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?