A diffusion process can be viewed as a scaled Brownian motion dropped into a vector field. And the set of vector fields that can be modeled via the transition matrix in the multivariate Ornstein-Uhlenbeck process can get quite interesting.
For example, consider the rotational vector field produced in a bivariate process when the diagonal elements of the transition matrix are zero and the off-diagonal elements are of identical magnitude but with opposite signs.
The paths of expected values are cyclical – shown here to have periods of one year.
The spread between the two variables is therefore also cyclical, as seen in the next graph.
These graphs are interesting – but are they useful for modeling the prices or yields of instruments we actually encounter in the market?
Well… consider the case of gasoline and diesel. Their prices have seasonal patterns, due in part to seasonal demand and in part to regulations that specify that gasoline must have lower vapor pressure in the summer. The differences in gasoline and diesel futures prices (RBOB and Ultra Low Sulfur Diesel) are shown in the graph below. Look familiar?
The next figure below shows the crack spreads along the futures curves (ie, RBOB-Brent and ULSD-Brent) as scatter plots for each of the next few calendar years. The cyclical patterns seen in these graphs are similar to the cyclical vector field seen in the first graph above.
So is this cyclical vector field a good candidate for modeling RBOB and ULSD crack spreads? As it happens, this representation falls short in one important respect. But I’ll save that for the next post.
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