Improving the Efficiency of Hedge Trading Using Higher-Order Standardized Weather Derivatives for Wind Power

概要

energies
Article

Improving the Efficiency of Hedge Trading Using Higher-Order
Standardized Weather Derivatives for Wind Power
Takuji Matsumoto 1, *

and Yuji Yamada 2, *
1
2

*

Faculty of Transdisciplinary Sciences for Innovation, Kanazawa University, Kanazawa 920-1192, Japan
Faculty of Business Sciences, University of Tsukuba, Tokyo 112-0012, Japan
Correspondence: mtakuji@staff.kanazawa-u.ac.jp (T.M.); yuji@gssm.otsuka.tsukuba.ac.jp (Y.Y.)

Abstract: Since the future output of wind power generation is uncertain due to weather conditions,
there is an increasing need to manage the risks associated with wind power businesses, which have
been increasingly implemented in recent years. This study introduces multiple weather derivatives
of wind speed and temperature and examines their effectiveness in reducing (hedging) the fluctuation risk of future cash flows attributed to wind power generation. Given the diversification of
hedgers and hedging needs, we propose new standardized derivatives with higher-order monomial payoff functions, such as “wind speed cubic derivatives” and “wind speed and temperature
cross-derivatives,” to minimize the cash flow variance and develop a market-trading scheme to
practically use these derivatives in wind power businesses. In particular, while demonstrating the
importance of standardizing weather derivatives regarding market liquidity and efficiency, we propose a strategy to narrow down the required number (or volume) of traded instruments and improve
trading efficiency by utilizing the least absolute shrinkage and selection operator (LASSO) regression.
Empirical analysis reveals that higher-order, multivariate standardized derivatives can not only
enhance the out-of-sample hedge effect but also help reduce trading volume. The results suggest
that diversification of hedging instruments increases transaction flexibility and helps wind power
generators find more efficient portfolios, which can be generalized to risk management practices in
other businesses.
Keywords: efficient frontier; generalized additive model; LASSO regression; optimal hedging;
weather derivatives; wind power
Citation: Matsumoto, T.; Yamada, Y.
Improving the Efficiency of Hedge
Trading Using Higher-Order
Standardized Weather Derivatives for
Wind Power. Energies 2023, 16, 3112.
https://doi.org/10.3390/en16073112
Academic Editor: François Vallée
Received: 27 February 2023
Revised: 23 March 2023
Accepted: 27 March 2023
Published: 29 March 2023

Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).

1. Introduction
In the context of achieving carbon neutrality goals, the penetration of wind power,
a typical clean energy source, has been steadily increasing worldwide. According to
the International Energy Agency [1], the global wind power capacity is increasing at an
accelerating rate and this trend is expected to continue. Small wind turbines, which can be
installed at residential sites, have especially attracted attention in recent years, and their
market is also expected to grow steadily [2]. With these forces in play, the number and
types of players in the wind power markets are also expected to increase worldwide. Then,
the need to manage the risk of fluctuations in wind power generation revenues becomes
increasingly vital for wind power businesses.
With regard to risk management in wind power generation businesses, in addition to
avoiding the risk of loss through accurate forecasting, “hedging” approaches play important
roles, which involve the various use of financial transactions/contracts to compensate for
inevitable loss risks. Risk management through forecasting approaches is often discussed
in the context of avoiding imbalance risk due to high-cost imbalance charges imposed on
forecast errors; for example, optimizing market transactions as proposed in [3,4]. However,
since it is extremely difficult to accurately forecast and compensate for losses, particularly
against the long-term ahead horizon, financial hedging transactions may be used to control

Energies 2023, 16, 3112. https://doi.org/10.3390/en16073112

https://www.mdpi.com/journal/energies

Energies 2023, 16, 3112

2 of 22

such cash flow fluctuation risks. Recently, a method combining both approaches has been
proposed, such as the use of derivatives based on prediction errors to hedge against losses
derived from prediction errors [5]. Thus, for effective risk management in the electricity
business, in addition to improving forecast accuracy, risk “hedging” to compensate for
financial losses is vitally essential [6], and a variety of hedging products exist in the
electricity market [7]; the focus of this study is also on the latter issue: the concept of
hedging.
For hedging “price” risk in the wind power business, financial instruments, such
as power futures and bilateral contracts, and support measures, including feed-in tariffs
(FIT), can be used. However, few financial instruments are available for hedging “volume”
risk in practice. For example, even if a wind power generator has a contract to supply a
certain amount of electricity at a fixed price, it may incur losses when electricity cannot be
generated owing to poor wind conditions. However, few instruments have been devised to
hedge these losses, leaving a room or need for research in developing practical and effective
trading schemes to manage the volume risks.
As one of the few instruments for treating these volume risks, Wind Power Futures
(WPF) have been listed on the European Energy Exchange (EEX; note that this was delisted
in 2020) and Nasdaq. WPFs are contracts with a weighted average of area-wide wind
power generation in Germany/Austria as the underlying index. As the underlying index is
the amount of electricity generated, the WPF may be effective for large-scale wind power
generators in the target area. However, more specific and refined products (derivatives
or forwards/futures) may be required for individual power producers, including smallscale wind farms or even households (prosumers) with small wind power generators
operating in a specific region. For instance, a more effective tool can be a flexible instrument
tailored to the specific weather conditions of the generator’s business area using only
observed weather values as the underlying index. These weather derivatives can help
ensure transparency as a financial instrument since the generator’s individual situation will
not affect the underlying index. Here, we propose an approach for using such “weather”
derivatives, while emphasizing transparency and flexibility in product design.
Several studies have examined derivatives for wind power, including not only
WPF [8,9] but also the European put-type quanto options [10], wind put barrier options [11],
and wind call options [12]. Typically, these derivatives have been studied in the context of
pricing rather than how effectively they hedge risk. Papers focusing on the hedge effect
of wind derivatives, although very limited, include [13], which analyzed the effectiveness
of the WPF, and [14], which showed the effectiveness of option-type derivatives on wind
speed. In contrast, Yamada [5] proposed a unique derivative for hedging wind power risk
and demonstrated its effectiveness based on non-parametric regression using the generalized additive model (GAM) [15]. The implication presented there is the fact that the hedge
effect can be enhanced by estimating arbitrary smooth functions as derivative payoffs
rather than using linear or conventional option-type functions (see [16] for the original
idea and a genealogy of prior works on this). Subsequently, this hedging approach was
extended to multivariate models [17] and also to retailers and solar power producers [18,19]
and demonstrated that it also has a high hedge effect for them. In these studies, hedgers
use non-parametric regression techniques to estimate smooth payoff functions and obtain
optimal weather derivatives. However, derivatives estimated by non-parametric regression
require bilateral contracts between insurers and hedgers in practice because the optimal
payoffs are hedger-specific. Therefore, there remained room to consider the use of market
principles from the perspective of improving the trading efficiency. Therefore, while the
derivatives proposed in this present study have the same motivation as these series of
studies in terms of constructing arbitrary payoff functions, we propose a new unique
“market trading scheme” in which weather derivatives of various types and dimensions
are introduced in advance and traded by multiple power utilities (hedgers). In other words,
instead of trading (contracting) arbitrary payoff functions, by introducing various types
(several to a dozen) of “standardized derivatives,” hedgers with different loss functions

Energies 2023, 16, 3112

3 of 22

(dependent on generation output patterns) can combine the necessary instruments and
trade them so that every hedger can replicate their own optimal payoff functions. This
would also lead to improve trading efficiency through liquid trading by multiple players
(details in Section 2).
For specific modeling, this study draws inspiration from previous research on wind
power forecasting models. In some previous studies, wind power generation can be generally modeled in the form of a cubic function of wind speed, and temperature is also
often taken into account as the next most influential weather variable after wind speed
(see Appendix A). Taking this into account, we apply their insights to design weather
derivatives based on wind speed and temperature for wind power businesses. In addition to estimating non-parametric payoffs using a two-dimensional tensor product spline
function [20] that can capture the complex interaction of wind speed and temperature on
wind power generation, and evaluating the hedge effect of previous methods, we introduce
new composite and standardized derivatives, such as the “wind speed cubic derivatives”
and “wind speed and temperature cross-derivatives,” and demonstrate the effectiveness of
derivative portfolios replicated with them.
As noted above, our primary methodological motivation is to design and replicate
optimal nonlinear payoffs for effective hedging, and there are relevant studies in this
context as well. For example, Oum et al. [21] and Oum and Oren [22] constructed optimal
nonlinear payoff functions on electricity prices for volume (demand) risk hedging in electric
utilities. Brick and Roncoroni [23] also derived nonlinear payoffs for temperature using
a similar approach. These results share analogous ideas with ours that optimal (and
effective) payoff functions are searched based on smooth nonlinear shapes and then are
replicated using standardized instruments for hedging volume risks. In particular, their
methods use the approach proposed by Carr and Madan [24], which showed that any
twice-differentiable non-linear payoff function can be replicated by a risk-less asset (the
bond), a single risky asset (the stock), and European options on the same underlying
asset with any strikes. However, their approach requires infinitely many standardized
derivatives (i.e., puts/calls with any strikes) in principle and may not be easy to achieve
a sufficient approximation using a restricted number of standardized derivatives (e.g.,
Oum and Oren [22] present a calculation case where dozens of different options are used
simultaneously, despite being univariate). Instead, our approach is to consolidate product
types needed to replicate the payoffs by creating new monomial-type derivatives and to
propose a trading scheme involving such derivatives rather than being limited to the idea
of using existing option-type derivatives. In this respect, this study also illustrates new
implications for the methodological motivations and perspectives of previous studies.
Moreover, while our recent studies have addressed the design of refined derivatives for
wind power [25,26], this study provides a new perspective by demonstrating the usefulness
of our proposed approach not only with respect to its hedge effect but also “trading
efficiency.” Simply trying to increase the hedge effect by using various types of derivatives
or combinations of products might not be original as an idea in hedge modeling. However,
as a new methodological contribution in hedging decision making, we propose the strategy
to estimate a hedging model for a standardized derivatives portfolio with the least absolute
shrinkage and selection operator (LASSO) regression, which can simultaneously achieve
both the hedge effect and the small transaction volume. In particular, we show that
adjusting the regularization parameters of LASSO can reduce the number of product types
that should be traded by exchanges/hedgers while maintaining hedge effect. We also find
that as the variety of derivatives offered by an insurer or exchange for trading increases,
hedgers can find more efficient derivatives portfolios using LASSO regression trading
strategies. Note that the hedging problem using a statistical approach, especially the most
straightforward ordinary least squares (OLS) method, has had several applications since
Ederington [27] proposed the method of minimum variance hedging in the futures market,
and it has been applied to various empirical studies in the context of cross hedging, etc.
(see, e.g., the papers [28,29] and the literature review therein.). Similar studies for the

Energies 2023, 16, 3112

4 of 22

electricity market include [30], which used OLS to estimate a hedging model consisting of
temperature (degree days), water flow, and gas prices against electricity spot prices. Other
related studies in terms of using LASSO regression include those that applied it to portfolio
selection for index tracking in the stock market and examined its performance [31,32].
However, to the best of our knowledge, the idea of using LASSO regression to enhance
trading efficiency in hedging problems has not been proposed before (perhaps this is
because most existing research on hedging takes as its starting point the idea of using
the existing limited types of hedge-related instruments.), and this study provides a new
perspective in this context as well. Thus, this study presents multi-layered novel ideas, not
only proposing a form of trading market for multiple derivative products in anticipation of
future changes, such as the decentralization of the electricity market environment, but also
presenting a LASSO regression-based trading strategy at the same time as a solution to the
problem of trading costs inherent in the diversification of financial products.
The remainder of this work is structured as follows. In Section 2, while developing
our market trading scheme, we also discuss the effects of trading multiple derivatives and
then formulate specific hedging models. In Section 3, we use empirical data to verify the
estimation results of the model and hedge effect, and in particular we reveal the mechanism
that the LASSO regression allows us to construct efficient portfolios out of using multiple
derivatives. Finally, Section 4 presents our conclusions.
2. Methods
2.1. Market Trading Model
We first develop a market model and discuss possible trading schemes for the proposed
weather derivatives. The overall picture of the transaction envisioned in this study is
illustrated in Figure 1. Assume that a wind power generator sells the electricity to an
off-taker under a power purchase agreement (PPA). The generator enters into a derivative
contract with an insurance company (or purchases derivatives on an exchange market)
in advance to hedge the risk of fluctuations in future electricity sales. Then, at delivery
time t, the generator delivers the volume of power output Vt to the off-taker and receives
(or pays if negative) a cash flow g(Vt ) each time as a settlement fee, where the function
g(·) is predetermined in the PPA. At the same time, due to the pre-contracted weather
derivatives, the power generator receives payo f f ( Mt ) that depends on the meteorological
index (calculated from the observed values) Mt (Mt may include wind speed, temperature,
etc., and the function payo f f (·) is predetermined when the derivatives contract), and the
generator’s loss is compensated to some extent. In this manner, the power generator can
mitigate the risk of cash flow fluctuations.

Figure 1. Overall picture of the transactions assumed in this study (from derivatives contracts to
settlements).

Energies 2023, 16, 3112

5 of 22

In designing such weather derivatives, this study deals with two types of derivatives:
non-parametric (made-to-order) and standardized derivatives. Figures 2 and 3 provide
a conceptual comparison of both derivative schemes, focusing on the market trading
models and payoff functions (with the problem to be solved), respectively. Non-parametric
derivatives use a bilateral contractual form where each hedger seeks an individual optimal
derivative payoff, and both the seller and buyer agree with the transaction. Meanwhile,
in the derivative trading scheme of our interest, the seller (i.e., an insurance company)
can offer derivatives with standardized payoff functions (i.e., monomials of the weather
indexes as the payoff functions) to any buyer, whereas buyers may agree to purchase
them and include them in their portfolio. The buyer should choose the types and trading
volumes of derivatives based on their own generation patterns and their correlation with
the weather indices used. This standardization will allow hedgers (buyers) to trade only
those instruments that are necessary according to their own risk tolerance. In addition,
unlike non-parametric derivatives where payoffs are determined on a made-to-order basis,
standardized derivatives allow multiple players to trade individual products. Similarly, if
a sufficient number of sellers sell standardized derivatives, the market principle will help
in better and fairer price discovery, and simultaneously many players may participate; this
can increase the liquidity of the market. Furthermore, this study also proposes a way to
use LASSO regression for decision-making on how to select effective derivative products
to be traded in the market (or which and how much products hedgers should trade from
the available products) in such a standardized derivatives market trading scheme, as will
be discussed in detail in Section 2.5.
Additionally, instead of setting the insurance company as a potential seller, a more
efficient trading scheme can be assuming that baseload power generators (who sell baseload
power generation in the wholesale market) are the sellers of derivatives (risk underwriters).
This is because when the derivatives’ payout is higher (i.e., when wind generation is lower),
the baseload generator’s revenue from the wholesale power price also tends to be higher
because of the reduced power supply, thereby conveniently offsetting cash flows. (For
example, Ketterer [33] showed logically and empirically that an increase (decrease) in the
supply of wind power affects the decrease (increase) in electricity prices.) In this transaction
scheme, an insurance company can serve as an intermediary rather than a simple seller.
Indeed, studies have considered similar schemes [34] for electricity, temperature, and/or
solar radiation derivatives. Here, we focus on standardized schemes that are highly
applicable to such transactions and that enhance liquidity market trading.

Figure 2. Market trading model for wind derivatives. Note: “W” represents wind speed index and
“T” represents temperature index.

Energies 2023, 16, 3112

6 of 22

Figure 3. Conceptual diagram of methods for determining standardized (non-parametric) derivatives’
trading volume (payoff functions). Note: this conceptual diagram is an example of using only wind
speed derivatives.

Note non-parametric derivatives are based on smoothing splines whereas standardized derivatives are based on (normal) polynomials, as will be discussed in detail in the next
section. The difference is that the former is expressed using a smoothing spline function
consisting of highly flexible piecewise (local) polynomials and is estimated by a GAM, a
type of non-parametric regression (more precisely, it is expressed as a linear sum of multiple
basis functions; see, e.g., [19] for the specific formulas). On the other hand, the latter is
represented by a limited number of monomial-type functions. In other words, the former
derivatives with payoffs represented by myriad polynomials do not fit into standardization,
and to distinguish them from monomial-type standardized derivatives, we refer to them as
non-parametric derivatives for convenience.
2.2. Minimum Variance Hedging Problem
In this study, we assume that the hedged risk is only the volume of wind power
generated, and we do not consider price risk. Hedging volume risk in electric business is a
challenging issue because it is generally difficult to completely hedge [7], and this is why
several recent studies, especially in the renewable energy business, have often addressed
the hedging problem for volume risk only [5,14,17,35]. These studies suggest that even if
price risk is excluded (i.e., even if price risk can be completely hedged), weather derivatives
are still effective because the volume risk of renewable energy is heavily linked to weatherrelated indices. With this in mind, we particularly consider the problem of mitigating
cash flow fluctuations derived from power generation variations using derivative payoffs
on wind speed and temperature. For example, if the hedger (wind power producer that
generates Vt at time t) has a fixed-rate and fixed-quantity supply obligation contract in
advance, where any shortage or surplus from the supply obligation amount v is settled
at a fixed unit price θ, the profit and loss function to be hedged (i.e., g(Vt ) in Figure 1)
is θ (Vt − v). Note that this situation of assuming only volume risk corresponds to a
power producer that has concluded a PPA with a general contract structure called “pay-asproduced,” and PPAs have been rapidly increasing in recent years as a new alternative to
the conventional FITs [36].

Energies 2023, 16, 3112

7 of 22

Then, the minimum variance hedging problem defined below can be modeled without
loss of generality by simply setting the hedge target to the generation volume Vt . In
this hedging problem (for a detailed exposition, see [34]), the hedger’s primary aim is to
minimize the variance of the net cash flows of the electricity sales revenue (wind power
generation volume) Vt and the derivative payoff, payo f f (Wt , Tt ), composed of wind speed
index Wt and temperature index Tt at each point in time:
minimize : Var[Vt − payo f f (Wt , Tt )]

(1)

where the term payo f f (Wt , Tt ) refers to the payoff function for non-parametric derivatives
or the net payoff of the derivative portfolio in our standardized derivative scheme. The
latter corresponds to the problem of finding the “contract volume” of standardized derivatives (detailed in the next subsections). Note that the units of each term in Equation (1) are
currencies (e.g., EUR), but in this study, which compares among hedging models (to be
constructed in the next and subsequent sections), the conversion of units and scales is not
considered because it does not affect the results of the analyses. For example, when the
unit price of electricity generation is θ 0 , the payoff function of either model estimated is
simply multiplied by θ 0 , and the same results (e.g., hedge effect, which will be discussed
later) will be obtained. Note also that Equation (1) has a degree of freedom with respect to
the payoff’s location (first-order moment); therefore, it is possible to impose constraints on
the mean value. For example, if a zero-mean constraint is imposed on the payoff itself (i.e.,
payo f f (Wt , Tt ) = 0), it can be designed as a derivative with zero expected payoff without
requiring an initial payment (or a premium) at contract time.
2.3. Non-Parametric Derivatives
In the case of the minimum variance hedging problem with non-parametric derivatives,
the problem of optimizing the derivatives’ “payoff function” payo f f (Wt , Tt ) := f (Wt , Tt )
in Equation (1) corresponds to estimating the following GAM:
Vt = f (Wt , Tt ) + ε t

(2)

where f is the two-dimensional tensor-product spline function [20] and ε t is the residual
with a mean of 0. Because the estimation of GAM involves minimizing the sum of squares
of the residual ε t under the smoothing constraint on f , it is synonymous with the variance
minimization in Equation (1). Note that in this study, the GAM is estimated using the
function gam() in the R package “mgcv” [20].
It is possible to design a “prediction error derivative” (i.e., a derivative with weather
prediction errors as the underlying index) by removing the time-dependent trend from
the underlying weather index (see, for example, [18]). However, we take an approach that
does not remove the time-dependent trend from the underlying index, prioritizing the
intuitive and straightforward cross-trend estimation of wind speed and temperature (e.g.,
the estimated payoff function shown in Section 3.1).
2.4. Standardized Derivatives
Next, in the case of minimum variance hedging problem with standardized derivatives,
the problem is to optimize “contract/trading volume” of standardized derivatives based on
the idea for approximating the payoff functions of non-parametric derivatives, f (Wt , Tt ),
using simple multivariate polynomials denoted by fe(Wt , Tt ). We consider the following
model:
Vt = fe(Wt , Tt ) + ε t ,
fe(Wt , Tt ) := ∑0≤i≤ I, 0≤ j≤ J β ij Wt i Tt j
(3)
where Wt i Tt j (i = 0, . . . I; j = 0, . . . , J ) are the payoff functions for different standardized
derivatives. By estimating Equation (3) using OLS, hedgers can estimate the optimal (in
the context of minimum variance hedging) contract volume β ij for each standardized

Energies 2023, 16, 3112

8 of 22

derivative. Note that when i, j = 0 is a constant, the term can be regarded as a discount
bond (with purchase unit β 00 ).
To consider the sufficient orders of derivatives, previous findings on wind power
models are helpful (see Appendix A). Wind power output is generally modeled using a
third-order formula for the wind speed [37]. In addition, wind turbines tend to reduce their
power output when the temperature becomes too hot or too cold owing to a decrease in
air pressure or freezing of components. This suggests an inverse U-shaped relationship
with respect to temperature (see, e.g., https://www.windturbinestar.com/wind-turbineannual-output.html, accessed on 15 June 2022). Based on the above considerations, this
study adopts I = 3 and J = 2 in the hedging model in Equation (3). Note that the validity
of assuming a nonlinear model was also verified by Ramsey’s RESET test [38] using our
empirical data (see Appendix B).
Note also that because the average of the observed values with respect to Wt and Tt is
positive, simply using the payoffs of second- or higher-order derivatives (i.e., when i or j is
greater than 2) as powers of the observed values may cause unnecessary large variations
in Wt i or Tt j . To avoid this problem, we design the initial (i.e., first-order) underlying
indexes by subtracting the historical average from the observed value in advance; that
is, Wt := Wobs, t − Wobs, t , Tt := Tobs, t − Tobs, t . This helps us in making derivatives more
manageable and in increasing trading efficiency. The specific mechanism is discussed later
(particularly in detail in Appendix D) using empirical data.
A recent study [26] has also shown that when designing hourly wind derivatives,
wind speed derivatives at not only the same hour but also an hour before and after are
effective. To apply this to our standardized derivative design, we set the wind speed index
at each hour as a three-hour average value, including an hour before (lagged) and after
(forward). That is, together with the above correction, the

wind speed index used here is
t +1
defined as follows: Wt := (1/3) ∑ττ =
W
−
W
obs, τ
obs, τ .
= t −1
2.5. Hedge Trading Strategy Using LASSO Regression
The more incorporated the standardized derivatives are, the better replicated the
nonlinear payoffs can be. However, using too many of them may reduce the out-ofsample period fit (i.e., hedge effect) due to overfitting. In addition, unnecessarily trading
multiple derivatives may lead to trading inefficiencies in the form of increased product
management costs or monitoring costs in the derivatives market. Similarly, reducing
trading volume may be more desirable for efficient hedging strategies (see Appendix C
for this discussion). Therefore, the strategy for estimating Equation (3) with the LASSO
regression [39] may be more effective and efficient. It should be noted that, when dealing
with conventional hedging problems, the standard OLS regressions were often applied for
estimating hedging models since only a limited number of (existing) hedging instruments
were mainly considered in the previous study. However, in this study, which introduces a
market scheme for multiple derivatives that can be customized for a variety of hedgers,
the use of LASSO regression should be particularly effective to ensure robustness and to
improve trading efficiency (choice of trading instruments).
In LASSO regressions, the residual sum of squares with the L1 norm constraint on
the regression coefficient vector is minimized. That is, each coefficient β is obtained as a
solution to the following optimization problem using the regularization parameter λ:
minimize :

T

∑ t =1



Vt − ∑0≤i≤ I, 0≤ j≤ J β ij Wt i Tt j

2

 
+ λ ∑0≤i≤ I, 0≤ j≤ J  β ij 

(4)

Energies 2023, 16, 3112

9 of 22

Solving (4) allows us to reduce or eliminate less relevant variables [39]. That is, the
LASSO regression will help sellers of derivatives (insurers or exchanges) identify products
that are inherently in high need. On the other side for the buyers, it would be more
efficient if they could select only the necessary (effective) derivatives while considering
their risk tolerance and reduce the volume of each derivative. Note that when estimating
the LASSO regression, it is desirable to standardize variables (i.e., “z-score” normalization)
to provide equivalent penalties for variables with different units; this is also followed
in this study. We estimate the LASSO regressions using the R package “glmnet,” where
standardization of variables before model estimation and rescaling of coefficients after
estimation are automatically performed as default settings [40,41].
For the regularization parameter λ, some values may be specified in advance. However, the cross-validation (CV) approach is an effective rational selection method. The most
common way is to select the λ that minimizes the mean CV error (λmin ); another method
is to use the “one-standard error rule” (1SE rule: λ1se ). Hastie et al. [40] defined the 1SE
rule as a method to choose the most parsimonious model (i.e., the largest λ), whose error
is no more than one standard error above the error of the minimized error model, and
suggested using the 1SE rule to select the best model. That is, the 1SE rule allows us to
select the simplest model with the same accuracy as the best one [42]. Here, the adoption
of this 1SE rule may help us in estimating a more efficient derivatives portfolio; that is,
a portfolio with sufficient hedge effect while narrowing down the required number (or
volume) of trading instruments as much as possible. Hence, our empirical analysis tests
the effectiveness of trading strategies with LASSO regression by using both λmin and λ1se ,
as well as by varying λ continuously.
3. Results
Next, we present the results of estimating the payoff function and calculating the
hedge effect. Here, we use the following observed values of wind power generation and
weather in Denmark, where wind power is very prevalent.

•

•

Wind power generation [MWh]: actual power generation in Eastern Denmark (DK2)
(downloaded from https://www.nordpoolgroup.com/en/Market-data1/, accessed
on 11 January 2021)
Wind speed [m/s] and temperature [◦ C]: observed values at Copenhagen Airport
(downloaded from http://rp5.ru/metar.php?metar=EKCH, accessed on 11 January
2021)

The estimation (in-sample) and validation periods (out-of-sample) are 2019 and 2020,
respectively, using hourly data for all 24 h days from January 1 to December 31. The
estimated trends and trading volumes (coefficients) were obtained from the in-sample
period data only, whereas the hedge effects were calculated using the out-of-sample period
data. Note that this study used data through 2020, which were available at the time the
data were obtained, but as of the time of this paper’s submission, the data, including the
most recent period, were no longer available to the public on that site.
Considering that sufficient historical data are often not available in practical situations,
we estimated the hedge models from only one-year of in-sample data and calculated
the hedge effect for the following year; this also helps in checking the robustness of the
proposed method. Specifically, the hedge effect uses the variance reduction ratio (VRR) as
1 − VRR. VRR is defined as follows:
VRR := Var[Vt − payo f f (Wt , Tt )]/Var[Vt ]

(5)

Energies 2023, 16, 3112

10 of 22

3.1. Estimated Trend (Non-Parametric Derivatives)
The trend of the estimated non-parametric derivative (i.e., f (Wt , Tt ) in Equation (2))
is shown in Figure 4. As mentioned in Section 2.3, a cubic form in the wind speed direction
and a near-quadratic form in the temperature direction were observed. For example, the
sensitivity of wind power generation to wind speed was low when the wind speed was
low, then increased as the wind speed increased, and finally decreased when the wind
speed became too high. Meanwhile, the effect of temperature peaked at approximately
5–10 ◦ C, and the power generation decreased at both higher and lower temperatures.

Figure 4. Estimated trends of non-parametric derivatives (left: perspective, right: contour).

3.2. Measurement of Hedge Effects
Figure 5 plots the cumulative hedge effect, and the hedging products included in each
hedging model are shown in Table 1 (note that the operators “+” and “*” are not strictly
defined but are used for convenience to correspond intuitively with the long regression
equation). The blue (orange) dots indicate the hedge effect of standardized (non-parametric)
derivatives. For reference, the light blue and light orange dots are plotted together, showing
the case where wind speeds before (lag) and after (forward) one hour are not included in
the underlying indexes, as described in Section 2.4. Evidently, including wind speeds at the
before and after hours in the underlying index is effective for improving the hedge effects;
almost 3% improvements are observed in all hedging models.
Next, for both non-parametric and standardized types, the hedge effects increasd as
the number of derivative combinations increased (i.e., the number of non-linear degrees of
freedom increased). The hedge effect of using only “wind speed forwards” (W1) was 65.9%,
while including “wind speed cubic derivatives” (W3) improved the hedge effect to 68.1%
(68.3% for wind speed non-parametric derivatives). For the wind-speed squared derivative
(W2), no improvement was observed in the hedge effect when combined with wind speed
forwards without wind speed cubic derivatives, indicating that both derivatives are meaningful only when used in combination. The effect of temperature standardized derivatives
was similar. The hedge effect can be increased to 69.3% when the “temperature squared
derivative” is included (69.8% when temperature non-parametric derivatives are used) and
70% when all “wind speed and temperature cross-derivatives” are included (70.5% when
the two-dimensional non-parametric derivative of wind speed and temperature is used).

Energies 2023, 16, 3112

11 of 22

Figure 5. Hedge effects of standardized derivatives and non-parametric derivatives.
Table 1. Components of hedging instruments by hedging model.
Non-Parametric
Derivatives Model

Standardized Derivatives Model

W
W2
Standardized
derivatives

W3
T + T2
WT + WT 2
W 2 T + W 2 T2 + W 3 T + W 3 T2

Nonparametric
derivatives

f W (W )
f T (T )
f te (W, T )

W1

W2

W3

W3+T2

W3+W*T2

W3*T2

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Wd

Wd+Td

X

X

Wd*Td

X
X
X

For confirmation, the actual wind power generation versus payoffs of the derivative
portfolio in the W3*T2 model is plotted in Figure 6. Because the payoff of the derivatives
may be thought of as the predicted wind power generation (assuming that wind speed
and temperature were observed), Figure 6 can be viewed as a fitting plot considering
Equation (3) as a forecast model. Notably, the fluctuations of the two were generally
consistent, except when power generation was extremely high/low. The wind speed and
temperature indices are based on data observed at only one location (Copenhagen Airport);
the model’s accuracy may improve by refining the indices.
The results of the above hedge effects are similar, even when viewed by month, as
shown in Table 2. Notably, the effect of the “wind speed cubic derivative” improves the
hedge effect in most months, and in some months, by more than 10% (e.g., 10.6% in March
and 64.2% in August) compared to the case where only wind speed forward was used.
While the hedge effect decreased in a few months, this decrease was limited.

Energies 2023, 16, 3112

12 of 22

The improvement in the hedge effect of “wind speed and temperature cross-derivatives”
is particularly pronounced in the summer months. This may be due to the temperature variable successfully explaining the relationship between the continuous decrease in generation
output (generation efficiency) with increasing temperature during the summer months, and
hence the increased effectiveness of the temperature (cross) derivatives. In particular, August had the strongest non-linearity and interaction; the hedge effect improved by 114.6%
when the temperature cross derivatives were included. Thus, the temperature-related
derivatives do affect the reduction of wind turbine output (cash flow loss) due to high
temperatures. Therefore, applying temperature derivatives or wind speed/temperature
cross-derivatives especially for the summer months may provide useful hedges.

Figure 6. Realized versus predicted values of wind power generation in the out-of-sample period
(Upper: 2020, Lower: February 2020). Note: the upper graph plots daily averaged values for visibility.
The realized data are the raw wind production data downloaded from https://www.nordpoolgroup.
com/en/Market-data1/ (accessed on 11 January 2021).
Table 2. Monthly hedge effect and improvement rate against the W1 model.
Monthly Hedge Effect

Hedge effect
(1-VRR)
Improvement
rate (to “W1”)

Jan

Feb

Mar

W1

68.6%

76.1%

W3

67.5%

79.4%

W3*T2

68.6%

W3

−1.6%

W3*T2

0.0%

Jun

Jul

Aug

Sep

Oct

Nov

Dec

All
Period

58.7%

40.9%

66.3%

15.4%

63.5%

62.8%

72.8%

60.9%

65.9%

59.5%

43.9%

69.1%

25.3%

68.8%

62.5%

75.9%

60.3%

68.1%

59.2%

61.1%

46.9%

74.0%

33.0%

68.8%

63.8%

76.0%

59.8%

69.3%

1.4%

1.4%

7.2%

4.3%

64.2%

8.3%

−0.4%

4.2%

−1.1%

3.4%

0.2%

4.0%

14.7%

11.6%

114.6%

8.4%

1.7%

4.4%

−1.8%

5.2%

Apr

May

65.4%

59.1%

72.3%

59.9%

80.7%

71.5%

4.4%

10.6%

6.1%

9.3%

Note: improvement rate is the ratio of the increase in the hedge effect of each hedging model relative to that of
the W1 model; the largest value among the models in each month for both hedge effect and improvement rate is
shown in bold.

Energies 2023, 16, 3112

13 of 22

3.3. Trading Efficiency Using LASSO Regression
Finally, we tested the effectiveness of the “trading product selection” strategy of
the LASSO regression described in Section 2.3. This strategy seems particularly effective
when one intends to trade multiple standardized derivatives. Therefore, here, we take the
“W3*T2” model (with an original hedge effect of 70.0%) as an example and estimate this
model with a LASSO regression instead of the normal OLS. The regularization parameter
λ in the LASSO regression is treated in two ways, λmin and λ1se , as described in Section 2.5.
The contract/trading volumes (i.e., absolute values of the coefficients of the explanatory variables) for each derivative estimated by this method, including other hedging
models using standardized derivatives, are shown in Table 3. The hedge effects shown
in Figure 5 are also described at the bottom of the table. First, comparing the W3*T2
model with the models estimated using the LASSO regression, the hedge effect decreased
slightly (by one percentage point) to 69.3% when using Lasso-λ1se and remained virtually
unchanged from the original 70.0% when using Lasso-λmin . However, there are notable
differences in the product selection and volumes of trading, as discussed below.
First, regarding the product selection, we see that the trading volume of W 3 T was zero
when Lasso-λ1se is used. This is because the L1 regularization term in LASSO, explained in
Section 2.5, reduced the variable’s coefficient (trading volume of the instrument) with the
lowest contribution to 0. Note that increasing λ just a bit more makes it possible to further
reduce the volume of multiple instruments traded to zero, which is discussed in detail later
with a figure at the very end of this section.
Table 3. Derivatives contract volume by hedging model, including LASSO regression-based trading
strategies.
W1

W2

W3

W3+T2

W3+W*T2

W3*T2

lasso.λmin

lasso.λ1se

Stdev of
Payoffs

Contract volumes of each derivative (absolute values of coefficients)
W

0.98465

0.97895

1.13905

1.11814

1.16114

1.18486

1.18426

1.05063

2.3

I(Wˆ2)

-

0.00378

0.06930

0.06541

0.06286

0.05867

0.05820

0.03367

8.3

I(Wˆ3)

-

-

0.01406

0.01369

0.01376

0.01460

0.01455

0.00728

62.3

T

-

-

-

0.03324

0.03283

0.04237

0.04244

0.03022

6.4

I(Tˆ2)

-

-

-

0.00569

0.00613

0.00694

0.00697

0.00579

46.3

I(W * T)

-

-

-

-

0.00355

0.00246

0.00229

0.00080

13.8

I(W * Tˆ2)

-

-

-

-

0.00113

0.00180

0.00177

0.00006

128.2

I(Wˆ2 * T)

-

-

-

-

-

0.00258

0.00262

0.00051

56.3

I(Wˆ2 * Tˆ2)

-

-

-

-

-

0.00016

0.00017

0.00011

428.9

I(Wˆ3 * T)

-

-

-

-

-

0.00015

0.00017

.

361.8

I(Wˆ3 * Tˆ2)

-

-

-

-

-

0.00004

0.00004

0.00003

2651.4

Contract volumes of derivatives portfolio
Simple sum

0.98465

0.98273

1.22241

1.23617

1.28140

1.31463

1.31348

1.12909

Weighted sum

2.22805

2.24632

4.02585

4.40015

4.69229

5.31285

5.30554

3.74113

0.68146

0.69343

0.69797

0.70006

0.70009

0.69348

Hedge effects of derivatives portfolio
1-VRR

0.65892

0.65929

In assessing trading volume, we examine not only the “simple sum” of the total
trading volume of the derivatives portfolio but also a “weighted sum” weighted by the
standard deviation of the payoffs of the individual derivatives. Note that the “weighted
sum” is a useful measure of overall trading inefficiency, considering that hedge tradingrelated costs, such as the risk premium that a seller may demand, can depend on the

Energies 2023, 16, 3112

14 of 22

instrument’s volatility (see Appendix C for more details). In both cases, the total volumes of
the derivatives portfolio with LASSO are significantly smaller than those with regular OLS.
For instance, the weighted sum using Lasso-λ1se decreased from 5.31 to 3.74, a reduction of
approximately 30%.
To verify the effectiveness of the LASSO regression to control trading volume, including the other hedging models (i.e., from W1 to W3+W*T2 models), we plot the relationship
between contract volume and the hedge effect using the hedging model in Figure 7; here,
the weighted sum in Table 3 is used as the trading volume on the vertical axis. First, as
shown by the blue line, the trading volume increased with each additional product type,
and simultaneously the hedge effect also increased. This result is consistent with practical
intuition, as it suggests a trade-off exists between the hedge effect and its cost. However, the
results of the trading strategies in the LASSO regression are more interesting. In Figure 7,
they are drawn as dashed curves extending from the plot of each hedging model (i.e., the
blue points where the trading volume is estimated by OLS, corresponding to λ = 0) to
the lower left as λ gradually increases. Notably, the trading strategy estimated by LASSO
for the W3*T2 model is generally below the line drawn by other models. That is, the
strategy of using LASSO to estimate the trading volume of all derivatives (including higherorder derivatives) creates a “Pareto optimal” portfolio. Furthermore, the “Pareto optimal”
frontier extends downward as the number of product types increases. This suggests that
the availability of multiple instruments (i.e., trading multiple hedge products) not only
increases the hedge effect but also contributes to reducing the hedger’s trading volume.
For example, a portfolio with the W3 model (using wind speed derivatives only) would be
an effective solution to increase the hedge effect as much as possible in a situation where
no temperature-related derivatives were available. However, if temperature derivatives
can be traded, one can construct a more efficient mixed portfolio in which only the trading
volume can be noticeably reduced with the same hedge effect.

Figure 7. Relationship between contract volume and hedge effect by hedging model. Note: the term
“lasso” refers to the W3*T2 model estimated by LASSO regression. Hedge effect is measured in
out-of-sample periods. Note: the blue dots represent the estimation results from OLS and the dashed
curves represent the estimation results obtained by varying λ in the LASSO regression.

Energies 2023, 16, 3112

15 of 22

Note that the above analysis results are for the case where the hedging model is
estimated using one year of in-sample data for 2019, as stated at the beginning of Section 3.
Here, in order to verify the robustness of the results, as reference scenarios, we also
calculated the results when using (A) two years of data for 2018–2019 and (B) half a year of
data for July-December 2019, as the in-sample period. The pairs of {hedge effect, contract
volume (weighted sum)} for the Lasso-λ1se strategy were calculated to be {0.692, 3.819}
for Case (A) and {0.682, 3.528} for Case (B). In both cases, the hedge effects were smaller
than that of the benchmark case using one year of in-sample data, i.e., 0.693. Even when
viewed in a two-dimensional plot that takes into account the trading volume, points A and
B are inside the efficient frontier shown in Figure 7. The reason for this may be that taking
too long of a learning period has the disadvantage of increasing the impact of changes
in generation patterns due to renewal or expansion of generation facilities, while taking
too short of a learning period makes robust estimation difficult. In general, for one-year
ahead hedging, the use of the most recent one-year in-sample period, as adopted for the
benchmark case, is considered appropriate from the perspective of ensuring a hedge effect.
For reference, Figure 8 shows the path diagram of the coefficients when the W3*T2
model is estimated by the LASSO regression with respect to the regularization parameter
λ. Notably, increasing λ generally reduces the absolute value of each coefficient (i.e.,
the volume of each derivative contract). Further, each coefficient shrinks to zero as λ
increases owing to the L1-norm penalty of the LASSO regression. If large transaction costs
arise depending on the types of products traded (e.g., monitoring costs of price changes
by product), then the decision of increasing λ can be made with reference to this path
diagram to reduce the number of product types. For example, from the red vertical line
in Figure 7, which represents λ1se , some product types can be excluded from the portfolio
by adjusting λ to slightly larger values. In any case, if λ is set too large, the coefficients
(derivative volume) converge to zero and the hedge effect shrinks. Overall, to construct
an efficient portfolio without impairing the hedge effect, the use of the 1SE rule λ is well
balanced, as demonstrated above, and may provide an excellent criterion in the hedger’s
decision-making process for the trading strategy.

Figure 8. Relationship between the regularization parameter λ and the coefficients in LASSO regression. The graph on the right transforms the scale of the left graph using the hyperbolic tangent
function (tanh(100x )) for ease of viewing.

Energies 2023, 16, 3112

16 of 22

4. Conclusions
This study demonstrated the effectiveness of hedging strategies using portfolios
of high-order standardized weather derivatives applied to the wind power generation
business. Our main contributions are as follows. First, we proposed new high-order
monomial-type standardized derivatives, such as wind speed cubic derivatives and wind
speed/temperature cross-derivatives, and demonstrated their effectiveness. Second, while
developing a specific market trading scheme, we also showed the practical benefits of
standardizing weather derivatives. Third, we proposed a “trading product selection”
strategy using LASSO regression and demonstrated its effectiveness in terms of both
increasing the efficiency of hedge trading and ensuring the hedge effect. Finally, we
visualized the latent mixed trend in wind power generation relative to temperature and
wind speed, and we noted the implications in the context of wind power forecasting
models.
Essentially, we have shown that the hedge effect can be enhanced by incorporating
various underlying hedge instruments and increasing the non-linearity of the portfolio. In
particular, our new implication is that constructing a standardized derivatives portfolio is
significant not only in hedge effect but also in hedge efficiency. By providing a variety of
standardized hedging instruments, insurance companies (or exchanges) can give hedgers
more flexible options for building their derivative portfolios. Estimating hedging models
using LASSO regressions can help insurers/exchanges make decisions to identify efficient
instruments. In addition, if hedgers adopt a trading strategy of LASSO regression, they can
find and construct more Pareto-optimal portfolios, which help deal with the trade-off of
reducing trading volume (transaction costs) and improving the hedge effect. Importantly,
this mechanism may have generalizable implications for risk management practices in
other businesses.
The primary idea provided by this study is in the original market scheme using new
standardized derivatives for granular risk hedging, which would be increasingly needed
in future decentralized electricity markets. Although relatively simple statistical methods
such as OLS and LASSO regression were used for the methodology, our approach should
have an advantage from the standpoint of practicality and applicability to the real world,
i.e., a market scheme on methodologies that are as simple and straightforward as possible
are superior. As a first step to examine the potential use of new higher-order standard
derivatives, this study deals only with volume risk, but it would be interesting to examine
the possibility of simultaneous hedging with price risk, which should be addressed in the
future. In addition, the perspective of this research can be versatile, for example, in building
a derivatives market for the distributed solar power generation business. The research
questions for such applications may include verifying the effectiveness at daily granularity
and further refining the hedging model by taking seasonality into account. Furthermore,
from the perspective of further reducing transaction costs, methods other than LASSO
regression could be used. LASSO regression has the advantage of reducing the number of
hedge products due to its ability to shrink the coefficients of variables to zero, but if the
costs associated with transaction volume (e.g., risk premiums) are more pronounced than
the costs associated with the number of products (e.g., monitoring costs), other regularized
regression methods, such as Ridge regression, might be superior. Further investigation of
such questions may also be a subject for future work.
Derivatives with finer granularity and more variety, such as those addressed here, may
have more demand once infrastructure, such as peer-to-peer power trading platforms, is in
place. For instance, a fully digital trading platform will most likely reduce transaction costs.
Moreover, a platform based on promising blockchain technology can enable immediate
settlement or payment. (For example, the EnergyWeb blockchain has been developed for
the energy sector and uses the “Proof-of-Authority” consensus algorithm. This blockchain
has transaction rates hundreds to thousands of times higher than the “Proof-of-Work”
conventional algorithm used by Bitcoin and other crypto applications, and it allows for
instantaneous settlement [43].) Granular derivatives can be particularly useful for small

Energies 2023, 16, 3112

17 of 22

players who wish to minimize fluctuations in their digital wallets. In any case, with the
diversification of power resources, the increasing variety of hedging needs is an emerging
trend. Then, naturally, hedging products should also be diversified. Given such future
changes, the market design concept of diversification of hedging instruments and efficient
trading scheme, as presented in this study, could be a potential basic idea to realize the
decentralized power trading that will be needed in the future. Design methodologies,
standardization schemes, and trading strategies for refined weather derivatives, such
as those proposed here, may have useful implications in the electricity market, where
transaction granularity is becoming increasingly finer in space and time.
Author Contributions: Conceptualization, T.M. and Y.Y.; methodology, T.M. and Y.Y.; software, T.M.;
validation, T.M.; data curation, T.M.; writing—original draft preparation, T.M.; writing—review and
editing, Y.Y.; visualization, T.M.; supervision, Y.Y.; project administration, Y.Y.; funding acquisition,
T.M. and Y.Y. All authors have read and agreed to the published version of the manuscript.
Funding: Grant-in-Aid for Scientific Research (A) 20H00285, Grant-in-Aid for Challenging Research
(Exploratory) 19K22024, and Grant-in-Aid for Young Scientists 21K14374 from the Japan Society for
the Promotion of Science (JSPS).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: All data used in this study were obtained from the following websites:
https://www.nordpoolgroup.com/en/Market-data1 (accessed on 11 January 2021); http://rp5.ru/
metar.php?metar=EKCH (accessed on 11 January 2021). Note that the publication of the data on the
former website has been terminated as of the time of submission of this paper, but the data for the
period covered in this study are available from the first author.
Acknowledgments: This paper is an extension of the one presented at ICEER 2022, and their proceedings are to appear in “Energy Reports” (see Matsumoto and Yamada (2023)). This work was
supported by Grant-in-Aid for Scientific Research (A) 20H00285, Grant-in-Aid for Challenging Research (Exploratory) 19K22024, and Grant-in-Aid for Young Scientists 21K14374 from the Japan
Society for the Promotion of Science (JSPS).
Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Research on Wind Power Output Forecasting
To develop an elaborate hedging model for wind power projects, we briefly reviewed
studies on wind power forecasting. Broadly, wind power forecasting methods can be classified into time-series and exogenous variable models [44]. In recent years, the latter have
emerged due to advances in computing technology and artificial intelligence, including several models that use machine learning (ML) methods such as artificial neural networks and
deep learning [45]. Wind power generation is strongly correlated with wind speed, where
intensity non-linearities exist. The relationship between wind turbine and wind speed has
been analyzed in detail, focusing on technical characteristics; for example, in [46–48]. In
general, the wind power curve is often described in the form of a cube of wind speed [37].
Marˇciukaitis et al. [49] also compared and validated various models of non-linear regression with wind speed as an explanatory variable. Temperature is the next most commonly
used explanatory variable after wind speed [45]. Some studies have demonstrated the
effect of using temperature to improve accuracy using ML methods [50–52]; however, these
studies do not clarify the structural effects of temperature and wind speed (including their
interaction) on wind-power generation. Therefore, we intuitively illustrate the interaction
between temperature and wind speed on wind power generation using a three-dimensional
plot (Figure 4). This can provide useful insights in the context of wind power forecasting
models.

Energies 2023, 16, 3112

18 of 22

Appendix B. Verification of Non-Linearity by Ramsey’s RESET Test
This study, which proposes the concept of trading schemes and strategies for high
order standard derivatives, does not enter into the problem of optimizing the order of
polynomial equations (3). That is, it is not the purpose of this study to determine whether
third or second order is optimal as a power of wind speed or temperature. However, to
verify the assumptions of the nonlinear models, Ramsey’s RESET test [38], a common test
for linear regression, was performed on several models. In this test, the order of diagnostic
variables for the linear function (null hypothesis) can be specified (see, for example, [53] as
an empirical study), and the test was performed using the R package “lmtest” [54]. The
results suggest the presence of strong nonlinearities, as shown in Table A1. In other words,
it is worthwhile to use derivatives with higher-order monomial payoffs in combination
with wind and temperature futures that have linear payoffs.
Table A1. The result of Ramsey’s RESET test.
Model

Diagnostic Variables

F-Statistic

p-Value

V~W
V~T
V~W+T

squares, cubes
squares
squares, cubes

264.97
305.21
294.66

< 2.2 × 1016
< 2.2 × 1016
< 2.2 × 1016

Note: this table is calculated using the function resettest() of the R package “lmtest” for the in-sample period
data for Denmark presented in Section 3. “Model” is the “formula” of the original linear model and “Diagnostic
variables” are the “power” of the fitted values of the linear model [54]. As the table shows, the p-values are
essentially zero, meaning that the linear models of the null hypothesis are strongly rejected. For more information
on this test, see, for example, “Chapter 9” in [55].

Appendix C. Hedge Trading Related Costs That Can Depend on Trading Volume
In general, an increase in hedging volume may result in an additional cost for a hedger
from the following reasons. First, the bid/ask spread (BAS) is often used as a proxy for
measuring the direct cost of transactions for market instruments [56]. Since a hedger must
pay a BAS-equivalent transaction cost per trading (see, e.g., [57,58]), an increase in the
trading volume leads to a rise in such a direct transaction cost. Another potential cost
(which, in some cases, may be greater than the BAS) is the “risk premium” that the seller
of a derivative contract adds to the selling price. The price of weather derivatives or
derivatives (futures) in electricity markets is often formulated as the expected price of
the underlying asset plus a risk premium (see, for example, [59] for weather derivatives
and [60] for electricity futures), and the risk premium is interpreted as compensation for
the seller’s risk burden (see, for example, [61,62]). That is, for the buyer, it corresponds to
the cost of lowering his exposure [63]. Thus, curtailing the trading volume reduces this
invisible cost corresponding to the risk premium.
In addition, both the BAS and the risk premium tend to increase as the volatility of the
asset increases. For example, it has been demonstrated that there is a positive correlation
between volatility and BAS [64,65] and volatility and risk premium [66] in various futures
markets. In other words, the values calculated in Section 3.3, where the trading volume
is weighted by the standard deviation of the payoffs of each instrument, have a certain
rationality as an indicator of hedge trading related costs.
Appendix D. The Case of Using Derivatives with No Average Value Correction for the
Underlying Index
In Section 2.4, we noted that when designing higher-order standardized derivatives, it
is desirable to subtract the mean from the first-order underlying (we refer to this case as a
“corrected case”); here, we show the reason for this. First, if such a correction is not made
(an “uncorrected case”), the volatility of higher-order derivatives will be unnecessarily
large, which will worsen the efficiency of the transaction. Figure 7 in Section 3.3, shows the
relationship between trading volume (“weighted sum” considering standard deviation) and
hedge effect for the “corrected” derivatives; the corresponding plot for the “uncorrected”

Energies 2023, 16, 3112

19 of 22

case is shown in Figure A1 below. While the absence of the correction has little effect
on the hedge effects as they are perfectly consistent in the ordinary least squares (OLS)
estimation case, the standard deviation-weighted trading volumes (trading costs) are
several times larger than those shown in Figure 7. Although the least absolute shrinkage
and selection operator (LASSO) regression strategy is effective in reducing trading volume
in this uncorrected case as well (the volume is reduced to approximately 1/3 using the 1SE
rule), the volume can be reduced much more when using corrected derivatives. Specifically,
even with the 1SE rule, trading volume decreased to 3.74 in the corrected case, while it only
decreased to 7.68 in the uncorrected case.

Figure A1. Relationship between contract volume and hedge effect by hedging model (cases in which
no mean value correction was made for the underlying indexes). Note: the term “lasso” refers to the
W3*T2 model estimated by LASSO regression. Hedge effect is measured in out-of-sample periods.
Note: the blue dots represent the estimation results from OLS and the dashed curves represent the
estimation results obtained by varying λ in the LASSO regression.

The path diagram of the estimated coefficients (derivative volume) from the LASSO
regression for the uncorrected case is also shown in Figure A2 (for comparison, see Figure 8
for the corrected case). Note that the coefficients are not stable with respect to the moving
λ. For example, the hedger’s position in wind futures has changed significantly from a sell
position to zero position and then to a buy position as λ increases. Further, the paths of
the other coefficients are also generally not smooth. This is irrational from the perspective
of efficient decision making in trading practice. In addition, assuming, for example, that
this derivative supported continuous trading, such an unstable volume path can lead to
unnecessary rebalancing of trading positions. This is not desirable from a transaction-cost
perspective.

Energies 2023, 16, 3112

20 of 22

Figure A2. Relationship between the regularization parameter λ and the coefficients in LASSO
regression (i.e., cases with no mean value correction for the underlying indexes). The graph on the
right transforms the scale of the left graph using the hyperbolic tangent function (tanh(100x )) for
ease of viewing.

References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.

IEA. Renewables 2021 Analysis and Forecast to 2026. Available online: https://www.iea.org/reports/renewables-2021 (accessed
on 15 June 2022).
Market Research Future. Small Wind Power Market. Available online: https://www.marketresearchfuture.com/reports/smallwind-power-market-4568 (accessed on 15 June 2022).
Botterud, A.; Wang, J.; Bessa, R.; Keko, H.; Miranda, V. Risk management and optimal bidding for a wind power producer. In
IEEE PES General Meeting; IEEE: Minneapolis, MN, USA, 2010; pp. 1–8. [CrossRef]
Bathurst, G.; Weatherill, J.; Strbac, G. Trading wind generation in short term energy markets. IEEE Trans. Power Syst. 2002, 17,
782–789. [CrossRef]
Yamada, Y. Optimal Hedging of Prediction Errors Using Prediction Errors. Asia-Pacific Financ. Mark. 2008, 15, 67–95. [CrossRef]
Eydeland, A.; Wolyniec, K. Energy and Power Risk Management: New Developments in Modeling, Pricing, and Hedging; John Wiley &
Sons: Hoboken, NJ, USA, 2002; Volume 97.
Deng, S.; Oren, S. Electricity derivatives and risk management. Energy 2006, 31, 940–953. [CrossRef]
Benth, F.E.; Pircalabu, A. A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures. Appl. Math. Financ. 2018,
25, 36–65. [CrossRef]
Gersema, G.; Wozabal, D. An equilibrium pricing model for wind power futures. Energy Econ. 2017, 65, 64–74. [CrossRef]
Benth, F.E.; Di Persio, L.; Lavagnini, S. Stochastic Modeling of Wind Derivatives in Energy Markets. Risks 2018, 6, 56. [CrossRef]
Rodríguez, Y.; Pérez-Uribe, M.; Contreras, J. Wind Put Barrier Options Pricing Based on the Nordix Index. Energies 2021, 14, 1177.
[CrossRef]
Kanamura, T.; Homann, L.; Prokopczuk, M. Pricing analysis of wind power derivatives for renewable energy risk management.
Appl. Energy 2021, 304, 117827. [CrossRef]
Christensen, T.S.; Pircalabu, A. On the spatial hedging effectiveness of German wind power futures for wind power generators. J.
Energy Mark. 2018, 11, 71–96. [CrossRef]
Masala, G.; Micocci, M.; Rizk, A. Hedging Wind Power Risk Exposure through Weather Derivatives. Energies 2022, 15, 1343.
[CrossRef]
Hastie, T.; Tibshirani, R. Generalized Additive Models; Chapman & Hall: Boca Raton, FL, USA, 1990.
Yamada, Y. Valuation and hedging of weather derivatives on monthly average temperature. J. Risk 2007, 10, 101–125. [CrossRef]
Matsumoto, T.; Yamada, Y. Cross Hedging Using Prediction Error Weather Derivatives for Loss of Solar Output Prediction Errors
in Electricity Market. Asia-Pacific Financ. Mark. 2018, 26, 211–227. [CrossRef]
Matsumoto, T.; Yamada, Y. Simultaneous hedging strategy for price and volume risks in electricity businesses using energy and
weather derivatives. Energy Econ. 2021, 95, 105101. [CrossRef]

Energies 2023, 16, 3112

19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.

21 of 22

Matsumoto, T.; Yamada, Y. Customized yet Standardized Temperature Derivatives: A Non-Parametric Approach with Suitable
Basis Selection for Ensuring Robustness. Energies 2021, 14, 3351. [CrossRef]
Wood, S.N. Generalized Additive Models: An Introduction with R; Chapman and Hall: New York, NY, USA, 2017.
Oum, Y.; Oren, S.; Deng, S. Hedging quantity risks with standard power options in a competitive wholesale electricity market.
Nav. Res. Logist. 2006, 53, 697–712. [CrossRef]
Oum, Y.; Oren, S.S. Optimal Static Hedging of Volumetric Risk in a Competitive Wholesale Electricity Market. Decis. Anal. 2010,
7, 107–122. [CrossRef]
Brik, R.I.; Roncoroni, A. Static mitigation of volumetric risk. J. Energy Mark. 2016, 9, 111–150. [CrossRef]
Carr, P.; Madan, D. Optimal positioning in derivative securities. Quant. Financ. 2001, 1, 19–37. [CrossRef]
Matsumoto, T.; Yamada, Y. Multivariate Weather Derivatives for Wind Power Risk Management: Standardization Scheme and
Trading Strategy. Energy Rep. 2023, in press.
Yamada, Y.; Matsumoto, T. Construction of Mixed Derivatives Strategy for Wind Power Producers; University of Tsukuba: Tokyo,
Japan, 2023; manuscript in preparation.
Ederington, L.H. The hedging performance of the new futures markets. J. Financ. 1979, 34, 157–170. [CrossRef]
Zainudin, A.D.; Mohamad, A. Cross hedging with stock index futures. Q. Rev. Econ. Finance 2021, 82, 128–144. [CrossRef]
Ong, T.S.; Tan, W.F.; Teh, B.H. Hedging effectiveness of crude palm oil futures market in Malaysia. World Appl. Sci. J. 2012, 19,
556–565.
Woo, C.-K.; Horowitz, I.; Hoang, K. Cross hedging and forward-contract pricing of electricity. Energy Econ. 2001, 23, 1–15.
[CrossRef]
Sant’Anna, L.R.; Caldeira, J.F.; Filomena, T.P. Lasso-based index tracking and statistical arbitrage long-short strategies. N. Am. J.
Econ. Finance 2020, 51, 101055. [CrossRef]
Chen, Q.-A.; Hu, Q.; Yang, H.; Qi, K. A kind of new time-weighted nonnegative lasso index-tracking model and its application.
N. Am. J. Econ. Financ. 2022, 59, 101603. [CrossRef]
Ketterer, J.C. The impact of wind power generation on the electricity price in Germany. Energy Econ. 2014, 44, 270–280. [CrossRef]
Yamada, Y.; Matsumoto, T. Going for Derivatives or Forwards? Minimizing Cashflow Fluctuations of Electricity Transactions on
Power Markets. Energies 2021, 14, 7311. [CrossRef]
Mosquera-López, S.; Uribe, J.M. Pricing the risk due to weather conditions in small variable renewable energy projects. Appl.
Energy 2022, 322, 119476. [CrossRef]
RE-Source. Risk Mitigation for Corporate Renewable PPAs. 2017. Available online: https://resource-platform.eu/wp-content/
uploads/files/statements/RE-Source%203.pdf (accessed on 3 February 2022).
Ko, W.; Hur, D.; Park, J.-K. Correction of wind power forecasting by considering wind speed forecast error. J. Int. Counc. Electr.
Eng. 2015, 5, 47–50. [CrossRef]
Ramsey, J.B. Tests for Specification Errors in Classical Linear Least-Squares Regression Analysis. J. R. Stat. Soc. Ser. B (Methodol.)
1969, 31, 350–371. [CrossRef]
Tibshirani, R. Regression Shrinkage and Selection Via the Lasso. J. R. Stat. Soc. Ser. B 1996, 58, 267–288. [CrossRef]
Hastie, T.; Tibshirani, R.; Friedman, J.H.; Friedman, J.H. The Elements of Statistical Learning: Data Mining, Inference, and Prediction;
Springer: New York, NY, USA, 2009; Volume 2, pp. 1–758.
Friedman, J.; Hastie, T.; Tibshirani, R.; Narasimhan, B.; Tay, K.; Simon, N.; Qian, J. Package ‘glmnet’. CRAN R Repositary.
Available online: https://cran.r-project.org/web/packages/glmnet/index.html (accessed on 15 June 2022).
Krstajic, D.; Buturovic, L.J.; Leahy, D.E.; Thomas, S. Cross-validation pitfalls when selecting and assessing regression and
classification models. J. Cheminform. 2014, 6, 10. [CrossRef] [PubMed]
Alladi, T.; Chamola, V.; Rodrigues, J.J.P.C.; Kozlov, S.A. Blockchain in Smart Grids: A Review on Different Use Cases. Sensors
2019, 19, 4862. [CrossRef] [PubMed]
Foley, A.M.; Leahy, P.G.; Marvuglia, A.; McKeogh, E.J. Current methods and advances in forecasting of wind power generation.
Renew. Energy 2012, 37, 1–8. [CrossRef]
Hanifi, S.; Liu, X.; Lin, Z.; Lotfian, S. A Critical Review of Wind Power Forecasting Methods—Past, Present and Future. Energies
2020, 13, 3764. [CrossRef]
Chaurasiya, P.K.; Kumar, V.K.; Warudkar, V.; Ahmed, S. Evaluation of wind energy potential and estimation of wind turbine
characteristics for two different sites. Int. J. Ambient Energy 2021, 42, 1409–1419. [CrossRef]
Chaurasiya, P.K.; Ahmed, S.; Warudkar, V. Wind characteristics observation using Doppler-SODAR for wind energy applications.
Resour.-Effic. Technol. 2017, 3, 495–505.
Werapun, W.; Tirawanichakul, Y.; Waewsak, J. Wind shear coefficients and their effect on energy production. Energy Procedia 2017,
138, 1061–1066. [CrossRef]
˙ I.; Martišauskas, L.; Jokšas, B.; Geceviˇcius, G.; Sfetsos, A. Non-linear regression model for wind
Marˇciukaitis, M.; Žutautaite,
turbine power curve. Renew. Energy 2017, 113, 732–741. [CrossRef]
Aksoy, B.; Selba¸s, R. Estimation of Wind Turbine Energy Production Value by Using Machine Learning Algorithms and
Development of Implementation Program. Energy Sources Part A Recover. Util. Environ. Eff. 2019, 43, 692–704. [CrossRef]

Energies 2023, 16, 3112

51.

52.
53.
54.

55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.

22 of 22

Bilal, B.; Ndongo, M.; Adjallah, K.H.; Sava, A.; Kébé, C.M.; Ndiaye, P.A.; Sambou, V. Wind turbine power output prediction
model design based on artificial neural networks and climatic spatiotemporal data. In Proceedings of the 2018 IEEE International
Conference on Industrial Technology, Buenos Aires, Argentina, 26–28 February 2020; pp. 1085–1092.
De Giorgi, M.G.; Ficarella, A.; Tarantino, M. Assessment of the benefits of numerical weather predictions in wind power
forecasting based on statistical methods. Energy 2011, 36, 3968–3978. [CrossRef]
Vaona, A. Spatial autocorrelation or model misspecification? The help from RESET and the curse of small samples. Lett. Spat.
Resour. Sci. 2009, 2, 53–59. [CrossRef]
Hothorn, T.; Zeileis, A.; Farebrother, R.W.; Cummins, C.; Millo, G.; Mitchell, D.; Zeileis, M.A. Package ‘lmtest’. Testing Linear
Regression Models. Available online: https://cran.r-project.org/web/packages/lmtest/lmtest.pdf (accessed on 3 February
2023).
Wooldridge, J.M. Introductory Econometrics: A Modern Approach; Cengage learning: Boston, MA, USA, 2015.
Demsetz, H. The Cost of Transacting. Q. J. Econ. 1968, 82, 33–53. [CrossRef]
Lobo, M.S.; Fazel, M.; Boyd, S. Portfolio optimization with linear and fixed transaction costs. Ann. Oper. Res. 2006, 152, 341–365.
[CrossRef]
Georgiev, K.; Kim, Y.S.; Stoyanov, S. Periodic portfolio revision with transaction costs. Math. Methods Oper. Res. (ZOR) 2015, 81,
337–359. [CrossRef]
Chen, F.Y.; Yano, C.A. Improving Supply Chain Performance and Managing Risk Under Weather-Related Demand Uncertainty.
Manag. Sci. 2010, 56, 1380–1397. [CrossRef]
Bunn, D.W.; Chen, D. The forward premium in electricity futures. J. Empir. Financ. 2013, 23, 173–186. [CrossRef]
Longstaff, F.A.; Wang, A.W. Electricity Forward Prices: A High-Frequency Empirical Analysis. J. Financ. 2004, 59, 1877–1900.
[CrossRef]
Gagliardini, P.; Ossola, E.; Scaillet, O. Time-Varying Risk Premium in Large Cross-Sectional Equity Data Sets. Econometrica 2016,
84, 985–1046. [CrossRef]
Woodard, J. Options and the Volatility Risk Premium; Pearson Education: London, UK, 2011.
Batchelor, R.A.; Alizadeh, A.H.; Visvikis, I.D. The relation between bid–ask spreads and price volatility in forward markets. Deriv.
Use Trading Regul. 2005, 11, 105–125. [CrossRef]
Bryant, H.L.; Haigh, M.S. Bid–ask spreads in commodity futures markets. Appl. Financ. Econ. 2004, 14, 923–936. [CrossRef]
Yoo, J.; Maddala, G.S. Risk premia and price volatility in futures markets. J. Futur. Mark. 1991, 11, 165–177. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.