Weather Derivatives for Quants

Weather derivatives transfer exposure to an objectively measured weather index rather than to physical damage. A utility worried about a warm winter can hedge heating demand with a Heating Degree Day (HDD) contract; a retailer may hedge a cold summer with Cooling Degree Days (CDD). The payoff is usually tied to observations at a named station over a fixed period, making these instruments a useful lesson in non-traded-underlying pricing.

The index is the contract

For daily average temperature \(T_d\) and base \(B\), common U.S. indexes are:

HDD = max(B - T_d, 0)
CDD = max(T_d - B, 0)
season index = sum of daily HDD or CDD

Contract choice	Quantitative implication
Station and source	Defines measurement and basis risk
Base temperature	Changes economic sensitivity
Month/season	Determines forecast uncertainty
Tick value	Converts index points to P&L
Cap/floor	Creates nonlinear payoff

The wording matters. A hedge on airport temperature may be poor protection for a regional load book affected by humidity, wind, and customer geography.

Price distributions, not a spot asset

There is no storage arbitrage connecting today’s temperature to a future temperature. Pricing therefore uses a physical probability distribution, adjusted for risk premium, capital, and market supply/demand. Start by removing seasonality and trend from daily temperature, model the residual process, simulate paths, calculate the index, then value the payoff.

import numpy as np

def hdd_index(simulated_temps, base=65.0):
    """simulated_temps: array shaped (paths, days)."""
    return np.maximum(base - simulated_temps, 0).sum(axis=1)

def capped_call_value(indexes, strike, tick_value, cap=None, discount=1.0):
    payoff = np.maximum(indexes - strike, 0) * tick_value
    if cap is not None:
        payoff = np.minimum(payoff, cap)
    return discount * payoff.mean()

This Monte Carlo calculation is only as credible as its temperature generator. Daily temperatures show strong seasonal means, autocorrelation, changing variance, geographic dependence, and occasional regime shifts. Fit on a rolling window, test forecast calibration by month, and avoid assuming normal residuals in tails.

Forecasts are information, not realized data

As the measurement period approaches, numerical weather predictions materially update the distribution. A practical model blends climatology, ensemble forecasts, and recent observations. Forecast ensembles are often underdispersed; calibrate them using historical forecast errors before using their raw spread as a risk estimate.

Horizon	Dominant input	Typical uncertainty
Seasonal	Climate trend and analog years	Very high
2–4 weeks	Ensemble and regime indicators	High
1–10 days	Numerical forecast ensemble	Moderate
Realized days	Official station readings	Measurement only

The trading signal is the difference between your calibrated fair value and the market price, after bid-ask, capital, and model-risk buffers—not a forecast that tomorrow is hot.

Basis risk and hedging

Weather risk is rarely isolated. A power utility’s P&L depends on temperature, electricity price, outages, fuel, customer behavior, and regulation. Estimate the business exposure directly:

P&L = a + b × HDD + c × CDD + d × power_price + error

Then size the derivative using the estimated \(b\) or \(c\), with conservative confidence intervals. This resembles beta hedging, except the hedge factor is non-tradable and nonlinear.

Risk controls

Station data revisions, missing observations, contract settlement rules, and over-the-counter counterparty exposure are operational risks. For an investment strategy, liquidity is usually the binding constraint: bespoke contracts can be hard to mark and impossible to exit. Stress historical extremes and synthetic tail scenarios, especially serial heatwaves or cold outbreaks that simple iid simulations miss.

Avoid treating a weather model as a broad commodity model. Temperature may affect natural gas demand, but the relationship changes with storage, supply shocks, and power generation. Use weather derivatives to hedge a defined weather exposure; use liquid markets separately for price risk.

Key takeaways

• Weather derivatives settle on a specified index, station, and observation period.
• Price them from calibrated physical distributions plus explicit risk-premium assumptions.
• Blend climatology and calibrated forecast ensembles as the season evolves.
• Basis risk between a station index and commercial P&L is usually the central risk.
• Model liquidity, data definitions, and tail dependence before relying on the hedge.