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TIME‐CHANGED ORNSTEIN–UHLENBECK PROCESSES AND THEIR APPLICATIONS IN COMMODITY DERIVATIVE MODELS
Authors:Lingfei Li  Vadim Linetsky
Institution:1. The Chinese University of Hong Kong;2. Northwestern University
Abstract:This paper studies subordinate Ornstein–Uhlenbeck (OU) processes, i.e., OU diffusions time changed by Lévy 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 processes. Further time changing by the integral of a Cox–Ingersoll–Ross 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.
Keywords:commodity derivatives  Ornstein–  Uhlenbeck  time change  Bochner subordination  mean reversion  jumps  stochastic volatility  commodity futures  commodity options  energy derivatives
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