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An Ambit Field Framework for the Full Panel of Day-ahead Electricity Prices

Thomas K. Kloster

TL;DR

The paper develops a tempo-spatial ambit-field framework for the full panel of European day-ahead electricity prices by indexing delivery periods on a cylinder, enabling a parsimonious yet rich dependence structure. It extends ambit-field theory to manifolds (the cylinder) with Lévy bases and Walsh integration, derives cumulant/moment formulas, and provides structure-preserving pricing for futures, spreads, and leverage, including a semi-parametric kernel estimation approach evaluated on German data. A practical simulation scheme is proposed to generate tempo-spatial price paths, and within-day spreads are introduced as novel derivatives that leverage the cross-sectional dependence across delivery periods. The framework supports pricing and hedging of highly granular electricity products and offers a path toward robust estimation and risk management in markets driven by volatile renewables.

Abstract

This paper considers the often overlooked fact that electricity spot prices in individual European generation zones evolve as a high dimensional panel structure. A general continuous time framework is developed by formulating the panel as an ambit field indexed by a cylinder surface, where the cross sectional dimension is represented by a circle. This requires a treatment of ambit fields on manifolds, but the departure from Euclidean space allows for embedding intrinsic dependence structures into the index set in a flexible and parameter-free way, where the daily delivery periods have a canonical mapping onto the circle. The model is a natural space-time extension of volatility modulated Lévy-driven Volterra processes, which have previously been studied in the context of energy markets, and the pricing of electricity derivatives turns out to be essentially as analytically tractable as in the null-spatial setting. The space-time framework extends the scope of possible derivatives to products written on individual delivery periods, where spreads between these constitute an interesting example. We establish useful formulas for the pricing of various derivatives along with a simulation scheme, and study specifications of the dependence structure in detail.

An Ambit Field Framework for the Full Panel of Day-ahead Electricity Prices

TL;DR

The paper develops a tempo-spatial ambit-field framework for the full panel of European day-ahead electricity prices by indexing delivery periods on a cylinder, enabling a parsimonious yet rich dependence structure. It extends ambit-field theory to manifolds (the cylinder) with Lévy bases and Walsh integration, derives cumulant/moment formulas, and provides structure-preserving pricing for futures, spreads, and leverage, including a semi-parametric kernel estimation approach evaluated on German data. A practical simulation scheme is proposed to generate tempo-spatial price paths, and within-day spreads are introduced as novel derivatives that leverage the cross-sectional dependence across delivery periods. The framework supports pricing and hedging of highly granular electricity products and offers a path toward robust estimation and risk management in markets driven by volatile renewables.

Abstract

This paper considers the often overlooked fact that electricity spot prices in individual European generation zones evolve as a high dimensional panel structure. A general continuous time framework is developed by formulating the panel as an ambit field indexed by a cylinder surface, where the cross sectional dimension is represented by a circle. This requires a treatment of ambit fields on manifolds, but the departure from Euclidean space allows for embedding intrinsic dependence structures into the index set in a flexible and parameter-free way, where the daily delivery periods have a canonical mapping onto the circle. The model is a natural space-time extension of volatility modulated Lévy-driven Volterra processes, which have previously been studied in the context of energy markets, and the pricing of electricity derivatives turns out to be essentially as analytically tractable as in the null-spatial setting. The space-time framework extends the scope of possible derivatives to products written on individual delivery periods, where spreads between these constitute an interesting example. We establish useful formulas for the pricing of various derivatives along with a simulation scheme, and study specifications of the dependence structure in detail.

Paper Structure

This paper contains 33 sections, 13 theorems, 107 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. A random field interpretation for the panel of prices
  3. Ambit fields on manifolds
  4. Lévy bases on manifolds
  5. Ambit fields
  6. Model cumulant function and moments
  7. Electricity derivatives
  8. Structure preserving change of measure
  9. Electricity futures
  10. Within-day spreads
  11. Incorporating leverage
  12. Specifying the kernel function
  13. A semi-parametric approximation scheme
  14. An empirical toy study of the kernel function
  15. A simulation scheme
  16. ...and 18 more sections

Key Result

Proposition 3.1

Let $M$ be a Riemannian manifold and let $L$ be a Lévy basis on $(M,\mathcal{M})$ with characteristic quadruplet $(\gamma ,\Sigma , \nu, c)$. Let also $G$ be a group of operations from $M$ onto $M$ with function composition being the group operation. Then $L$ is stationary with respect to $G$ if and

Figures (8)

  • Figure 1: Correlation matrix of hourly spot prices on the German market from October 1st 2018 to October 1st 2024 (left), and regression of spot price during the first delivery period on the last spot price of the preceding day (right). Prices are shown in Euros (EUR) per megawatt hour (MWh).
  • Figure 2: Visualizations of the index set $\mathcal{C}$ (left), and of $H=24$ observations along slices of $\mathcal{C}$ represented by blue dots (right).
  • Figure 3: Fitted isotropic kernel function (left) and fitted spectral density at mode $n=0$ against the empirical counterpart (right).
  • Figure 4: Generic realizations of simulated Gaussian (left) and NIG (right) ambit fields without stochastic volatility.
  • Figure 5: Price of within-day spread as computed via Monte Carlo simulation, with the increasing sample size on the horizontal axis. The Gaussian model is represented as a blue line, and the NIG model is represented as a red line. Corresponding $95\%$ confidence intervals are indicated as dotted lines.
  • ...and 3 more figures

Theorems & Definitions (35)

  • Remark 2.1
  • Definition 3.1: Lévy basis
  • Definition 3.2: Stationary Lévy basis
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Example 3.1: Gaussian Lévy basis
  • Example 3.2: Normal inverse Gaussian Lévy basis
  • Remark 3.1
  • Definition 3.3: Ambit set
  • ...and 25 more