# Introducing stylized facts on electricity futures through a market impact model

## Jakob Krause

#### Need to know

• Quantity risk considerations are used to introduce stylized facts.
• Electricity markets exhibit high frequency features on low frequencies.
• Liquidation behavior leads to the stylized facts of electricity futures.

#### Abstract

This paper provides an alternative way to introduce the stylized facts on electricity futures. For nonstorable commodities, forward-looking information is not necessarily incorporated in the price history. In contrast to contemporary electricity finance models, we introduce a mechanism based on the trading behavior of market participants and their corresponding market impact by exploiting characteristic initial positions and quantity risk considerations. For long times to maturity we end up with a market influenced by hedging pressure, whereas for short times to maturity quantity risk comes into play and yields an increased volatility. In addition, prices can also be negative. The model is accompanied by an empirical analysis showing that parameters that are typically only relevant on small timescales have a significant dynamic in daily returns over a period of years. This allows us to access the toolbox usually reserved for high-frequency modeling.

## Abstract

This paper provides an alternative way to introduce the stylized facts on electricity futures. For nonstorable commodities, forward-looking information is not necessarily incorporated in the price history. In contrast to contemporary electricity finance models, we introduce a mechanism based on the trading behavior of market participants and their corresponding market impact by exploiting characteristic initial positions and quantity risk considerations. For long times to maturity we end up with a market influenced by hedging pressure, whereas for short times to maturity quantity risk comes into play and yields an increased volatility. In addition, prices can also be negative. The model is accompanied by an empirical analysis showing that parameters that are typically only relevant on small timescales have a significant dynamic in daily returns over a period of years. This allows us to access the toolbox usually reserved for high-frequency modeling.

## 1 Introduction

The debate over the origin of futures risk premiums has been intense ever since Keynes connected a downward-sloping futures curve (backwardation) with the desire of producers to hedge their production (see Keynes 1930). The counter-position is taken by a long list of authors, starting with Working (1949), who relate the risk premium to issues of storage and inventories rather than hedging pressure and risk transfer. This position mostly implies an upward-sloping futures curve (contango).

The underlying assumption in mathematical finance models is that we have access to the underlying and that it is possible to store the underlying in some form. Consequently, these models are built in the spirit of the theory by Working (1949). In some commodity markets, such as precious metals or agricultural products, this assumption is mostly satisfied, while in others it is not and it is dubious to use classical spot models in the description of futures markets on nonstorable commodities. In fact, it has been noted that it is fundamentally wrong to do so (see Benth and Meyer-Brandis 2009). However, use of spot price models in, for example, electricity markets is still very prominent since they are able to produce the market-specific stylized facts we would like to model (see Benth et al 2008a). The objective of this paper is to introduce those stylized facts in an alternative way by exploiting nonstorability and the symptoms of the related problem of continuous supply-and-demand matching.

Electricity futures exhibit a variety of characteristic stylized facts: prices can be negative; they can experience a significant drift; and their risk profile and volatility may change over their life span (see Benth et al 2008a). So far, those features have been modeled using either long- or short-term spot models (see Schwartz and Smith 2000) or Heath–Jarrow–Morton approaches. However, the Heath–Jarrow–Morton approach has a low explanatory power in electricity markets (see Koekebakker and Ollmar 2005), and for spot price models the drawback mentioned above applies.

In order to proceed, we must ensure that the qualitative assumptions of the Keynes (1930) theory are satisfied, ie, we argue for limited liquidity and against market efficiency. Qualitatively, we can say that the documented stylized fact of hedging pressure in electricity markets (see, for example, Benth et al 2008b) is not compatible with market efficiency and, more specifically, requires limited liquidity: in order for the hedging pressure argument to hold, we need to assume market participants are not able to achieve their desired position size in arbitrarily short time frames, ie, hedging pressure only works in a world where liquidity and market efficiency are limited and the visible asset prices are the outcome of an optimization between hedging premium (market impact) and portfolio risk considerations, ie, optimal liquidation (see the seminal paper by Almgren and Chriss (2001)). This qualitative argument leads to the hypothesis that instead of borrowing models from other commodity and financial markets it might be helpful to look at the electricity market through the lens of high-frequency finance, ie, to check whether the parameters typically discussed in this field of literature are present in electricity futures markets and whether the returns of futures can be related to the position size. The first objective of this paper is to check this hypothesis empirically; this is investigated in Section 2. Subsequently, in Section 3, we set up a model that is able to capture the known stylized facts on electricity futures using market-specific information and nonstorability. In addition, we provide a strategy for using this model to infer aspects of the trading behavior of market participants. Conclusions are given in Section 4.

## 2 Liquidity and daily returns

This section provides evidence for an analogy between intraday block trading and electricity portfolio liquidation over a time frame of several years. We relate the return anomalies of a buy-and-hold trading strategy to the time dynamics of liquidity metrics. In principle, electricity-producing companies are faced with the problem that, for a date in the far future, the electricity price risk is concentrated entirely in the hands of a small number of energy-producing companies via their production facilities. Consequently, they have enormous positions and should have an incentive to reduce their exposure by giving out discounts. Finding a balance between position reduction and market impact is a complex problem, but in the market microstructure literature there are models that deal with this type of problem (see, for example, the seminal paper by Almgren and Chriss (2001)). This position reduction can be expected to influence the prices more severely for long times to maturity than for short times to maturity, and since the energy-producing companies own the large positions, they want to find buyers. Consequently, we expect prices to be lower for longer times to maturity. One issue that makes the problem more difficult in the context of electricity markets is that we encounter severe quantity risk (Pérez-González and Yun 2013), ie, the amount that is liquidated is constantly changing and we have to trade toward a moving target. This observation implies that liquidity concerns again play a role for short times to maturity. Since limited liquidity is typically related to increasing risk, we can expect the effect on the prices to be negative and we have to compensate the counterparty in order to trade, which implies lower prices.

### 2.1 Data description

The data set used in this study consists of price and volume data from the Norwegian Energy Exchange (NASDAQ OMX Commodities) and its predecessor, Nord Pool. The futures contracts are financially settled quarterly contracts stretching from 2004 to 2013. To be precise, the handles of the contracts are ENOQ$X$-$\mathrm{YY}$, where $X\in\{1,2,3,4\}$ denotes the relevant quarter and $\mathrm{YY}\in\{06,07,08,09,10,11,12,13\}$. The data set also includes yearly contracts ENOYR-$\mathrm{YY}$, where $\mathrm{YY}$ refers to the years from 1999 to 2013.

Throughout this paper, we adopt the language of the electricity literature and speak about “futures” instead of “swaps” and use “time to maturity” instead of “time to delivery” (see Benth et al 2008a).

### 2.2 Returns of liquidity exploitation strategies

The simple strategy below is based on the following two ideas. First, Benth et al (2008b) show that their notion of “market power” changes over the life span of the contracts and document first-order price effects. Thus, if we are always on the right side of market power, we should be able to earn a return. It is interesting to test whether, since the publication of their paper in 2008, the effect has been robust to revelation, which is known to be an issue in financial markets (see the seminal paper on the “wandering weekday effect” by Doyle and Chen (2009)). In addition, since Benth et al (2008b) use data from Germany between 2002 and 2006, a time period that included a significant change in legislative circumstances targeting the phase out of nuclear power, it is important to test our hypothesis on a different data set. We find that the effect is persistent in the sense that there are profitable liquidity exploitation strategies. This indicates that this finding is not a “statistical arbitrage” but serves an economic purpose that is prevalent in electricity markets in general.

The second notion is the asymmetry in the initial distribution of market participants laid out in Section 1. A party liquidating a large block of assets has to trade off between their market impact and the risk on their book. This typically involves a high market impact at the start of the liquidation period, and therefore it would be beneficial to buy the contracts at inception and earn a premium for liquidity provision; this is in line with the hedging pressure arguments by Benth et al (2008b).

Consequently, the strategy exploiting these ideas is the following:

1. (i)

2. (ii)

sell the contract at some time to maturity (twice);

3. (iii)

buy the electricity in the spot market.

The mean profitability of this strategy is depicted in Figure 1(a), depending on the time to maturity, where the position is liquidated and switched against a short position. Only the quarterly contracts are shown; due to cascading effects, the pattern for yearly contracts looks similar. As mentioned before, the findings are consistent with our intuition. It is, however, very interesting to observe that the contracts for the different quarters show very similar behavior that is not connected to a time-to-maturity-dependent event, but rather is connected to a date, namely January 1 in the year prior to maturity, which is highlighted with a vertical line. Note that the time to maturity of January 1 is different for the different classes of quarterly contracts: for the Q1 contracts it is, typically, and in line with a variety of implied option pricing models, 252 trading days. For the respective other quarters the times to maturity are $252+(0.25\times 252)=315$ days for the Q2 contracts, $252+(0.5\times 252)=378$ days for the Q3 contracts and $252+(0.75\times 252)=441$ days for the Q4 contracts. When we turn to possible explanatory variables, we find that there is a similar delayed shape in a variety of (il)liquidity metrics. Liquidity is an elusive concept, and in the following we use a variety of different liquidity metrics. While it is clear what is meant by trading volume and bid–ask spreads, an additional measure of liquidity we employ in the following is the so-called Amihud illiquidity metric (see Amihud 2002). In its simplest form it is defined as

 $\mathrm{amihud}_{t}=\frac{\lvert r_{t}\lvert}{\mathrm{Vol}_{t}},$ (2.1)

where $r_{t}$ denotes the return at time $t$ and $\mathrm{Vol}_{t}$ is the corresponding trading volume. Consequently, liquid markets are markets in which the absolute values of returns are low and trading volume is high, while illiquid markets are characterized by either high absolute returns or low trading volumes. The Amihud illiquidity metric is a prominent measure of liquidity that can be easily calculated from data for long time frames, which are readily available for most markets. It has been used extensively in the literature on market microstructure (see, for example, Foucault et al (2013) and the references therein). We take the average five-day Amihud illiquidity, which is a common adaption proposed in the original study by Amihud, and we use the log of the trading volume instead of the absolute number, since in electricity markets the trading volume changes by several orders of magnitude throughout the life of the contracts.

Given the liquidity metrics, we can analyze their dynamics and relate them to the returns of the liquidity exploitation strategy. In Figure 1, we can see that the leveling out of the Amihud illiquidity metric aligns with the peak of the return of the strategy in column (a). In Figure 2, it can be seen that the average and median autocorrelations change sign throughout the life of the contracts. For long times to maturity, autocorrelation is negative, which is consistent with the market microstructure literature (see, for example, Zhang et al 2005), and for short times to maturity it is positive. Since positive autocorrelation is not compatible with the stylized facts on asset returns (see Cont 2001), this observation indicates that liquidity might play a role for short times to maturity as well.

Those observations point to the assessment that electricity futures not only exhibit time-to-maturity-dependent behavior (a feature that can be introduced by means of a short-term component in spot price models, as in Schwartz and Smith (2000) and its generalizations) but also show aspects of seasonal behavior, ie, informative signals located around certain dates with corresponding similar behavior in liquidity metrics. This point is insufficiently addressed by the existing electricity finance models.

### 2.3 Test design

In this section, we are interested in the qualitative statement of whether liquidity metrics show a significant directional dynamics throughout the life spans of the contracts. In order to detect nonstationary behavior, we need to have a time series of the parameters in question. For returns and trading volume, the original time series can be used. However, the centralized moments and liquidity measures have to be estimated. A rolling window of fifty sequential data points is used for this purpose.11 1 The results are stable with respect to the length of the rolling window; seventy-five and one-hundred data points yield similar results (not shown).

First, we test whether the time series of the parameter in question reveals a drift by employing a (modified) Mann–Kendall test. This is a drift test, which can be modified to achieve robustness with respect to the existence of autocorrelation (see Hamed and Rao 1998). The number of contracts that show a significant drift for the parameter in question is counted, and we test whether the distribution of positive and negative drifts is symmetric by using a binomial test with parameter $p=0.5$. The intuition behind this last step is that if there is no time-to-maturity-dependent behavior, then the number of paths that show a positive drift (as determined by Mann–Kendall) should be roughly equal to the number of negative paths. Anything else indicates that parameters will more likely grow or decline as a function of time to maturity. To sum up, we carry out the following steps.

1. (i)

Calculate time-to-maturity-dependent time series for the first four centralized moments and the autocorrelation for every contract based on a fifty-data-point rolling window.

2. (ii)

Apply a modified Mann–Kendall test to detect possible drifts in these time series.

3. (iii)

Count the number of positive and negative drifts and test whether it is reasonable to assume that they are $B_{0.5}$-distributed.

Table 1 shows the results for this test procedure applied to the return time series for the electricity forward data introduced earlier.

#### 2.3.1 Results

The following conclusions can be drawn from the data in Table 1.

1. (i)

Autocorrelation decreases with increasing time to maturity.

2. (ii)

Although the signal is not very significant, we find a drift in the mean of the returns. The mean return becomes more positive for longer time to maturity. This result is compatible with the idea that for longer times to maturity the market for electricity is a buyers’ market due to hedging pressure.

3. (iii)

We find that the variance of returns (volatility) decreases with increasing time to maturity, ie, the method is able to detect the so-called Samuelson effect. In this context we note that the volatility increase is less pronounced for the yearly contracts. This is reasonable, since the relative amount of short-term information is greater for quarterly contracts than for yearly contracts.

4. (iv)

Skewness decreases with increasing time to maturity.

5. (v)

Kurtosis increases with increasing time to maturity. This result is particularly interesting since kurtosis can be seen as a measure for extreme events (see Westfall 2014).

6. (vi)

All the (il)liquidity metrics point to a gradual worsening of the liquidity situation for longer maturities. However, this assessment is not yet plausible for the autocorrelation, since only negative autocorrelations can be associated with a bad state of liquidity. In order to support the claim, we have to check whether the autocorrelation is negative for long times to maturity. This is indeed the case: Lau (2015) reports a negative first autocorrelation coefficient for the same data set. The result in Table 1 shows that autocorrelation becomes more negative for longer times to maturity. Therefore, we can conclude that autocorrelation is negative for longer times to maturity (see also Figure 1).

All the reported results are compatible with the assertion that electricity futures exhibit severe nonstationary features, especially for the quarterly contracts, where even the returns show an informative change of behavior. So far, the qualitative assessment of this fact has been at the heart of our analysis. Next, we present the characteristic returns of a buy-and-hold strategy. Note that the version of the Amihud illiquidity metrics used here is based on the traded log volume, since the volume grows by several orders of magnitude throughout the life span, ie, the results in Table 1 would be more drastic when using the unmodified volume data.

### 2.4 Price dynamics and cumulative trading volume

The rationale for the results was given in Section 1: it stems from the highly imbalanced position sizes and different structures of the buy and sell sides. Hence, it is plausible that the adjustment of positions is an explanatory variable, and it is therefore reasonable to turn to using the cumulative trading volume since inception as a possible explanatory variable for course movements. However, it is also reasonable to assume that there is a decline in the informativeness of the trading volume, since there is no longer much of an incentive to affect the market after the positions are reduced due to hedging pressure. Hence, it is reasonable to use a concave function of the trading volume $f(\int_{0}^{t}V_{s}\mathrm{d}s)$ as the explanatory variable. The results of a price regression on time to maturity and log trading volume can be found in Table 2. Since it is known that electricity futures exhibit nonstationary behavior, we control for the time to maturity.

The results support our intuition fully for the Q2, Q3 and Q4 contracts, which also display stable behavior when we compare the returns of the strategy against the Amihud illiquidity metric (a volume-based measure) in Figure 2. However, for the Q1 contracts this relationship is violated (see Figure 2) and we can also see that, in the regression, the results are disturbed to some degree.

### 2.5 Liquidity for short times to maturity

The conceptual difference from the setup usually encountered in a market microstructure environment comes into play for short times to maturity: in equity or foreign exchange markets the number of assets we want to sell or buy is typically known, whereas for electricity markets the demand is random and expected future production is a stochastic process changing over time. With the data set used above it is not possible to isolate the corresponding signal, since it would involve building a model for expected future production and a corresponding liquidation strategy. However, note that liquidity plays a role for long and short times to maturity and the model in Section 3 introduces both effects with only one tool.

## 3 A limited liquidity model for stylized facts on electricity futures

The objective of this section is to set up a mathematical model able to capture the results of our empirical analysis and to provide a strategy for how this model can be used to answer questions on the trading behavior of market participants.

### 3.1 A simple liquidity-based model

We assume the price of a future with maturity at $T$, denoted by $F_{t}^{T}$, is given by

 $F_{t}^{T}=S_{t}^{T}+L_{t}^{T},$ (3.1)

where $S_{t}^{T}$ denotes the current expectation of the future spot price under a risk-neutral measure, ie,

 $S_{t}^{T}=E^{Q}[S(T)\mid I^{T}(t)].$ (3.2)

For a nonstorable commodity such as electricity it is fundamentally wrong to use only the information filtration generated by the asset (Benth and Meyer-Brandis 2009). One way to circumvent this problem is to model the information set on a per-contract basis (here expressed by the superscript $T$ in $I^{T}$), ie, we assume every future has its own price-determination process. Since this paper is solely concerned with the effects of liquidity on the qualitative properties of the futures price, we focus solely on $L_{t}^{T}$, and the spot price component will be ignored in the following. However, it can be very elegant to think of the pricing equation (3.1) as a representation splitting the futures price into one component that can be analyzed using classical mathematical finance techniques (ie, concepts that are based on the concept of storability) on the spot price (eg, through structural models and hedging through the fuel markets (Aïd et al 2009)) and another component representing a distortion that captures the effects stemming from the nonstorable nature of electricity.

Nonstorability implies that the energy produced at one point in time has to be consumed at the same time. Since electricity demand is random, this gives rise to a challenging trading problem where we have to trade toward a moving target in a notoriously illiquid environment. The aim of this paper is to use this problem-specific information to set up a model that produces stylized facts on electricity futures.

In the following, we take a hedging pressure perspective, where the position imbalance of the market participants and their corresponding trading activity to achieve risk management goals or supply–demand balancing has a direct influence on the price. We assume the price of the liquidity component can be modeled as

 $L_{t}^{T}:=-\frac{\alpha}{T-t}P^{T}_{t},$ (3.3)

where $P_{t}^{T}$ is the difference to the desired position size of electricity-producing companies or, alternatively, the difference from the equilibrium of the market. In this context, $\alpha$ can be interpreted as an illiquidity parameter, since $\alpha=0$ implies that $L_{t}^{T}=0$ and that the futures price is not influenced by any market impact.

In order to proceed, we need to specify the process $P_{t}^{T}$. We assume that this process is Brownian with an initial condition $P_{0}^{T}=\eta$ (the total amount of the current estimate of future demand). In addition, we assume the market has to clear at maturity, ie, the difference from the desired position is zero, since supply has to match demand due to nonstorability, ie, $P^{T}_{T}=0$. Augmenting this information into a Brownian motion equation leads to Brownian bridge dynamics, ie,

 $\mathrm{d}P_{t}^{T}=\frac{-P_{t}^{T}}{T-t}\mathrm{d}t+\mathrm{d}B_{t}^{T}.$ (3.4)

This solution can be derived by a standard variations-of-parameters argument and is given by

 $P_{t}^{T}=\frac{T-t}{T}P_{0}^{T}+\int_{0}^{t}\frac{T-t}{T-s}\mathrm{d}B_{s}^{T}.$ (3.5)

Note that this is an implicit assumption on the trading behavior of the market participants, ie, the drift in this process can be interpreted as the solution to some optimal liquidation problem that balances risk management needs with market impact. Consequently, it is also an assumption on limited liquidity, since in a perfectly liquid market the optimal liquidation speed is infinite and there is no trade-off between market impact and risk management considerations.

For the market impact component we obtain

 $\displaystyle L_{t}^{T}$ $\displaystyle:=-\frac{\alpha}{T-t}P^{T}_{t}$ $\displaystyle\phantom{:}=-\frac{\alpha}{T-t}\bigg{(}\frac{T-t}{T}P_{0}^{T}+% \int_{0}^{t}\frac{T-t}{T-s}\mathrm{d}B_{s}^{T}\bigg{)}$ $\displaystyle\phantom{:}=-\frac{\alpha}{T}P_{0}^{T}-\int_{0}^{t}\frac{1}{T-s}% \mathrm{d}B_{s}^{T}.$ (3.6)

The process $L_{t}^{T}$ displays all the stylized facts that we are interested in.

1. (i)

The stochastic integral is added, and consequently the sign of $L_{t}^{T}$ is not determined, which implies the overall prices can be negative since the futures price was modeled by adding a spot price process and the liquidity premium.

2. (ii)

The volatility of the process is increasing throughout the life span, due to the functional form of the integrand in the stochastic integral, which exhibits a singularity at $T$.

3. (iii)

The process $L_{t}^{T}$ exhibits a drift stemming from the initial condition. Thus, this process captures the hedging pressure phenomenon.

This model can easily be simulated. Figure 3(a) shows the expected future production (black line) and the filled position (gray line) with an initial expectation of “100”. Figure 3(b) shows the corresponding market impact and Figure 3(c) shows the price of a future assuming a drift-free geometric Brownian motion for the spot model. For the seed value of the simulation we picked the first seed that led to a negative futures price close to maturity.

### 3.2 Empirical strategy

While the primary objective of this paper is to introduce stylized facts (ie, qualitative aspects) of electricity futures, the model introduced here can be used for empirical (ie, quantitative) considerations as well. Starting from the setup, ie, $\smash{F_{t}^{T}}=\smash{S_{t}^{T}}+\smash{L_{t}^{T}}$, we can specify appropriate models for all the individual parts, ie, a spot price model for $\smash{S_{t}^{T}}$, a futures model for $\smash{F_{t}^{T}}$ and a model for the open position and the corresponding trading behavior impact $\smash{L_{t}^{T}}$. In this case, the specific model parameters of $\smash{F_{t}^{T}}$ and $\smash{S_{t}^{T}}$ can be estimated from individual time series data of spot and futures prices. After subtraction we are in a situation where we have access to the process $\smash{L_{t}^{T}}$ and can estimate its properties. Since $\smash{L_{t}^{T}}$ is a combination of trading behavior and the difference in the desired position size, knowledge on either component can be used to infer knowledge on the other, ie, if we can correctly model future supply and demand shocks, eg, through weather forecasts or power plant shutdown schedules, we could use the framework introduced above to infer aspects of trading behavior or vice versa.

For the qualitative argument presented here, we chose a Brownian model for all the components due to its simplicity and because this choice leads to an analytical solution for the dynamics from which we can immediately obtain the stylized facts. Conceptually, however, this argument is not limited to diffusion processes, and we can simply use, eg, a jump–diffusion method for the spot price model or, since the spot is not a traded asset and cannot interfere with no-arbitrage principles, even non-semi-martingale processes such as fractional processes. A similar breadth of choices is available for all the other components. Since this makes the space of possible models very large, the corresponding model selection problem is a complex one and is beyond the scope of this paper.

## 4 Conclusion

This paper investigates an alternative way to introduce some of the stylized facts on electricity futures. The disadvantage of most models used in electricity finance is that they are built on the premise of access to the underlying, which is an assumption that is currently not valid in electricity markets. Since mathematical models have to be judged by the credibility of their assumptions, it is useful to provide a mechanism able to introduce the stylized facts using market-specific information. Due to the interplay between price risk management and quantity risk, the liquidation problem in electricity markets is a complex one and the large trading departments found in electricity-producing companies are a testimony to this statement. Consequently, it is useful to have a model that incorporates those market-specific properties and translates them into the stylized facts on the products traded in these markets.

This paper also uses the natural synergy between models in high-frequency finance and electricity markets, since both fields use additive models. Further exploration of this synergy potential yields a host of interesting questions. In addition, the mechanism provided here is a natural extension to any spot price model, due to its additive structure and the appearance of a liquidity parameter that can be tuned to the properties of the market. In addition, in contrast to the setup in high-frequency finance, using a model that can yield negative prices is a useful property in the context of electricity markets.

Throughout our argument, we have used an ad-hoc assumption regarding the liquidation speed market participants use. From a theoretical perspective it would be interesting to find the corresponding optimal liquidation problem that yields the assumed behavior as its solution.

## Declaration of interest

The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.

## References

• Aïd, R., Campi, L., Huu, A. N., and Touzi, N. (2009). A structural risk-neutral model of electricity prices. International Journal of Theoretical and Applied Finance 12(7), 926–947 (https://doi.org/10.1142/s021902490900552x).
• Almgren, R., and Chriss, N. (2001). Optimal execution of portfolio transactions. The Journal of Risk 3(2), 5–39 (https://doi.org/10.21314/JOR.2001.041).
• Amihud, Y. (2002). Illiquidity and stock returns: cross-section and time-series effects. Journal of Financial Markets 5(1), 31–56 (https://doi.org/bvbjkd).
• Benth, F. E., and Meyer-Brandis, T. (2009). The information premium for non-storable commodities. The Journal of Energy Markets 2(3), 111–140 (https://doi.org/10.21314/JEM.2009.021).
• Benth, F. E., Benth, J. S., and Koekebakker, S. (2008a). Stochastic Modeling of Electricity and Related Markets. World Scientific, Singapore (https://doi.org/dz4xcz).
• Benth, F. E., Cartea, A., and Kiesel, R. (2008b). Pricing forward contracts in power markets by the certainty equivalence principle: explaining the sign of the market risk premium. Journal of Banking and Finance 32(10), 2006–2021 (https://doi.org/10.1016/j.jbankfin.2007.12.022).
• Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1(2), 223–236 (https://doi.org/10.1088/1469-7688/1/2/304).
• Doyle, J. R., and Chen, C. H. (2009). The wandering weekday effect in major stock markets. Journal of Banking and Finance 33(8), 1388–1399 (https://doi.org/10.1016/j.jbankfin.2009.02.002).
• Foucault, T., Pagano, M., and Röell, A. (2013). Market Liquidity. Oxford University Press (https://doi.org/10.1093/acprof:oso/9780199936243.001.0001).
• Hamed, K. H., and Rao, A. R. (2009). A modified Mann–Kendall trend test for autocorrelated data. Journal of Hydrology 204(1–4), 182–196 (https://doi.org/brxmzz).
• Keynes, J. M. (1930). Treatise on Money. Harcourt, Brace and Co, San Diego, CA.
• Koekebakker, S., and Ollmar, F. (2005). Forward curve dynamics in the Nordic electricity market. Managerial Finance 31(6), 73–94 (https://doi.org/fnjrcm).
• Lau, C. (2015). Non-normality in financial markets and the measurement of risk. PhD Thesis, Martin-Luther-Universität Halle-Wittenberg (https://doi.org/10.25673/1585).
• Pérez-González, F., and Yun, H. (2013). Risk management and firm value: evidence from weather derivatives. Journal of Finance 68(5), 2143–2176 (https://doi.org/10.1111/jofi.12061).
• Schwartz, E., and Smith, J. E. (2000). Short-term variations and long-term dynamics in commodity prices. Management Science 46(7), 893–911 (https://doi.org/10.1287/mnsc.46.7.893.12034).
• Westfall, P. H. (2014). Kurtosis as peakedness, 1905–2014. RIP. American Statistician 68(3), 191–195 (https://doi.org/10.1080/00031305.2014.917055).
• Working, H. (1949). The theory of price of storage. American Economic Review 39(6), 1254–1262. URL: http://www.jstor.org/stable/1816601.
• Zhang, A., Mykland, P. A., and Aït-Sahalia, Y. (2005). A tale of two time scales. Journal of the American Statistical Association 100(472), 1394–1411 (https://doi.org/10.1198/016214505000000169).

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