# Contango and backwardation in VIX futures

I understand the meanings of contango and backwardation, but I'm trying to better understand the theory behind what creates each. For future readers of this question, here are the examples from the CME website:

Contango:

Backwardation:

I have the following questions about the theory behind these two phenomena:

1. Since contracts on physical goods have associated costs, it makes sense that the term structure curve would be upward sloping. Since there is no cost associated with delivery for the VIX and contango is considered to exist in healthy markets, is the upward slope simply accounting for the greater potential for the market to become unhealthy over longer periods of time?
2. In periods of backwardation, does the existence of historical mean reversion in the VIX cause later contracts to be cheaper than near-term contracts (i.e., mean reversion is the driving factor behind the slopes in both contango and backwardation)?
3. Is there a true default state? Would it be a straight line or would it be contango? If contango, is there an established formula that describes contango for VIX purposes, maybe a log curve?

I'm mainly interested in the last question. I'd like to use the term structure as inputs to a machine learning model, but due to the variability in days to expiry, I think it will require some kind of preprocessing. Understanding how a given term structure relates to the default, assuming there is one, will help to determine how to go about that.

Since contracts on physical goods have associated costs, it makes sense that the term structure curve would be upward sloping. Since there is no cost associated with delivery for the VIX and contango is considered to exist in healthy markets, is the upward slope simply accounting for the greater potential for the market to become unhealthy over longer periods of time?

Yes. The more time to expiry, the greater the chance of something happening.

In periods of backwardation, does the existence of historical mean reversion in the VIX cause later contracts to be lower than near-term contracts (i.e., mean reversion is the driving factor behind the slopes in both contango and backwardation)?

Yes. VIX backwardation, historically, mean reverts within a few weeks. Though it has lasted quite a bit longer recently.

Is there a true default state? Would it be a straight line or would it be contango? If contango, is there an established formula that describes contango, maybe a log curve?

In his presentation at the IAQF Thalesian Series a few years ago, Andrew Papanicolaou discusses exactly this. He discusses Bergomi's Model, rolling contracts, the stationary state, and the "Dull" or "Most Likely State" as he puts it in the presentation--this is what I believe you are asking about. He defines it as a curve of the modes of each contract:

$$mode(V^T_t) = V^\infty exp \left(-\frac{1}{2}\sum_{i,j=1}^d\frac{\rho_{ij}\bar{\sigma}_i\bar{\sigma}_j}{k_i+k_j}e^{-(k_i+k_j)\tau}\right)$$

and that it should be a contango.

I think you will find his presentation slides to be informative for what you are trying to do. Presentation slides can be found here and the mode formula on slide 21.

• Thanks for the links. – ilovevolatility Jun 29 at 20:08