The options-implied probability of a Tesla crash by January 2020 (part 2)

Or: More about options-implied densities

Our earlier post on the options-implied probability that Tesla will be dramatically lower in Jan 2020 than it was recently generated a good amount of excitement and follow-up work.

To be sure, our main aim was/is not to make claims about the actual probability of Tesla disappearing, but to showcase the ease and robustness with which non-trivial information from the options market can be unlocked using the Vola analytics library.

Let’s start with Tesla again. In our earlier post we used our 10-parameter curve for the last term, since something like this curve is necessary to fit the most liquid TSLA expiries. Our infrastructure makes it easy to use the curve needed for the most liquid terms also in the potentially much less liquid terms (even in the extreme case where they have less strikes than curve parameters!).

But the last TSLA term can definitely be fitted with less than 10 parameters, e.g. with our 6 and 8 parameters curves. The next three plots show the fits of the 20180710-1559000 snapshot for the last term with the C6, C8 and C10 curves. The fits are of comparable quality (C6 is slightly worse), as indicated by the fit metrics in the top right of each plots’ title (the “chi” and “avE5” metrics are explained e.g. on the bottom of this page).

To avoid “plot overload” we will from now on mainly concentrate on the C6 and C8 fits. Here are their cumulative densities:

[NOTE: We label the x-axis with the strike label “K”, but it is of course really the underlier spot price.]

In quantitative terms, the cumulative implied densities (aka quantiles) for K<$10 and K<$100 (called p10 and p100 in the earlier post) are 6.8%, 6.6%, 6.8% and 15.0%, 15.2%, 15.6%, respectively, for C6, C8 and C10. In other words, the different cumulative densities agree very well, as is also clear from the plots. As mentioned in our earlier post, these values are unusually high; the almost immediate jump in the cumulative density for very small spot values is very uncommon for a firm of anything like Tesla's market cap.

Another interesting question is how the implied densities (“pdf” not “cdf”) look and compare across curve types. They are shown below for C6 and C8, as a function of strike (really spot again):

A couple of comments are in order. First, implied densities, just like local volatilities, amplify small differences between vol curves, i.e. they have effectively much larger “error bars”. So, within limits (see below), one shouldn’t worry too much about some of the fine details of the behavior of the implied densities. Secondly, in cases with significant extreme-downside probabilities, like here, the density as a function of spot will have an (integrable) singularity at the origin. This to some extent obscures the “true shape” of the density, since this is essentially just an artifact of a bad choice of coordinates. If instead we plot the density in normalized strike space, we get this:

It might not be obvious to those who haven’t tried, but these are actually very well behaved densities. Note that we’re plotting them over a significantly larger range than the set of fitted strikes (the smallest and largest of which are indicated by dotted lines). Even outside the fitted range do they agree quite well; especially considering the above mentioned “error bars”.

We attribute this to the careful and financially well motivated design of our vol curves. For comparison, we refer to Fabien Le Floc’h’s interesting blog for some examples of the kind of implied densities one gets if one uses non-parametric or at least “less-financially-well-motivated parametric” vol curves (see e.g. here, here and here). They tend to have numerous “wiggles”, mostly of unclear financial significance, i.e. they are not the wiggles corresponding to bimodal distributions around earnings or other significant events (many examples of which can be found at VolaDynamics.com).

We show a couple more examples of vol fits and densities for the Jan 2020 expiry, again as of 20180710 at 15:59 ET. These examples are mainly chosen to allow comparison with the examples here.

For Netflix (NFLX) the C6 vol fit looks like this:

The C6 fit for this term is good in absolute terms, given the data set, and as good as C8, C10 and C12 fits (not shown). Next, here the cumulative density:

Note the qualitative difference to the behavior of the TSLA cumulative density at small spots.

Finally, here the implied density in normalized strike space:

Note that we are again showing all densities way beyond the fitted/listed range; the various non/semi-parametric approaches in use typically do not behave well there.

Snapchat (SNAP) is a somewhat less liquid tech name, and here the S5 (aka SVI) curve might have some hope of being competitive, at least for the last term. Here are the S5 and C6 fits:

Alas, the C6 fit here is significantly better than the S5 one, as is clear from both the plot and the metrics shown.

Here the cumulative densities for S5 and C6:

And, finally, the S5 and C6 implied densities in normalized strike space:

There are many more interesting further questions to explore here, e.g. related to time series of “crash” quantiles, their term-structure, both within names and across names. There are also applications of implied densities in risk management, of course.

But we will stop here with a few more remarks about the ease of doing this kind of research in the Vola Dynamics infrastructure. Note that the above fits and densities are produced with default settings in the fitter. One just has to choose the vol curve type, everything else has good defaults (no fiddling with splines nodes, regularizations constants, kernel bandwidths, etc, as in other approaches). It is also super-fast: 10 million implied density calculations take about one second for any curve type. (Given a fitted vol curve; the vol fitting itself of course takes longer, but is also super-fast: the whole OPRA universe can be fit on one box, with a fitting frequency of, say, once per second for liquid names, and a bit slower for less liquid names. The fits can be saved down in a suitable volsurface object at any desired sampling frequency for future use.)

Having a battle-tested, super-fast and robust options analytics infrastructure to easily answer any number of non-trivial questions within and across underliers has, in our experience, a transformative effect on the operation of an options trading desk.