We introduce a multi-factor stochastic volatility model based on the CIR/Heston stochastic volatility process. In order to capture the Samuelson effect displayed by commodity futures contracts, we add expiry-dependent exponential damping factors to their volatility coefficients. The pricing of single underlying European options on futures contracts is straightforward and can incorporate the volatility smile or skew observed in the market. We calculate the joint characteristic function of two futures contracts in the model  and use the two-dimensional FFT method of Hurd and Zhou (SIFIN 2010 ) to price calendar spread options. The model leads to stochastic correlation between the returns of two futures contracts. We illustrate the distribution of this correlation in an example  .
We introduce a multi-factor stochastic volatility model based on the CIR/Heston stochastic volatility process. In order to capture the Samuelson effect displayed by commodity futures contracts, we add expiry-dependent exponential damping factors to their volatility coefficients. The pricing of single underlying European options on futures contracts is straightforward and can incorporate the volatility smile or skew observed in the market. We calculate the joint characteristic function of two futures contracts in the model in analytic form and use the one-dimensional Fourier inversion method of Caldana and Fusai (JBF 2013 ) to price calendar spread options. The model leads to stochastic correlation between the returns of two futures contracts. We illustrate the distribution of this correlation in an example . We then propose analytical expressions to obtain the copula and copula density directly from the joint characteristic function of a pair of futures. These expressions are convenient to analyze the term-structure of dependence between the two futures produced by the model. In an empirical application, we calibrate the proposed model to volatility surfaces of vanilla options on WTI and provide evidence that the model is able to produce the desired stylized facts in terms of volatility and dependence. In the two appendices, we give proofs of our main results and guidance for the implementation of the proposed model and the Fourier inversion results by means of one- and two-dimensional FFT methods .
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in analytic form
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two-dimensional FFT method of Hurd and Zhou (SIFIN 2010
one-dimensional Fourier inversion method of Caldana and Fusai (JBF 2013
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. We then propose analytical expressions to obtain the copula and copula density directly from the joint characteristic function of a pair of futures. These expressions are convenient to analyze the term-structure of dependence between the two futures produced by the model. In an empirical application, we calibrate the proposed model to volatility surfaces of vanilla options on WTI and provide evidence that the model is able to produce the desired stylized facts in terms of volatility and dependence. In the two appendices, we give proofs of our main results and guidance for the implementation of the proposed model and the Fourier inversion results by means of one- and two-dimensional FFT methods