Introduction
In Precio-Temporal Spread Model we investigated the seasonality and front month dependence of calendar spreads. Subsequently we have build models for all of the calendar spreads we are interested in that take as input features the value of the front month and days until expiry of the near dated contract. The plot below shows an example of one of these models for the C UZ spread. The x- and y-axis represent days to expiry and value of the near dated contract respectively. The more yellow the tile the greater (more backwardated) the spread. The more blue the tile the smaller (more contangoed) the spread.
These models can be useful because the model explores areas of the feature space that the actual values of the spread have not fully explored. Below we show the aggregated spreads of the actual data since 2000. Notice that the full feature space has not been explored. The model generalises the results to fill in all the white spaces.These models create surfaces in 3D space. In the following we aim to create a statistical measure to identify calendar spreads that deviate substantially from these model surfaces.
Surface Distance Plots
In this section we create a fictional model to illustrate the process. Suppose we can write the spread as a function of days to expiry and the value of the front month as
\[ S(d,f) = d^{2/3} + f^{1/3} - 20 \frac{d}{f} \] where \(d \in [1, 360]\) and \(f \in [200, 500]\) denote the days to expiry and front month price respectively. This toy model has the general features we have seen to persist in calendar spread models. The first is that many spreads tend to become more contango (decrease) as we approach expiry. The second is that the spread tends to become more contango with decreasing front month prices. The plot below shows the surface generated by this model. The orange dot above the surface represents a fictional value in \((d, f, S)\)-space that we want to compare to the surface. The second orage dot on the surface is the point on the surface that is the closest to the first one.
We are looking for a statistical measure that can be used to compare any calendar spread across our universe of commodities to highlight those that deviate most from the model surface. To do this we need to scale the input features and spreads. For this we use a min-max scaler,
\[ \text{scaled value} = \frac{\text{original value} - min}{max - min}. \] This scaling reduces all the model values to the \([0,1]\) range. Within this rescaled space we again determine the closest distance of our point of interest to the rescaled surface. The plot below show the rescaled surface and point closest to the surface.
We propose to use this distance to the surface measure as a statistical metric to rank calendar spreads in term of their deviation from model predictions.
Remarks
In this model we propose a simple scaled distance measure to identify calendar spreads with large deviations from model spread surfaces. In this way we can identify calendar spreads where the value of the near dated contract and that of the spread are disjointed.
