Time-Changed Ornstein-Uhlenbeck Processes And Their Applications In Commodity Derivative Models

TL;DR

This paper introduces subordinate Ornstein-Uhlenbeck processes with mean-reverting jumps, develops their mathematical properties, and applies them to create flexible commodity derivative models that capture observed market features.

Contribution

It constructs and analyzes subordinate OU processes with jumps, and proposes a new commodity modeling framework incorporating stochastic volatility and seasonality.

Findings

Models fit initial futures curves accurately

Capture Samuelson's maturity effect

Reproduce implied volatility smiles

Abstract

This paper studies subordinate Ornstein-Uhlenbeck (OU) processes, i.e., OU diffusions time changed by L\'{e}vy subordinators. We construct their sample path decomposition, show that they possess mean-reverting jumps, study their equivalent measure transformations, and the spectral representation of their transition semigroups in terms of Hermite expansions. As an application, we propose a new class of commodity models with mean-reverting jumps based on subordinate OU process. Further time changing by the integral of a CIR process plus a deterministic function of time, we induce stochastic volatility and time inhomogeneity, such as seasonality, in the models. We obtain analytical solutions for commodity futures options in terms of Hermite expansions. The models are consistent with the initial futures curve, exhibit Samuelson's maturity effect, and are flexible enough to capture a variety of implied volatility smile patterns observed in commodities futures options.