Power Prices: GBM vs Mean-Reversion (OU) for Valuation & Risk

Risk Models 12 min read Technical Guide

Executive Summary

Power prices exhibit both trending and mean-reverting behavior, making model selection critical for accurate valuation and risk assessment. Geometric Brownian Motion (GBM) captures trending behavior but ignores mean reversion. Ornstein-Uhlenbeck (OU) models mean reversion but can underestimate long-term uncertainty.

Bottom line: Use GBM for long-term investments with structural price trends. Use OU for short-term positions and markets with strong fundamental anchors. For complex projects, combine both through regime-switching or forward curve integration.

Table of Contents

Understanding Power Price Dynamics

Power prices exhibit unique characteristics that challenge traditional financial modeling approaches. Unlike equity prices, electricity prices display:

The choice between GBM and mean-reversion models fundamentally affects investment valuation, risk assessment, and hedging strategies. Each model captures different aspects of price behavior, leading to dramatically different results for long-term energy investments.

Why Model Choice Matters

A renewable energy project valued with GBM might show an NPV of €50M, while the same project valued with OU mean-reversion shows €35M. The difference comes from how each model projects long-term price evolution and volatility decay.

Geometric Brownian Motion (GBM) for Power Prices

GBM assumes power prices follow a log-normal random walk with constant drift and volatility:

dS = μS dt + σS dW

Where:

GBM Characteristics

Price Evolution: Prices can reach any level over time with no upper or lower bounds. Volatility remains constant regardless of price level.

Long-term Behavior: Expected future price grows exponentially at rate μ. Price uncertainty increases with the square root of time.

Mathematical Properties: Log prices are normally distributed. Prices cannot go negative. Volatility scales with price level.

GBM Advantages for Power Modeling

  • Captures trending behavior from fuel cost inflation, carbon pricing
  • Simple calibration from historical price data
  • Analytical solutions for many option types
  • Market standard for equity-like assets
  • Handles structural shifts in price regimes

GBM Limitations for Power Modeling

  • Ignores mean reversion from supply-demand fundamentals
  • Overestimates long-term volatility for mean-reverting prices
  • No price spikes or capacity-constrained behavior
  • Constant volatility doesn't reflect market conditions
  • Poor short-term fit for fundamentally-driven prices

Ornstein-Uhlenbeck (OU) Mean-Reversion Model

The OU model assumes power prices mean-revert to a long-term level with speed of reversion α:

dS = α(θ - S) dt + σ dW

Where:

OU Model Characteristics

Mean Reversion: Prices are "pulled back" toward the long-term mean θ. Deviations from θ decay exponentially at rate α.

Long-term Behavior: Price uncertainty approaches a steady-state level σ²/(2α). Long-term forecasts converge to θ regardless of starting price.

Mathematical Properties: Prices are normally distributed at all times. Can theoretically go negative (limitation for power prices).

OU Advantages for Power Modeling

  • Captures mean reversion from supply-demand fundamentals
  • Realistic volatility decay for long-term projections
  • Economic intuition matches fundamental analysis
  • Bounded long-term uncertainty around fundamental value
  • Fast calibration from price time series

OU Limitations for Power Modeling

  • Can produce negative prices (unrealistic for power)
  • Constant mean level doesn't capture structural trends
  • No jump/spike behavior during scarcity events
  • Linear mean reversion may be too simplistic
  • Underestimates tail risk in constrained markets

Head-to-Head Model Comparison

Characteristic GBM Ornstein-Uhlenbeck
Long-term trend Exponential growth/decline at rate μ Converges to constant level θ
Volatility over time Increases with √t (no bound) Approaches steady state σ²/(2α)
Mean reversion None Exponential decay with speed α
Price bounds Always positive, no upper bound Can go negative, unbounded
Fundamental anchor None (pure momentum) Long-term level θ
Volatility scaling Scales with price level (σS) Constant absolute volatility (σ)
Option values Higher for long-dated options Lower due to volatility decay
Risk measures VaR increases with time horizon VaR bounded at long horizons

Real Market Examples

German Power (Day-Ahead): Shows strong mean reversion (α ≈ 2-4) around fuel cost plus carbon price. GBM overestimates long-term volatility by 2-3x.

ERCOT Real-Time: Exhibits both trending (renewable penetration) and mean-reverting behavior. Pure OU underestimates extreme price events.

Nordic Hydro Markets: Strong seasonal mean reversion around reservoir levels, but long-term climate trends favor GBM components.

Calibration from Historical Data

GBM Calibration

For GBM, estimate drift μ and volatility σ from log price returns:

μ = mean(log(St+1/St)) × 252
σ = std(log(St+1/St)) × √252

Data Requirements: Daily price series, minimum 2-3 years for stable estimates. Remove obvious outliers and market disruptions.

Parameter Stability: Test for regime changes using rolling windows. Structural breaks often require separate calibration periods.

OU Model Calibration

OU parameters can be estimated using maximum likelihood or regression methods:

ΔS = α(θ - S) + ε
α = -log(correlation(St, St+1))
θ = mean(S) in steady state

Long-term Mean (θ): Can be set to fundamental value (fuel costs + carbon + margin) or estimated as sample mean of detrended prices.

Mean Reversion Speed (α): Typical values range from 0.5 (slow reversion, 2-year half-life) to 5.0 (fast reversion, 2-month half-life).

Calibration Best Practices

  • Use appropriate frequency: Daily data for short-term modeling, monthly for long-term trends
  • Handle seasonality: Detrend or use seasonal adjustment before calibration
  • Test stability: Check parameters across different time periods
  • Market regime awareness: Recalibrate after major market structure changes
  • Forward-looking adjustment: Consider structural changes not in historical data

Impact on Option Values

Model choice dramatically affects option valuations, especially for long-dated instruments:

Call Options (Upside Exposure)

GBM: Option values increase with maturity due to growing volatility. Out-of-the-money calls retain significant value even at long horizons.

OU: Option values peak at intermediate maturities, then decline as mean reversion dominates. Very long-dated options approach zero value if current price equals long-term mean.

Put Options (Downside Protection)

GBM: Put values increase with time, providing growing downside protection. Deep out-of-the-money puts can become valuable at long horizons.

OU: Put values decay for long-dated options if underlying price is below long-term mean. Mean reversion reduces downside risk over time.

Real Options Example

Project Context: Option to build 100MW wind farm, current power price €50/MWh, strike price €45/MWh, 5-year decision window.

Model Parameters Option Value Key Driver
GBM μ=3%, σ=40% €15M Growing volatility over 5 years
OU α=2, θ=€48, σ=€15 €8M Mean reversion caps upside

Interpretation: GBM values flexibility higher due to unbounded upside potential. OU recognizes that extreme prices are temporary, reducing option value.

Risk Measurement Implications

Value at Risk (VaR) Calculations

Short-term VaR (1-day to 1-month): Both models give similar results. Historical volatility dominates the calculation.

Long-term VaR (1-year+): Dramatic differences emerge. GBM VaR grows with √t, while OU VaR approaches steady-state bound.

VaR_GBM(T) = |μT - 1.645σ√T| × Current Price
VaR_OU(T) ≈ 1.645σ/√(2α) × √(1-e^(-2αT))

Stress Testing Scenarios

GBM Stress Tests: Focus on trending scenarios (persistent fuel price increases, carbon tax escalation). Model naturally generates extreme price paths.

OU Stress Tests: Emphasize shocks to long-term mean (stranded assets, demand destruction). Model bounds extreme scenarios through mean reversion.

Risk Management Implications

Portfolio Hedging: GBM suggests buying long-term options for tail protection. OU suggests focusing on short-term hedges and fundamental analysis.

Capital Allocation: GBM supports flexible, scalable projects with embedded options. OU favors projects with strong fundamental value.

Forward Curve Integration

Modern power modeling often combines stochastic processes with forward curve information:

Forward-Adjusted GBM

Replace constant drift μ with time-varying forward curve slope:

dS = [∂ln(F(t,T))/∂T] S dt + σS dW

This approach preserves GBM's volatility structure while matching market forward prices exactly.

Forward-Adjusted OU

Allow time-varying long-term mean θ(t) based on forward curve:

dS = α(θ(t) - S) dt + σ dW

This captures both mean reversion and forward market expectations of fundamental value evolution.

Forward Curve Calibration

Liquid Markets: Use exchange-traded futures and forwards directly. Calibrate volatility to at-the-money option prices.

Illiquid Markets: Construct synthetic curves from fuel costs, carbon prices, and capacity factors. Apply risk premiums based on market liquidity.

Practical Model Selection Guide

When to Use GBM

When to Use OU Mean-Reversion

Hybrid Approaches

Regime-Switching Models: Use OU during normal periods, GBM during crisis periods. Markov chains govern regime transitions.

Multi-Factor Models: Separate short-term (OU) and long-term (GBM) components. Allows independent calibration of different time scales.

Jump-Diffusion Extensions: Add price spikes to either model using Poisson jumps. Critical for markets with capacity constraints.

Model Selection Framework

Step 1: Analyze price data for trending vs. mean-reverting behavior using ADF tests and ACF analysis.

Step 2: Consider investment horizon - longer horizons often favor GBM, shorter horizons favor OU.

Step 3: Assess market maturity - mature markets with stable fundamentals suit OU better.

Step 4: Test sensitivity - run valuations under both models to understand the range of outcomes.

Model Power Prices with Confidence

CapexEdge supports both GBM and OU modeling with automated calibration from historical data. Compare option values, risk metrics, and forward curve integration in one platform.

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Conclusion

The choice between GBM and mean-reversion models for power prices isn't just academic—it fundamentally affects investment decisions, risk assessments, and hedging strategies.

For energy finance professionals:

The key insight is that power prices exhibit both behaviors at different time scales and market conditions. Successful modeling requires understanding when each mechanism dominates and choosing tools that match the specific investment context.

As power markets continue evolving with renewable penetration and carbon pricing, model selection becomes even more critical. The ability to switch between modeling approaches—and understand their implications—is essential for navigating energy transition investments.