Forecasting of volatility and risk premia in electricity markets

Technical Summary: Forecasting of Volatility and Risk Premia in Electricity Markets

Problem Statement
The paper addresses the challenge of forecasting the realized covariation (RCV) of electricity spot prices in European generation zones. Unlike traditional equity or commodity markets, electricity markets exhibit unique characteristics: prices are determined daily via auctions for specific delivery periods, representing averages of a latent, infinite-dimensional "true" price process. This structure introduces specific biases in volatility estimation, including drift effects and mean-reversion/propagation effects. Furthermore, existing literature on volatility estimation in electricity markets is sparse. The authors aim to determine if the RCV—a matrix-valued representation of the latent covariance operator—is predictable over short horizons and whether such forecasts can improve the prediction of risk premia in forward markets.

Methodology
The study builds upon the abstract modeling framework established in Kloster and Benth (2026), where the latent price process $X_t(\cdot)$ evolves in a Hilbert space $L^2(S^1)$ (the unit circle representing the day). The observed spot prices are integrals of this latent process over specific delivery periods.

1. Estimation of Realized Covariation (RCV):

   • The authors first de-trend the observed price series to isolate a stationary component $Y_n$.
   • They estimate a semigroup matrix $\hat{S}_{\delta, N}$ to account for mean-reversion dynamics.
   • Residuals are computed by adjusting for this semigroup, and the annualized RCV ($RCV^w_n$) is constructed as the sum of outer products of these residuals over a window $w$. This estimator serves as a noisy proxy for the "effective variance" of the price innovations, acknowledging that the pure integrated variance operator cannot be fully identified due to the smoothing effects of the mean-reversion mechanism.

2. Forecasting Framework (Matrix-HAR):

   • The forecasting target is the one-week-ahead RCV matrix.
   • The authors propose a parsimonious Matrix-Heterogeneous Autoregressive (HAR) model, inspired by Quiroz et al. (2024) and the classical HAR framework of Corsi (2009).
   • Decomposition: The RCV matrix is decomposed via $RCV^w_t = D^w_t R^w_t D^w_t$, where $D^w_t$ contains the marginal realized volatilities (diagonal) and $R^w_t$ contains the conditional cross-sectional correlations.
   • Modeling $D^w_t$: A log-linear HAR model is applied to the diagonal elements, incorporating daily, weekly, monthly, and quarterly (91-day) horizons. The quarterly component is included to capture long-range dependencies and seasonality specific to electricity markets.
   • Modeling $R^w_t$: A scalar HAR model is applied to the half-vectorization of the correlation matrix, propagating lagged correlations uniformly.
   • Extensions: The baseline model is extended with covariates including:
     • Current price levels (linear and quadratic).
     • Renewable generation shares (wind and solar).
     • Time-varying spillover effects (regime-dependent variables).
   • Estimation: Models are estimated using a rolling window approach on German spot price data (October 2018 – October 2025).

3. Risk Premia Application:

   • The study investigates the prediction of spread risk premia between peak-load and baseload forward contracts.
   • Following the structural model of Bessembinder and Lemmon (2002), the risk premium is a function of the conditional variance and skewness of the spot price.
   • The authors use the forecasted RCV to proxy the conditional variance of the spread (peak minus baseload), comparing this against standard backward-looking rolling variance estimates.

Key Results

• Predictability: The realized covariation is predictable over short horizons. Simple matrix-HAR models effectively capture the conditional second-order structure of electricity prices.
• Model Performance:
  • The inclusion of a quarterly horizon significantly improves forecasts compared to models with only daily, weekly, and monthly components.
  • Models incorporating renewable generation information (wind/solar shares) and quadratic price controls dominate the performance metrics.
  • The best-performing models (e.g., DRD-HAR-P2+ren+tvSpill) achieve a Mincer-Zarnowitz $R^2$ of 40–45% for hourly realized variances, indicating that nearly half the variation is explained by the forecasts.
  • The models are well-calibrated to the cross-sectional dependence structure (off-diagonal elements), as evidenced by variance-targeting exercises on net-zero portfolios (e.g., peak vs. off-peak, solar shape).
  • A known limitation is that the models tend to over-predict realized volatility on "calm" days due to the heavy right-skew of the conditional variance distribution.
• Risk Premia Prediction:
  • Using the forecasted RCV as a variance proxy in the risk premium regression significantly outperforms standard backward-looking methods.
  • The $R^2$ statistic for predicting spread risk premia increases from 24.4% (using rolling variance) to 47.2% (using the DRD-HAR-P2 RCV forecast).
  • The coefficient for the variance term remains highly significant, while the skewness term (approximated by backward-looking skewness) is not statistically significant in the tested specifications.

Significance and Claims
The paper claims that while the realized covariation arises from abstract infinite-dimensional considerations, it carries highly useful, actionable information for market participants. The primary contributions are twofold:

1. Demonstrating Predictability: Showing that the RCV of electricity prices is predictable over short horizons using simple, matrix-HAR inspired models that incorporate long-term horizons and renewable generation data.
2. Improving Risk Premium Forecasts: Proving that forecasts of the realized covariation substantially improve the prediction of spread risk premia in electricity forward markets compared to standard methods relying solely on backward-looking volatility.

The authors conclude that these findings validate the utility of the RCV framework for practical applications such as hedging ratios, confidence bands for price forecasts, and risk management in electricity markets, despite the structural challenges of right-skewed volatility.