# On the Probability of a TSLA Stock Crash

Timothy Klassen had an interesting post on linkedin recently, with the title “the options market thinks there is a 16% chance that Tesla will not exist in January 2020”. As I was also recently looking at the TSLA options, I was a bit intrigued. I looked at the option chain on July 10th, and implied the European volatility from the American option prices. I then fit a few of my favorite models: Andreasen-Huge with Tikhonov regularization, the lognormal mixture, and a polynomial collocation of degree 7. This results in the following graph

The shape is not too difficult to fit, and all models give nearly the same fit, within the bid-ask spread as long as we include an inverse relative bid-ask spread weighting in the calibration. Note that I removed the quote for strike 30, as the bid-ask spread was, unusually extremely large and would only introduce noise.

Like Timothy Klassen, we can look at the implied cumulative density of each model.

More explicitely, we obtain

Model probability of TSLA < 100 probability of TSLA < 15
Collocation 15.4% 7.2%
Lognormal Mixture 15.2% 8.0%
Andreasen-Huge regularized 15.0% 7.7%

It would have been great of Timothy Klassen had shown the implied density as well, in the spirit of my earlier posts on a similar subject.

In particular, the choice of model has a much stronger impact on the implied density. Even though Andreasen-Huge has more modes, its fit in terms of volatilities is not better than the lognormal mixture model. We could reduce the number of modes by increasing the constant of the Tikhonov regularization, at the cost of a slightly worse fit.

The collocation produces a significantly simpler shape. This is not necessarily a drawback, since the fit is quite good and more importantly, it does not have a tendency to overfit (unlike the two other models considered).

Model weighted root mean square error in vols
Collocation 0.00484
Lognormal Mixture 0.00432
Andreasen-Huge regularized 0.00435

The most interesting from T. Klassen plot, is the comparison with the stock price across time. It is expected that the probability of going under $100 will be larger when the stock price moves down, which is what happen around March 27, 2018. But it is not so expected that when it comes back up higher (above$350 in mid June), the probability of going under $100 stays higher than it was before March 27, 2018, where the stock price was actually lower. In fact, there seems to be a signal in his time-serie, where the probability of going under$100 increases significantly, one week before March 27, that is one week before the actual drop, suggesting that the drop was priced in the options.