What Taleb might mean November 20, 2012 at 7:35 am

It is perhaps to my discredit that I don’t have the interest, stamina, or bloody mindedness to go through Taleb’s technical papers (HT FT Alphaville*). Still, a glance at them and hefty dose of imagination quickly gave me what I think he might mean in the first few pages, at least from 50,000 feet.

The first thing to note is that options have vega: they are sensitive to changes in implied volatility. For our purposes vega can be thought of as a measure of sensitivity to being wrong about the asset return distribution. Naively it’s `I thought it was log normal with a 30% volatility, and it turned out to be log-normal with a 31% volatility – how much did I screw up?’

The second point is that vega is highly strike dependent. A 1% change in vol for a five year ATM option might make only a 2% difference in option value: it makes nearly an 8% difference for a 200% strike on the same underlying. This is (I think) what Taleb calls tail vega. It is simply the observation that being wrong about the distribution matters more for out of the money options. Pricing far OTM options is fragile: one knock of the distribution and you are screwed. Far OTM puts are particularly bad because vols rise as the markets crash, and you get crushed by both the gamma and the vega.

In contrast there are some products that are not very sensitive to being wrong about the distribution. What a savvy tail risk hedger tries to do is either buy cheap protection – i.e. find implied vols that are ‘too low’ – or find a product which is relatively vol insensitive yet is still long the downside.

The hard part is that ideally you want to do this analysis not with respect to a mis-parameterised distribution – it’s log normal but we don’t know the vol – but rather with respect to some class of distributions. What class do you pick, though? Too general, and you can’t prove anything, plus there are lots of distributions that really don’t occur in nature that you are trying to talk about; too narrow, and you risk missing the `real’** one. So the whole game in work like this is figuring out what class is general enough to capture some interesting fat tailed distributions, yet narrow enough you can prove something***. This is grandiosely termed meta-probability, but all it really is is hunting for a tractable class of distributions to do the meta-theory on.

*You would get a precise link except my session on the FT website has timed out and I can’t be bothered to login again. Honestly you would think the FT is trying to reduce readership…

**To the extent that even makes sense (which isn’t far).

***I am pretty sure I can come up with return `distributions’ (under a sufficiently loose construction of the term) that break all of Taleb’s theorems, but they would be insanely pathological – something Lebesgue integrable but not Borel measurable would be a good start. What, you wanted something continuous?

11 Responses to “What Taleb might mean”

  1. […] It is perhaps to my discredit that I don’t have the interest, stamina, or bloody mindedness to go through Taleb’s technical papers (HT FT Alphaville*). Still, a glance at them and hefty dose of imagination quickly gave me what I think he might mean in the first few pages, at least from 50,000 feet.  […]

  2. As a preface when Black Swan came out in 2007 I was a big Taleb fan. Since then I’ve totally reversed my opinion of him. His primary theses (related to financial markets) is that the belief that options are systematically overpriced is an illusion. He believes the historical tendency of realized vol to fall below implied vol is due to the lack of tail events from the historical records. That in fact if you traded the strategy of selling vol for long enough eventually your expected value would be zero or negative as you captured tail events.

    Luckily we had a test of this not too long after. The 2008 financial crisis was the most volatile time for markets since the great depression. And predictably at the start of the crisis those who sold vol did take big losses. But as long as you Kelly-sized appropriately and survived the first two months you made a fortune over the next two years as implied vol stayed at a huge premium to realized vol. (Just flip this chart upside down http://finance.yahoo.com/echarts?s=VXX+Interactive#symbol=VXX;range=5y)

    Now does this completely invalidate Taleb’s hypothesis? No, maybe we’ll find those options are fairly risk-neutral priced once we capture even bigger tail event. But I find it unlikely. The much better explanation to me is that short-vol is simply a risk factor that earns an expected premium, not much different than value or momentum. There are systematic natural buyers of vol, particularly tail vol. There are few systematic sellers, and this drives the premium above risk-neutral pricing (even taking into account tail events).

    Okay, that being said, here are my particular criticisms of a (very) cursory glance at those Taleb papers. First off it’s completely off-base to use the term “Vega” instead of “Gamma” in the “Fragility” section. It’s a toy model, but as it relates back to actual options, vega has nothing to do with asset price return distributions. It is simply the sensitivity to the market price of vol, which can (and does) remain completely divorced from realized vol. Implied vol can be whipsawed around by the market while the asset’s return distribution stays perfectly in line. So to talk use vega to talk about asset price returns is fundamentally wrong.

    At it’s core this seems driven by Taleb’s unshakable belief that given enough iterations market implied vol is the truer measure of forward-looking vol than any historical measure. (As I’ve mentioned above history has not been kind to this view.) In option price calibrations gamma usually comes from historical measures, whereas vega comes from market implied pricing, so using this term is (not-so) subtle way for Taleb to push his agenda.

    Secondly “Fragility” is already completely described by sensitivity to higher order moments. This is simple to demonstrate because any probability distribution is completely described by its higher order moments, and hence any function on that distribution (i.e. value of a derivative on an asset’s returns) can be too. The concept of fragility gives us nothing that we can’t already get from highly well established, studied, and tractable vol/skew/kurtosis/etc.

    Finally “meta-probabilities”… Huh?! This makes no sense whatsoever. Taleb initially tries to claim that the whole concept of probability is worthless. That if we have a random variable X, and a function F, then having a fully specified probability distribution on X is insufficient to specify F(X). This claim is trivially absurd.

    Of course, this isn’t what he shows when he actually does the math. Instead he throws out some formulas showing that if X is mis-specified then the distribution of F(X) will also be mis-specified. Well, duh! The case he focuses on is when there’s unaccounted for error in the estimation of X’s distribution leading to under-estimated variance.

    Well ignoring the fact that integrating error in moment estimation into a probability distribution was solved more than a century ago, Taleb decides that the proper thing to do is try to account for the error through the distribution of F(X). So the solution to a mis-specified random variable isn’t, to you know, correct the distribution on the variable itself.

    Instead Taleb builds some sort of grotesque system where there’s now two layers of probability distributions, one for the original variable and one for every output point of F.

  3. Doug – I agree with almost everything you said. Implied is indeed enough above historic enough of the time that you can, with prudent leverage, monetise the difference. Taleb is wrong about implieds; it is simply in my view a phenomenon of supply and demand (which reverses in areas like inflation caps where there is over-supply).

    Where I would slightly disagree with you is on the meta-theory. There is a reading of this that I think isn’t foolish (although whether this is what Taleb meant is unclear). Namely, that one might want to work with some class of distributions that one can’t calibrate exactly, and know how wrong one might be. Thus for instance hedging vega in a pure Black-Scholes world is crazy because in a B/S world vol can’t change – but in the real world it can, which is why we vega hedge even though our deltas come from a model that assumes it can’t. At a more sophisticated level I might want to know if my vol of vol estimate is wrong, say, what product combo is relatively insensitive to that. So read as advice on not being really short higher order convexities, Taleb isn’t foolish – which is not to say it is worth paying up to be long them of course…

  4. Taleb is starting to remind me of Wolfram, in that he had some great and lucid ideas a while ago, but has since spiraled inward into self-referential theories that are largely wanking.

  5. It seems to me that the conventional math answer for the class of distributions you’re looking for is “partition functions” or “kernel parametrizations.” The assumption of local normality makes the math better (‘tractable,’ perhaps, if not ‘clean’), and of course you can approximate any crazy shape you like if you’re willing to rely on a lot of partitions.

    The space of ‘people trying to do complicated math to make market hypothesis about extreme tail events’ has for one reason or another it seems to me always been full of very quirky characters. One strange, borderline insane but quite smart and entertaining book on the subject is Iceberg Risk by Kent Osband, a former hedge fund risk manager whose current whereabouts I can’t quite determine. It can be bought on amazon: http://www.amazon.com/Iceberg-Risk-Adventure-Portfolio-Theory/dp/1587990687 and is maybe worth it for the oddball writing style alone.

  6. Thank you mg. I think I might like Osband. The lines “markets measure the beliefs about risk rather than risk itself. And risk changes often enough that beliefs rarely have time to converge on the truth.” and “Twentieth-century finance theory focused too much on risk, too little on changes in risk, and hardly at all on beliefs about changing risk.” seem absolutely on the money.

  7. I think it’s f’ing wonderful to see lebesgue / borel discussed in anything since I left graduate school.

  8. “Taleb is starting to remind me of Wolfram, in that he had some great and lucid ideas a while ago, but has since spiraled inward into self-referential theories that are largely wanking.”

    Hofstadter too, except his self-referential theories about self-reference were always intended to reference himself. 😉

  9. I agree most with your footnote on FTAV’s login policy. Boooh for that.

  10. This is all fascinating, but is it a book on the maths of options pricing, of which I already have many, including ‘Dynamic Hedging’, or a book of Taleb’s philosophical writings, which I can’t justify spending money on, or some hodge-podge of the two? 😉

    Statistics is mostly boring, but it does have its moments.

  11. As a follow-up to mg, Osband has also written another book called Pandora’s Risk (http://cup.columbia.edu/book/978-0-231-15172-6/pandoras-risk) and is currently a managing director at Fortress Investment Group.