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 into the evaluation of commodity derivatives. We choose to model the instantaneous variance employing the exponential decay parameterizations of the Samuelson effect. We develop efficient calibration techniques utilizing historical futures data and conduct an analysis of statistical errors to provide a benchmark for model performance. The study employs 15 years of data for WTI, Brent, and NG, producing excellent results, with the fitting error consistently inside the statistical error, except for the 2020 crisis period. We assess the stability of the fitted parameters via cross-validation techniques and examine the model’s out-of-sample efficacy. The approach is generalized to encompass seasonal commodities, such as natural gas and electricity. We illustrate the application of the calibrated model of instantaneous variance for the evaluation of commodity derivatives, including swaptions, as well as in the evaluation of power purchase agreements (PPAs). We demonstrate a compelling application of the Samuelson effect to a widely utilized auto-callable equity derivative known as the snowball.