Ornstein-Uhlenbeck Temperature Model
The stochastic temperature model used for derivative pricing — mean-reverting process with seasonal components.
Cliff Horizon's derivative pricing module uses the Alaton-Djehiche-Stillberger (2002) Ornstein-Uhlenbeck temperature model — the standard framework for weather derivative pricing.
Why O-U for Temperature?
Temperature is a mean-reverting process. If today is unusually hot, tomorrow is likely to be closer to the seasonal average — not even hotter. This mean-reversion property makes the Ornstein-Uhlenbeck process a natural fit, unlike geometric Brownian motion (used for stock prices) which has no such tendency.
Black-Scholes is inapplicable to weather derivatives because weather is not a tradeable asset. There is no risk-neutral hedging argument. Derivative pricing must rely on actuarial/statistical methods rather than no-arbitrage pricing theory.
The Model
The temperature process follows this stochastic differential equation:
dT_t = { dT_m/dt + a(T_m - T_t) } dt + σ_t dW_t
Where:
| Symbol | Meaning |
|---|---|
| T_t | Temperature at time t |
| T_m | Deterministic seasonal mean: T_m = A + Bt + C·sin(ωt + φ) |
| a | Mean-reversion speed (how quickly temperature returns to seasonal mean) |
| σ_t | Piecewise-constant monthly volatility |
| W_t | Standard Brownian motion |
Parameters
Each location requires estimation of:
| Parameter | Description | Estimation Method |
|---|---|---|
| A | Baseline temperature level | Least squares on historical daily means |
| B | Warming trend (°C per year) | Least squares |
| C | Seasonal amplitude | Least squares |
| φ | Phase shift (timing of seasonal peak) | Least squares |
| a | Mean-reversion speed | Martingale estimation (Bibby & Sørensen method) |
| σ₁–σ₁₂ | Monthly volatility | Quadratic variation or regression residuals |
Reference values (Stockholm, from Alaton et al.):
- â ≈ 0.237 (≈24% daily mean-reversion)
- σ_January ≈ 3.4°C, σ_July ≈ 1.7°C (winter volatility roughly double summer)
These parameters must be estimated for each target location using a minimum of 5 years of historical daily temperature data.
Market Price of Risk (λ)
Since temperature is non-tradeable (incomplete market), unique derivative prices require specifying a market price of risk λ. Under the risk-neutral measure Q:
E^Q[T_t] = E^P[T_t] - (λσ/a)(1 - e^{-a(t-s)})
Design Decision
Cliff Horizon does not need to solve for λ independently. The engine's responsibility is producing well-calibrated P(event). Ensuro's capital pool economics determine the effective risk price:
- Engine provides P(event) → the
lossProbparameter - Ensuro applies:
purePremium = payout × lossProb × MoC - Full premium adds:
jrCoc + srCoc + ensuroCommission + partnerCommission - Ensuro's LPs decide whether the return compensates for the risk
λ is set by the people bearing the risk (Ensuro's capital providers), not by the pricing engine. The engine's edge is calibration accuracy — the better the P(event), the tighter the uncertainty loading Ensuro needs, and the more competitive the premium.
Derivative Pricing
Semi-closed-form pricing formulae exist for HDD/CDD calls and puts. Monte Carlo validation shows ~2% pricing accuracy — sufficient for real-time dashboard pricing.
Cliff Horizon's enhancement over the base model: Alaton et al. note (Section 4.3.2) that near the contract period, meteorological forecasts should be incorporated by adjusting model parameters. The engine does this systematically — updating A, σ, and C with forward-looking NWP ensemble data rather than relying solely on historical calibration.
This forecast-adjusted O-U model is the hybrid approach that generates pricing alpha over pure burn analysis.
Burn Analysis Fallback
For simpler pricing (or as a cross-check), the engine also implements burn analysis: applying the derivative's payout function to historical weather data to calculate what the contract would have paid over past periods.
Burn analysis is transparent and intuitive, but limited: it assumes past weather patterns predict future weather — increasingly questionable under climate change. The O-U model captures dynamics (mean-reversion, trend, seasonality) that pure historical lookback misses.