Volume 49 Number 3

Simplified analytical pricing of moment swaps in commodity markets

Pattira Ruengsinsub^a, Kim Pluemjai^b, Kittisak Chumpong^b,c,d,*

 
ABSTRACT:     Moment swaps allow market participants to manage risks and take positions on higher-order realized moments of asset log-returns, such as variance, skewness, and kurtosis. Although well studied in equity markets, their application to commodity markets remains limited because commodity prices exhibit mean reversion, seasonality, and storage-related effects. This paper proposes a closed-form approach for pricing discretely sampled moment swaps under a time-dependent mean-reverting commodity price model, in which the log-price follows a trending Ornstein Uhlenbeck (O?U) process. We derive the conditional moment-generating function of the standard O?U process and use complete Bell polynomials with recurrence relations to obtain closed-form conditional moments. These results are transformed to the trending log-price process through a centered auxiliary O?U process. We then derive conditional central moments, mixed moments, covariance, and correlation, and rederive the fair delivery prices of variance, skewness, and kurtosis swaps directly from the realized log-return payoff using the one-step innovation representation. The resulting formulas are explicit, interpretable, and computationally efficient. Monte Carlo simulations based on the same log-price formulation show strong agreement with the closed-form values, supporting the accuracy and practical usefulness of the proposed method for hedging, volatility forecasting, and higher-order risk management in commodity markets.

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* Corresponding author, E-mail: kittisak.ch@psu.ac.th

Received 12 Sep 2025, Accepted 20 Jun 2026