Journal of Energy Markets

Pricing crude oil options using Lévy processes

Akbar Shahmoradi

  • Application of Levy processes to crude oil futures options addresses fat tails and skewness in crude oil price returns.
  • The Jump Diffusion model tells us that the volatility of size of the jumps in crude oil market is bigger than volatility of the diffusion part.
  • The Variance Gamma approach results in slightly smaller volatility than Jump diffusion model.
  • Skewness parameter of Variance Gamma and Jump diffusion models indicate existence of right-skew in crude oil price returns.
  • Valuation of crude oil options on futures based on Levy processes results in very good fit for 5% out of money options.


Crude oil prices exhibit significant volatility over time. The distribution of returns on crude oil prices shows fat tails and skewness that barely follow a normal distribution. For this reason, we use the normal Gaussian process, jump diffusion process and variance gamma process as three Lévy processes that do not have these drawbacks. Their tails also carry a heavier mass than in a normal distribution. We employ the fractional fast Fourier transform to calibrate parameters in an optimization setup, using data about European-style options on crude oil futures in the New York Mercantile Exchange for a settlement date of April 24, 2015. Our results indicate that these three Lévy processes have very good out-of-sample results for near at-the-money options compared with others.


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