exists if spot price is greater that variable cost, even if the uniqueness of the equilibrium point is still to be investigated. The multi period model represents the natural extension of the single period model, and the random variables already introduced are labeled with the temporal notation and represented as a (𝑁 + 1)-dimension stochastic process: ̃1 (𝑛), 𝑄 ̃2 (𝑛), … , 𝑄 ̃ 𝑁 ( 𝑛) ) 𝜋̃(𝑛) = (𝑝̃ 𝑠 (𝑛), 𝑄 𝑛≥1
(1)
̃𝑖 (𝑛) represents the KWh where 𝑝̃ 𝑠 (𝑛) represents the spot price of 1 KWh at time 𝑛, while 𝑄 quantity of energy requested by agent 𝑖 at time 𝑛. The filtration generated by the process, ℱ 𝜋̃ , represents the information generated by the process and an equilibrium is defined as a process adapted to the filtration ℱ 𝜋̃ (𝑝∗ , 𝑞1∗ (𝑛), 𝑞2∗ (𝑛), … , 𝑞𝑁∗ (𝑛))𝑛≥1 such that any couple (𝑝∗ , 𝑞𝑖∗ (𝑛)) maximizes 𝑖-th agent expected utility at time 𝑛 ≥ 1, given the information available at the former date (𝑛 − 1).
3. Numerical Implementation The model is implemented† in its multi-period version and in the hypothesis we formulate there are no differences among the agents, so that they sustain the same costs and have the same probability distribution of the energy request. With these hypothesis the equilibrium price is the following: 𝑝∗ (𝑛) = 𝑝𝑣 − 𝔼 [(𝑝𝑣 − 𝑝̃ 𝑠 (𝑛 + 1))+ ] + 𝔼[(𝑝̃ 𝑠 (𝑛 + 1)−𝑝𝑣 )+ ]ℙ (𝑄̃ (𝑛 + 1) ≥ 𝑐|𝜋̃(𝑛)). The series 𝑎̃𝑛 : = 𝑝𝑣 − 𝔼 [(𝑝𝑣 − 𝑝̃ 𝑠 (𝑛 + 1))+ ]
†
(3)
Fortran 90 language, Geany compiler, Ubuntu OS.
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(2)
𝑏̃𝑛 : = 𝑝𝑣 − 𝔼 [(𝑝𝑣 − 𝑝̃ 𝑠 (𝑛 + 1))+ ] + 𝔼[(𝑝̃ 𝑠 (𝑛 + 1)−𝑝𝑣 )+ ]
(4)
represent the equilibrium price variability interval. The series 𝑎̃𝑛 and 𝑏̃𝑛 are identified once we observe the spot price realization 𝑝̃ 𝑠 (𝑛) and we can forecast the possible evolution at the following time (𝑛 + 1). In order to numerically implement the model, we have to make hypothesis regarding the spot price 𝑝̃ 𝑠 and demand 𝑄̃𝑖 dynamics. In figure 1 the flow chart of the numerical scheme is proposed. The spot price series 𝑝̃ 𝑠 (𝑛) is obtained by applying the model Fanelli et al. [10]. Indeed, we consider as input the spot price determined by applying the HJM model. The currency change represents the technical instrument to make the HJM model suitable in the electricity market [20]. According to the HJM model, we assume that the forward price dynamics 𝑓𝑡 (𝑇), 𝑡 < 𝑇 is given by the stochastic differential equation 𝑑𝑓𝑡 (𝑇) = 𝛼𝑡 (𝑇)𝑑𝑡 + 𝜎𝑡 (𝑇)𝑑𝑊(𝑡). By definition, the price process of a forward contract is: 𝑇
𝑝𝑡 (𝑇) = 𝑒 − ∫𝑡
𝑓𝑡 (𝑠)𝑑𝑠
.
(5)
In order to avoid arbitrage opportunities, 𝛼𝑡 (𝑇) is function of the volatility 𝑇
𝛼𝑡 (𝑇) = 𝜎𝑡 (𝑇) ∫ 𝜎𝑠 (𝑇)𝑑𝑠 𝑡
and the forward rate dynamics in its integral form becomes 𝑡
𝑇
𝑡
𝑓𝑡 (𝑇) = 𝑓0 (𝑇) + ∫0 (𝜎𝑠 (𝑇) ∫𝑠 𝜎𝑢 (𝑢) 𝑑𝑢) 𝑑𝑠 + ∫0 𝜎𝑠 (𝑇)𝑑𝑊(𝑠).
(6)
The curve evolution in completely determined by the initial curve 𝑓0 (𝑇) = 𝜂0 − 𝜂1 𝑒 𝜂2 𝑡 and the volatility function 𝜎𝑡 (𝑇). In order to construct the currency change market model we have to introduce a risky asset with the following price dynamics: 𝑑𝑁𝑡 = 𝑁𝑡 (𝑓𝑡 (𝑡)𝑑𝑡 + 𝑣𝑁𝑡 𝑑𝑊(𝑡)) where 𝑣 is a constant volatility function. We consider the currency change 𝐹𝑡 (𝑇): =
𝑝𝑡 (𝑇) 𝑁𝑡
and obtain the forward price of electricity. The spot price is obtained with 𝑇 = 𝑡 8
𝑝̃𝑡𝑠 = 𝐹𝑡 (𝑡).
(7)
The HJM model features make it highly versatile because of the volatility function choice 𝜎𝑡 (𝑇) . We consider the following forward rate volatility function (6): 𝜎𝑡 (𝑇) ≔ 𝑒 −𝜆(𝑇−𝑡) [𝑎𝑓 𝑓(𝑡, 𝑇)]
(8)
and we estimate the parameters 𝜆 and 𝑎𝑓 .The price process is obtained by simulating the daily entire forward curve stochastic evolution over 1 year period. The second fundamental input to implement the equilibrium model is the demand process. The process is given by the discrete sequence of random variables 𝑄̃𝑖 (𝑛 + 1) = 𝜇 𝑒 𝛼𝑥(𝑛−1)+𝛽𝜐𝑖 (𝑛)
(9)
where 𝜇, 𝛼, 𝛽 > 0, 𝜐𝑖 (𝑛) is, for each n, a standard normal variable and 𝑥(𝑛) is ARMA(2,1) process 𝑥(𝑛 + 1) = 𝜑1 𝑥(𝑛) + 𝜑2 𝑥(𝑛 − 1) + 𝜗1 𝜀(𝑛) + 𝜗2 𝜀(𝑛 + 1)
(10)
|𝜑𝑖 | < 1, |𝜗𝑖 | < 1, 𝑖 = 1,2 , 𝜀(𝑛)~𝑁(0,1) driving the electricity demand addressed to each agent in the market. We suppose that the electricity requests received by the agents have the same probability distribution, and 𝑄̃ (𝑛) is the random variable representing the electricity demand addressed to each agent. The demand path is simulated with the following parameters
The condition 𝑐 = 1,5 > 𝜇 = 1.0 indicates that a large part of demand is met with in-house production rather than by purchasing electricity on the market. Finally, the agent capability of forecasting demand levels for the following day becomes crucial in order to define the equilibrium price on the day ahead market. This ability is deeply influenced by the demand volatility and it is possible to take it into account simply by fixing parameter 𝛽 > 𝜇. As it is shown in Fanelli et al. [12], the high predictability of demand level determines lower equilibrium prices, while low predictability level causes high volatile prices showing that agents accept to pay prices closer to the maximum boundary 𝑏𝑛 .
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The equilibrium model gives the opportunity to take into account the observed volatility function in the electricity market, simulate electricity prices according to the CCEM model, integrating the data with the electricity demand. The calibration of the volatility function in the HJM model and the use of the ARMA model for the demand evolution, represents the key instruments to take the physical characteristics of the electricity into the model implementation.
4. Numerical Results The data set consists of daily time series of the term structure of the Eonia rates, spanning from January 2007 to mid May 2010. These data are used to estimate parameters of the forward rate volatility function in the EEX electricity market. The parameters of the initial curve 𝑓0 (𝑇) = 𝜂0 − 𝜂1 𝑒 𝜂2𝑡 have been calibrated on historical data, obtaining 𝜂0 = 0,004 , 𝜂1 = 0,003 and 𝜂2 = −0,2. The volatility function parameters in (8) have been
estimated by applying the nonlinear least squares method to the volatility time series referring to the observed period , obtaining the following significant values: 𝜆 = −0.0399219 and 𝑎𝑓 = 0,0182197. The Monte Carlo simulation method (with 𝑁 paths) has been applied by simulating the day ahead price over a 1 year horizon time ( 1 < 𝑛 < 30 ). The European Call and Put have 1 year maturity and the Asian Call and Put option have been priced averaging the final price over the last month daily observation of the year. For each path i , 1 < i < N, the last 30 observed prices have been averaged to obtain the payoff of the arithmetic Asian call option, by comparison with the strike price, while the last observed price (𝑛 = 360) gives the 𝑖 − 𝑡ℎ payoff for the European call option (see figure 1). According to the Monte Carlo method, the expected values over the 𝑁 simulated paths are obtained both for the Asian and the European call payoff. Option prices are finally obtained by discounting the payoffs at the risk free rate up to the initial time 0. The main feature of the equilibrium model is the opportunity to determine the time series of the day ahead electricity, starting from the spot price time series. The key instrument to evaluate the day ahead price at time 𝑛 is to make hypothesis on the realization of the spot price on the following day, time (𝑛 + 1) , in relation with the expected demand level at time (𝑛 + 1), 𝑄̃ (𝑛 + 1). In Table 3 and Table 4 are reported the Call option prices with strike price of 90. In Table 3 we have supposed the hypothesis of low capability of forecasting demand level by the agent, setting the demand function parameter 𝛽 < 𝜇 , 𝛽 = 1 and 𝜇 = 7 . In Table 4 the parameters values are 10
switched, and the high predictability hypothesis is obtained with 𝛽 = 7 and 𝜇 = 1. The call option values are obtained by using different number of paths (first column) in both Table 3 and Table 4, and the European option value is reported in the second column. The Asian Call Option value has been estimated averaging the daily prices over the last one month period before option maturity, with arithmetic average (column 3). Table 5 and 6 report the analogous estimation numerically obtained for the Put option with strike price 110. In Table 5 results are obtained with low predictability level, while in Table 6 the numerical simulations have been carried forward with the hypothesis of high predictability of the demand level by the agents acting in the market. In columns 3 the put option values with arithmetic average is shown. For every estimate the 95% confidence interval is reported. Asian options are options in which the underlying variable is the average price over a period of time. On account of this fact, Asian options have a lower volatility and hence rendering them cheaper relative to their European counterparts ([24], [34]). By comparing the estimates, we can observe that Asian options have smaller values compared to the correspondent European ones, and the predictability hypothesis makes the option values in most of the cases lower when the agents are able to forecast, in an adequate manner, the demand level for the following day. These results confirm how the production scheduling remains one of the key strategic instruments in order to have an efficient market. As we were expecting, the European options have a greater price respect to Asian options, because the averaging over the 30 days of the last months makes the option price lower because reduces the payoff variability at maturity time.
5. Conclusions Electricity producers participating in electricity market, face risks inherent to prices they are exposed to, both as seller and as buyers of energy. Moreover, the availability risk is the aspect producers have to deal with, if a producing unit shuts down. Among electricity derivatives, options represent an adequate instrument to manage these two risks. By gathering this important issue faced by Pineda and Conejo [33], we have implemented a numerical method to price European and Asian options in the electricity market. The numerical implementation we propose of the Hinz [20] equilibrium model, allows us to take into account the electricity price variability by simply calibrating the volatility parameters as input of the model. The spot price path is, in this analysis, 11
determined by taking into account the historical volatilities observed and it allows, therefore, to simulate in a realistic way the spot price evolution, once the demand evolution is given as input. The demand is supposed to be driven by an ARMA (2,1) process. Furthermore, Hinz [21] is the first researcher dealing with the use of classical no arbitrage models in the energy market price modeling: the term structure model appears, therefore, coherent in order to determine a spot price as input to obtain the day-ahead price in the equilibrium model. The implementation gives the opportunity to evaluate different hypothesis about demand process and to verify how different hypothesis reflects in the option price levels. We have priced both European and Asian options, put and call type, considering daily prices over one month period, with arithmetic average. By comparing the estimates obtained for the European and Asian options, we conclude that the numerical method gives results coherent with the financial definition of these two types of derivatives and can be considered as a valid instrument to price derivative that can be very useful in managing financial risk. In particular, Asian Options are path dependent options and for this reason, we are confident that this type of model, with a path dependent volatility and an explicit hypothesis about the demand process, is adequate in estimating this kind of electricity derivatives.
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15
Figure 1: Flowchart of the i-th payoff algorithm
16
Table 1: Variables and parameters of the equilibrium model
𝑝
1 MWh day-ahead price (variable)
𝑞𝑖
Amount of energy purchased today by the agent i on the day ahead market (control variable for the agent)
𝑄̃𝑖
Amount of energy that the agent i will be asked for at time 𝑇1 (variable)
𝑝̃ 𝑠
Tomorrow 1 MWh spot price (variable)
𝑝𝑖𝑟
Selling price the agent i will obtain by selling 1 MWH of electricity (fixed)
𝑝𝑏
“Back supply price”, price obtained selling 1 MWh (for the production exceeding the request) (fixed)
𝑃𝑖
𝑓
General cost of production for 1 MWh of energy (fixed)
𝑝𝑖𝑣
General cost of production for 1 MWh of energy (fixed)
𝑐𝑖
Production capacity for agent i (fixed)
17
Table 2: Demand process parameters
𝜑1
𝜑2
𝜗1
𝜗2
𝛼
𝑐
0,95 -0,7 0,2 0,2 1,5 1,5
Table 3
CALL Option Pricing - LOW predictable Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
11.2893 (8.88176-13.6968) 11.0502 (8.7669-13.3334) 10.3943 (8.2559-12.5327) 10.1215 (7.9818-12.2613)
10. 8675 (9.4864-11.1310) 10.0859 (9.3064-10.8653) 10.1924 (9.3774-11.0073) 10.2737 (9.43737-11.1100)
Table 4
CALL Option Pricing - HIGH predictable Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
10,3440 (9,0398-11,6483) 10,2658 (9,0837-11,4479) 10,1470 (9,0660-11,2280) 10,0197 (8,9381-11,1013)
10.1240 (9,4329-10,8152) 10,0704 (9,4391-10,7018) 10,0975 (9,4481-10,7470) 10,1231 (9,4600-10,7862)
18
Table 5
PUT Option Pricing - LOW predictability Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
5.9937 (4.4417-7.5457) 5.9179 (4.3952-7.4406) 6.2092 (4.7259-7.6926) 6.4058 (4.9154-7.8962)
4.6413 (3.8773-5.4053) 4.8364 (4.1010-5.5717) 4.7346 (3.9623-5.5069) 4.6660 (3.8781-5.4540)
Table 6
PUT Option Pricing - HIGH predictability Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
5,0472 (4,0001-6,0943) 4,9955 (4,0188-5,9721) 4,9863 (4,0516-5,9210) 5,0928 (4,1509-6,0346)
4,7757 (4,1165-5,4349) 4,8055 (4,1921-5,4189) 4,7648 (4,1231-5,4065) 4,7406 (4,0856-5,3955)
19
S2210-6707(16)30137-8 http://dx.doi.org/doi:10.1016/j.scs.2016.06.025 SCS 459
To appear in: Received date: Revised date: Accepted date:
3-9-2015 20-4-2016 29-6-2016
Please cite this article as: Fanelli, Viviana., Maddalena, Lucia., & Musti, Silvana., Asian options pricing in the day-ahead electricity market.Sustainable Cities and Society http://dx.doi.org/10.1016/j.scs.2016.06.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Asian options pricing in the day-ahead electricity market Viviana Fanelli1 Lucia Maddalena2 Silvana Musti3
1
Dipartimento di Economia, Management e Diritto d'Impresa, Università degli Studi di Bari, [email protected] Dipartimento di Economia, Università degli Studi di Foggia, [email protected]
2
3
Dipartimento di Economia, Università degli Studi di Foggia, [email protected]
1
Highlights
We implement an electricity market equilibrium model. We model electricity prices by using the interest rate theory. We price Asian options in the day-ahead electricity market. We verify that agent provisional ability enhances the market efficiency by decreasing prices.
Abstract The purpose of this paper is to show that by using the toolkit of interest rate theory, it is possible to evaluate option prices through an electricity market equilibrium model. Options represent an adequate instrument to manage price risk faced by electricity producers that are related to both pool price and unit availability uncertainty. We have priced path dependent put and call Asian options by using arithmetic average, verifying that the possibility of market agents to forecast the future demand contributes to the market efficiency making option price level lower. Keywords: Electricity price modeling; Asian options; electricity producer; risk; equilibrium model.
1. Introduction During the last decades, dramatic changes to the structure of the power sector have taken place: a growing number of countries have deregulated their monopolistic electricity markets. They become, therefore, competitive markets, where energy prices are derived by the interaction of supply and demand. A fundamental role is played by the electricity demand side: the electric load management (ELM) . Many researchers have focused on the demand side management in order to develop techniques and methods to improve the efficiency of the power infrastructure by controlling and coordinating the operation of power loads (see [28], [29], [30], [31]). A complete and exhaustive literature review on this topic is given by Benetti et al. ([3]). This new context, along with the physical characteristics of the electricity, has generated new price dynamics, never seen before, neither on the financial markets nor on the commodities markets. The most notable electricity characteristic is its non-storability, or at least the relevant costs associated with the storage. Therefore, with the exception of hydropower, electricity should be generated exactly at the time of its request for consumption. The offer turns out to be completely inelastic to price changes, and also prices show very high volatilities and sudden changes of price levels, which are called spikes. Market operators, both producers and consumers, therefore suffer the exposure to energy price uncertainty. In Valenzuela, Thimmapuram, and Kim [37], a multi-agent architecture is used to 2
simulate the energy market, and to study the behaviour of customers under dynamic-pricing conditions. Several approaches have been proposed in the literature, focusing on demand side management and demand response, in order to control and change the power consumption habits: Gelazanskas and Gamage [16], Strbac [35], Aalami et al. [1], Ashok and Banerjee [2], Finn et al. [13] Risk management techniques become fundamental tools to quantify and cope with price risk generated from such uncertainty. Electricity derivatives aiming at handling the exposure of market agents to different kinds of risk, have emerged in the restructured electricity production market. The financial derivatives contribute to control risks through adequate hedging strategies. In particular, forward contracts are agreements in order to buy or sell a fixed amount of electricity at a settled price throughout a future time period. On one hand, this contract allows power producers to hedge price risk; on the other hand, the forward contract is mandatory and the power producer is obliged to supply the agreed quantity throughout the established time interval. Therefore, the forward contract reduces the price risk, but increases the probability of financial losses due to unexpected unit shutdowns, exposing the producer to the so called availability risk [33]. An alternative to these derivatives is offered to the producers by selling or buying their production through electricity options. An option is a contract that gives the holder of the option the right to buy (call option) or sell (put option) e specified energy amount during a certain future time period and at a fixed price (strike). The holder can decide whether or not to exercise the option, depending on the price behavior and the availability of the generating units. Nonetheless, while a forward contract is costless, the subscriber of the option has to pay an initial adequate cost, option price. In [33] the authors describe electricity options as instruments to manage the two main risks faced by power producers: price and production availability risk. They describe a multi-stage stochastic programming model that enables a risk averse power producer to decide its optimal portfolio of forward contracts and options taking into account the pool price volatility. The option price obviously plays a crucial role in these hedging strategies and in this paper we evaluate numerically European and Asian call and put options. Asian options are derivative contracts which have, as underlying variable, the average price of the asset over a period of time. Due to this definition, Asian options display a lower volatility compared to the standard European ones and therefore they show lower prices ([24],[34]). These path dependent options appear to be particularly appropriate to the electricity market, where the 3
contracts are written to give electrical power continuously over a period of time. It appears therefore reasonable for this market to refer to the average price over a period of time instead of price at the maturity of the option. The pricing of Asian Options cannot be obtained with analytical methods but
solely by developing numerical approximations [23]. The procedure is
computationally intensive following different methods: the authors in [36], [32] and [27] approximate the average distribution by fitting integer moments. Rogers and Shi [34] and Vecer [38] give a numerical solution of a rescaled version of the pricing partial differential equation, while Geman and Yor [17] apply numerical inversion of the Laplace transform, and Fusai [14] and Cai and Kou [5] apply numerical inversion of the double transform. Linetsky [27] makes use of the eigenfunction method and Zhang [40] applies the perturbation approach. The values are monitored on the time horizon between the settlement day and the contract maturity. The first option, depending on a price average, has been traded in Tokyo in 1987, giving the name of this option type. For this reason, both researchers and practitioners have concentrated on the electricity price evolution study, by setting up models able to catch the main characteristics of price evolution, in order to price derivative products and manage price risk. The present work follows the research line which refers to the modeling framework of Heath, Jarrow and Morton (hereafter HJM) [19] that, using only few stochastic factors and the initial price curve as given, models futures prices under some equivalent martingale measure in a no-arbitrage environment. Clewlow and Strickland [8] have been the first researchers to introduce this approach to the energy market. Bjerksund et al. [4] and Koekebakker et al. [25] model a continuum of instantaneous-delivery forward contracts under risk neutral probability measure. Fanelli et al. [10] model electricity prices using seasonal path dependent volatility. Hinz [18] gives an interesting interpretation of the electricity market and demonstrates that it is possible to create a market framework where the existence of a risk neutral measure is guaranteed: the energy cannot be stored, but it can be produced. Therefore, the producer can put himself in the condition of having the ability to produce electricity, creating a sort of “electricity storability”. According to this perspective, the electricity market becomes more complex and has to be considered as composed of both power electricity and agreements on power production capacities. The market reaches the equilibrium and determines the price process for all tradable assets both physical (production capacity agreements) and financial (future electricity prices). The equilibrium existence gives an economic interpretation of the martingale measure Q: it is 4
equivalent to the market measure P, such that equilibrium asset prices under P are given by their future revenues, expected with respect to Q. Price dynamics can be, therefore, described directly under the equivalent martingale measure Q and it becomes possible to price all contracts by using classical no arbitrage models of financial mathematics. In addition, The HJM [19] interest rate framework we have developed, does not need the market price of risk as input, but only the volatility function ([7], [8], [9], [10], [11]). This important feature allows us to bypass the market price of risk as input to price electricity derivatives (Weron [39]): we have obtained from market data the volatility parameters representing the input of the equilibrium model. According to this approach, Fanelli et al. ([12]) have obtained the electricity price evolution in the electricity market by applying the HJM term structure model to the Currency Change Electricity Market. The option pricing has never been obtained in the existing literature by combining the HJM framework with the equilibrium model approach. The advantage of this research is to combine the benefits of the free volatility formulation of the HJM model, with the stochastic electricity demand dynamics of the equilibrium model. Market structure is analyzed in both aspects, allowing us to efficiently price options whose evaluation in most cases results numerically difficult to handle. Our contribution to the existing literature consists in furnishing a range of average option pricing in order to complete former studies about hedging strategies in the electricity market. In the present work, we further develop our research on electricity price evolution in the no arbitrage HJM approach, by connecting it with the Hinz ([20]) equilibrium model. The explicit dynamics of the spot price is obtained with a precise parameters evaluation by collecting market data as input in the HJM model. The calibration of the model on the market information is therefore twofold: on one side the explicit dynamics of the spot price is calibrated on the market with the market data by use of the volatility parameters. This represents the contribution of our work to the existing literature on the equilibrium model ([15],[20]). On the other side, the equilibrium model allows us to price options taking into account the electricity level request at each single day of the time interval of the implementation. To our knowledge, there are no published papers pricing Asian options taking account both of the market price volatility and the daily demand fluctuation: the Asian Option prices confirm that the demand function reflects in the results level and in most cases the forecast capability of the agents contributes to market efficiency. Indeed, having as input the spot prices obtained with the methodology introduced by Fanelli et al. ([12]), we numerically implement the Hinz ([20]) equilibrium model to numerically estimate Asian 5
and European Options on the day-ahead electricity market. The rest of the paper is organized as follows: in paragraph 2 we describe the equilibrium model that we are implementing in its singleperiod and multi-period version. In paragraph 3 we expose the methodology to obtain the day ahead price series with the equilibrium model. In paragraph 4 we expose the input data and the numerical estimates obtained. Results and conclusions are exposed in paragraph 5.
2. Equilibrium model We consider at this stage the one period version of the model. The market is composed by N agents acting simultaneously as retailer as well as producers of electricity. In the single period model there are two dates: 𝑇0 represents today and 𝑇1 represents tomorrow, the delivery time. At time 𝑇0 each agent does not know the energy request he will be asked for the following day 𝑇1 . This amount represents an exogenous random variable. The agent energy request can be satisfied both by purchasing today on the day ahead market with price 𝑝 and by tomorrow production, by using his own power plants. The choice about how to satisfy the request depends on the economic valuation based on prices and quantities. The profit and loss function for agent i is the following (Hinz [20]):
𝑓 𝐺𝑖 (𝑝, 𝑞𝑖 , 𝑄̃𝑖 , 𝑝̃ 𝑠 ) ≔ 𝑝𝑖𝑟 𝑄̃𝑖 + 𝑝𝑏 (𝑞𝑖 − 𝑄̃𝑖 )+ − 𝑃𝑖 − 𝑝𝑞𝑖 − 𝑚𝑖𝑛{𝑝𝑖𝑣 , 𝑝̃ 𝑠 }𝑚𝑖𝑛 {(𝑄̃𝑖 − 𝑞𝑖 )+ , 𝑐𝑖 } +
−𝑝̃ 𝑠 (𝑄̃𝑖 − 𝑞𝑖 − 𝑐𝑖 )+ whose variables are defined in Table 1.
Each agent at time 𝑇0 establishes the quantity of energy to buy on the day-ahead market and the interaction among decisions of the N agents determines the price 𝑝. The random variables are ̃𝑖 and 𝑝̃ 𝑠 . The profit and loss function is concave in respect to the represented by the quantities 𝑄 control variable 𝑞𝑖 and it guarantees the optimal solution for every agent maximizing his expected utility 𝔼[𝒰𝑖 (𝐺𝑖 (𝑝, 𝑞𝑖 ,∙,∙))] subject to the global production capacity constraints. Indeed, every agent can sell at least his production, buy at least the all system production and the net balance of all quantities has to be null: 6
𝑞𝑖 ∈ [−𝑐𝑖 , ∑ 𝑐𝑗 ] 𝑗≠𝑖
∑𝑁 𝑖=1 𝑞𝑖 = 0. The author in [20] demonstrates that the equilibrium price exists if spot price is greater that variable cost, even if the uniqueness of the equilibrium point is still to be investigated. The multi period model represents the natural extension of the single period model, and the random variables already introduced are labeled with the temporal notation and represented as a (𝑁 + 1)-dimension stochastic process: ̃1 (𝑛), 𝑄 ̃2 (𝑛), … , 𝑄 ̃ 𝑁 ( 𝑛) ) 𝜋̃(𝑛) = (𝑝̃ 𝑠 (𝑛), 𝑄 𝑛≥1
(1)
̃𝑖 (𝑛) represents the KWh where 𝑝̃ 𝑠 (𝑛) represents the spot price of 1 KWh at time 𝑛, while 𝑄 quantity of energy requested by agent 𝑖 at time 𝑛. The filtration generated by the process, ℱ 𝜋̃ , represents the information generated by the process and an equilibrium is defined as a process adapted to the filtration ℱ 𝜋̃ (𝑝∗ , 𝑞1∗ (𝑛), 𝑞2∗ (𝑛), … , 𝑞𝑁∗ (𝑛))𝑛≥1 such that any couple (𝑝∗ , 𝑞𝑖∗ (𝑛)) maximizes 𝑖-th agent expected utility at time 𝑛 ≥ 1, given the information available at the former date (𝑛 − 1).
3. Numerical Implementation The model is implemented† in its multi-period version and in the hypothesis we formulate there are no differences among the agents, so that they sustain the same costs and have the same probability distribution of the energy request. With these hypothesis the equilibrium price is the following: 𝑝∗ (𝑛) = 𝑝𝑣 − 𝔼 [(𝑝𝑣 − 𝑝̃ 𝑠 (𝑛 + 1))+ ] + 𝔼[(𝑝̃ 𝑠 (𝑛 + 1)−𝑝𝑣 )+ ]ℙ (𝑄̃ (𝑛 + 1) ≥ 𝑐|𝜋̃(𝑛)). The series 𝑎̃𝑛 : = 𝑝𝑣 − 𝔼 [(𝑝𝑣 − 𝑝̃ 𝑠 (𝑛 + 1))+ ]
†
(3)
Fortran 90 language, Geany compiler, Ubuntu OS.
7
(2)
𝑏̃𝑛 : = 𝑝𝑣 − 𝔼 [(𝑝𝑣 − 𝑝̃ 𝑠 (𝑛 + 1))+ ] + 𝔼[(𝑝̃ 𝑠 (𝑛 + 1)−𝑝𝑣 )+ ]
(4)
represent the equilibrium price variability interval. The series 𝑎̃𝑛 and 𝑏̃𝑛 are identified once we observe the spot price realization 𝑝̃ 𝑠 (𝑛) and we can forecast the possible evolution at the following time (𝑛 + 1). In order to numerically implement the model, we have to make hypothesis regarding the spot price 𝑝̃ 𝑠 and demand 𝑄̃𝑖 dynamics. In figure 1 the flow chart of the numerical scheme is proposed. The spot price series 𝑝̃ 𝑠 (𝑛) is obtained by applying the model Fanelli et al. [10]. Indeed, we consider as input the spot price determined by applying the HJM model. The currency change represents the technical instrument to make the HJM model suitable in the electricity market [20]. According to the HJM model, we assume that the forward price dynamics 𝑓𝑡 (𝑇), 𝑡 < 𝑇 is given by the stochastic differential equation 𝑑𝑓𝑡 (𝑇) = 𝛼𝑡 (𝑇)𝑑𝑡 + 𝜎𝑡 (𝑇)𝑑𝑊(𝑡). By definition, the price process of a forward contract is: 𝑇
𝑝𝑡 (𝑇) = 𝑒 − ∫𝑡
𝑓𝑡 (𝑠)𝑑𝑠
.
(5)
In order to avoid arbitrage opportunities, 𝛼𝑡 (𝑇) is function of the volatility 𝑇
𝛼𝑡 (𝑇) = 𝜎𝑡 (𝑇) ∫ 𝜎𝑠 (𝑇)𝑑𝑠 𝑡
and the forward rate dynamics in its integral form becomes 𝑡
𝑇
𝑡
𝑓𝑡 (𝑇) = 𝑓0 (𝑇) + ∫0 (𝜎𝑠 (𝑇) ∫𝑠 𝜎𝑢 (𝑢) 𝑑𝑢) 𝑑𝑠 + ∫0 𝜎𝑠 (𝑇)𝑑𝑊(𝑠).
(6)
The curve evolution in completely determined by the initial curve 𝑓0 (𝑇) = 𝜂0 − 𝜂1 𝑒 𝜂2 𝑡 and the volatility function 𝜎𝑡 (𝑇). In order to construct the currency change market model we have to introduce a risky asset with the following price dynamics: 𝑑𝑁𝑡 = 𝑁𝑡 (𝑓𝑡 (𝑡)𝑑𝑡 + 𝑣𝑁𝑡 𝑑𝑊(𝑡)) where 𝑣 is a constant volatility function. We consider the currency change 𝐹𝑡 (𝑇): =
𝑝𝑡 (𝑇) 𝑁𝑡
and obtain the forward price of electricity. The spot price is obtained with 𝑇 = 𝑡 8
𝑝̃𝑡𝑠 = 𝐹𝑡 (𝑡).
(7)
The HJM model features make it highly versatile because of the volatility function choice 𝜎𝑡 (𝑇) . We consider the following forward rate volatility function (6): 𝜎𝑡 (𝑇) ≔ 𝑒 −𝜆(𝑇−𝑡) [𝑎𝑓 𝑓(𝑡, 𝑇)]
(8)
and we estimate the parameters 𝜆 and 𝑎𝑓 .The price process is obtained by simulating the daily entire forward curve stochastic evolution over 1 year period. The second fundamental input to implement the equilibrium model is the demand process. The process is given by the discrete sequence of random variables 𝑄̃𝑖 (𝑛 + 1) = 𝜇 𝑒 𝛼𝑥(𝑛−1)+𝛽𝜐𝑖 (𝑛)
(9)
where 𝜇, 𝛼, 𝛽 > 0, 𝜐𝑖 (𝑛) is, for each n, a standard normal variable and 𝑥(𝑛) is ARMA(2,1) process 𝑥(𝑛 + 1) = 𝜑1 𝑥(𝑛) + 𝜑2 𝑥(𝑛 − 1) + 𝜗1 𝜀(𝑛) + 𝜗2 𝜀(𝑛 + 1)
(10)
|𝜑𝑖 | < 1, |𝜗𝑖 | < 1, 𝑖 = 1,2 , 𝜀(𝑛)~𝑁(0,1) driving the electricity demand addressed to each agent in the market. We suppose that the electricity requests received by the agents have the same probability distribution, and 𝑄̃ (𝑛) is the random variable representing the electricity demand addressed to each agent. The demand path is simulated with the following parameters
The condition 𝑐 = 1,5 > 𝜇 = 1.0 indicates that a large part of demand is met with in-house production rather than by purchasing electricity on the market. Finally, the agent capability of forecasting demand levels for the following day becomes crucial in order to define the equilibrium price on the day ahead market. This ability is deeply influenced by the demand volatility and it is possible to take it into account simply by fixing parameter 𝛽 > 𝜇. As it is shown in Fanelli et al. [12], the high predictability of demand level determines lower equilibrium prices, while low predictability level causes high volatile prices showing that agents accept to pay prices closer to the maximum boundary 𝑏𝑛 .
9
The equilibrium model gives the opportunity to take into account the observed volatility function in the electricity market, simulate electricity prices according to the CCEM model, integrating the data with the electricity demand. The calibration of the volatility function in the HJM model and the use of the ARMA model for the demand evolution, represents the key instruments to take the physical characteristics of the electricity into the model implementation.
4. Numerical Results The data set consists of daily time series of the term structure of the Eonia rates, spanning from January 2007 to mid May 2010. These data are used to estimate parameters of the forward rate volatility function in the EEX electricity market. The parameters of the initial curve 𝑓0 (𝑇) = 𝜂0 − 𝜂1 𝑒 𝜂2𝑡 have been calibrated on historical data, obtaining 𝜂0 = 0,004 , 𝜂1 = 0,003 and 𝜂2 = −0,2. The volatility function parameters in (8) have been
estimated by applying the nonlinear least squares method to the volatility time series referring to the observed period , obtaining the following significant values: 𝜆 = −0.0399219 and 𝑎𝑓 = 0,0182197. The Monte Carlo simulation method (with 𝑁 paths) has been applied by simulating the day ahead price over a 1 year horizon time ( 1 < 𝑛 < 30 ). The European Call and Put have 1 year maturity and the Asian Call and Put option have been priced averaging the final price over the last month daily observation of the year. For each path i , 1 < i < N, the last 30 observed prices have been averaged to obtain the payoff of the arithmetic Asian call option, by comparison with the strike price, while the last observed price (𝑛 = 360) gives the 𝑖 − 𝑡ℎ payoff for the European call option (see figure 1). According to the Monte Carlo method, the expected values over the 𝑁 simulated paths are obtained both for the Asian and the European call payoff. Option prices are finally obtained by discounting the payoffs at the risk free rate up to the initial time 0. The main feature of the equilibrium model is the opportunity to determine the time series of the day ahead electricity, starting from the spot price time series. The key instrument to evaluate the day ahead price at time 𝑛 is to make hypothesis on the realization of the spot price on the following day, time (𝑛 + 1) , in relation with the expected demand level at time (𝑛 + 1), 𝑄̃ (𝑛 + 1). In Table 3 and Table 4 are reported the Call option prices with strike price of 90. In Table 3 we have supposed the hypothesis of low capability of forecasting demand level by the agent, setting the demand function parameter 𝛽 < 𝜇 , 𝛽 = 1 and 𝜇 = 7 . In Table 4 the parameters values are 10
switched, and the high predictability hypothesis is obtained with 𝛽 = 7 and 𝜇 = 1. The call option values are obtained by using different number of paths (first column) in both Table 3 and Table 4, and the European option value is reported in the second column. The Asian Call Option value has been estimated averaging the daily prices over the last one month period before option maturity, with arithmetic average (column 3). Table 5 and 6 report the analogous estimation numerically obtained for the Put option with strike price 110. In Table 5 results are obtained with low predictability level, while in Table 6 the numerical simulations have been carried forward with the hypothesis of high predictability of the demand level by the agents acting in the market. In columns 3 the put option values with arithmetic average is shown. For every estimate the 95% confidence interval is reported. Asian options are options in which the underlying variable is the average price over a period of time. On account of this fact, Asian options have a lower volatility and hence rendering them cheaper relative to their European counterparts ([24], [34]). By comparing the estimates, we can observe that Asian options have smaller values compared to the correspondent European ones, and the predictability hypothesis makes the option values in most of the cases lower when the agents are able to forecast, in an adequate manner, the demand level for the following day. These results confirm how the production scheduling remains one of the key strategic instruments in order to have an efficient market. As we were expecting, the European options have a greater price respect to Asian options, because the averaging over the 30 days of the last months makes the option price lower because reduces the payoff variability at maturity time.
5. Conclusions Electricity producers participating in electricity market, face risks inherent to prices they are exposed to, both as seller and as buyers of energy. Moreover, the availability risk is the aspect producers have to deal with, if a producing unit shuts down. Among electricity derivatives, options represent an adequate instrument to manage these two risks. By gathering this important issue faced by Pineda and Conejo [33], we have implemented a numerical method to price European and Asian options in the electricity market. The numerical implementation we propose of the Hinz [20] equilibrium model, allows us to take into account the electricity price variability by simply calibrating the volatility parameters as input of the model. The spot price path is, in this analysis, 11
determined by taking into account the historical volatilities observed and it allows, therefore, to simulate in a realistic way the spot price evolution, once the demand evolution is given as input. The demand is supposed to be driven by an ARMA (2,1) process. Furthermore, Hinz [21] is the first researcher dealing with the use of classical no arbitrage models in the energy market price modeling: the term structure model appears, therefore, coherent in order to determine a spot price as input to obtain the day-ahead price in the equilibrium model. The implementation gives the opportunity to evaluate different hypothesis about demand process and to verify how different hypothesis reflects in the option price levels. We have priced both European and Asian options, put and call type, considering daily prices over one month period, with arithmetic average. By comparing the estimates obtained for the European and Asian options, we conclude that the numerical method gives results coherent with the financial definition of these two types of derivatives and can be considered as a valid instrument to price derivative that can be very useful in managing financial risk. In particular, Asian Options are path dependent options and for this reason, we are confident that this type of model, with a path dependent volatility and an explicit hypothesis about the demand process, is adequate in estimating this kind of electricity derivatives.
12
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Figure 1: Flowchart of the i-th payoff algorithm
16
Table 1: Variables and parameters of the equilibrium model
𝑝
1 MWh day-ahead price (variable)
𝑞𝑖
Amount of energy purchased today by the agent i on the day ahead market (control variable for the agent)
𝑄̃𝑖
Amount of energy that the agent i will be asked for at time 𝑇1 (variable)
𝑝̃ 𝑠
Tomorrow 1 MWh spot price (variable)
𝑝𝑖𝑟
Selling price the agent i will obtain by selling 1 MWH of electricity (fixed)
𝑝𝑏
“Back supply price”, price obtained selling 1 MWh (for the production exceeding the request) (fixed)
𝑃𝑖
𝑓
General cost of production for 1 MWh of energy (fixed)
𝑝𝑖𝑣
General cost of production for 1 MWh of energy (fixed)
𝑐𝑖
Production capacity for agent i (fixed)
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Table 2: Demand process parameters
𝜑1
𝜑2
𝜗1
𝜗2
𝛼
𝑐
0,95 -0,7 0,2 0,2 1,5 1,5
Table 3
CALL Option Pricing - LOW predictable Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
11.2893 (8.88176-13.6968) 11.0502 (8.7669-13.3334) 10.3943 (8.2559-12.5327) 10.1215 (7.9818-12.2613)
10. 8675 (9.4864-11.1310) 10.0859 (9.3064-10.8653) 10.1924 (9.3774-11.0073) 10.2737 (9.43737-11.1100)
Table 4
CALL Option Pricing - HIGH predictable Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
10,3440 (9,0398-11,6483) 10,2658 (9,0837-11,4479) 10,1470 (9,0660-11,2280) 10,0197 (8,9381-11,1013)
10.1240 (9,4329-10,8152) 10,0704 (9,4391-10,7018) 10,0975 (9,4481-10,7470) 10,1231 (9,4600-10,7862)
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Table 5
PUT Option Pricing - LOW predictability Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
5.9937 (4.4417-7.5457) 5.9179 (4.3952-7.4406) 6.2092 (4.7259-7.6926) 6.4058 (4.9154-7.8962)
4.6413 (3.8773-5.4053) 4.8364 (4.1010-5.5717) 4.7346 (3.9623-5.5069) 4.6660 (3.8781-5.4540)
Table 6
PUT Option Pricing - HIGH predictability Demand Paths 100 200 500 1000
European
Asian Arithmetic Average
5,0472 (4,0001-6,0943) 4,9955 (4,0188-5,9721) 4,9863 (4,0516-5,9210) 5,0928 (4,1509-6,0346)
4,7757 (4,1165-5,4349) 4,8055 (4,1921-5,4189) 4,7648 (4,1231-5,4065) 4,7406 (4,0856-5,3955)
19