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@article{Andersen2003,

	abstract = {We provide a framework for integration of high–frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency return volatilities and return distributions. Building on the theory of continuous–time arbitrage–free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen/Dollar spot exchange rates, we find that forecasts from a simple long–memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal–normal mixture distribution produces well–calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation, and financial risk management applications.},

	author = {Torben G. Andersen and Tim Bollerslev and Francis X. Diebold and Paul Labys},

	file = {AndersenBollerslevDieboldLabys2003.pdf:AndersenBollerslevDieboldLabys2003.pdf:PDF},

	journal = {Econometrica},

	number = {2},

	owner = {dimpflth},

	pages = {529–626},

	timestamp = {2007.04.17},

	title = {Modeling and Forecasting Realized Volatility},

	volume = {71},

	year = {2003}

}

@article{Schneider12,

	author = {Stefan Schneider},

	doi = {10.21314/JEM.2011.079},

	journal = {Journal of Energy Markets},

	number = {4},

	pages = {77–102},

	title = {Power spot price models with negative prices},

	volume = {4},

	year = {2012}

}

@book{Cochrane2005,

	address = {Princeton and Oxford},

	author = {John H. Cochrane},

	edition = {Revised},

	isbn = {9780691121376},

	pagetotal = {552},

	ppn_gvk = {839022409},

	publisher = {Princeton University Press},

	subtitle = {(Revised Edition)},

	title = {Asset Pricing},

	year = {2005}

}

@article{AihountonH21,

	author = {Ghislain B D Aihounton and Arne Henningsen},

	doi = {10.1093/ectj/utaa032},

	journal = {The Econometrics Journal},

	number = {2},

	pages = {334–351},

	title = {Units of measurement and the inverse hyperbolic sine transformation},

	volume = {24},

	year = {2021}

}

@article{SimsSW90,

	author = {Christopher A. Sims and James H. Stock and Mark W. Watson},

	journal = {Econometrica},

	number = {1},

	pages = {113–144},

	title = {Inference in Linear Time Series Models with Some Unit Roots},

	volume = {58},

	year = {1990}

}

@comment{jabref-meta: databaseType:bibtex;}

@article{AndersenBDL03,

	abstract = {We provide a framework for integration of high–frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency return volatilities and return distributions. Building on the theory of continuous–time arbitrage–free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen/Dollar spot exchange rates, we find that forecasts from a simple long–memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal–normal mixture distribution produces well–calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation, and financial risk management applications.},

	author = {Torben G. Andersen and Tim Bollerslev and Francis X. Diebold and Paul Labys},

	file = {AndersenBollerslevDieboldLabys2003.pdf:AndersenBollerslevDieboldLabys2003.pdf:PDF},

	journal = {Econometrica},

	number = {2},

	owner = {dimpflth},

	pages = {529–626},

	timestamp = {2007.04.17},

	title = {Modeling and Forecasting Realized Volatility},

	volume = {71},

	year = {2003}

}

@article{GarmanKlass80,

	abstract = {Improved estimators of security price volatilities are formulated. These estimators employ data of the type commonly found in the financial pages of a newspaper: the high, low, opening, and closing prices and the transaction volume. The new estimators are seen to have relative efficiencies that are considerably higher than the standard estimators.},

	author = {Mark B. Garman and Michael J. Klass},

	file = {GarmanKlass1980.pdf:/home/dimpflth/research/literature/GarmanKlass1980.pdf:PDF},

	journal = {The Journal of Business},

	number = {1},

	owner = {dimpflth},

	pages = {67–78},

	timestamp = {2007.04.02},

	title = {On the Estimation of Security Price Volatilities from Historical Data},

	volume = {53},

	year = {1980}

}

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\begin{document}

\onehalfspacing

\vspace*{-0.5cm}\hspace*{-0.75cm}\includegraphics[width=6.5cm]{RZ_UniHohenheim_Logo_4C_Blau_EN.png}

\vspace*{0.5cm}

\textbf{\Large \textcolor{UHpurple}{Proposal for Capstone Project}}\\[6pt]

\textbf{\LARGE \textcolor{UHblau}{New ways to modelling power prices

}}\\[6pt]

The European Union's Third Energy Package, adopted in 2009, was a

significant step in further liberalizing the EU energy market. The

package set the foundation for more flexible electricity pricing, including the occurrence of

negative prices during periods of low demand and high renewable energy generation. Since then,

the number of days where negative prices occur has increased substantially which leads to serious

econometric problems when modelling these price time series.

This project aims to explore theoretical and/or econometric models to address the challenges posed

by negative prices in power markets.

The traditional way to transform nonstationary prices into stationary time series is to

calculate log-returns. This approach is  commonly used with stock market

data. However, it is impossible to use when prices are negative or zero leading to undefined

values as $\log(P_t)$ is only defined for $P_t>0$.

Nevertheless, the issue of modeling non-stationary price time series remains also for power prices.

The crucial question is how to transform these prices such that one obtains a stationary time

series. The calculation of arithmetic returns is also infeasible as prices can be zero.

Stationarity and ergodicity of a time series are fundamental to the application of all existing

statistical methods.

Simultaneously, the financial economic background that underlies the established methods relies

heavily on the calculation of returns. For example, the basic asset pricing equation and the

deduction of the stochastic discount factor \cite[see][]{Cochrane2005} can only

be modeled by transforming the price series to returns.

Furthermore, pure statistical solutions to the problem like \cite{Schneider12} neglect this

economic background. Instead of taking the first difference of the price time series,

\cite{Schneider12} suggests to transform the price time series with a hyperbolic-sine

transformation.

The resulting series resembles the approximate distribution of logarithmic returns, but the entire

interpretation as \emph{price changes} does not apply anymore. Furthermore, the units of

measurement (euro or cents) have a decisive impact on the distribution \citep{AihountonH21}.

Simple solutions that come to mind could be to add a constant to the price series so that

all prices are positive. However, the relative increase and decrease of the time series are

thereby distorted.

Another way would be to treat the zero bound as a reflecting boundary and only model

the absolute values. This again leaves out the possibility of a zero price and also distorts the

modeling. It is also plausible that the occurrence of negative prices changes the

price dynamics (different distribution, in particular different volatility).

A first attempt to classify days with positive and negative prices (see Figure

\ref{fig:vola}) shows that these are remarkably different in terms of volatility (as approximated

by squared price differences or the high-low range).

\begin{figure}[h!]

\centering

 \includegraphics[width=.95\linewidth]{descriptive_Germany.pdf}

 \caption{Power price volatility approximation}

 \label{fig:vola}

 \begin{minipage}{\linewidth}

   \footnotesize The figure contrasts days with negative prices (yellow) and days with positive

prices only (blue). Volatility is approximated as the squared difference of prices (akin to the

calculation of realized volatility) and the price range is the highest minus the lowest price on a

particular day. Calculations are based on data obtained from

\href{https://ember-climate.org/data-catalogue/european-wholesale-electricity-price-data/}{EMBER}.

 \end{minipage}

\end{figure}

\newpage

\textbf{\Large \textcolor{UHpurple}{Possible research questions}}\\[6pt]

In the proposed project, one could tackle various aspects that can advance the modelling of power

prices.

\begin{enumerate}

\item How should we model the rate of change if prices can be negative?

 \item Starting from the diffusion model in \cite{Schneider12}, is it possible to arrive at an

estimator of the volatility similar to the realized volatility measure as in \cite{AndersenBDL03}?

\item Can range-based estimators for the volatility \cite[e.g.][]{GarmanKlass80} be applied in the

modelling of power price volatility?

\end{enumerate}

\vfill

\begin{flushright}

\fboxsep14pt

\colorbox{UHblau!20}{\begin{minipage}{8cm}

  \textbf{\Large Contact:}\\[6pt]

  Prof.\ Dr.\ Thomas Dimpfl\\

  University of Hohenheim, Germany \\[6pt]

  thomas.dimpfl@uni-hohenheim.de

\end{minipage}}\end{flushright}

%

% we would like to investigate new modeling approaches that maintain the

% financial

% economic theory and simultaneously allow for the statistically sound modeling of the negative

% prices. The project's objectives are as follows.

%

% First, we want to propose a way to model power prices in line with finance theory that allows to

% accommodate negative prices. This may either mean to find a transformation that can approximate the

% calculation of arithmetic returns (just like the log-transformation with positive only prices) or

% an attempt to model the nonstationary series directly \cite[in the spirit of][]{SimsSW90}.

%

% Second, we want to investigate under which circumstances it is likely that prices become negative

% such that a prediction of these occurrences is facilitated. The study will focus on identifying the

% key drivers of negative prices, including supply-demand imbalances, renewable energy integration,

% and regulatory impacts.

%

% Third, we want to investigate how traditional volatility models (such as realized volatility models

% Andersen (2003)) that work well with financial data can be adjusted to incorporate the peculiarities

% of power markets. By constructing robust models that accurately reflect the occurrence and frequency

% of negative prices, the research will provide insights into market dynamics and risk management

% strategies.

%

%

% The outcomes of this project will contribute to better understanding and forecasting of price

% behaviors in power markets, aiding market participants in optimizing their trading strategies and

% improving the design of market regulation to improve the allocation mechanism to increase the

% penetration of renewable energy sources.

\newpage

\bibliography{literature}

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