From the Samuelson Volatility Effect to a Samuelson Correlation Effect: Evidence from Crude Oil Calendar Spread Options

摘要

We introduce a multi-factor stochastic volatility model based on theCIR/Heston stochastic volatility process. In order to capture the Samuelsoneffect displayed by commodity futures contracts, we add expiry-dependentexponential damping factors to their volatility coefficients. The pricing ofsingle underlying European options on futures contracts is straightforward andcan incorporate the volatility smile or skew observed in the market. Wecalculate the joint characteristic function of two futures contracts in themodel in analytic form and use the one-dimensional Fourier inversion method ofCaldana and Fusai (JBF 2013) to price calendar spread options. The model leadsto stochastic correlation between the returns of two futures contracts. Weillustrate the distribution of this correlation in an example. We then proposeanalytical expressions to obtain the copula and copula density directly fromthe joint characteristic function of a pair of futures. These expressions areconvenient to analyze the term-structure of dependence between the two futuresproduced by the model. In an empirical application we calibrate the proposedmodel to volatility surfaces of vanilla options on WTI. In this application weprovide evidence that the model is able to produce the desired stylized factsin terms of volatility and dependence. In a separate appendix, we give guidancefor the implementation of the proposed model and the Fourier inversion resultsby means of one and two-dimensional FFT methods.