In Pursuit of Samuelson for Commodity Futures: How to Parameterize and Calibrate the Term Structure of Volatilities

The phenomenon of rising forward price volatility, both historical and implied, as maturity approaches, is referred to as the Samuelson} effect or maturity effect. Disregarding this effect leads to significant mispricing of early exercise options, extendible options, or other path-dependent options. The primary objective of the research is to identify a practical way to incorporate the Samuelson effect in the evaluation of commodity derivatives. We opt to represent the instantaneous variance and utilize the exponential decay parameterizations of the Samuelson effect. We develop efficient calibration techniques utilizing historical data from nearby futures and conduct analytics on statistical mistakes to establish a baseline for model performance. Utilizing 15 years of data for WTI, Brent, and NG, the study yields excellent results with the fitting error remaining well within the statistical error, except for the 2020 crisis period. We validate the stability of the fitted parameters by cross-validation approaches and assess the model's out-of-sample performance. The technique is applicable to seasonal commodities, including natural gas and electricity. We illustrate the application of the seasonal Samuelson model in extrapolating implied volatilities, which is crucial for the valuation and risk assessment of long-term contracts, such as widely utilized Power Purchase Agreements (PPAs)