German wind lost 15.1 EUR/MWh against the last pre-delivery baseload future in 2025, and futures bias explains only 1.0. Solar lost 26.0 EUR/MWh on the same comparison, with another 1.0 attributable to the baseload premium. Across the twelve complete 2025 delivery months, the remaining capture discount averaged 14.05 EUR/MWh for wind and 24.94 for solar. Those figures come straight from observed EEX settlements and realized hourly prices. They are the most useful result in Chatziandreou and Karbach's paper, independent of whether the model convinces you.
The decomposition comes first
Under a pay-as-produced PPA, the offtaker receives realized generation at a fixed strike. Its payoff integrates q(S-K) over delivery. Conditioning yields E[qS] = E[q]E[S] + Cov(q,S). The fair strike therefore equals the production-weighted expected spot price.
Subtract the flat baseload forward and two pieces remain. The deterministic profile term measures how expected output aligns with the hourly forward curve. The stochastic covariance term is the pricing expression of cannibalisation. This split holds without a model. The rest of the paper depends on one.
For the empirical work, the authors fit a bivariate Lévy-driven MCARMA state-space model, meaning a continuous-time ARMA system expressed in state-space form. Lower-triangular coefficient blocks allow wind shocks to propagate into prices while preventing price shocks from feeding back dynamically. The paper is precise about the restriction: the triangular structure removes dynamic feedback only. Contemporaneous dependence stays through the off-diagonal of the driver covariance. A multivariate normal inverse Gaussian law, recovered from the fitted state, supplies the driver. The specification also includes a mean-reverting spike component whose negative-jump intensity activates with renewable penetration.
The dataset contains 43,819 hourly observations from January 2021 to December 2025. It combines DE-LU day-ahead prices and DE-LU actual total load from ENTSO-E with the ENWEX Germany wind and solar utilisation indices. The indices are nationwide weather-based measures. The paper instructs readers to treat them as country-level capacity utilisation, rather than metered plant output. Estimation uses 2023-2024 only. The contract delivers a 50 MW notional through the twelve months of 2025 at zero rates.
Hedging proceeds through a two-stage variance-optimal projection. The first stage projects the payoff onto gains from the active monthly futures stack, allowing each contract to enter and leave the tradable set on its own dates. The orthogonal residual then goes onto a fixed catalog of 360 auxiliary claims (356 for solar), each held from inception. Four families fill that catalog: power vanillas, renewable-index vanillas, orthant quantos on centred price and centred capacity factor, and capture-spread options on monthly baseload minus achieved price.
Two families lack listed products. Section 4.5 acknowledges this before presenting results. It opens with "Two qualifications are essential", then describes power-renewable orthant quantos and capture-spread options as instruments that "should be interpreted as potential OTC hedging products that would need to be issued bilaterally". Across 3,000 held-out simulated paths, the multi-month futures hedge reduces payoff standard deviation by 36.4% for wind and 64.8% for solar. The static overlay raises those reductions to 87.3% and 96.3%, without premiums, spreads or margin.
We did not reproduce any of it. Our data covers US equities, ETFs, crypto and listed options. It contains no German hourly day-ahead prices, no EEX delivery-period power futures and no renewable utilisation index. A delivery-period average-price future has no substitute in that universe. The discussion below therefore rests on the authors' numbers.
Wind covariance, solar shape
The realized monthly panel assigns wind an out-of-sample covariance term of -14.34 EUR/MWh and a profile term of +0.29. Solar shows the opposite pattern: -3.37 from covariance and -21.57 from profile. Prior work in this line prices energy quantos and hedges joint price-volume exposure through standard power options and copula models. This paper applies one set of definitions to both technologies, producing a clean separation.
For wind, the profile term remains economically zero in every window, never exceeding 0.5 EUR/MWh in absolute value. Covariance carries essentially the entire discount, at -16.69 across 2023-2024 and -14.34 in 2025. Solar reverses the composition. Its covariance term ranges from -2.2 to -3.4, while the profile term worsens from -6.4 (2021-2022) to -12.3 (2023-2024), then -21.6 in 2025. Model strikes tell the same story. Wind prices at 80.01 EUR/MWh against an expected discounted baseload of 88.90; solar prices at 63.89 against 88.46. The volume-capture covariance correction is 0.001 EUR/MWh.
Solar's 24.57 EUR/MWh model discount is overwhelmingly about midday.
The solar profile term has one listed proxy. On the monthly panel, it moves with the traded peak-base spread at +0.49 correlation and a slope of +0.35 per EUR/MWh of spread. Wind's profile term loads at +0.03. Yet the simulated hedge uses the peak block as one buy-and-hold volume for the full year, improving solar standard-deviation reduction only from 41.3% to 43.5%. A monthly loading and a twelve-month static position are different exposures. The +0.49 correlation and +0.35 slope make the monthly loading the more tradable one.
What does the futures stack contribute?
One contract supplies nearly the entire futures reduction. For wind, the front-month dynamic hedge reaches 36.0% (230.2 kEUR). A variance-optimal front-month position reaches 36.4% (228.6 kEUR), while the complete six-maturity stack stays at 36.4%. The joint model adds nothing on solar's futures leg. Selling expected model volume forward delivers 64.6%, compared with 64.5% for the variance-optimal position. Wind is the case where covariance adjustment matters, adding about four points from 32.2% to 36.4%.
The positions themselves diverge sharply.
February's variance-optimal front-month position is 7.99 MW, less than the naive volume delta of 16.10 MW. Across wind months, their ratio ranges from 0.50 to 1.34; solar spans 0.95 to 1.36. A hindsight delta based on realized volumes performs worse than either, reaching 30.4%. The authors interpret this as evidence that covariance, rather than volume forecasting, drives the correction. Daily rebalancing through delivery fares worse again for solar, producing 56.2% versus 64.8%. Estimation noise in the intramonth positions is their proposed explanation. The solar daily estimates also use heavier regularization than the monthly roll.
Every percentage here carries the same caveat. Futures marks are model-implied conditional projections of simulated settlements, rather than observed EEX prices. The paper describes the dynamic strategy as "a model-implied benchmark rather than a directly implementable market hedge".
A market that does not exist
Claims unavailable on European exchanges generate the leap to 87.3% and 96.3%. The paper prices them from training-sample Monte Carlo expectations under its own model. Its conclusion calls the figures "within-model results, not an independent market validation" and says they "likely overstate feasible risk reduction once executable premiums, bid-ask spreads, margin requirements and transaction costs are included". The authors defend the exercise as a frictionless benchmark intended to rank claim types. Judged on that basis, it produces a clear ordering.
Used alone and without constraints, capture spreads remove 45.6% of the post-futures wind residual with 23 claims. For solar, 25 claims remove 66.9%. Orthant quantos have the sign theorem behind them and achieve a 100% weighted sign match in every wind month. Alone, they remove 12.3% for wind and 1.1% for solar. Power vanillas manage 3.6% and 0.1%. In the wind convexity regression, they reduce payoff dispersion by 15.3% while removing only 4.4% of the nonlinear residual. Much of their apparent contribution therefore repurchases futures exposure.
The sparse tables show the version an offtaker might plausibly request from a bank. Long-only selection, with no short options, gives wind a 66.8% total standard-deviation reduction using 31 claims and gross weight 27.83. Solar reaches 78.9% using 16 claims and gross weight 27.10. The unconstrained sparse books rise to 85.1% and 93.8%. Gross weights are 72.69 and 74.38, spread across 75 claims for wind and 77 for solar, against a 50 MW contract. Every figure comes from the paper's own tables.
Calibrated on 2023-2024, missing 2025
From March to June 2025, realized solar capture discounts reached 40.9 to 44.8 EUR/MWh. Model-implied expectations were roughly 16 to 23. May produced a 22.60 EUR/MWh capture price against 67.34 baseload. The authors place realized 2025 summer value factor in the adverse tail of the distribution fitted on 2023-2024. At the model strikes, realized PPA value was -735.7 kEUR for solar and -460.6 kEUR for wind.
Anyone setting a strike with this framework should dwell on those figures. Two years of hourly calibration priced solar at 63.89, and realized delivery came in far below.
The raw data reveals the mechanism. In 2025, solar's share of MWh in the bottom price quartile exceeded its demand share by 36.2 percentage points, versus 1.8 in 2022. Strictly negative prices accounted for 24.5% of solar MWh.
Seasonality relies on several hand-set constants, among them a 0.98 clear-sky quantile and a 15-day bandwidth. They are fixed using 2023-2024 and carried forward into 2025, which preserves the correct direction of evaluation. Unlike the hindsight-chosen radius discussed in an earlier note on Wasserstein-ball allocation, none is tuned on the evaluation window. Sparse cardinality is selected on validation paths. Each reported statistic is then calculated once on the test paths.
What survives the model
The realized decomposition can be built in a spreadsheet from EEX settlements and hourly ENTSO-E prices. For a German buyer, it separates the quoted discount into intraday shape, which peak-base structures can address on the solar side, and covariance, which fixed-volume instruments leave behind. If a bilateral capture hedge is available, the ranking favors a capture spread on baseload minus achieved price. The quanto book works better as part of a combination trade.
The paper does not supply an executable price. Across 2019-2024, the quoted German PPA index stood 52.7 EUR/MWh below front-month futures and 61.0 below front-year futures. The authors attribute those gaps to delivery tenor, production profile, price-volume covariance exposure, contract structure, and credit and liquidity conditions. Values in the simulated hedge are training-sample expectations, carrying neither a bid nor a credit charge.
A dealer quote for a monthly capture spread would change my mind. Until one appears, the 96.3% remains a description of the model.