Seasonal Stochastic Volatility and Correlation together with the Samuelson Effect in Commodity Futures Markets

TL;DR

This paper develops a multi-factor stochastic volatility model for commodity futures that incorporates seasonality and the Samuelson effect, providing analytical tools and numerical illustrations for volatility and correlation patterns.

Contribution

It introduces a novel multi-factor model with seasonality and Samuelson effect, deriving conditions for well-defined volatility factors and an analytical joint characteristic function.

Findings

Model produces volatility smiles and seasonal volatility term-structure.

Correlation exhibits seasonal patterns.

Numerical illustrations demonstrate model's ability to replicate observed market features.

Abstract

We introduce a multi-factor stochastic volatility model based on the CIR/Heston volatility process that incorporates seasonality and the Samuelson effect. First, we give conditions on the seasonal term under which the corresponding volatility factor is well-defined. These conditions appear to be rather mild. Second, we calculate the joint characteristic function of two futures prices for different maturities in the proposed model. This characteristic function is analytic. Finally, we provide numerical illustrations in terms of implied volatility and correlation produced by the proposed model with five different specifications of the seasonality pattern. The model is found to be able to produce volatility smiles at the same time as a volatility term-structure that exhibits the Samuelson effect with a seasonal component. Correlation, instantaneous or implied from calendar spread option prices via a Gaussian copula, is also found to be seasonal.