Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

Analytically pricing variance swaps in commodity derivative markets under stochastic convenience yields

Pages: 111 – 146



Sanae Rujivan (Center of Excellence in Data Science for Health Study, Division of Mathematics and Statistics, School of Science, Walailak University, Thailand)


In this paper we present an analytical formula for pricing discretely-sampled variance swaps with the realized variance being defined in terms of squared log return of the underlying asset. The dynamics of the underlying asset price follows the Schwartz’s two-factor model which can be used to describe commodity prices and allows the convenience yields to be stochastic. A partial differential equation formulated for the pricing $n$th moment swaps is analytically solved in real space to obtain our solution as a special case for variance swaps when $n=2$. Interestingly, we successfully manage to establish an interrelationship equation for variance swap prices and futures prices that would be beneficial for market practitioners who prefer to hedge price volatility risk using futures contracts. We further discuss the validity of our solution as well as propose a methodology to characterize a feasible parameter subspace, ensuring the model parameters estimated from market data produce finiteness and strict positiveness of variance swap prices when they are applied to our solution. We also demonstrate that our solution approach is quite versatile and can be adopted for pricing new generation of variance and volatility derivatives as well. Numerical tests are provided to confirm the correctness and efficiency of our solution as well as investigate how sensitive our solution is to the change of the model parameters. Finally, we apply our solution to quantify variance risk premia in gold using historical price data of gold futures obtained from the Thailand Futures Exchange.


variance swaps, discrete sampling, the Schwartz’s two-factor model, commodity prices, convenience yields

2010 Mathematics Subject Classification


Received 5 December 2019

Accepted 31 July 2020

Published 24 March 2021