As you know, not just many quants, but also the Fed and ECB love playing with options-implied densities, for what they might reveal about future underlier evolution (and various other reasons). So take a look at this:

https://www.linkedin.com/pulse/options- ... t-klassen/

Any comments are appreciated.

when you say Tesla crash, do you mean the stock price or one of the cars?

Very interesting, Tim. I can remember trying this exercise with some of the financial stocks during the 2008-2009 Crisis. Tesla is certainly the best candidate to try nowdays.

If I understand, your fit is to the form implied by the (original) Breeden-Litzenberger paper: some [$]p(K)[$] with support on [$](0,\infty)[$] such that [$] \int_{0^+}^\infty p(K) \, dK = 1[$]. So, as you say, if [$]p(K)[$] diverges as [$]K \downarrow 0[$], it will still be integrable there. 

But, for a single-name equity, which can go bankrupt and truly disappear, one can generalize somewhat, and allow support on [$][0,\infty)[$] with

[$] p(K) = A \, \delta(K) + 1_{K>0} \, p_{reg}(K)[$],

using the Dirac delta. Now the regular part, [$]p_{reg}(K)[$], is integrable at zero (norm < 1), and the complete density has a Dirac mass at zero. Then, one can fit to this and estimate [$]A[$] directly. Might be an interesting variation to try ... 

Thanks, Alan.

To be sure, in our approach the implied density is derived from the fitted parametric vol curve rather than being the primary object to be calibrated.

And in practice it is a rather tricky problem to try to detect a mass at 0, i.e. a non-zero probability of default p0, in the detailed asymptotic behavior of the implied vol curve in the wings. 

The argument is this. As we all know, the total variance w := vol^2 * T   (in hopefully obvious notation…) can’t diverge worse than 2*|y| in terms of log-moneyness y = log(K/F), as y goes to negative infinity (in the sense that the ratio w/|y| can’t be > 2 in the limit…). With a non-zero probability of default, p0 > 0, the leading term is exactly 2*|y|. But the leading term can be 2*|y| even when p0 = 0; one just needs a sufficiently fat left tail. Note that the actual value of p0 only enters in the sub-leading term(s) to this leading behavior of the total variance (there is a nice paper by DeMarco/Hillairet/Jacquier with all the mathematical details). Reliably extracting the sub-leading terms would be very difficult in practice. 

[ And even if one could, it’s not clear it would give the “right” answer, i.e. the market’s “true” opinion of what p0 is, even on average. This question is tied to the question of what equity options market makers actually do to come up with their vol curves and prices. Do they mark/fit vol curves? What kind of parametrizations do they use? Do they know how p0 > 0 should be reflected in the asymptotic put wing behavior? (Most likely not, in my experience…). Do they do something different if they think p0 > 0, i.e. work in price space? (Or add some term related to default in price- rather than vol-space?). ]

Another way to say this is in terms of the density in spot-space: it will be very hard to tell whether there is “just” an integrable singularity at small spot values or an actual delta function at 0. And you could have both too!

It’s not really easier in price space. One could try to get p0 as the coefficient in the (undiscounted) put price: put(K) = p0 * K for very small strike K. But it is hard to isolate a purely linear term from the higher order terms that could in principle be arbitrarily close to linear. And even if one somehow knew that the higher order terms are significantly different from the linear ones, there is no a priori reason there should be any (good) data where the linear term dominates. 

One final technical note: Since for me vol curves/surfaces are the primary objects, and for speed reasons, I prefer to calculate the implied density directly from the vol curve, rather than going through prices, as in Breeden-Litzenberger. The formula is nowadays well-known (it’s derived e.g. in my SSVI/S3 paper, or in the Gatheral/Jacquier arb-free SVI paper). It’s similar to the calculation of local vols in terms of implied vols (see Jim Gatheral’s book).

Yes, we are on the same wavelength re method. More specifically, my suggestion is to fit IV(K) with a smooth, arb-free form consistent with a positive mass at zero and extract the mass and the (regular part of the) implied density from that fit in the usual way. (The usual way meaning just Black-Scholes + chain rule for [$]C_{KK}[$] -- probably what you refer to, haven't checked). 

Anyway, it's my suggestion, so a good project for me; will try it at some point and post results somewhere -- likely on my blog at the link below.

It turns out that I have been looking at this problem as well https://chasethedevil.github.io/post/ts ... obability/. Initially this was partly to evaluate the stochastic collocation technique to imply the probability density (it is very simple) as I had written a paper (not yet published) on this and more.

In general, I find that techniques that start with the density or related quantities to be easier to handle (with regards to arbitrage free constraints) than techniques that work directly with the vols.

Now it is straightforward to include a probability of absorption in the stochastic collocation technique.

Note that the absorption probability only corresponds roughly to a theoretical probability of default within the model considered. The effective probability of default is not aligned with the theoretical probability of default: it is very likely that TSLA will disappear before going below $1. 
And my threshold of $1 is very arbitrary. This way you can still compute simili probability of default with models without absorption. Below I consider the probability of going below $1 for the TSLA stock according to a few different models

absorption probability: 6.6% (quintic collocation) 6.1% (septic collocation)

probability of going below $1:
Quintic collocation with absorption: 6.7%
Andreasen-Huge regularized: 3%
lognormal mixture: 2.7%

I had never thought that absorption at zero implied a linear wing in total variance, even though I am quite familiar with Lee's moment formula. I'll definitely look into that.

But you probably forgot to include point mass at 420.

But you probably forgot to include point mass at 420.

Assuming the scenario plays out where the Saudi sovereign wealth fund proposes to acquire a significant stake in Tesla, I think it will trigger an interesting CFIUS review. 

According to press reports, the rationale for the Saudi's to want such a stake is a hedge against declining oil revenues. Makes sense. Presumably, to grossly pay up (over current market) to $420, they will insist on a couple board seats and proportional voting rights. Say they end up with 15% of the company. Musk only owns 20%. Suppose Musk falters or gets hit by a bus or whatever and the Saudi's gain control of Tesla. I'm not saying this is probable -- just not impossible. Then, they would be both in control of a significant source of oil and what could become the largest non-petrochemical-based car manufacturer in the US (in the world?) Is there a US national security risk in Aramco-Tesla? 

I don't know, but if Elon Musk changed his name to Jeff Bezos, I'm pretty sure the answer would turn out to be yes!    

But you probably forgot to include point mass at 420.

Assuming the scenario plays out where the Saudi sovereign wealth fund proposes to acquire a significant stake in Tesla, I think it will trigger an interesting CFIUS review. 

According to press reports, the rationale for the Saudi's to want such a stake is a hedge against declining oil revenues. Makes sense. Presumably, to grossly pay up (over current market) to $420, they will insist on a couple board seats and proportional voting rights. Say they end up with 15% of the company. Musk only owns 20%. Suppose Musk falters or gets hit by a bus or whatever and the Saudi's gain control of Tesla. I'm not saying this is probable -- just not impossible. Then, they would be both in control of a significant source of oil and what could become the largest non-petrochemical-based car manufacturer in the US (in the world?) Is there a US national security risk in that? 

I don't know, but if Elon Musk changed his name to Jeff Bezos, I'm pretty sure the answer would turn out to be yes!    

If there is sny truth to the Saudi interest, it would be intriguing to see what value they would put to Solarcity

Very good point. Especially since California has now mandated that every new home constructed in the state, starting next year (with a few exceptions) must have a solar roof!

BTW, just went googling and found:
 Musk's Saudi funding could hit hurdle if US government deems Tesla 'critical technology'

I’ve heard even the Russia Electoral Commission makes fun of Saudi’s maths

Although today's move in Tesla (-$30 before the close), costs Musk over $1 billion, it may be the best $1 billion he ever lost. I will guess he can actually take the company private now for, say $350. Crazy as a fox?

Cold comfort if he goes to prison for market manipulation, which is not preposterous.

Musk to board: Nevermind

Most reliable cars in US


Subaru #1
...
Tesla #17

Of course.